Site du GDR Structuration de la théorie des nombres

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Bienvenue sur le site du GDR Structuration de la théorie des nombres


Le GDR Structuration de la théorie des nombres est une unité du CNRS (GDR 2251). Il fédère les recherches en théorie des nombres en France. Il s’investit dans plusieurs activités de caractère national ou international (soutien de rencontres, participation de chercheurs du réseau à des colloques internationaux, mobilité de chercheurs membres ou invités entre pôles du réseau,...), avec une attention particulière aux jeunes chercheurs.

La liste de diffusion du GDR est gdrstn@listes.math.cnrs.fr

Directeur : Guillaume RICOTTA, maître de conférences habilité à diriger des recherches Guillaume.Ricotta@math.u-bordeaux.fr

Nouveaux articles en théorie des nombres

[hal-00528714] Représentations galoisiennes p-adiques et (phi,tau)-modules

24 mai 2017

Let p be an odd prime number and K be a p-adic field. In this paper, we develop an analogue of Fontaine's theory of (phi,Gamma)-modules replacing the p-cyclotomic extension by the extension K_infty obtained by adding to K a compatible system of p^n-th roots of a fixed uniformizer pi of K. As a (...)

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[hal-00211173] Polygones de Hodge, de Newton et de l'inertie modérée des représentations semi-stables

23 mai 2017

Let k be a perfect field, and K be a totally ramified extension of K_0 = Frac W(k) of degree e. To a semi-stable p-adic representation of G_K (the absolute Galois group of K), one can classicaly associate two polygons : the Hodge polygon et the Newton polygon. It is well known that the former (...)

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[hal-00338806] F_p-représentations semi-stables

23 mai 2017

Torsion semi-stable representations can be constructed and studied using Breuil modules. In this paper, we define the notion of pylonet and prove that some categories of Breuil modules naturally define pylonets. As a consequence, we are able to define full subcategories of Breuil's categories (...)

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[hal-01522391] A proof of Syracuse conjecture

17 mai 2017

The conjecture of Syracuse or Collatz's conjecture is an old conjecture relating to natural numbers. It was discovered by the German mathematician Lothar Collatz in 1930. Since then, many mathematicians have sought to explain why this conjecture is true, but today no one has yet arrived. We (...)

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[hal-01524140] Nemo/Hecke: Computer Algebra and Number Theory Packages for the Julia Programming Language

17 mai 2017

We introduce two new packages, Nemo and Hecke, written in the Julia programming language for computer algebra and number theory. We demonstrate that high performance generic algorithms can be implemented in Julia, without the need to resort to a low-level C implementation. For specialised (...)

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[hal-01283042] Series Representation of Power Function

17 mai 2017

This paper presents the way to make expansion for the next form function: $y=x^n, \ \forall(x,n) \in \mathbbN$ to the numerical series. The most widely used methods to solve this problem are Newton’s Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are (...)

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[tel-01518362] Sûreté : de l'analyse à l'instrumentation et à la synthèse de code

9 mai 2017

Les machines multiprocesseurs, multi-cœurs et les accélérateurs de type GPU se généralisent et pourtant il devient de plus en plus difficile pour les programmeurs de tirer profit de leurs capacités. La compilation source-à-source des applications permet de faciliter le développement d’implémentations (...)

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[hal-01465483] Simple connexité des fibres d'une application d'Abel-Jacobi et corps de classe local

9 mai 2017

Cet article traite du cas abélien de la conjecture de géométrisation de la correspondance de Langlands locale formulée par l'auteur. On y montre qu'un certain morphisme d'Abel-Jacobi et une fibration pro-étale localement triviale en diamants simplement connexes. On en déduit la conjecture pour (...)

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[hal-01519169] On the locus of 2-dimensional crystalline representations with a given reduction modulo p

6 mai 2017

We consider the family of irreducible crystalline representations of dimension $2$ of $\rm Gal(\overline\bf Q_p/\bf Q_p)$ given by the $V_k,a_p$ for a fixed weight integer $k\geq 2$. We study the locus of the parameter $a_p$ where these representations have a given reduction modulo $p$. We give (...)

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[hal-01516012] Low-discrepancy sequences: Atanassov’s methods revisited

28 avril 2017

We revisit Atanassov’s methods for Halton sequences, (t,s)-sequences, and View the MathML source-sequences by providing a unifying framework enlightening the power and the efficiency of these methods for the study of low-discrepancy sequences. In this context, we obtain new improved explicit (...)

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[hal-00836097] Une famille d'applications linéaires liée à l'hypothèse de Riemann généralisée

27 avril 2017

We exhibit a family of linear operators related to the almost-periodic approach for the generalized Riemann hypothesis.

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[hal-01177065] Fonctions complètement multiplicatives de somme nulle

27 avril 2017

Completely multiplicative functions whose sum is zero ($CMO$). The paper deals with $CMO$, meaning completely multiplicative ($CM$) functions $f$ such that $f(1)=1$ and $\sum\limits_1^\infty f(n)=0$. $CM$ means $f(ab)=f(a)f(b)$ for all $(a,b)\in \N^*2$, therefore $f$ is well defined by the (...)

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[hal-01512648] Möbius disjointness for models of an ergodic system and beyond

24 avril 2017

We give a necessary and sufficient condition (called the strong MOMO property) for a uniquely ergodic model of an ergodic measure-preserving system to have all uniquely ergodic models of the system Möbius disjoint. It follows that all uniquely ergodic models of: ergodic unipotent diffeomorphisms (...)

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[hal-01309134] Are monochromatic Pythagorean triples unavoidable under morphic colorings ?

24 avril 2017

A Pythagorean triple is a triple of positive integers a, b, c ∈ N⁺ satisfying a² + b² = c². Is it true that, for any finite coloring of N⁺ , at least one Pythagorean triple must be monochromatic? In other words, is the Dio-phantine equation X²+ Y² = Z² regular? This problem, recently solved for (...)

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[hal-01510045] Courbes et fibrés vectoriels en théorie de Hodge p-adique

21 avril 2017

Dans ce travail nous définissons et étudions la courbe fondamentale en théorie de Hodge p-adique. Nous démontrons un théorème de classification des fibrés vectoriels sur celle-ci et nous en déduisons de nouvelles preuves des deux théorèmes fondamentaux de la théorie de Hodge p-adique: faiblement (...)

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[hal-01499011] A note on some constants related to the zeta–function and their relationship with the Gregory coefficients

18 avril 2017

In this article, new series for the first and second Stieltjes constants (also known as generalized Eu-ler's constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the so–called Gregory coefficients, which are also known as (...)

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[hal-00686065] A Diophantine duality applied to the KAM and Nekhoroshev theorems

15 avril 2017

In this paper, we use geometry of numbers to relate two dual Diophantine problems. This allows us to focus on simultaneous approximations rather than small linear forms. As a consequence, we develop a new approach to the perturbation theory for quasi-periodic solutions dealing only with (...)

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[hal-01499011] A note on some constants related to the zeta–function and their relationship with the Gregory coefficients

14 avril 2017

In this article, new series for the first and second Stieltjes constants (also known as generalized Eu-ler's constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the so–called Gregory coefficients, which are also known as (...)

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[hal-01503749] On the number of perfect lattices

7 avril 2017

We show that the number $p_d$ of non-similar perfect $d$-dimensional lattices satisfies eventually the inequalities $e^d^1-\epsilon

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[hal-01503560] Twists of non-hyperelliptic curves

7 avril 2017

In this paper we present a method for computing the set of twists of a non-singular projective curve defined over an arbitrary (perfect) field k. The method is based on a correspondence between twists and solutions to a Galois embedding problem. When in addition, this curve is non-hyperelliptic (...)

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[hal-01501694] Sur les plus grands facteurs premiers d'entiers consécutifs

6 avril 2017

Let §P^+ (n) §denote the largest prime factor of the integer §n§ and §P ^+ y (n)§ denote the largest prime factor §p§ of §n§ which satisfies §p≤ y§. In this paper, firstly we show that the triple consecutive integers with the two patterns §P^+ (n − 1) > P^+ (n) < P ^+ (n + 1)§ and §P^+ (n−1) < P^+ (n) (...)

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[hal-01502378] Sufficient conditions for the dirichlet property

5 avril 2017

The effectivity up to R-linear equivalence (Dirichlet property) of pseudoeffective adelic R-Cartier divisors is a subtle problem in arithmetic geometry. In this article, we propose sufficient conditions for the Dirichlet property by using the dynamic system in the classic Arakelov geometry (...)

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[hal-01502051] Random Sampling Revisited: Lattice Enumeration with Discrete Pruning

5 avril 2017

In 2003, Schnorr introduced Random sampling to find very short lattice vectors, as an alternative to enumeration. An improved variant has been used in the past few years by Kashiwabara et al. to solve the largest Darmstadt SVP challenges. However, the behaviour of random sampling and its (...)

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[tel-01501685] Blocs des chiffres des nombres premiers

4 avril 2017

Au cours de cette thèse nous nous intéressons à des orthogonalités asymptotiques (au sens ou le produit scalaire dans le tore discret de taille N tend vers 0 lorsque N tend vers l’infini) entre certaines fonctions liées aux blocs des chiffres des entiers et la fonction de Möbius (ainsi qu’avec la (...)

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[hal-01499011] A note on some constants related to the zeta–function and their relationship with the Gregory coefficients

4 avril 2017

In this article, new series for the first and second Stieltjes constants (also known as generalized Eu-ler's constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the so–called Gregory coefficients, which are also known as (...)

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[hal-01444560] The p-adic Kummer-Leopoldt constant -- Normalized p-adic regulator

4 avril 2017

The p-adic Kummer--Leopoldt constant kappa_K of a number field K is (assuming the Leopoldt conjecture) the least integer c such that for all n >> 0, any global unit of K, which is locally a p^(n+c)th power at the p-places, is necessarily the p^nth power of a global unit of K. This (...)

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[hal-01444560] The p-adic Kummer-Leopoldt constant -- Normalized p-adic regulator

3 avril 2017

The p-adic Kummer--Leopoldt constant kappa_K of a number field K is (assuming the Leopoldt conjecture) the least integer c such that for all n >> 0, any global unit of K, which is locally a p^(n+c)th power at the p-places, is necessarily the p^nth power of a global unit of K. This (...)

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[hal-01225061] On the order modulo p of an algebraic number (for p large enough)

3 avril 2017

Let K/Q be Galois, and let eta in K* whose conjugates are multiplicatively independent. For a prime p, unramified, prime to eta, let np be the residue degree of p and gp the number of P I p, then let o_P(eta) and o_p(eta) be the orders of eta modulo P and p, respectively. Using Frobenius (...)

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[hal-01404933] Approche p-adique de la conjecture de Greenberg (cas totalement réel p-décomposé)

3 avril 2017

Let k be a totally real number field ant let k∞ be its cyclotomic Zp-extension for a prime p>2. We give (Theorem 3.2) a sufficient condition of nullity of the Iwasawa invariants lambda, mu, when p totally splits in k, and we obtain important tables of quadratic fields and p for which we can (...)

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[tel-01495417] Sur les représentations automorphes non ramifiées des groupes linéaires sur Q de petits rangs.

31 mars 2017

Cette thèse est consacrée à l'étude des représentations automorphes algébriques des groupes linéaires découvertes par Chenevier-Renard. On s'intéresse plus particulièrement à leurs paramètres de Satake. Pour cela, nous utilisons la théorie d'Arthur afin de faire apparaître ces représentations par le biais (...)

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[hal-01497127] Study on the digitization dual combinatorics and convex case

29 mars 2017

The action of a translation on a continuous object before its digitization generates several digitizations. The dual, introduced by the authors in a previous paper, stands for these digitizations in function of the translation parameters. This paper focuses on the combinatorics of the dual by (...)

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[hal-01492074] La cohomologie des espaces de Lubin-Tate est libre

24 mars 2017

Le résultat principal de ce travail est l'absence de torsion dans la $\overline\mathbbZ_l$-cohomologie de la tour de Lubin-Tate. La stratégie est globale et repose sur l'étude du complexe des cycles évanescents de certaines variétés de Shimura de type Kottwitz-Harris-Taylor. Nous reprenons les (...)

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[hal-01494793] On the linear bounds on genera of pointless hyperelliptic curves

24 mars 2017

An irreducible smooth projective curve over $\mathbbF_q$ is called \textitpointless if it has no $\mathbbF_q$-rational points. In this paper we study the lower existence bound on the genus of such a curve over a fixed finite field $\mathbbF_q$. Using some explicit constructions of hyperelliptic (...)

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[hal-01493550] Values of globally bounded G-functions

23 mars 2017

In this paper we define and study a filtration (G_s) on the algebra of values at algebraic points of analytic continuations of G-functions: G_s is the set of values at algebraic points in the disk of convergence of all G-functions sum_n a_n z^n for which there exist some positive integers b and (...)

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[hal-01494169] On Harder-Narasimhan filtrations and their compatibility with tensor products

23 mars 2017

We attach buildings to modular lattices and use them to develop a metric approach to Harder-Narasimhan filtrations. Switching back to a categorical framework, we establish an abstract numerical criterion for the compatibility of these filtrations with tensor products. We finally verify our (...)

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[hal-01407300] Note sur la conjecture de Greenberg

23 mars 2017

Nous étudions la conjecture de Greenberg sur les ℓ-invariants d'Iwasawa des corps totalement réels, en termes de classes logarithmiques. Sous la conjecture de Leopoldt, et donc en particulier pour les corps abéliens, nous montrons que la conjecture de Greenberg postule tout simplement la (...)

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[hal-01493577] Problème de Lehmer sur les courbes elliptiques à multiplications complexes

22 mars 2017

Nous étudions le problème de minoration de la hauteur canonique sur les courbes elliptiques. Notre résultat principal utilise des méthodes d’intersection arithmétique pour retrouver un résultat de Laurent, qui démontrait la conjecture de Lehmer pour les courbes elliptiques à multiplications complexes à (...)

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[hal-01490725] The Stern Sequence and Moments of Minkowski's Question Mark Function

21 mars 2017

1 : We use properties of the Stern Sequence for numerical computations of moments 1 0 t n d?(t) associated to Minkowski's Question Mark function.

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[hal-01266567] Differential uniformity and second order derivatives for generic polynomials

20 mars 2017

For any polynomial $f$ of $\mathbb F_2^n[x]$ we introduce the following characteristic of the distribution of its second order derivative, which extends the differential uniformity notion: $$\delta^2(f):=\max_\substack \alpha \in \mathbb F_2^n^\ast ,\alpha' \in \mathbb F_2^n^\ast ,\beta \in (...)

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[cea-01491820] The algebraic structure of cut Feynman integrals and the diagrammatic coaction

17 mars 2017

We study the algebraic and analytic structure of Feynman integrals by proposing an operation that maps an integral into pairs of integrals obtained from a master integrand and a corresponding master contour. This operation is a coaction. It reduces to the known coaction on multiple (...)

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[hal-00483259] Estimation des dimensions de certaines variétés de Kisin

16 mars 2017

In this paper, we study dimensions of some varieties, that were introduced recently by Kisin in order to prove modularity of some Galois representations. In fact, we mainly consider a special case for which we obtain an estimation of the dimension we are interested in. Then, based on this (...)

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[hal-00202447] Computation of 2-groups of narrow logarithmic divisor classes of number fields

16 mars 2017

We present an algorithm for computing the 2-group of narrow logarithmic divisor classes of degree 0 for number fields F. As an application, we compute in some cases the 2-rank of the wild kernel WK2(F).

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[hal-00202450] Sur le sous-groupe des éléments de hauteur infinie du K2 d'un corps de nombres

16 mars 2017

By using the logarithmic approach of the classical kernels for the K2 of number fields, we compute the 2-rank of the wild kernel WK2(F) and the 2-rank of the subgroup of infinite heigh elements in K2(F) in terms of positive class groups for any number field (...)

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[hal-00202456] A new Algorithm for the Computation of logarithmic l-Class Groups of Number Fields

16 mars 2017

We present an algorithm for the computation of logarithmic l-class groups of number fields. Our principal motivation is the effective determination of the l-rank of the wild kernel in the K-theory of number fields.

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[hal-00203111] Computation of 2-groups of positive classes of exceptional number fields

16 mars 2017

We present an algorithm for computing the 2-group of the positive divisor classes of a number field F in case F has exceptional dyadic places. As an application, we compute the 2-rank of the wild kernel WK2(F) in K2(F) for such number (...)

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[hal-00140540] New blow-up rates for fast controls of Schrödinger and heat equations

15 mars 2017

We consider the null-controllability problem for the Schrödinger and heat equations with boundary control. We concentrate on short-time, or fast, controls. We improve recent estimates of Miller on the norm of the operator associating to any initial state the minimal norm control driving the (...)

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[hal-00091966] Une approche séquentielle de l'hypothèse de Riemann généralisée

15 mars 2017

Une généralisation du théorème de Beurling et Nyman établit que l'hypothèse de Riemann pour une fonction F de la classe de selberg est équivalente à l'appartenance de la fonction indicatrice de l'intervalle ]0,1[ à l'adhérence d'un sous-espace de fonctions B dans l'espace L^2(R+). Dans cet article, nous (...)

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[hal-00091189] Sur l'écart quadratique moyen des diviseurs d'un entier normal, 2

15 mars 2017

As in the first part of this work with A. Raouj [Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 3, 399--415], this work is devoted to a measure of the propensity of an integer's divisors to agglutinate around certain values. Here, the function $D^*(n,t)$ counting the number of pairs of (...)

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[hal-01348576] On a two-valued sequence and related continued fractions in power series fields

15 mars 2017

We explicitly describe a noteworthy transcendental continued fraction in the field of power series over Q, having irrationality measure equal to 3. This continued fraction is a generating function of a particular sequence in the set 1, 2. The origin of this sequence, whose study was initiated (...)

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[hal-01372952] A sum-shuffle formula for zeta values in Tate algebras

14 mars 2017

We prove a sum-shuffle formula for multiple zeta values in Tate algebras (in positive characteristic), introduced in \citePEL3. This follows from an analog result for double twisted power sums, implying that an $\mathbbF_p$-vector space generated by multiple zeta values in Tate algebras is an (...)

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[hal-01332997] On a variant of Schanuel conjecture for the Carlitz exponential

13 mars 2017

We introduce and discuss a variant of Schanuel conjecture in the framework of the Carlitz exponential function over Tate algebras and allied functions. Another purpose of the present paper is to widen the horizons of possible investigations in transcendence and algebraic independence in (...)

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[hal-01488318] Exceptional planar polynomials

13 mars 2017

Planar functions are special functions from a finite field to itself that give rise to finite projective planes and other combinatorial objects. We consider polynomials over a finite field that induce planar functions on infinitely many extensions of ; we call such polynomials exceptional (...)

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[hal-01488043] On the average of some arithmetical functions under a constraint on the sum of digits of squares

13 mars 2017

Let q >= 2 be an integer and S-q(n) denote the sum of the digits in base q of the positive integer n. We look for an estimate of the average of some multiplicative arithmetical functions defined by sums over divisors d of n satisfying S-q(d(2)) r mod m for some integers r and (...)

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[hal-01487856] Combinatorial and probabilistic properties of systems of numeration

13 mars 2017

Let G = (G(n))(n) be a strictly increasing sequence of positive integers with G(0) = 1. We study the system of numeration defined by this sequence by looking at the corresponding compactification K-G of N and the extension of the addition-by-one map tau on K-G (the 'odometer'). We give (...)

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[hal-01484740] Collatz conjecture The trivial cycle is unique (because a Collatz sequence that becomes periodic converges)

10 mars 2017

A Collatz sequence that is periodic after a certain rank reaches the value 1 (or converges). Therefore the trivial cycle is unique, and a Collatz sequence that has an upper bound becomes periodic, and then converges.

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[hal-01447059] Computation of Euclidean minima in totally definite quaternion fields

10 mars 2017

We describe an algorithm that allows to compute the Euclidean minimum (for the norm form) of any order of a totally definite quaternion field over a number field K of degree strictly greater than 1. Our approach is a generalization of previous work dealing with number fields. The algorithm was (...)

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[hal-01485516] The geometry of non-unit Pisot substitutions

9 mars 2017

Let $\sigma$ be a non-unit Pisot substitution and let $\alpha$ be the associated Pisot number. It is known that one can associate certain fractal tiles, so-called \emphRauzy fractals, with $\sigma$. In our setting, these fractals are subsets of a certain open subring of the ad\`ele ring (...)

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[hal-01485418] A (doubly) elementary formula for prime numbers

8 mars 2017

In this paper the author gives formulas for the prime counting function and for the n-th prime numbers. These formulas only use elementary functions. Moreover their proof requires only elementary facts of number theory, so that the title of the paper is (...)

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[hal-01283042] Series Representation of Power Function

8 mars 2017

This paper presents the way to make expansion for the next form function: $y=x^n, \ \forall(x,n) \in \mathbbN$ to the numerical series. The most widely used methods to solve this problem are Newton's Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are (...)

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[hal-01288069] Discriminants of $\mathop{\mathfrak{S}}\nolimits_n$-orders

8 mars 2017

Let alpha be an algebraic integer of degree n >= 2. Let alpha(1),..., alpha(n) be the n complex conjugate of alpha. Assume that the Galois group Gal(Q(alpha(1),..., alpha(n))/Q) is isomorphic to the symmetric group S-n. We give a Z-basis and the discriminant of the order Z[alpha(1),..., (...)

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[hal-01260203] On short sums of trace functions

8 mars 2017

We consider sums of oscillating functions on intervals in cyclic groups of size close to the square root of the size of the group. We first prove non-trivial estimates for intervals of length slightly larger than this square root (bridging the "Poly\'a-Vinogradov gap" in some sense) (...)

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[hal-01478267] Families of elliptic curves with non-zero average root number

7 mars 2017

We consider the problem of finding $1$-parameter families of elliptic curves whose root number does not average to zero as the parameter varies in $\mathbbZ$. We classify all such families when the degree of the coefficients (in the parameter $t$) is less than or equal to $2$ and we compute the (...)

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[hal-01484257] Cinq preuves de l'Hypothèse de Riemann

7 mars 2017

Pour explorer l'univers des particules, les physiciens se sont servis des accélérateurs des particules pour les faire entrer en collision et étudier les particules élémentaires générées au cours de cette collision. Par analogie, pour étudier les zéros de la fonction zeta de Riemann que j'ai assimilés à (...)

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[hal-01483768] Montgomery curves and their arithmetic

6 mars 2017

Three decades ago, Montgomery introduced a new elliptic curve model for use in Lenstra's ECM factorization algorithm. Since then, his curves and the algorithms associated with them have become foundational in the implementation of elliptic curve cryptosystems. This article surveys the theory (...)

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[hal-01445353] Persistance d'un nombre

2 mars 2017

Je démontre qu’en base 10, il n'existe aucun nombre dont la persistance multiplicative est supérieure ou égale à 12.

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[hal-00425088] Periodic binary harmonic functions

1er mars 2017

A function on a (generally infinite) graph $\G$ with values in a field $K$ of characteristic 2 will be called \it harmonic if its value at every vertex of $\G$ is the sum of its values over all adjacent vertices. We consider binary pluri-periodic harmonic functions $f: \Z^s\to\F_2=\GF(2)$ on (...)

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[hal-00323512] Periodic binary harmonic functions

1er mars 2017

A function on a (generally infinite) graph $\G$ with values in a field $K$ of characteristic 2 will be called \it harmonic if its value at every vertex of $\G$ is the sum of its values over all adjacent vertices. We consider binary pluri-periodic harmonic functions $f: \Z^s\to\F_2=\GF(2)$ on (...)

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[hal-00323515] Convolution equations on lattices: periodic solutions with values in a prime characteristic field

1er mars 2017

These notes are inspired by the theory of cellular automata. A linear cellular automaton on a lattice of finite rank or on a toric grid is a discrete dinamical system generated by a convolution operator with kernel concentrated in the nearest neighborhood of the origin. In the present paper we (...)

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[hal-01479205] Lois de répartition des diviseurs des entiers friables

28 février 2017

According to a general probabilistic principle, the natural divisors of friable integers (i.e.~free of large prime factors) should normally present a Gaussian distribution. We show that this indeed is the case with conditional density tending to 1 as soon as the standard necessary conditions (...)

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[hal-01479088] Ergodic measures on spaces of infinite matrices over non-Archimedean locally compact fields

28 février 2017

Let $F$ be a non-discrete non-Archimedean locally compact field and $\mathcalO_F$ the ring of integers in $F$. The main results of this paper are Theorem 1.2 that classifies ergodic probability measures on the space $\mathrmMat(\mathbbN, F)$ of infinite matrices with enties in $F$ with respect (...)

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[tel-01266158] Représentations galoisiennes et groupe de Mumford-Tate associé à une variété abélienne

28 février 2017

Soient $K$ un corps de nombres et $A$ une variété abélienne sur $K$ dont nous notons $g$ la dimension. Pour tout premier $ell$, le module de Tate $ell$-adique de $A$ nous fournit une représentation $ell$-adique du groupe de Galois absolu de $K$, et c'est à l'image de ces représentations galoisiennes (...)

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[hal-01476843] Remarques sur les moyennes des fonctions de Piltz sur les entiers friables

26 février 2017

We study the mean value of generalized divisor functions $\tau_\kappa(n)$ over integers without large prime factors (here $\kappa>0$). We relate this problem to the computation of the ratio $\Psi(x^1/\kappa, y)^\kappa/\Psi(x, y)$ , involving the y-smooth numbers counting function. We (...)

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[hal-01476629] Nonarchimedean dynamical systems and formal groups

25 février 2017

We confirm an observation of Lubin concerning families of $p$-adic power series that commute under composition. We prove that under certain conditions, there is a formal group such that the power series in the family are either endomorphisms of this group, or at least semi-conjugate to (...)

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[hal-01475616] On M-functions associated with modular forms

24 février 2017

Let $f$ be a primitive cusp form of weight $k$ and level $N,$ let $\chi$ be a Dirichlet character of conductor coprime with $N,$ and let $\mathfrakL(f\otimes \chi, s)$ denote either $\log L(f\otimes \chi, s)$ or $(L'/L)(f\otimes \chi, s).$ In this article we study the distribution of the values (...)

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[hal-00172014] Quasi-semi-stable representations

24 février 2017

Fix K a p-adic field and denote by G_K its absolute Galois group. Let K_infty be the extension of K obtained by adding (p^n)-th roots of a fixed uniformizer, and G_\infty its absolute Galois group. In this article, we define a class of p-adic torsion representations of G_\infty, named (...)

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[hal-00708733] Formes différentielles réelles et courants sur les espaces de Berkovich

24 février 2017

We define a theory of real $(p,q)$-forms and currents on Berkovich spaces which is parallel to the theory of differential forms on complex spaces: integrals, Poincaré-Lelong formula, plurisubharmonic functions, positive currents, Chern forms of metrized line (...)

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[hal-00136039] Lifting Galois Representations, and a Conjecture of Fontaine and Mazur

23 février 2017

Soit F un corps de nombres. Une représentation p-adique de dimension 8 du groupe de Galois absolu de F est appelé une représentation à la Mumford si l'algèbre de Lie de son image est c + sl(2)^3 et si cette algèbre de Lie agit via le produit tensoriel des représentations standard. Quitte à replacer F (...)

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[hal-00256409] Arithmetic $\D$-modules and Representations

23 février 2017

We propose in this paper an approach to Breuil's conjecture on a Langlands correspondence between $p$-adic Galois representations and representations of $p$-adic Lie groups in $p$-adic topological vector spaces. We suggest that Berthelot's theory of arithmetic $D$-modules should give a $p$-adic (...)

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[hal-01300328] Valeurs moyennes effectives de fonctions multiplicatives complexes

23 février 2017

Nous établissons des estimations de valeurs moyennes pour une large classe de fonctions arithmétiques multiplicatives, fournissant ainsi des versions quantitatives essentiellement optimales des évaluations classiques de Wirsing et étendant celles de Halász. Plusieurs applications sont explicitées, (...)

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[hal-01469972] On function field Mordell-Lang: the semiabelian case and the socle theorem

22 février 2017

In this paper we complete our " second generation " model-theoretic account of the function field Mordell-Lang conjecture, where we avoid appeal to dichotomy theorems for (generalized) Zariski geometries. In the current paper we reduce the semiabelian case to the abelian case using (...)

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[hal-01464714] Rational points on linear slices of diagonal hypersurfaces

22 février 2017

An asymptotic formula is obtained for the number of rational points of bounded height on the class of varieties described in the title line. The formula is proved via the Hardy-Littlewood method, and along the way we establish two new results on Weyl sums that are of some independent (...)

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[hal-01472833] A new simple recursive algorithm for finding prime numbers using Rosser's theorem

21 février 2017

In our previous work (The distribution of prime numbers: overview of n.ln(n), (1) and (2)) we defined a new method derived from Rosser's theorem (2) and we used it in order to approximate the nth prime number. In this paper we improve our method to try to determine the next prime number if the (...)

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[hal-00720823] Imaginärquadratische Einbettung von Ordnungen rationaler Quaternionenalgebren, und die nichtzyklischen endlichen Untergruppen der Bianchi-Gruppen

21 février 2017

Seien k ein imaginaerquadratischer Zahlkoerpemit Hauptordnung o, F eine rationale Quaternionenalgebra und M eine Erweiterung von F zur k-Algebra. Wir klassifizieren die F-Ordnungen G, die sich optimal in eine M-Maximalordnung N einbetten lassen, d. h. fuer die G = F geschnitten N gilt. Der (...)

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[hal-01469950] ADDITION THEOREMS IN Fp VIA THE POLYNOMIAL METHOD

21 février 2017

In this article, we use the Combinatorial Nullstellensatz to give new proofs of the Cauchy-Davenport, the Dias da Silva-Hamidoune and to generalize a previous addition theorem of the author. Precisely, this last result proves that for a set A ⊂ Fp such that A ∩ (−A) = ∅ the cardinality of the set (...)

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[hal-00854664] Dirichlet series associated to quartic fields with given cubic resolvent

21 février 2017

Let $k$ be a cubic field. We give an explicit formula for the Dirichlet series $\sum_K|\Disc(K)|^-s$, where the sum is over isomorphism classes of all quartic fields whose cubic resolvent field is isomorphic to $k$. Our work is a sequel to an unpublished preprint of Cohen, Diaz y Diaz, and (...)

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[hal-01260806] Effective arithmetic in finite fields based on Chudnovsky's multiplication algorithm

20 février 2017

Thanks to a new construction of the Chudnovsky and Chudnovsky multiplication algorithm, we design efficient algorithms for both the exponentiation and the multiplication in finite fields. They are tailored to hardware implementation and they allow computations to be parallelized, while (...)

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[hal-01174800] Comptage de représentations cuspidales congruentes

17 février 2017

Let $\F$ be a non-Archimedean locally compact field of residue characteristic $p$, $\G$ be an inner form of $\GL_n(\F)$, $n\>1$, and $\ell$ be a prime number different from $p$. We give a numerical criterion for an integral $\ell$-adic irreducible cuspidal representation $\rt$ of $\G$ to (...)

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[hal-01288587] Sur l’exemple d’Euler d’une fonction complètement multiplicative de somme nulle

17 février 2017

Euler a publié une formule que nous écrivons aujourd'hui ∞ 1 λ(n) n = 0, λ étant la fonction complètement multiplicative qui vaut −1 sur les nombres premiers. Ainsi λ(n) n est un exemple de fonction CM O (complètement mul-tiplicative à somme nulle). Nous étendons cette formule au cas où λ est dé-nie sur (...)

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[in2p3-01470087] Monodromy Relations in Higher-Loop String Amplitudes

17 février 2017

New monodromy relations of loop amplitudes are derived in open string theory. We particularly study N-point one-loop amplitudes described by a world-sheet cylinder (planar and non-planar) and derive a set of relations between subamplitudes of different color orderings. Various consistency (...)

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[hal-01467925] The distribution of prime numbers: overview of n.ln(n)

16 février 2017

The empirical formula giving the nth prime number p(n) is p(n) = n.ln(n) (from ROSSER (2)). Other studies have been performed (from DUSART for example (1)) in order to better estimate the nth prime number. Unfortunately these formulas don't work since there is a significant difference between (...)

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[hal-01361276] Normes cyclotomiques naïves et unités logarithmiques

16 février 2017

Nous déterminons le rang du sous-groupe des éléments du groupe multiplicatif d'un corps de nombres K qui sont normes à chaque étage fini de sa Zℓ-extension cyclotomique ; et nous comparons son ℓ-adifié avec le ℓ-groupe des unités logarithmiques. Nous donnons à cette occasion une preuve très facile de la (...)

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[hal-01465483] Simple connexité des fibres d'une application d'Abel-Jacobi et corps de classe local

15 février 2017

Cet article traite du cas abélien de la conjecture de géométrisation de la correspondance de Langlands locale formulée par l'auteur. On y montre qu'un certain morphisme d'Abel-Jacobi et une fibration pro-étale localement triviale en diamants simplement connexes. On en déduit la conjecture pour (...)

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[hal-01456773] Double spend races

15 février 2017

We correct the double spend race analysis given in Nakamoto's foun-dational Bitcoin article and give a closed-form formula for the probability of success of a double spend attack using the Regularized Incomplete Beta Function. We give the first proof of the exponential decay on the number of (...)

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[hal-01216763] FROM LOCAL CLASS FIELD TO THE CURVE AND VICE VERSA

14 février 2017

We begin by reviewing our joint work with J.-M. Fontaine about the fundamental curve of p-adic Hodge theory. We then explain our results obtained in [4] about the classification of G-bundles on this curve and its link with local class field theory. We finish by formulating conjectures that (...)

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[hal-01464831] On the van der Corput's $k$-th derivative test for exponential sums

14 février 2017

We give an overview of van der Corput's method for exponential sums, with a particular interest for the simplest estimates with the $k$-thderivative test. We study the optimality of the results and we present recent improvements.

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[hal-01139920] G-torseurs en théorie de Hodge p-adique

14 février 2017

Etant donné un groupe réductif G sur une extension de degré fini de Qp on classifie les G-fibrés sur la courbe introduite dans nos travaux en commun avec Fontaine. Le résultat est interprété en termes de l'ensemble B(G) de Kottwitz. On calcule également la cohomologie étale de la courbe à coefficients (...)

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[hal-01445353] Persistance d'un nombre

14 février 2017

Je démontre qu’en base 10, il n'existe aucun nombre dont la persistance multiplicative est supérieure ou égale à 12.

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[hal-01236582] Right-jumps and pattern avoiding permutations

13 février 2017

We study the iteration of the process "a particle jumps to the right" in permutations. We prove that the set of permutations obtained in this model after a given number of iterations from the identity is a class of pattern avoiding permutations. We characterize the elements of the (...)

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[hal-01445353] Persistance d'un nombre

13 février 2017

Je démontre qu’en base 10, il n'existe aucun nombre dont la persistance multiplicative est supérieure ou égale à 12.

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[hal-00863161] Computing p-adic L-functions of totally real number fields

13 février 2017

We prove explicit formulas for the p-adic L-functions of totally real number fields and show how these formulas can be used to compute values and representations of p-adic L-functions.

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[hal-00863162] A Brumer-Stark Conjecture for non-abelian Galois extensions

13 février 2017

The Brumer-Stark conjecture deals with abelian extensions of number fields and predicts that a group ring element, called the Brumer-Stickelberger element constructed from special values of L-functions associated to the extension, annihilates the ideal class group of the extension under (...)

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[hal-01463665] Periodic balanced binary triangles

13 février 2017

A binary triangle of size $n$ is a triangle of zeroes and ones, with $n$ rows, built with the same local rule as the standard Pascal triangle modulo $2$. A binary triangle is said to be balanced if the absolute difference between the numbers of zeroes and ones that constitute this triangle is (...)

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[hal-01463366] A local model for the trianguline variety and applications

11 février 2017

We describe the completed local rings of the trianguline variety at certain points of integral weights in terms of completed local rings of algebraic varieties related to Grothendieck's simultaneous resolution of singularities. We derive several local consequences at these points for the (...)

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[hal-01464788] On the fourth derivative test for exponential sums

10 février 2017

We give an upper bound for the exponential sum $\sum_m=1^Mexp(2i\pi f(m))$ where $f$ is a real-valued function whose fourth derivative has the order of magnitude $\lambda>0$ small. Van der Corput's classical bound, in terms of $M$ and $\lambda$ only, involves the exponent $1/14$. We show how (...)

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[hal-01205177] A classification of ${\mathbb C}$-Fuchsian subgroups of Picard modular groups

10 février 2017

Given an imaginary quadratic extension $K$ of $\mathbb Q$, we give a classification of the maximal nonelementary subgroups of the Picard modular group $PSU_1,2(\mathcal O_K)$ preserving a complex geodesic in the complex hyperbolic plane $\mathbb H^2_\mathbb C$. Complementing work of Holzapfel, (...)

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[hal-00486680] Relaxed algorithms for p-adic numbers

10 février 2017

Current implementations of p-adic numbers usually rely on so called zealous algorithms, which compute with truncated p-adic expansions at a precision that can be specified by the user. In combination with Newton-Hensel type lifting techniques, zealous algorithms can be made very efficient from (...)

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[hal-01217995] Distributions of traces of Frobenius for smooth plane curves over finite fields

9 février 2017

In a previous article, we obtained data on the distribution of traces of Frobenius of non-hyperelliptic genus 3 curves over small finite fields. In the present one, we give a heuristic explanation of these data, by extrapolating from results on the distribution of traces of Frobenius for plane (...)

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[hal-00828153] New uniform and asymptotic upper bounds on the tensor rank of multiplication in extensions of finite fields

9 février 2017

We obtain new uniform upper bounds for the tensor rank of the multiplication in the extensions of the finite fields $\mathbbF_q$ for any prime power $q$; moreover these uniform bounds lead to new asymptotic bounds as well. In addition, we also give purely asymptotic bounds which are (...)

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[hal-00689746] Describing A Cyclic Code by Another Cyclic Code

9 février 2017

A new approach to bound the minimum distance of $q$-ary cyclic codes is presented. The connection to the BCH and the Hartmann--Tzeng bound is formulated and it is shown that for several cases an improvement is achieved. We associate a second cyclic code to the original one and bound its minimum (...)

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[hal-01459826] Multiplicity one at full congruence level

7 février 2017

Let F be a totally real field in which p is unramified. Let r be a modular representation of G_F on a two dimension mod p vector space satisfying the Taylor--Wiles hypotheses and tamely ramified and generic at a place v above p. Let m be the corresponding Hecke eigensystem. We describe the (...)

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[hal-01216803] Smoothness and Classicality on eigenvarieties

7 février 2017

Let p be a prime number and f an overconvergent p-adic automorphic form on a definite unitary group which is split at p. Assume that f is of "classical weight" and that its Galois representation is crystalline at places dividing p, then f is conjectured to be a classical automorphic (...)

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[hal-00909045] Density of potentially crystalline representations of fixed weight

7 février 2017

Let K be a finite extension of Qp. We fix a continuous absolutely irreducible representation of the absolute Galois group of K over a finite dimensional vector space with coefficient in a finite field of characteristic p and consider its universal deformation ring R. If we fix a regular set of (...)

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[hal-00661644] Sur la présentation des représentations supersinguliéres de $\mathrm{GL}_2(F)$

7 février 2017

Soit $F$ une extension quadratique de $\mathbbQ_p$. Nous prouvons que les représentations lisses irréductibles supersingulières à caractère central de $\mathrmGL_2(F)$ ne sont pas de présentation finie.

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[hal-01458027] Characteristic polynomials of p-adic matrices

7 février 2017

We analyze the precision of the characteristic polynomial of an n×n p-adic matrix A using differential precision methods developed previously. When A is integral with precision O(p^N), we give a criterion (checkable in time O˜(n^omega)) for χ(A) to have precision exactly O(p^N). We also give a (...)

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[hal-01455679] Discriminants of complete intersection space curves

7 février 2017

In this paper, we develop a new approach to the discriminant of a complete intersection curve in the 3-dimensional projective space. By relying on the resultant theory, we first prove a new formula that allows us to define this discriminant without ambiguity and over any commutative ring, in (...)

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[hal-01445353] Persistance d'un nombre

6 février 2017

Je démontre qu’en base 10, il n'existe aucun nombre dont la persistance multiplicative est supérieure ou égale à 12.

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[hal-01455036] Plane quartics over Q with complex multiplication

3 février 2017

We give examples of smooth plane quartics over $Q$ with complex multiplication over $\overlineQ$ by a maximal order with primitive CM type. We describe the required algorithms as we go, these involve the reduction of period matrices, the fast computation of Dixmier-Ohno invariants, and (...)

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[hal-01450032] Linear independence of values of G-functions

2 février 2017

Given any non-polynomial $G$-function $F(z)=\sum_k=0^\infty A_k z^k$ of radius of convergence $R$, we consider the $G$-functions $F_n^[s](z)=\sum_k=0^\infty \fracA_k(k+n)^sz^k$ for any integers $s\geq 0$ and $n\geq 1$. For any fixed algebraic number $\alpha$ such that $0 < \vert \alpha \vert (...)

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[hal-00709802] The Tate conjecture for K3 surfaces over finite fields

2 février 2017

Artin's conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin's conjecture over fields of characteristic p>3. This implies Tate's conjecture for K3 surfaces over finite fields of characteristic p>3. Our results also yield (...)

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[hal-00600373] The Theorem of Jentzsch--Szegö on an analytic curve. Application to the irreducibility of truncations of power series

2 février 2017

The theorem of Jentzsch--Szeg\Ho describes the limit measure of a sequence of discrete measures associated to the zeroes of a sequence of polynomials in one variable. Following the presentation of this result by Andrievskii and Blatt in their book, we extend this theorem to compact Riemann (...)

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[hal-01447059] Computation of Euclidean minima in totally definite quaternion fields

1er février 2017

We describe an algorithm that allows to compute the Euclidean minimum (for the norm form) of any order of a totally definite quaternion field over a number field K of degree strictly greater than 1. Our approach is a generalization of previous work dealing with number fields. The algorithm was (...)

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[hal-01433370] Brisure de symétrie et nombres de lychrel

31 janvier 2017

Nous montrons que l'existence des nombres de Lychrel est due à une brisure de symétrie dans l'espace des nombres.

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[hal-01264150] Some more functions that are not APN infinitely often. The case of Gold and Kasami exponents

29 janvier 2017

We prove a necessary condition for some polynomials of Gold and Kasami degree to be APN over F q n for large n.

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[hal-00828070] Shimura modular curves and asymptotic symmetric tensor rank of multiplication in any finite field

29 janvier 2017

We obtain new asymptotical bounds for the symmetric tensor rank of multiplication in any finite extension of any finite field~$\F_q$. In this aim, we use the symmetric Chudnovsky-type generalized algorithm applied on a family of Shimura modular curves defined over $\F_q^2$ attaining the (...)

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[hal-00145890] Nonlinarity of Boolean functions and hyperelliptic curves

29 janvier 2017

We study the nonlinearity of functions defined on a finite field with 2^m elements which are the trace of a polynomial of degree 7 or more general polynomials of binary degree equal to 3. We show that for m odd such functions have rather good nonlinearity properties. We use for that recent (...)

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[hal-00151591] Non linéarité des fonctions booléennes données par des traces de polynômes de degré binaire 3

29 janvier 2017

We study the nonlinearity of functions defined on a finite field with 2^m elements which are the trace of a polynomial of degree 7 or more general polynomials of binary degree equal to 3. We show that for m odd such functions have rather good nonlinearity properties. We use for that recent (...)

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[hal-01448316] Heights on square of modular curves

27 janvier 2017

We develop a strategy for bounding from above the height of rational points of modular curves with values in number fields, by functions which are polynomial in the level. Our main technical tools come from effective arakelovian descriptions of modular curves and jacobians. We then fulfill this (...)

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[hal-01444183] Computations with p-adic numbers

27 janvier 2017

This document contains the notes of a lecture I gave at the "Journées Nationales du Calcul Formel" (JNCF) on January 2017. The aim of the lecture was to discuss low-level algorithmics for p-adic numbers. It is divided into two main parts: first, we present various implementations of (...)

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[hal-01433370] BRISURE DE SYMÉTRIE ET NOMBRES DE LYCHREL

26 janvier 2017

Nous montrons que l'existence des nombres de Lychrel est due à une brisure de symétrie dans l'espace des nombres.

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[hal-00772014] Rational points on X_0^+ (p^r)

26 janvier 2017

We show how the recent isogeny bounds due to É. Gaudron and G. Rémond allow to obtain the triviality of X_0^+ (p^r)(Q), for r>1 and p a prime exceeding 2.10^11. This includes the case of the curves X_split (p). We then prove, with the help of computer calculations, that the same holds true for p (...)

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[hal-01445353] Persistance d'un nombre

24 janvier 2017

Je démontre qu’en base 10, il n'existe aucun nombre dont la persistance multiplicative est supérieure ou égale à 12.

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[hal-01281334] Sur la concentration de certaines fonctions additives

24 janvier 2017

Improving on estimates of Erdős, Halász and Ruzsa, we provide new upper and lower bounds for the concentration function of the limit law of certain additive arithmetic functions under hypotheses involving only their average behaviour on the primes. In particular we partially confirm a conjecture (...)

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[hal-01444560] The p-adic Kummer-Leopoldt constant -- Normalized p-adic regulator

24 janvier 2017

The p-adic Kummer--Leopoldt constant kappa(K) of a number field K is (under the Leopoldt conjecture for p in K) the least integer c such that for all n large enough, any global unit epsilon of K, which is locally (at the p-places) a p^n+cth power, is necessarily a p^nth power of a global unit (...)

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[hal-01441140] Computer Visualization of the Riemann Zeta Function

20 janvier 2017

Various representations of the multi-dimensional Riemann zeta function ζ(s) are provided herein. The representations include 3D plots in the xyz space as well as contour maps where the third dimension is represented by color or iso-line contours. Some polar projections of the zeta function are (...)

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[tel-01438559] On the 16-rank of class groups of quadratic number fields

20 janvier 2017

We prove two new density results about 16-ranks of class groups of quadratic number fields. The first of the two is that the class group of Q(sqrt-p) has an element of order 16 for one-fourth of prime numbers p that are of the form a^2+c^4 with c even. The second is that the class group of (...)

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[hal-01433370] BRISURE DE SYMÉTRIE ET NOMBRES DE LYCHREL

19 janvier 2017

Nous montrons que l'existence des nombres de Lychrel est due à une brisure de symétrie dans l'espace des nombres.

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[hal-00118511] On the spacing distribution of the Riemann zeros: corrections to the asymptotic result

17 janvier 2017

It has been conjectured that the statistical properties of zeros of the Riemann zeta function near $z = 1/2 + \ui E$ tend, as $E \to \infty$, to the distribution of eigenvalues of large random matrices from the Unitary Ensemble. At finite $E$ numerical results show that the nearest-neighbour (...)

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[hal-00814356] Rankin-Selberg methods for closed strings on orbifolds

17 janvier 2017

In recent work we have developed a new unfolding method for computing one-loop modular integrals in string theory involving the Narain partition function and, possibly, a weak almost holomorphic elliptic genus. Unlike the traditional approach, the Narain lattice does not play any role in the (...)

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[hal-00639799] A new look at one-loop integrals in string theory

17 janvier 2017

We revisit the evaluation of one-loop modular integrals in string theory, employing new methods that, unlike the traditional 'orbit method', keep T-duality manifest throughout. In particular, we apply the Rankin-Selberg-Zagier approach to cases where the integrand function grows at most (...)

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[hal-00678471] One-Loop BPS amplitudes as BPS-state sums

17 janvier 2017

Recently, we introduced a new procedure for computing a class of one-loop BPS-saturated amplitudes in String Theory, which expresses them as a sum of one-loop contributions of all perturbative BPS states in a manifestly T-duality invariant fashion. In this paper, we extend this procedure to all (...)

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[hal-00487420] Rigid Calabi-Yau threefolds, Picard Eisenstein series and instantons

17 janvier 2017

Type IIA string theory compactified on a rigid Calabi-Yau threefold gives rise to a classical moduli space that carries an isometric action of U(2,1). Various quantum corrections break this continuous isometry to a discrete subgroup. Focussing on the case where the intermediate Jacobian of the (...)

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[hal-01430637] On the erdös flat polynomials problem, chowla conjecture and riemann hypothesis

16 janvier 2017

There are no square L 2-flat sequences of polynomials of the type 1 √ q (ǫ 0 + ǫ 1 z + ǫ 2 z 2 + · · · + ǫ q−2 z q−2 + ǫqz q−1), where for each j, 0 ≤ j ≤ q −1, ǫ j = ±1. It follows that Erdös's conjectures on Littlewood polynomials hold. Consequently, Turyn-Golay's conjecture is true, that is, there are only (...)

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[hal-01298060] Weyl sums, mean value estimates, and Waring's problem with friable numbers

12 janvier 2017

In this paper we study Weyl sums over friable integers (more precisely $y$-friable integers up to $x$ when $y = (\log x)^C$ for a large constant $C$). In particular, we obtain an asymptotic formula for such Weyl sums in major arcs, nontrivial upper bounds for them in minor arcs, and moreover a (...)

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[hal-01430637] On the erdös flat polynomials problem, chowla conjecture and riemann hypothesis

12 janvier 2017

There are no square L 2-flat sequences of polynomials of the type 1 √ q (ǫ 0 + ǫ 1 z + ǫ 2 z 2 + · · · + ǫ q−2 z q−2 + ǫqz q−1), where for each j, 0 ≤ j ≤ q −1, ǫ j = ±1. It follows that Erdös's conjectures on Littlewood polynomials hold. Consequently, Turyn-Golay's conjecture is true, that is, there are only (...)

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[hal-01430637] On the erdös flat polynomials problem, chowla conjecture and riemann hypothesis

11 janvier 2017

There are no square L 2-flat sequences of polynomials of the type 1 √ q (ǫ 0 + ǫ 1 z + ǫ 2 z 2 + · · · + ǫ q−2 z q−2 + ǫqz q−1), where for each j, 0 ≤ j ≤ q −1, ǫ j = ±1. It follows that Erdös's conjectures on Littlewood polynomials hold. Consequently, Turyn-Golay's conjecture is true, that is, there are only (...)

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[hal-01429498] Local theta-regulators of an algebraic number -- p-adic Conjectures

10 janvier 2017

Let K/Q be Galois and let eta in K* be such that the multiplicative Z[G]-module generated by eta is of Z-rank n. We define the local theta-regulators Delta_p^theta(eta) in F_p for the Q_p-irreducible characters theta of G=Gal(K/Q). Let V_theta be the theta-irreducible representation. A linear (...)

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[hal-01426832] Certification of modular Galois representations

9 janvier 2017

We show how the output of the algorithm to compute modular Galois representations described in our previous article can be certified. We have used this process to compute certified tables of such Galois representations obtained thanks to an improved version of this algorithm, including (...)

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[hal-00133410] Mesures et équidistribution sur les espaces de Berkovich

6 janvier 2017

The proof by Ullmo and Zhang of Bogomolov's conjecture about points of small height in abelian varieties made a crucial use of an equidistribution property for ``small points'' in the associated complex abelian variety. We study the analogous equidistribution property at $p$-adic places. Our (...)

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[hal-01426349] On the Prym variety of genus 3 covers of genus 1 curves

6 janvier 2017

Given a generic degree-2 cover of a genus 1 curve D by a non hyperelliptic genus 3 curve C over a field k of characteristic different from 2, we produce an explicit genus 2 curve X such that Jac(C) is isogenous to the product of Jac(D) and Jac(X). This construction can be seen as a degenerate (...)

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[hal-00694113] A faster pseudo-primality test

6 janvier 2017

We propose a pseudo-primality test using cyclic extensions of $Z/nZ$. For every positive integer $k$ this test achieves the security of $k^2$ Miller-Rabin tests at the cost of $k^1+o(1)$ Miller-Rabin tests.

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[hal-00133407] Mesures de Mahler et équidistribution logarithmique

6 janvier 2017

Let X be a projective integral normal scheme over a number field F; let L be a ample line bundle on X together with a semi-positive adelic metric in the sense of Zhang. The main results of this article are 1) an analogue of a formule of Mahler which allows to compute the height (relative to L) (...)

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[hal-00145173] Schémas en groupes et poids de Diamond-Serre

6 janvier 2017

This note is a correction of (statement and proof of) proposition 3.3.1 of Toby Gee's preprint intitled *On the weights of mod p Hilbert modular forms*. The aim is to compare Galois representations arising from extensions of some group schemes (over the ring of integers of a p-adic field) (...)

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[hal-00373511] Galois invariant smoothness basis

6 janvier 2017

This text answers a question raised by Joux and the second author about the computation of discrete logarithms in the multiplicative group of finite fields. Given a finite residue field K, one looks for a smoothness basis for K* that is left invariant by automorphisms of K. For a broad class of (...)

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[hal-00456456] Fast construction of irreducible polynomials over finite fields

6 janvier 2017

We present a randomized algorithm that on input a finite field $K$ with $q$ elements and a positive integer $d$ outputs a degree $d$ irreducible polynomial in $K[x]$. The running time is $d^1+o(1) \times (\log q)^5+o(1)$ elementary operations. The $o(1)$ in $d^1+o(1)$ is a function of $d$ that (...)

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[hal-00455575] Dynamique p-adique (d'après des exposés de Jean-Christophe Yoccoz)

6 janvier 2017

This paper is an introduction to the dynamics of rational fractions over a p-adic field. It follows the lectures that Jean-Christophe Yoccoz gave during the États de la recherche in May 2006, and develops certain points. Most of the results to be explained here are due to Benedetto, Bézivin, (...)

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[hal-01102024] “Chinese & Match”, an alternative to Atkin's “Match and Sort” method used in the SEA algorithm

6 janvier 2017

A classical way to compute the number of points of elliptic curves defined over finite fields from partial data obtained in SEA (Schoof Elkies Atkin) algorithm is a so-called ``Match and Sort'' method due to Atkin. This method is a ``baby step/giant step'' way to find the number of points among (...)

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[hal-00630394] Galois invariant smoothness basis

6 janvier 2017

This text answers a question raised by Joux and the second author about the computation of discrete logarithms in the multiplicative group of finite fields. Given a finite residue field $\bK$, one looks for a smoothness basis for $\bK^*$ that is left invariant by automorphisms of $\bK$. For a (...)

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[hal-01281383] An application of a theorem of Emerton to mod $p$ representations of $\mathrm{GL}_2$

6 janvier 2017

Let $p$ be a prime and $L$ be a finite extension of $\mathbbQ_p$. We study the ordinary parts of $\mathrmGL_2(L)$-representations arised in the mod $p$ cohomology of Shimura curves attached to indefinite division algebras which splits at a finite place above $p$. The main tool of the proof is a (...)

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[hal-01176189] Correspondance de Jacquet-Langlands locale et congruences modulo ℓ

6 janvier 2017

Let F be a non-Archimedean local field of residual characteristic p, and ℓ be a prime number different from p. We consider the local Jacquet-Langlands correspondence between ℓ-adic discrete series of GL(n,F) and an inner form GL(m,D). We show that it respects the relationship of congruence modulo (...)

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[hal-00860947] Weil representation and metaplectic groups over an integral domain

6 janvier 2017

Given F a locally compact, non-discrete, non-archimedean field of characteristic different from 2 and R an integral domain such that a non-trivial smooth F-character with values in the multiplicative group of R exists, we construct the (reduced) metaplectic group attached to R. We show that it (...)

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[hal-01390143] CONGRUENCES AUTOMORPHES ET TORSION DANS LA COHOMOLOGIE D'UNE VARIÉTÉ DE SHIMURA UNITAIRE SIMPLE

5 janvier 2017

Nous donnons d'une part un procédé relativement souple de construction de classes de cohomologie de torsion dans la cohomologie d'une variété de Shimura de type Kottwitz-Harris-Taylor à coefficients dans un système local pas trop régulier. D'autre part nous montrons qu'associée à toute classe de (...)

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[hal-01029331] On the Power Series Expansion of the Reciprocal Gamma Function

5 janvier 2017

Using the reflection formula of the Gamma function, we derive a new formula for the Taylor coefficients of the reciprocal Gamma function. The new formula provides effective asymptotic values for the coefficients even for very small values of the indices. Both the sign oscillations and the (...)

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[inria-00386309] Modular Curves of Composite Level

5 janvier 2017

We examine a class of functions on $X_0 (N)$ where $N$ is the product of two arbitrary primes. The functions are built as products of Dedekind's $\eta$-function and play a role in the construction of elliptic curves with complex multiplication. We show how to determine the modular polynomials (...)

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[inria-00386311] Provably Secure Non-Interactive Key Distribution Based on Pairings

5 janvier 2017

We define a security notion for non-interactive key distribution protocols. We identify an apparently hard computational problem related to pairings, the Bilinear Diffie--Hellman problem (BDH). After extending Sakai, Ohgishi, and Kasahara's pairing based protocol to a slightly more general (...)

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[hal-00697614] Exemples de comptage de courbes sur les surfaces

5 janvier 2017

Let X be a surface whose Cox ring has a single relation satisfying moreover a kind of linearity property. Under a simple assumption, we show that the geometric Manin's conjectures hold for some degrees lying in the dual of the effective cone of X (in particular, for those degrees the moduli (...)

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[hal-00578750] An algorithm to compute relative cubic fields

5 janvier 2017

Let k be an imaginary quadratic number field (with class number 1). We describe a new, essentially linear-time algorithm, to list all isomorphism classes of cubic extensions L/k up to a bound X on the norm of the relative discriminant ideal. The main tools are Taniguchi's generalization of (...)

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[hal-00129368] Galois representations, Mumford-Tate groups and good reduction of abelian varieties

5 janvier 2017

Let K be a number field and A an abelian variety over K. We are interested in the following conjecture of Morita: if the Mumford-Tate group of A does not contain unipotent Q-rational points then A has potentially good reduction at any discrete place of K. The Mumford-Tate group is an object of (...)

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[hal-00672889] Smooth representations of GL_m(D) VI : semisimple types

5 janvier 2017

We give a complete description of the category of smooth complex representations of the multiplicative group of a central simple algebra over a locally compact nonarchimedean local field. More precisely, for each inertial class in the Bernstein spectrum, we construct a type and compute its (...)

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[hal-00661700] Endomorphism algebras of admissible p-adic representations of p-adic Lie groups

5 janvier 2017

Building on recent work of Ardakov and Wadsley, we prove Schur's lemma for absolutely irreducible admissible p-adic Banach space (respectively locally analytic) representations of p-adic Lie groups. We also prove finiteness results for the endomorphism algebra of an irreducible admissible (...)

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[hal-00661701] The Jordan-Hölder series of the locally analytic Steinberg representation

5 janvier 2017

We determine the composition factors of a Jordan-Hölder series including multiplicities of the locally analytic Steinberg representation. For this purpose we prove the acyclicity of the evaluated locally analytic Tits complex giving rise to the Steinberg representation. Further we describe some (...)

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[hal-00138538] Facteurs {$\bold Q$}-simples de {$J\sb 0(N)$} de grande dimension et de grand rang

5 janvier 2017

By improvements of J.-P Serre's methods, we prove lower bounds of the largest dimension of simple abelian subvarieties of J0(N), with or without conditions on the algebraic rang (these conditions are : rankX=0 or rankX=dimX).

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[hal-00138540] Petits zéros de fonctions {$L$} de formes modulaires

5 janvier 2017

We study the density conjecture of Katz and Sarnak about the zeros of the L functions of modular forms on the critical strip. We introduce new families of modular forms carcterized by the eigenvalues of the Atkin-Lehner operators. We prove that these families satisfy a weak form of the density (...)

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[hal-00138543] Statistique de la variable aléatoire L(sym^2f,1)

5 janvier 2017

We study the statistic of the value at 1 of the L function of the symmetric square of a primitive form computing his postive and negative moments. We deduce properties of the distribution of these values.

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[hal-00085619] Sur la complexité des nombres algébriques

5 janvier 2017

Let bgreater-or-equal, slanted2 be an integer. We prove that real numbers whose b-ary expansion satisfies some given, simple, combinatorial condition are transcendental. This implies that the b-ary expansion of any algebraic irrational number cannot be generated by a finite (...)

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[hal-00605397] Sur quelques représentations potentiellement cristallines de GL_2(Q_p)

5 janvier 2017

To each 2-dimensional irreducible p-adic representation of Gal(Qpbar/Qp) which becomes crystalline over an abelian extension of Q_p, we associate a Banach space B(V) endowed with a linear continuous unitary action of GL_2(Q_p). When V is moreover phi-semi-simple, we use the (phi,Gamma)-module (...)

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[hal-00616277] Périodes entières de groupes p-divisibles sur une base générale

5 janvier 2017

Faltings a démontré un théorème de comparaison à coefficients entiers entre module de Tate et module de Dieudonné filtré des groupes p-divisibles sur un anneau de valuation discrète p-adique complet. Dans cet article, nous généralisons son résultat sur une base plus générale, plus précisément, sur une (...)

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[hal-00617986] Presentation of an Iwasawa algebra : the case of \Gamma_1 SL(2,Z_p)

5 janvier 2017

Assume G is a semi-simple Chevalley group, so G(Zp ) ⊂ G(Qp ) is a max- imal compact subgroup. Both the p-adic representation theory of G(Qp ) and non-commutative Iwasawa theory involve the Iwasawa algebra of G(Zp ) or suitable congruence subgroups. It seems to have been assumed that explicit (...)

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[inria-00420605] Families of Explicit Isogenies of Hyperelliptic Jacobians

5 janvier 2017

We construct three-dimensional families of hyperelliptic curves of genus 6, 12, and 14, two-dimensional families of hyperelliptic curves of genus 3, 6, 7, 10, 20, and 30, and one-dimensional families of hyperelliptic curves of genus 5, 10 and 15, all of which are equipped with an an explicit (...)

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[hal-00880994] New identities relating wild Goppa codes

5 janvier 2017

For a given support $L\in \mathbbF_q^m^n$ and a polynomial $g\in \mathbbF_q^m[x]$ with no roots in $\mathbbF_q^m$, we prove equality between the $q$-ary Goppa codes $\Gamma_q(L,N(g)) = \Gamma_q(L,N(g)/g)$ where $N(g)$ denotes the norm of $g$, that is $g^q^m-1+\cdots +q+1.$ In particular, for (...)

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[inria-00547454] Construction of rational surfaces yielding good codes

5 janvier 2017

In the present article, we consider Algebraic Geometry codes on some rational surfaces. The estimate of the minimum distance is translated into a point counting problem on plane curves. This problem is solved by applying the upper bound "à la Weil" of Aubry and Perret together with the (...)

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[hal-01074153] Additive properties of sequences of pseudo s-th powers

5 janvier 2017

In this paper, we study (random) sequences of pseudo s-th powers, as introduced by Erdös and Rényi in 1960. In 1975, Goguel proved that such a sequence is almost surely not an asymptotic basis of order s. Our first result asserts that it is however almost surely a basis of order s + x for any x (...)

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[inria-00386299] Building curves with arbitrary small MOV degree over finite prime fields

5 janvier 2017

We present a fast algorithm for building ordinary elliptic curves over finite prime fields having arbitrary small MOV degree. The elliptic curves are obtained using complex multiplication by any desired discriminant.

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[hal-01069510] An upper bound on the number of rational points of arbitrary projective varieties over finite fields

4 janvier 2017

We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field. This bound depends only on the dimensions and degrees of the irreducible components and holds for very general varieties, even reducible and non (...)

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[hal-00833291] Sur la correspondance de Simpson p-adique. II : aspects globaux

4 janvier 2017

We develop a new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0. This second article is devoted to the global aspects of the (...)

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[hal-00725800] Formes modulaires de Hilbert modulo p et valeurs d'extensions galoisiennes

4 janvier 2017

Soit F un corps totalement réel, v une place de F non ramifiée divisant p et rhobar : Gal(Fbar/F) --> GL2(Fpbar) une représentation continue irréductible dont la restriction à un sous-groupe de décomposition Gal(Fvbar/Fv) en v est réductible et suffisamment générique. Si rhobar est modulaire (et (...)

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[hal-00009286] Caractérisation des formes modulaires par des valeurs de fonctions $L$

4 janvier 2017

We give an application of the theory of periods to the determination of modular forms by values of some of their $L$ functions. On pr\'esente une application de la th\'eorie des p\'eriodes \`a la d\'etermination des formes modulaires par les valeurs de certaines de leurs fonctions (...)

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[hal-00664709] Concerning the semistability of tensor products in Arakelov geometry

4 janvier 2017

We study the semistability of the tensor product of hermitian vector bundles by using the $\varepsilon$-tensor product and the geometric (semi)stability of vector subspaces in the tensor product of two vector spaces.

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[hal-00874925] Easy scalar decompositions for efficient scalar multiplication on elliptic curves and genus 2 Jacobians

4 janvier 2017

The first step in elliptic curve scalar multiplication algorithms based on scalar decompositions using efficient endomorphisms---including Gallant--Lambert--Vanstone (GLV) and Galbraith--Lin--Scott (GLS) multiplication, as well as higher-dimensional and higher-genus constructions---is to (...)

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[hal-00136082] Points rationnels et groupes fondamentaux : applications de la cohomologie $p$-adique

4 janvier 2017

In this talk, I report on three theorems concerning algebraic varieties over a field of characteristic $p>0$. a) over a finite field of cardinal $q$, two proper smooth varieties which are geometrically birational have the same number of rational points modulo $q$ (cf. Ekedahl, 1983). b) over (...)

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[hal-01419948] Conjecture de Syracuse : avancées inédites

4 janvier 2017

Bon nombre de résultats connus et nouveaux relatifs à la conjecture de Syracuse.

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[hal-01404933] Approche p-adique de la conjecture de Greenberg (cas totalement réel p-décomposé)

4 janvier 2017

Let k be a totally real number field ant let k∞ be its cyclotomic Zp-extension for a prime p > 2. We give (Theorem 3.4) a sufficient condition of nullity of the Iwasawa invariants λ, µ, when p totally splits in k, and we obtain important tables giving quadratic fields and various p for which we (...)

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[hal-01293573] A note on certain representations in characteristic p and associated functions

27 décembre 2016

We discuss certain representations of GL 2 Fq[T] in equal characteristic and associated vectorial modular forms

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[hal-01421031] Isogenies for point counting on genus two hyperelliptic curves with maximal real multiplication

27 décembre 2016

Schoof's classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, (...)

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[hal-01421211] Equidistribution and counting under equilibrium states in negatively curved spaces and graphs of groups. Applications to non-Archimedean Diophantine approximation

27 décembre 2016

In this book, we study equidistribution and counting problems concerning locally geodesic arcs in negatively curved spaces endowed with potentials, and we deduce, from the special case of tree quotients, various arithmetic applications to equidistribution and counting problems in (...)

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[ensl-01420345] Syntomic cohomology and p-adic regulators for varieties over p-adic fields.

23 décembre 2016

We show that the logarithmic version of the syntomic cohomology of Fontaine and Messing for semistable varieties over p-adic rings extends uniquely to a cohomology theory for varieties over p-adic fields that satisfies h-descent. This new cohomology-syntomic cohomology-is a Bloch-Ogus (...)

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[hal-00720823] Imaginärquadratische Einbettung von Ordnungen rationaler Quaternionenalgebren, und die nichtzyklischen endlichen Untergruppen der Bianchi-Gruppen

22 décembre 2016

Seien k ein imaginaerquadratischer Zahlkoerper, F eine rationale Quaternionenalgebra und M eine Erweiterung von F zur k-Quaternionenalgebra. Wir klassifizieren die F-Ordnungen, die als Durchschnitt einer M-Maximalordnung mit F auftreten. Wir zeigen, dass der Isomorphietyp einer M-Maximalordnung (...)

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[hal-01421031] Isogenies for point counting on genus two hyperelliptic curves with maximal real multiplication

21 décembre 2016

Schoof's classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, (...)

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[ensl-01420357] On syntomic regukators I: constructions.

21 décembre 2016

We show that classical Chern classes from higher (p-adic) K-theory to syntomic cohomology extend to logarithmic syntomic cohomology. These Chern classes are compatible – in a suitable sense – with addition, products, and λ-operations. They are also compatible with the canonical Gysin sequences (...)

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[ensl-01420358] SYNTOMIC COMPLEXES AND p-ADIC NEARBY CYCLES

21 décembre 2016

We compute syntomic cohomology of semistable affinoids in terms of cohomology of (ϕ, Γ)-modules which, thanks to work of Fontaine-Herr, Andreatta-Iovita, and Kedlaya-Liu, is known to compute Galois cohomology of these affinoids. For a semistable scheme over a mixed characteristic local ring this (...)

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[ensl-01420353] GEOMETRIC SYNTOMIC COHOMOLOGY AND VECTOR BUNDLES ON THE FARGUES-FONTAINE CURVE

21 décembre 2016

We show that geometric syntomic cohomology lifts canonically to the category of Banach-Colmez spaces and study its relation to extensions of modifications of vector bundles on the Fargues-Fontaine curve. We include some computations of geometric syntomic cohomology Spaces: they are finite rank (...)

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[ensl-01420349] On p-adic absolute Hodge cohomology and syntomic coefficients, I.

21 décembre 2016

We interpret syntomic cohomology defined in [49] as a p-adic absolute Hodge cohomology. This is analogous to the interpretation of Deligne-Beilinson cohomology as an absolute Hodge cohomol-ogy by Beilinson [8] and generalizes the results of Bannai [6] and Chiarellotto, Ciccioni, Mazzari [15] in (...)

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[ensl-01420347] SYNTOMIC COHOMOLOGY AND p-ADIC MOTIVIC COHOMOLOGY

21 décembre 2016

We prove a mixed characteristic analog of the Beilinson-Lichtenbaum Conjecture for p-adic motivic cohomology. It gives a description, in the stable range, of p-adic motivic cohomology (defined using algebraic cycles) in terms of differential forms. This generalizes a result of Geisser [10] from (...)

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[hal-01420485] Discrete Logarithms

21 décembre 2016

The Discrete Logarithm Problem (DLP) is one of the most used mathematical problems in asymmetric cryptography design, the other one being the integer factorization. It is intrinsically related to the Diffie-Hellman problem (DHP). DLP can be stated in various groups. It must be hard in (...)

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[hal-01419870] The logarithmic class group package in PARI/GP

20 décembre 2016

This note presents our implementation in the PARI/GP system of the various arithmetic invariants attached to logarithmic classes and units of number fields. Our algorithms simplify and improve on works of Diaz y Diaz, Pauli, Pohst, Soriano and the second (...)

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[hal-01352764] A formula for ζ(2n + 1) and some related expressions

14 décembre 2016

Using a polylogarithmic identity, we express the values of $\zeta$ at odd integers $2n+1$ as integrals over unit $n-$dimensional hypercubes of simple functions involving products of logarithms. We also prove a useful property of those functions as some of their variables are raised to a power. (...)

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[inria-00001040] The complexity of class polynomial computation via floating point approximations

13 décembre 2016

We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one (...)

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[inria-00440223] An L(1/3) algorithm for ideal class group and regulator computation in certain number fields

13 décembre 2016

We analyse the complexity of the computation of the class group structure, regulator, and a system of fundamental units of a certain class of number fields. Our approach differs from Buchmann's, who proved a complexity bound of L(1/2,O(1)) when the discriminant tends to infinity with fixed (...)

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[inria-00246115] Computing Hilbert Class Polynomials

13 décembre 2016

We present and analyze two algorithms for computing the Hilbert class polynomial $H_D$ . The first is a p-adic lifting algorithm for inert primes p in the order of discriminant D < 0. The second is an improved Chinese remainder algorithm which uses the class group action on CM-curves over (...)

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[hal-01135202] On some Euclidean properties of matrix algebras

13 décembre 2016

Let $\mathfrakR$ be a commutative ring and $n \in \mathbfZ_>1$. We study some Euclidean properties of the algebra $\mathrmM_n(\mathfrakR)$ of $n$ by $n$ matrices with coefficients in $\mathfrakR$. In particular, we prove that $\mathrmM_n(\mathfrakR)$ is a left and right Euclidean ring if and (...)

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[hal-01183284] Big image of Galois representations associated with finite slope $p$-adic families of modular forms

13 décembre 2016

We consider the Galois representation associated with a finite slope $p$-adic family of modular forms. We prove that the Lie algebra of its image contains a congruence Lie subalgebra of a non-trivial level. We describe the largest such level in terms of the congruences of the family with (...)

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[hal-01352764] A formula for ζ(2n + 1) and a proof of their irrationality

12 décembre 2016

Using a polylogarithmic identity, we express the values of $\zeta$ at odd integers $2n+1$ as integrals over unit $n-$dimensional hypercubes of simple functions involving products of logarithms. We also prove a useful property of those functions as some of their variables are raised to a power. (...)

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[tel-01408445] Uniformisation des variétés de Laumon-Rapoport-Stuhler et application à la correspondance de Langlands locale

8 décembre 2016

Cette thèse se compose de deux parties plus ou moins indépendantes. La partie I traite de l'uniformisation "à la Cerednik" des variétés dites "de Laumon, Rapoport et Stuhler". Dans la partie II, les résultats obtenus sont appliqués à la correspondance de Langlands locale afin de (...)

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[hal-01408447] LOWER BOUNDS FOR MAASS FORMS ON SEMISIMPLE GROUPS

6 décembre 2016

Let G be an anisotropic semisimple group over a totally real number field F. Suppose that G is compact at all but one infinite place v 0. In addition, suppose that G v0 is R-almost simple, not split, and has a Cartan involution defined over F. If Y is a congruence arithmetic manifold of (...)

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[hal-01408448] LARGE VALUES OF CUSP FORMS ON GL n

6 décembre 2016

We establish lower bounds on the sup norm of Hecke–Maass cusp forms on congruence quotients of GLn(R). The argument relies crucially on uniform estimates for Jacquet-Whittaker functions. These purely local results are of independent interest, and are valid in the more general context of split (...)

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[hal-01408064] Galois level and congruence ideal for $p$-adic families of finite slope Siegel modular forms

6 décembre 2016

We consider $p$-adic families of Siegel eigenforms of genus $2$ and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a (...)

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[tel-01408061] Big Galois image for $p$-adic families of positive slope automorphic forms

6 décembre 2016

Let $g=1$ or $2$ and $p>3$ be a prime. For the symplectic group $\mathrmGSp_2g$ the Hecke eigensystems appearing in the spaces of classical automorphic forms, of a fixed tame level and varying weight, are $p$-adically interpolated by a rigid analytic space, the $\mathrmGSp_2g$-eigenvariety. (...)

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[hal-01407334] A New and Deterministic Scheme for Characterizing The Organization of Prime Numbers

5 décembre 2016

The fundamental theorem of arithmetic states that any composite natural integer can be expressed in one and only one way as a product of prime numbers. This sets the understanding of the organization of prime numbers at the core of number theory. In this work we present a simple, (...)

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[hal-01407300] Note sur la conjecture de Greenberg

2 décembre 2016

Nous étudions la conjecture de Greenberg sur les ℓ-invariants d'Iwasawa des corps totalement réels, en termes de classes logarithmiques. Nous faisons apparaître un critère suffisant de validité très simple dont nous montrons qu'il est également suffisant dans le contexte de la conjecture faible. Dans (...)

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[hal-01404884] Twisted characteristic $p$ zeta functions

1er décembre 2016

We propose a " twisted " variation of zeta functions introduced by David Goss in 1979.

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[hal-01406713] Algebraic points, non-anticanonical heights and the Severi problem on toric varieties

1er décembre 2016

In this article, we apply counting formulas for the number of morphisms from a curve to a toric variety to three different though related contexts (the first two are to be understood over global function fields): Manin’s problem for rational points of bounded non-anticanonical height, (...)

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[hal-00491561] Intégration numérique par la méthode double-exponentielle

30 novembre 2016

On démontre la convergence de la méthode d'intégration double-exponentielle pour des fonctions holomorphes, et on étudie son application à l'intégration sur divers domaines.

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[hal-01313548] On certain recurrent and automatic sequences in finite fields

30 novembre 2016

In this work we extend our study on a link between automaticity and certain algebraic power series over finite fields. Our starting point is a family of sequences in a finite field of characteristic $2$, recently introduced by the first author in connection with algebraic continued fractions. (...)

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[tel-01403849] Explicit computation of the Abel-Jacobi map and its inverse

30 novembre 2016

The Abel-Jacobi map links the short Weierstrass form of a complex elliptic curve to the complex torus associated to it. One can compute it with a number of operations which is quasi-linear in the target precision, i.e. in time O(M(P) log P). Its inverse is given by Weierstrass's p-function, (...)

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[hal-01404933] Approche p-adique de la conjecture de Greenberg (cas totalement réel p-décomposé)

30 novembre 2016

Let k be a totally real number field ant let k∞ be its cyclotomic Zp-extension for a prime p > 2. We give (Theorem 3.4) a sufficient condition of nullity of the Iwasawa invariants λ, µ, when p totally splits in k, and we obtain important tables giving quadratic fields and various p for which we (...)

lire la suite

[hal-01389421] Fast Computation of the $N$th Term of an Algebraic Series over a Finite Prime Field

29 novembre 2016

We address the question of computing one selected term of an algebraic power series. In characteristic zero, the best algorithm currently known for computing the~$N$th coefficient of an algebraic series uses differential equations and has arithmetic complexity quasi-linear in~$\sqrtN$. We show (...)

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[hal-00396375] Explicit unifor estimation of rational points II. Hypersurface coverings

29 novembre 2016

By using the slope method, we obtain an explicit uniform estimation for the density of rational points in an arithmetic projective variety with given degree and dimension, embedded in a given arithmetic projective space.

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[hal-00605387] Théorie d'Iwasawa des représentations cristallines II

25 novembre 2016

Let $K$ be a finite unramified extension of $\Qp$ and let $V$ be a crystalline representation of $\mathrmGal(\Qpbar/K)$. In this article, we give a proof of the $C_\mathrmEP(L,V)$ conjecture for $L \subset \Qp^\mathrmab$ as well as a proof of its equivariant version $C_\mathrmEP(L/K,V)$ for $L (...)

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[hal-00605400] La correspondance de Langlands locale p-adique pour GL_2(Q_p)

25 novembre 2016

La correspondance de Langlands locale p-adique pour GL_2(Q_p) est une bijection entre certaines representations de dimension 2 de Gal(Q_p^bar/Q_p) et certaines representations de GL_2(Q_p). Cette bijection peut en fait etre construite en utilisant la theorie des (phi,Gamma)-modules et des (...)

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[hal-00605404] Local constancy for the reduction mod p of 2-dimensional crystalline representations

25 novembre 2016

Irreducible crystalline representations of dimension 2 of Gal(Qpbar/Qp) depend up to twist on two parameters, the weight k and the trace of frobenius a_p. We show that the reduction modulo p of such a representation is a locally constant function of a_p (with an explicit radius) and a locally (...)

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[hal-00605966] Central characters for smooth irreducible modular representations of GL_2(Q_p)

25 novembre 2016

We prove that every smooth irreducible F_p^alg-linear representation of GL_2(Q_p) admits a central character.

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[hal-00605401] A p-adic family of dihedral (phi,Gamma)-modules

25 novembre 2016

The goal of this article is to construct explicitly a p-adic family of representations (which are dihedral representations), to construct their associated (phi,Gamma)-modules by writing down explicit matrices for phi and for the action of Gamma, and finally to determine which of these are (...)

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[hal-00840755] Lifting the field of norms

25 novembre 2016

Let K be a finite extension of Q_p. The field of norms of a p-adic Lie extension K_infty/K is a local field of characteristic p which comes equipped with an action of Gal(K_infty/K). When can we lift this action to characteristic 0, along with a compatible Frobenius map? In this note, we (...)

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[hal-00671261] Irreducible modular representations of the Borel subgroup of GL_2(Q_p)

25 novembre 2016

Let E be a finite extension of Fp. Using Fontaine's (phi,Gamma)-modules, Colmez has shown how to attach to any irreducible E-linear representation of Gal(Qpbar/Qp) an infinite dimensional smooth irreducible E-linear representation of B_2(Qp) that has a central character. We prove that every (...)

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[hal-01116294] Iterated extensions and relative Lubin-Tate groups

25 novembre 2016

Let K be a finite extension of Q_p with residue field F_q and let P(T) = T^d + a_d-1T^d-1 + ... +a_1 T, where d is a power of q and a_i is in the maximal ideal of K for all i. Let u_0 be a uniformizer of O_K and let u_n_n \geq 0 be a sequence of elements of Q_p^alg such that P(u_n+1) = u_n for (...)

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[hal-00920290] Multivariable (phi,Gamma)-modules and locally analytic vectors

25 novembre 2016

Let K_\infty/K be a Galois extension such that K_\infty contains the cyclotomic extension and such that Gamma=Gal(K_\infty/K) is a p-adic Lie group. We construct (phi,Gamma)-modules over the ring of locally analytic vectors (for the action of Gamma) of some of Fontaine's rings. When K_\infty is (...)

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[hal-00754931] Multivariable Lubin-Tate (\phi,\Gamma)-modules and filtered \phi-modules

25 novembre 2016

We define some rings of power series in several variables, that are attached to a Lubin-Tate formal module. We then give some examples of (\phi,\Gamma)-modules over those rings. They are the global sections of some vector bundles on the p-adic open unit polydisk, that are constructed from a (...)

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[hal-00996074] Théorie de Sen et vecteurs localement analytiques

25 novembre 2016

We generalize Sen theory to extensions $K_\infty/K$ whose Galois group is a $p$-adic Lie group of arbitrary dimension. To do so, we replace Sen's space of $K$-finite vectors by Schneider and Teitelbaum's space of locally analytic vectors. One then gets a vector space over the field of locally (...)

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[hal-01402391] Potentially semi-stable deformation rings for discrete series extended types

24 novembre 2016

We define deformation rings for potentially semi-stable deformations of fixed discrete series extended type in dimension 2. In the case of representations of the Galois group of ℚ p , we prove an analogue of the Breuil-Mézard conjecture for these rings. As an application, we give some results on (...)

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[hal-01402388] Reduction of Galois Representations of slope 1

24 novembre 2016

We compute the reductions of irreducible crystalline two-dimensional representations of GQp of slope 1, for primes p≥5, and all weights. We describe the semisimplification of the reductions completely. In particular, we show that the reduction is often reducible. We also investigate whether the (...)

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[hal-01402378] An algorithm for computing the reduction of 2-dimensional crystalline representations of $\text{Gal}(\overline\mathbf{Q}_p/\mathbf{Q}_p)$

24 novembre 2016

We describe an algorithm to compute the reduction modulo p of a crystalline Galois representation of dimension 2 of $\textGal(\overline\mathbfQ_p/\mathbfQ_p)$ with distinct Hodge-Tate weights via the semi-simple modulo p Langlands correspondence. We give some examples computed with an (...)

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[hal-01401869] Sums of the digits in bases 2 and 3

24 novembre 2016

Let b ≥ 2 be an integer and let s b (n) denote the sum of the digits of the representation of an integer n in base b. For sufficiently large N , one has Cardn ≤ N : |s 3 (n) − s 2 (n)| ≤ 0.1457205 log n > N 0.970359. The proof only uses the separate (or marginal) distributions of the values of s (...)

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[hal-01320496] Solving discrete logarithms on a 170-bit MNT curve by pairing reduction

24 novembre 2016

Pairing based cryptography is in a dangerous position following the breakthroughs on discrete logarithms computations in finite fields of small characteristic. Remaining instances are built over finite fields of large characteristic and their security relies on the fact that the embedding field (...)

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[hal-01399890] Revêtements du demi-plan de Drinfeld et correspondance de Langlands p-adique

24 novembre 2016

Nous décrivons le complexe de de Rham des revêtements du demi-plan de Drinfeld pour GL_2(Q_p). Cette description, conjecturée par Breuil et Strauch, fournit une réalisation géométrique de la correspondance de Langlands locale p-adique pour certaines représentations de de Rham de dimension de (...)

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[hal-01348576] From a quartic continued fraction in $\mathbb{F}_3((T^-1))$ to a transcendental continued fraction in $\mathbb{Q}((T^-1))$ through an infinite word over {1,2}

23 novembre 2016

We explicitly describe a noteworthy transcendental continued fraction in the field of power series over Q, having irrationality measure equal to 3. This continued fraction is a generating function of a particular sequence in the set 1, 2. The origin of this sequence, whose study was initiated (...)

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[hal-01400869] Local variational study of 2d lattice energies and application to Lennard-Jones type interactions

23 novembre 2016

In this paper, we focus on finite Bravais lattice energies per point in two dimensions. We compute the first and second derivatives of these energies. We prove that the Hessian at the square and the triangular lattice are diagonal and we give simple sufficient conditions for the local (...)

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[hal-01398905] SYMMETRIC POWER CONGRUENCE IDEALS AND SELMER GROUPS

22 novembre 2016

We prove for all $n$ less than or equal to 8 and under certain assumptions the equality predicted by the Iwasawa-Greenberg main conjecture between the characteristic power series of the Selmer group of the $2n$-th symmetric power twisted by $det^-n$ of the Galois representation associated to a (...)

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[hal-01337040] A Modular Method for Computing the Splitting Field of a Polynomial

21 novembre 2016

We provide a modular method for computing the splitting field $K_f$ of an integral polynomial $f$ by suitable use of the byproduct of computation of its Galois group $G_f$ by $p$-adic Stauduhar’s method. This method uses the knowledge of $G_f$ with its action on the roots of $f$ over a $p$-adic (...)

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[hal-01398062] About Jarník's-type relation in higher dimension

19 novembre 2016

Using the Parametric Geometry of Numbers introduced recently by W.M. Schmidt and L. Summerer [17, 18] and results by D. Roy [13, 14], we show that German's transference inequalities between the two most classical exponents of uniform Diophantine approximation are optimal. Further, we establish (...)

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[hal-01398468] Generators of the pro-p Iwahori and Galois representations

18 novembre 2016

For an odd prime p, we determine a minimal set of topological generators of the pro-p Iwahori subgroup of a split reductive group G over Z_p. In the simple adjoint case and for any sufficiently large regular prime p, we also construct Galois extensions of Q with Galois group between the pro-p (...)

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[hal-01398025] There is no analogue to Jarník's relation for twisted Diophantine approximation

18 novembre 2016

Jarník gave a relation between the two most classical uniform exponents of Diophantine approximation in dimension 2. In this paper we consider a twisted case, between the classical and the multiplicative one, and we show that no analogue to Jarník's relation (...)

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[hal-01393875] Conditions pour que les entiers de Beurling aient une densité

14 novembre 2016

En 1977 H.G.Diamond donna une condition portant sur les nombres premiers généralisés de Beurling qui entraîne que les entiers cor-respondants aient une densité. Nous donnons une nouvelle preuve que cette condition est susante (Théorème 1) et nous montrons qu'elle n'est pas né-cessaire (Théorème 2 et (...)

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[tel-01395483] Dualité et principe local-global sur les corps de fonctions

14 novembre 2016

Dans cette thèse, nous nous intéressons à l'arithmétique de certains corps de fonctions. Nous cherchons à établir dans un premier temps des théorèmes de dualité arithmétique sur ces corps, pour les appliquer ensuite à l'étude des points rationnels sur certaines variétés algébriques. Dans les trois premiers (...)

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[hal-01393513] COHOMOLOGIE DES ESPACES DE LUBIN-TATE : UNE NOUVELLE PREUVE GÉOMÉTRIQUE NATURELLE

10 novembre 2016

In my paper at Inventiones 2009, we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle Ψ of some Shimura variety of Kottwitz-Harris-Taylor type. The most (...)

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[hal-01361725] A NUMERICAL NOTE ON UPPER BOUNDS FOR B 2 [g] SETS

9 novembre 2016

Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a satisfactory way. Their natural generalization, called B 2 (...)

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[hal-01390775] SUR LA PROPAGATION DE LA PROPRIÉTÉ MILD AU-DESSUS D'UNE EXTENSION QUADRATIQUE IMAGINAIRE DE Q

7 novembre 2016

Nous nous intéressons dans ce travail aux pro-p groupes G_S , groupes de Galois de pro-p extensions maximales de corps de nombres non ramifiées en dehors d'un ensemble fini S de places ne divisant pas p, et plus particulièrement à la propagation de la propriété mild au-dessus d'une extension (...)

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[cel-01391907] Algebraicity and transcendence of power series: combinatorial and computational aspects

4 novembre 2016

From ancient times, mathematicians are interested in the following question: is a given real number "algebraic" (that is, a root of a nonzero univariate polynomial with rational number coefficients), or is it "transcendental"? Although almost all real numbers are (...)

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[hal-01321043] Computation of the Similarity Class of the $p$-Curvature

4 novembre 2016

The $p$-curvature of a system of linear differential equations in positive characteristic $p$ is a matrix that measures how far the system is from having a basis of polynomial solutions. We show that the similarity class of the $p$-curvature can be determined without computing the $p$-curvature (...)

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[hal-01390143] CONGRUENCES AUTOMORPHES ET TORSION DANS LA COHOMOLOGIE D'UNE VARIÉTÉ DE SHIMURA UNITAIRE SIMPLE

4 novembre 2016

Nous montrons que l'existence d'une classe de cohomologie de torsion dans la cohomologie d'une variété de Shimura de type Kottwitz-Harris-Taylor à coefficients dans un système local, s'accompagne de l'existence d'une infinité dénombrable de représenta-tions irréductibles automorphes cohomologiques, en (...)

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[hal-01137273] SUR LA TORSION DANS LA COHOMOLOGIE DES VARI ETES DE SHIMURA DE KOTTWITZ-HARRIS-TAYLOR

4 novembre 2016

Lorsque le niveau en $l$ d'une variété de Shimura de Kottwitz-Harris-Taylor n'est pas maximal, sa cohomologie à coefficients dans un $\overline\mathbb Z_l$-système local n'est en général pas libre. Afin d'obtenir des énoncés d'annulation de la torsion, on localise en un idéal maximal $\mathfrak m$ de (...)

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[hal-01389696] FACTORIZATION OF SPARSE POLYNOMIALS OVER A FUNCTION FIELD

2 novembre 2016

We present a structure theorem for the non-constant irreducible factors appearing in the family of of all univariate polynomials with a given set of coefficients in a function field and varying exponents. Roughly speaking, this result shows that the non-constant irreducible irreducible factors (...)

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[hal-01238906] Sur la $\overline{\mathbb F}_l$-cohomologie des variétés de Shimura unitaires simples.

31 octobre 2016

On étudie les classes de cohomologie de torsion des variétés de Shimura de type Kottwitz-Harris-Taylor et on montre qu'elles se relèvent " à une place arbitraire près " en une représentation automorphe. En application à tout système de valeurs propres de Hecke modulo $l$ apparaissant dans la (...)

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[hal-01352764] A new formula for ζ(2n + 1) (and how not to prove that ζ(5) is irrational)

31 octobre 2016

Using a new polylogarithmic identity, we express the values of ζ at odd integers 2n + 1 as integrals over unit n−dimensional hypercubes of simple functions involving products of logarithms. We then make several conjectures, based on numerical evidence, on the behaviour of those functions as some (...)

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[hal-01389421] Fast Computation of the Nth Term of an Algebraic Series over a Finite Prime Field

29 octobre 2016

We address the question of computing one selected term of an algebraic power series. In characteristic zero, the best algorithm currently known for computing the~$N$th coefficient of an algebraic series uses differential equations and has arithmetic complexity quasi-linear in~$\sqrtN$. We show (...)

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[hal-01389421] Fast Computation of the Nth Term of an Algebraic Series over a Finite Prime Field

28 octobre 2016

We address the question of computing one selected term of an algebraic power series. In characteristic zero, the best algorithm currently known for computing the~$N$th coefficient of an algebraic series uses differential equations and has arithmetic complexity quasi-linear in~$\sqrtN$. We show (...)

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[hal-01387011] Projective convergence of inhomogeneous $2 \times 2$ matrix products

25 octobre 2016

Each digit in a finite alphabet labels an element of a set M of 2 x 2 column-allowable matrices with nonnegative entries; the right inhomogeneous product of these matrices is made up to rank n, according to a given one-sided sequence of digits; then, the n-step matrix is multiplied by a fixed (...)

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[hal-01387010] On a class of sofic affine invariant subsets of the 2-torus related to an Erdős problem

25 octobre 2016

Let 1 β G be the closed projection on the 2-torus of the (modified) Rademacher graph in base β. The smallest compact containing G and left invariant by the diagonal endomorphism $$(x,y)\mapsto(2x,\beta y)$$ (mod 1) is denoted by K. For β a simple Parry number of PV-type, K is proved to be a sofic (...)

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[inria-00330567] Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions

21 octobre 2016

This paper studies tilings and representation sapces related to the β-transformation when β is a Pisot number (that is not supposed to be a unit). The obtained results are applied to study the set of rational numbers having a purely periodic β-expansion. We indeed make use of the connection (...)

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[inria-00181997] Purely Periodic beta-Expansions in the Pisot Non-unit Case

21 octobre 2016

It is well known that real numbers with a purely periodic decimal expansion are rationals having, when reduced, a denominator coprime with $10$. The aim of this paper is to extend this result to beta-expansions with a Pisot base beta which is not necessarily a unit. We characterize real numbers (...)

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[inria-00539742] Topological properties of Rauzy fractals

21 octobre 2016

Substitutions are combinatorial objects (one replaces a letter by a word) which produce sequences by iteration. They occur in many mathematical fields, roughly as soon as a repetitive process appears. In the present monograph we deal with topological and geometric properties of substitutions, (...)

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[hal-00968922] Models of $\mu_{p^2,K}$ over a discrete valuation ring

21 octobre 2016

Let R be a discrete valuation ring with residue field of characteristic p>0. Let K be its fraction field. We prove that any finite and flat R-group scheme, isomorphic to \mu_p^2,K on the generic fiber, is the kernel in a short exact sequence which generically coincides with the Kummer (...)

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[hal-00678646] Decoding interleaved Reed-Solomon codes beyond their joint error-correcting capability

21 octobre 2016

A new probabilistic decoding algorithm for low-rate interleaved Reed-Solomon (IRS) codes is presented. This approach increases the error correcting capability of IRS codes compared to other known approaches (e.g. joint decoding) with high probability. It is a generalization of well-known (...)

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[hal-01383821] From exploration to theory-driven tables (and back again). A History of Tables in Number Theory.

20 octobre 2016

The article first discusses the earliest extant instances of such tables in Antiquity, and then moves on to the Modern period, dwelling at some length on the table projects of Guldin, Pell and Lambert. It is shown that the numer theoretical tables have incluenfed the development of theory (and (...)

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[hal-01383243] TURBULENT FUNCTIONS AND SOLVING THE NAVIER-STOKES EQUATION BY FOURIER SERIES

20 octobre 2016

I give a resolution of the Navier-Stokes equation by using the series of Fourier.

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[hal-01381425] Hausdorff dimension and uniform exponents in dimension two

20 octobre 2016

In this paper we prove the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponent µ ∈ (1/2, 1) is 2(1 − µ) when µ ≥ √ 2/2, whereas for µ < √ 2/2 it is greater than 2(1 − µ) and at most (3 − 2µ)(1 − µ)/(1 + µ + µ 2). We also establish that this dimension (...)

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[hal-00843094] La conjecture de Tate entiére pour les cubiques de dimension quatre

20 octobre 2016

We prove the Tate conjecture for integral degree 4 classes on a smooth cubic hypersurface X of dimension 4 over an algebraic closure of a field finitely generated over its prime subfield.

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[hal-01102034] The Number Field Sieve in the Medium Prime Case

20 octobre 2016

In this paper, we study several variations of the number field sieve to compute discrete logarithms in finite fields of the form $\GFp^n$, with $p$ a medium to large prime. We show that when $n$ is not too large, this yields a $L_p^n(1/3)$ algorithm with efficiency similar to that of the (...)

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[hal-00133406] Analytic curves in algebraic varieties over number fields

20 octobre 2016

We establish algebraicity criteria for formal germs of curves in algebraic varieties over number fields and apply them to derive a rationality criterion for formal germs of functions, which extends the classical rationality theorems of Borel-Dwork and Pólya-Bertrandias valid over the projective (...)

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[hal-00695590] New solutions to the Hurwitz problem on square identities

20 octobre 2016

The Hurwitz problem of composition of quadratic forms, or of "sum of squares identity" is tackled with the help of a particular class of $(\mathbbZ_2)^n$-graded non-associative algebras generalizing the octonions. This method provides an explicit formula for the classical (...)

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[hal-00294978] Some bounds for ramification of p^n-torsion semi-stable representations

20 octobre 2016

Let p be an odd prime, K a finite extension of Q_p , G_K = Gal(Kbar/K) its absolute Galois group and e = e(K/Q_p) its absolute ramification index. Suppose that T is a p^n-torsion representation of G_K that is isomorphic to a quotient of G_K -stable Z_p -lattices in a semi-stable representation (...)

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[hal-01383243] Fonctions turbulentes et la résolution de l'équation de Navier-Stokes par les séries de Fourier

20 octobre 2016

Je donne une résolution de l'équation de Navier-Stokes par les séries de Fourier.

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[tel-01383309] The CM class number one problem for curves

19 octobre 2016

The main subject of this thesis is the CM class number one problem for curves of genus g, in the cases g=2 and g=3. The problem asks for which CM fields of degree 2g with a primitive CM type are the corresponding CM curves of genus g defined over the reflex field. Chapter 1 is an introduction (...)

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[hal-01382899] PRINCIPAL QUARTIC EXTENSIONS AND ELLIPTIC CURVES

19 octobre 2016

We associate to an S 4 extension, M/K, a Brauer-Severi variety, whose K-rational points correspond to quartic polynomials of the form u 4 + Au + B with splitting field M. The condition that M/K is generated by such a polynomial is a necessary and sufficient condition for M ⊆ K(E[4]) for some (...)

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[hal-00906113] The Breuil-Mezard conjecture for non-scalar split residual representations

18 octobre 2016

We prove the Breuil-Mezard conjecture for split non-scalar residual representations of Gal(Qp/Qp) by local methods. Combined with the cases previously proved in [18] and [24], this completes the proof of the conjecture (when p>3). As a consequence, the local restriction in the proof of the (...)

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[hal-01383243] Fonctions turbulentes et la résolution de l'équation de Navier-Stokes par les séries de Fourier

18 octobre 2016

Je donne une résolution de l'équation de Navier-Stokes par les séries de Fourier.

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[hal-01383161] Le théorème des nombres premiers

18 octobre 2016

Le présent fascicule de la collection Nano est une introduction à l'application des méthodes de l'analyse réelle à l'étude de la répartition des nombres premiers. Le texte a pour fil conducteur l'énoncé emblématique de la théorie analytique des nombres : le théorème des nombres premiers, qui affirme que (...)

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[lirmm-00739245] Odd Perfect Numbers are Greater than 10^1500

17 octobre 2016

Brent, Cohen, and te Riele proved in 1991 that an odd perfect number N is greater than 10^300. We modify their method to obtain N > 10^1500. We also obtain that N has at least 101 not necessarily distinct prime factors and that its largest component (i.e. divisor p^a with p prime) is greater (...)

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[lirmm-00739245] Odd Perfect Numbers are Greater than 10^1500

17 octobre 2016

Brent, Cohen, and te Riele proved in 1991 that an odd perfect number N is greater than 10^300. We modify their method to obtain N > 10^1500. We also obtain that N has at least 101 not necessarily distinct prime factors and that its largest component (i.e. divisor p^a with p prime) is greater (...)

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[hal-01382385] Counting Co-Cyclic Lattices

17 octobre 2016

There is a well-known asymptotic formula, due to W. M. Schmidt [Duke Math. J., 35 (1968), pp. 327--339], for the number of full-rank integer lattices of index at most $V$ in $\mathbbZ^n$. This set of lattices $L$ can naturally be partitioned with respect to the factor group $\mathbbZ^n/L$. (...)

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[hal-01230858] Groupes de Galois et nombres automatiques

13 octobre 2016

Nous montrons dans le cadre de la théorie de Mahler, que les relations de dépendance algébrique sur Q entre les valeurs de fonctions solutions d'un système d'équations fonctionnelles proviennent, par spé-cialisation, des relations entre les fonctions elle-mêmes. Nous en dédui-sons quelques résultats (...)

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[hal-01332997] On a variant of Schanuel conjecture for the Carlitz exponential

13 octobre 2016

We introduce and discuss a variant of Schanuel conjecture in the framework of the Carlitz exponential function over Tate algebras and allied functions. Another purpose of the present paper is to widen the horizons of possible investigations in transcendence and algebraic independence in (...)

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[hal-01372952] A sum-shuffle formula for zeta values in Tate algebras

13 octobre 2016

We prove a sum-shuffle formula for double zeta values in Tate algebras. This follows from an analog result for double twisted power sums, implying that an F_p-vector space generated by multiple zeta values in Tate algebras is an (...)

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[hal-01379484] Le grand théorème de Fermat

13 octobre 2016

grand théorème de Fermat

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[hal-00407999] Fractal tiles associated with shift radix systems

12 octobre 2016

Shift radix systems form a collection of dynamical systems depending on a parameter $\mathbfr$ which varies in the $d$-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. (...)

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[hal-00400799] Rational numbers with purely periodic $\beta$-expansion

12 octobre 2016

We study real numbers $\beta$ with the curious property that the $\beta$-expansion of all sufficiently small positive rational numbers is purely periodic. It is known that such real numbers have to be Pisot numbers which are units of the number field they generate. We complete known results due (...)

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[hal-01379473] On $D_\ell$ extensions of odd prime degree $\ell$

12 octobre 2016

Generalizing the work of A. Morra and the authors, we give explicit formulas for the Dirichlet series generating function of $D_ℓ$-extensions of odd prime degree $ℓ$ with given quadratic resolvent. Over the course of our proof, we explain connections between our formulas and the (...)

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[hal-00649954] On fields with the Property (B)

12 octobre 2016

Let K be a number field and let L/K be an infinite Galois extension with Galois group G. Let us assume that G/Z(G) has finite exponent. We show that L has the Property (B) of Bombieri and Zannier: the absolute and logarithmic Weil height on L^* (outside the set of roots of unity) is bounded (...)

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[hal-00004206] Points de petite hauteur sur une sous-variété d'un tore

12 octobre 2016

Nous obtenons une minoration quasi optimale pour le dernier des minimums dits "géométriques" d'une sous-variété $V$ d'une puissance du groupe multiplicatif $G_m^n$. Plus précisément, nous montrons que si $x$ est un point de $V(\Q)$ n'appartenant pas à un translaté d'un sous-tore de $G_m^n$ (...)

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[hal-00672822] Models of the group schemes of roots of unity

12 octobre 2016

Let O_K be a discrete valuation ring of mixed characteristics (0,p), with residue field k. Using work of Sekiguchi and Suwa, we construct some finite flat O_K-models of the group scheme \mu_p^n,K of p^n-th roots of unity, which we call Kummer group schemes. We set carefully the general (...)

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[hal-00672824] Sekiguchi-Suwa theory revisited

12 octobre 2016

We present an account of the construction by S. Sekiguchi and N. Suwa of a cyclic isogeny of affine smooth group schemes unifying the Kummer and Artin-Schreier-Witt isogenies. We complete the construction over an arbitrary base ring. We extend the statements of some results in a form adapted to (...)

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[hal-00581694] Dynamics of lambda-continued fractions and beta-shifts

11 octobre 2016

For a real number $0<\lambda1$. Some properties of the map $\lambda\mapsto\beta(\lambda)$ are established: It is increasing and continuous from ]0, 2[ onto ]1,\infty[ but non-analytic.

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[hal-00824216] Quelques courbes de Hecke se plongent dans l'espace de Colmez

11 octobre 2016

Let p be a prime, C the p-adic Eigencurve (with tame level 1) and Z the blow-up of the Fredholm hypersurface of the U_p - operator at the special points. We show that for p = 2, 3, 5 and 7, the natural map C -> Z is a rigid-analytic (...)

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[hal-01278713] The sum of digits of polynomial values in arithmetic progressions

11 octobre 2016

Let q, m ≥ 2 be integers with (m, q − 1) = 1. Denote by s_q (n) the sum of digits of n in the q-ary digital expansion. Further let p(x) ∈ Z[x] be a polynomial of degree h ≥ 3 with p(N) ⊂ N. We show that there exist C = C(q, m, p) > 0 and N_0 = N_0(q, m, p) ≥ 1, such that for all g ∈ Z and all N ≥ N 0 (...)

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[tel-01378003] Algorithmes pour les anneaux finis

11 octobre 2016

In this thesis we are interested in describing algorithms that answer questions arising in ring and module theory. Our focus is on deterministic polynomial-time algorithms and rings and modules that are finite. The first main result of this thesis concerns the module isomorphism problem: we (...)

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[tel-01378003] Algorithmes pour les anneaux finis

10 octobre 2016

In this thesis we are interested in describing algorithms that answer questions arising in ring and module theory. Our focus is on deterministic polynomial-time algorithms and rings and modules that are finite. The first main result of this thesis concerns the module isomorphism problem: we (...)

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[hal-01377376] Schertz style class invariants for quartic CM fields

10 octobre 2016

A class invariant is a CM value of a modular function that lies in a certain unram-ified class field. We show that Siegel modular functions over $Q$ for $Γ^0 (N) ⊆ Sp_4 (Z)$yield class invariants under some splitting conditions on N. Small class invariants speed up constructions in explicit class (...)

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[hal-01376507] Une preuve relativiste du Théorème de Fermat-Wiles

7 octobre 2016

Abstract : I give a relativistic proof to the Fermat's last theorem : no three positive integers a, b, and c satisfy the equation $a^n + b^n = c^n$ for any integer value of n greater than two.

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[hal-01329432] Graded structures and differential operators on nearly holomorphic and quasimodular forms on classical groups

4 octobre 2016

We wish to use graded structures [KrVu87], [Vu01] on dffierential operators and quasimodular forms on classical groups and show that these structures provide a tool to construct p-adic measures and p-adic L-functions on the corresponding non-archimedean weight spaces. An approach to (...)

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[hal-01374286] Shidlovsky's multiplicity estimate and Irrationality of zeta values

30 septembre 2016

In this paper we follow the approach of Bertrand-Beukers (and of later work of Bertrand), based on differential Galois theory, to prove a very general version of Shidlovsky's lemma that applies to Padé approximation problems at several points, both at functional and numerical levels (i.e., (...)

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[hal-00264057] Landau's function for one million billions

30 septembre 2016

Let $\mathfrak S_n$ denote the symmetric group with $n$ letters, and $g(n)$ the maximal order of an element of $\mathfrak S_n$. If the standard factorization of $M$ into primes is $M=q_1^\al_1q_2^\al_2\ldots q_k^\al_k$, we define $\ell(M)$ to be $q_1^\al_1+q_2^\al_2+\ldots +q_k^\al_k$; one (...)

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[hal-01081277] On the representation of friable integers by linear forms

29 septembre 2016

Let $P^+(n)$ denote the largest prime of the integer $n$. Using the \beginalign* \Psi_F_1\cdots F_t\left(\mathcalK\cap[-N,N]^d,N^1/u\right):= \#\left\\mathcalK\in \mathbfN\cap[-N,N]^d:\vphantomP^+(F_1(\boldsymboln)\cdots F_t(\boldsymboln))\leq N^1/u\right. \left.P^+(F_1(\boldsymboln)\cdots (...)

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[hal-01372952] A sum-shuffle formula for zeta values in Tate algebras

29 septembre 2016

We prove a sum-shuffle formula for double zeta values in Tate algebras. This follows from an analog result for double twisted power sums, implying that an F_p-vector space generated by multiple zeta values in Tate algebras is an (...)

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[hal-00269095] Zero-sum free sequences with small sum-set

29 septembre 2016

Let A be a zero-sum free subset of Z_n with |A|=k. We compute for k\le 7 the least possible size of the set of all subset-sums of A.

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[hal-00013042] Hauteur asymptotique des points de Heegner

28 septembre 2016

The asymptotic behaviour of the Neron-Tate height of Heegner points on a rational elliptic curve attached to an arithmetically normalized new cusp form f of weight 2, level N and trivial character is studied in this paper. By Gross-Zagier formula, this height is related to the special value at (...)

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[hal-00418299] Theta lifting for some cohomologicaly induced representations

28 septembre 2016

In his paper 'Theta lifting for representations with non zero cohomlogy', Jian-Shu Li proved that a certain kind of cohomological representations of $U(a,b)$ is automorphic. In this paper, this result is generalized to a more general class of cohomological representations of this group. It (...)

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[hal-00675682] Eigenvarieties for classical groups and complex conjugations in Galois representations

28 septembre 2016

The goal of this paper is to remove the irreducibility hypothesis in a theorem of Richard Taylor describing the image of complex conjugations by $p$-adic Galois representations associated with regular, algebraic, essentially self-dual, cuspidal automorphic representations of $\GL_2n+1$ over a (...)

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[hal-01280111] Improving Roth's theorem in the primes

28 septembre 2016

Let A be a subset of the primes. Let δP (N) = |n ∈ A : n ≤ N |/ |n prime : n ≤ N | be the relative density of A in the primes. We prove that if δP(N) ≥ C(log log log N)/(log log N)^1/3 for N ≥ N0, where C and N0 are absolute constants, then A ∩ [1, N] contains a non-trivial three-term arithmetic (...)

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[hal-00010037] S\\éries Gevrey de type arithm\\étique, I. Th\\éor\èmes de puret\\é et de dualit\\é

28 septembre 2016

Gevrey series are ubiquitous in analysis; any series satisfying some (possibly non-linear) analytic differential equation is Gevrey of some rational order. The present work stems from two observations: 1) the classical Gevrey series, e.g. generalized hypergeometric series with rational (...)

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[hal-00010038] S\\éries Gevrey de type arithm\\étique, II. Transcendance sans transcendance

28 septembre 2016

In this second part, we study the Diophantine properties of values of arithmetic Gevrey series of non-zero order at algebraic points. We rely on the fact, proved in the first part, that the minimal differential operator (with polynomial coefficients) which annihilates such a series has no (...)

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[halshs-00119444] Matrimonial Ring Structures

28 septembre 2016

The paper deals with matrimonial rings, a particular kind of cycles in kinship networks which result when spouses are linked to each other by ties of consanguinity or affinity. By taking a network-analytic perspective, the paper endeavours to put this classical issue of structural kinship (...)

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[hal-00661713] Représentations p-adiques de GL2(L) et catégories dérivées

28 septembre 2016

We construct some locally ℚ p -analytic representations of GL2(L), L a finite extension of ℚ p , associated to some p-adic representations of the absolute Galois group of L. We prove that the space of morphisms from these representations to the de Rham complex of Drinfel'd's upper half space has (...)

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[hal-00186207] Inductive Methods and zero-sum free sequences

28 septembre 2016

We obtain a decidability result for the Davenport constant.

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[pastel-01066463] Deux applications arithmétiques des travaux d'Arthur

28 septembre 2016

Nous proposons deux applications à l'arithmétique des travaux récents de James Arthur sur la classification endoscopique du spectre discret des groupes symplectiques et orthogonaux. La première consiste à ôter une hypothèse d'irréductibilité dans un résultat de Richard Taylor décrivant l'image des (...)

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[hal-00290583] Asymptotique des nombres de Betti des variétés arithmétiques

28 septembre 2016

We study the question of the growth of Betti numbers of certain arithmetic varieties in tower of congruence coverings. In fact, our results are about Siegel varieties and varieties associated to orthogonal groups. We explain how a theorem of Waldspurger can be used to obtain lower and upper (...)

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[hal-00694296] Geometric and differential Galois theories

28 septembre 2016

On March 29-April 2, 2010, a meeting was organized at the Luminy CIRM (France) on geometric and differential Galois theories, witnessing the close ties these theories have woven in recent years. The present volume collects the Proceedings of this meeting. Although it may be viewed as a (...)

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[hal-00014041] A logarithm law for automorphism groups of trees

28 septembre 2016

Let G be a geometrically finite tree lattice. We prove a Khintchine-Sullivan type theorem for the Hausdorff measure of the points at infinity of the tree that are well approximated by the parabolic fixed points of G. Using Bruhat-Tits trees, an application is given for the Diophantine (...)

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[hal-01361725] A NUMERICAL NOTE ON UPPER BOUNDS FOR B 2 [g] SETS

28 septembre 2016

Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a satisfactory way. Their natural generalization, called B 2 (...)

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[hal-01174146] La relativité et la théorie des nombres

26 septembre 2016

I use the theory of relativity to prove the Riemann hypothesis, Goldbach's conjecture, De Polignac's conjecture, the Legendre's conjecture, the Syracuse problem, the problems of Mersenne and Fermat primes, and the Fermat's last (...)

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[hal-01370273] Equidistribution in S-arithmetic and adelic spaces

23 septembre 2016

We give an introduction to adelic mixing and its applications for mathematicians knowing about the mixing of the geodesic flow on hyperbolic surfaces. We focus on the example of the Hecke trees in the modular surface.

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[hal-01370209] EXISTENCE ET ÉQUIDISTRIBUTION DES MATRICES DE DÉNOMINATEUR n DANS LES GROUPES UNITAIRES ET ORTHOGONAUX

23 septembre 2016

We study some subsets of rational points in an algebraic groups defined by open conditions on their projection in the finite adeles points. Using adelic mixing we are able to prove an equidistribution's result for the projection of these sets in the real points. As an application, we study the (...)

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[inria-00560258] Computing endomorphism rings of elliptic curves under the GRH

22 septembre 2016

We design a probabilistic algorithm for computing endomorphism rings of ordinary elliptic curves defined over finite fields that we prove has a subexponential runtime in the size of the base field, assuming solely the generalized Riemann hypothesis. Additionally, we improve the asymptotic (...)

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[hal-01277169] Computing theta functions in quasi-linear time in genus 2 and above

22 septembre 2016

We outline an algorithm to compute θ(z, τ) in genus 2 in quasi-optimal time, borrowing ideas from the algorithm for theta constants and the one for θ(z, τ) in genus 1. Our implementation shows a large speedup for precisions as low as a few thousand decimal digits. We also lay out a strategy to (...)

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[hal-01366792] The optimal solution for the Knapsack problem find by The electronic algorithm

19 septembre 2016

In [8] , the electronic algorithm has allowed to find the optimal solutions for the travelling salesman problem. In this article, I demonstrate that the same algorithm will allow to find the optimal solutions for the knapsack (...)

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[hal-00932275] On a conjecture of G. Rémond

16 septembre 2016

We provide an example which gives some new evidence to a recent conjecture of G. Rémond on lower bounds for the height.

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[inria-00180239] Boundary of central tiles associated with Pisot beta-numeration and purely periodic expansions

16 septembre 2016

This paper studies tilings related to the $\beta$-transformation when $\beta$ is a Pisot number (that is not supposed to be a unit). Then it applies the obtained results to study the set of rational numbers having a purely periodic $\beta$-expansion. Secial focus is given to some quadratic (...)

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[hal-00959764] GÉNÉRALISATIONS QUANTITATIVES DU CRITÈRE D'INDÉPENDANCE LINÉAIRE DE NESTERENKO

16 septembre 2016

In this paper we extend Fischler's quantitative generalization of Nesterenko's linear independence criterion, by weakening the hypotheses on the divisors of the coe cients of the linear forms and allowing (to some extent) the linear forms not to tend to 0. Another version of this result is (...)

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[hal-00448031] Pairing the Volcano

16 septembre 2016

Isogeny volcanoes are graphs whose vertices are elliptic curves and whose edges are $\ell$-isogenies. Algorithms allowing to travel on these graphs were developed by Kohel in his thesis (1996) and later on, by Fouquet and Morain (2001). However, up to now, no method was known, to predict, (...)

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[inria-00091441] Fast algorithms for computing isogenies between elliptic curves

16 septembre 2016

We survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime. We introduce a new algorithm that computes an isogeny of degree $\ell$ ($\ell$ different from the characteristic) in time quasi-linear with respect to $\ell$. (...)

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[hal-00576946] Multiplicity theorems modulo p for GL2(Qp)

16 septembre 2016

Let F a nonarchimedean local field, π, ρ admissible irreducible representations of GL_n (F ), GL_n−1 (F) respectively. Aizenbud, Gourvitch, Rallis and Schiffmann have recently proved that the space of Hom_GL_n−1 (F) (π, ρ) is at most 1-dimensional and by the works of Waldspurger and Moeglin such (...)

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[hal-00776680] Universal Gauss-Thakur sums and L-series

16 septembre 2016

In this paper we study the behavior of the function omega of Anderson-Thakur evaluated at the elements of the algebraic closure of the finite field with q elements F_q. Indeed, this function has quite a remarkable relation to explicit class field theory for the field K=F_q(T). We will see that (...)

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[tel-00765982] Théorie algorithmique des nombres et applications à la cryptanalyse de primitives cryptographiques

12 septembre 2016

Le problème de la factorisation et celui du logarithme discret sont deux fondements essentiels de nombreux algorithmes de la cryptographie à clé publique. Dans le champ des algorithmes pour attaquer ces problèmes éminemment ardus, le crible algébrique et ses algorithmes cousins occupent une place de (...)

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[hal-01361725] A NUMERICAL NOTE ON UPPER BOUNDS FOR B 2 [g] SETS

9 septembre 2016

Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a satisfactory way. Their natural generalization, called B 2 (...)

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[hal-01300328] Valeurs moyennes effectives de fonctions multiplicatives complexes

9 septembre 2016

Nous établissons des estimations de valeurs moyennes pour une large classe de fonctions arithmétiques multiplicatives, fournissant ainsi des versions quantitatives essentiellement optimales des évaluations classiques de Wirsing et étendant celles de Halász. Plusieurs applications sont explicitées, (...)

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[hal-01361276] Normes cyclotomiques naïves et unités logarithmiques

7 septembre 2016

Nous déterminons le rang du sous-groupe des éléments du groupe multiplicatif d'un corps de nombres K qui sont normes à chaque étage fini de sa Zℓ-extension cyclotomique ; et nous comparons son ℓ-adifié avec le ℓ-groupe des unités logarithmiques.

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[hal-01357331] LE «LEMME D'ABHYANKAR» PERFECTOIDE

7 septembre 2016

Nous étendons le théorème de «presque-pureté» de Faltings-Scholze-Kedlaya-Liu sur les extensions étales finies d'algèbres perfectoïdes au cas des extensions ramifiées, sans restriction sur le discriminant. Le point clé est une version perfectoïde du théorème d'extension de Riemann. Au préalable, nous (...)

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[hal-01357338] LA CONJECTURE DU FACTEUR DIRECT

7 septembre 2016

RÉSUMÉ. M. Hochster a conjecturé que pour toute extension finie S d'un anneau com-mutatif régulier R, la suite exacte de R-modules 0 → R → S → S/R → 0 est scindée. En nous appuyant sur sa réduction au cas d'un anneau local régulier R complet non rami-fié d'inégale caractéristique, nous proposons une (...)

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[hal-00957580] Cryptage et décryptage : communiquer en toute sécurité

3 septembre 2016

La sécurisation de nos cartes bleues, ainsi que d'autres procédés de cryptages utilisés couramment, se basent sur l'impossibilité, en pratique, de factoriser de très grands nombres. Ce type de cryptage pourrait cependant être détrôné par d'autres méthodes, sa fiabilité étant sans cesse remise en cause (...)

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[hal-00957580] Cryptage et décryptage : communiquer en toute sécurité

2 septembre 2016

La sécurisation de nos cartes bleues, ainsi que d'autres procédés de cryptages utilisés couramment, se basent sur l'impossibilité, en pratique, de factoriser de très grands nombres. Ce type de cryptage pourrait cependant être détrôné par d'autres méthodes, sa fiabilité étant sans cesse remise en cause (...)

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[hal-01357007] Nombres de résidus quadratiques des nombres premiers ou composés

2 septembre 2016

On propose d'établir le caractère de primalité d'un entier n en comparant le nombre de ses résidus quadratiques non nuls à sa moitié.

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[hal-00687289] Classes logarithmiques et capitulation

1er septembre 2016

We study a logarithmic version of the classical result of Artin-Furwängler on principalization of ideal classes in the Hilbert class-field by applying the group theoretic description of the transfert map to logarithmic class-groups of degree (...)

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[hal-01214814] Sur la capitulation pour le module de Bertrandias-Payan

1er septembre 2016

Nous déterminons l'ordre du sous-groupe de capitulation pour le module de Bertrandias-Payan dans une $\ell$-extension arbitraire de corps de nombres qui satisfait la conjecture de Leopoldt. Nous relions en particulier la question de sa trivialisation au problème des tours localement (...)

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[hal-01358091] On the residue of Eisenstein classes of Siegel varieties

31 août 2016

Eisenstein classes of Siegel varieties are motivic cohomology classes defined as pull-backs by torsion sections of the polylogarithm prosheaf on the universal abelian scheme. By reduction to the Hilbert-Blumenthal case, we prove that the Betti realization of these classes on Siegel varieties of (...)

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[hal-01358089] Foliated dynamical systems associated to ordinary abelian varieties over finite fields

31 août 2016

We interpret the "explicit formula" in the sense of analytic number theory for the zeta function of an ordinary abelian variety of dimension g over a finite field as a transversal index theorem on a (2g+1)-dimensional Riemannian foliated space. This generalizes a work of Deninger for (...)

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[hal-01357538] On higher regulators of Siegel threefolds II: the connection to the special value

30 août 2016

We establish a connection between motivic cohomology classes over the Siegel threefold and special values of the degree four L -function of some cuspidal automorphic representations of GSp(4) .

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[hal-01356663] Nombre de résidus quadratiques d'un nombre entier inférieurs à sa moitié

27 août 2016

On propose un critère qui permet de distinguer les nombres premiers des nombres composés. Ce critère est basé sur le comptage des petits résidus quadratiques.

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[hal-01070551] Geometry, dynamics, and arithmetic of $S$-adic shifts

26 août 2016

This paper studies geometric and spectral properties of $S$-adic shifts and their relation to continued fraction algorithms. These shifts are symbolic dynamical systems obtained by iterating infinitely many substitutions in an adic way. Pure discrete spectrum for $S$-adic shifts and tiling (...)

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[hal-01356663] Nombre de résidus quadratiques d'un nombre entier inférieurs à sa moitié

26 août 2016

On propose un critère qui permet de distinguer les nombres premiers des nombres composés. Ce critère est basé sur le comptage des petits résidus quadratiques.

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[hal-01356061] DES APPLICATIONS GÉNÉRATRICES DES NOMBRES PREMIERS ET CINQ PREUVES DE L'HYPOTHÈSE DE RIEMANN

26 août 2016

Je démontre qu'il existe une application $\psi(\psi^-,\psi^+)$ définie sur $\mathbbR^2$ et telle que $\mathcalP = \\pm2,\pm3 \ \cup 6\times\mathcalF^-+1 \cup6\times\mathcalF^+-1$ où : $\mathcalP$ est l'ensemble des nombres relatifs premiers, $\mathcalF^- = \mathbbZ\cap( \psi^+ ( \mathbbZ^*\times (...)

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[hal-01355926] Short addition sequences for theta functions

24 août 2016

The main step in numerical evaluation of classical Sl2 (Z) modular forms and elliptic functions is to compute the sum of the first N nonzero terms in the sparse q-series belonging to the Dedekind eta function or the Jacobi theta constants. We construct short addition sequences to perform this (...)

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[hal-01355277] The Riemann Hypothesis: A Qualitative Characterization of the Nontrivial Zeros of the Riemann Zeta Function Using Polylogarithms

24 août 2016

We formulate a parametrized uniformly absolutely globally convergent series of ζ(s) denoted by Z(s, x). When expressed in closed form, it is given by Z(s, x) = (s − 1)ζ(s) + 1 x Li s z z − 1 dz, where Li s (x) is the polylogarithm function. As an immediate first application of the new parametrized (...)

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[hal-01273045] Collecting relations for the number field sieve in $GF(p^6)$

18 août 2016

In order to assess the security of cryptosystems based on the discrete logarithm problem in non-prime finite fields, as are the torus-based or pairing-based ones, we investigate thoroughly the case in GF(p^6) with the Number Field Sieve. We provide new insights, improvements, and comparisons (...)

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[hal-00919367] Root optimization of polynomials in the number field sieve

18 août 2016

The general number field sieve (GNFS) is the most efficient algorithm known for factoring large integers. It consists of several stages, the first one being polynomial selection. The quality of the chosen polynomials in polynomial selection can be modelled in terms of size and root properties. (...)

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[hal-00823745] Computing class polynomials for abelian surfaces

18 août 2016

We describe a quasi-linear algorithm for computing Igusa class polynomials of Jacobians of genus 2 curves via complex floating-point approximations of their roots. After providing an explicit treatment of the computations in quartic CM fields and their Galois closures, we pursue an approach due (...)

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[inria-00560256] A low-memory algorithm for finding short product representations in finite groups

17 août 2016

We describe a space-efficient algorithm for solving a generalization of the subset sum problem in a finite group G, using a Pollard-rho approach. Given an element z and a sequence of elements S, our algorithm attempts to find a subsequence of S whose product in G is equal to z. For a random (...)

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[inria-00383155] Computing the endomorphism ring of an ordinary elliptic curve over a finite field

17 août 2016

We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field F_q. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q, while our bound for the second (...)

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[hal-00932377] Explicit methods in number theory. Rational points and Diophantine equations.

17 août 2016

This volume contains a selection of seven short courses in number theory taught during a special trimester at Institut Henri Poincaré (from September to December 2004), centered on Diophantine equations and how to effectively solve them. The lectures, targeted at second year graduate students, (...)

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[hal-01052449] Improvements to the number field sieve for non-prime finite fields

17 août 2016

We propose various strategies for improving the computation of discrete logarithms in non-prime fields of medium to large characteristic using the Number Field Sieve. This includes new methods for selecting the polynomials; the use of explicit automorphisms; explicit computations in the number (...)

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[hal-00756838] Square root algorithms for the number field sieve

17 août 2016

We review several methods for the square root step of the Number Field Sieve, and present an original one, based on the Chinese Remainder Theorem.

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[hal-00671948] Finding ECM-friendly curves through a study of Galois properties

17 août 2016

In this paper we prove some divisibility properties of the cardinality of elliptic curves modulo primes. These proofs explain the good behavior of certain parameters when using Montgomery or Edwards curves in the setting of the elliptic curve method (ECM) for integer factorization. The ideas of (...)

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[hal-00580855] Multi-precision computation of the complex error function

17 août 2016

We give a very simple algorithm to compute the error and complementary error functions of complex argument to any given accuracy.

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[hal-00835446] A heuristic quasi-polynomial algorithm for discrete logarithm in finite fields of small characteristic

17 août 2016

In the present work, we present a new discrete logarithm algorithm, in the same vein as in recent works by Joux, using an asymptotically more efficient descent approach. The main result gives a quasi-polynomial heuristic complexity for the discrete logarithm problem in finite field of small (...)

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[inria-00598029] Counting Points on Genus 2 Curves with Real Multiplication

17 août 2016

We present an accelerated Schoof-type point-counting algorithm for curves of genus 2 equipped with an efficiently computable real multiplication endomorphism. Our new algorithm reduces the complexity of genus 2 point counting over a finite field \(\F_q\) of large characteristic from (...)

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[hal-00935050] Symmetrized summation polynomials: using small order torsion points to speed up elliptic curve index calculus

10 août 2016

Decomposition-based index calculus methods are currently efficient only for elliptic curves E defined over non-prime finite fields of very small extension degree n. This corresponds to the fact that the Semaev summation polynomials, which encode the relation search (or “sieving”), grow (...)

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[hal-01352764] A new formula for ζ(2n + 1) (and how not to prove that ζ(5) is irrational)

9 août 2016

Using a new polylogarithmic identity, we express the values of ζ at odd integers 2n + 1 as integrals over unit n−dimensional hypercubes of simple functions involving products of logarithms. We then make several conjectures, based on numerical evidence, on the behaviour of those functions as some (...)

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[hal-01352498] Almost all non-archimedean Kakeya sets have measure zero

9 août 2016

We study Kakeya sets over local non-archimedean fields with a probabilistic point of view: we define a probability measure on the set of Kakeya sets as above and prove that, according to this measure, almost all non-archimedean Kakeya sets are neglectable according to the Haar measure. We also (...)

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[hal-00976540] SL(2,Z)-invariance and D-instanton contributions to the D^6R^4 interaction

9 août 2016

The modular invariant coefficient of the $D^6R^4$ interaction in the low energy expansion of type IIB string theory has been conjectured to be a solution of an inhomogeneous Laplace eigenvalue equation, obtained by considering the toroidal compactification of two-loop Feynman diagrams of (...)

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[hal-01064255] The Q-curve construction for endomorphism-accelerated elliptic curves

8 août 2016

We give a detailed account of the use of $\mathbbQ$-curve reductions to construct elliptic curves over $\mathbbF_p^2$ with efficiently computable endomorphisms, which can be used to accelerate elliptic curve-based cryptosystems in the same way as Gallant--Lambert--Vanstone (GLV) and (...)

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[hal-00976540] SL(2,Z)-invariance and D-instanton contributions to the D^6R^4 interaction

8 août 2016

The modular invariant coefficient of the $D^6R^4$ interaction in the low energy expansion of type IIB string theory has been conjectured to be a solution of an inhomogeneous Laplace eigenvalue equation, obtained by considering the toroidal compactification of two-loop Feynman diagrams of (...)

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[hal-00470562] Eisenstein series for higher-rank groups and string theory amplitudes

8 août 2016

Scattering amplitudes of superstring theory are strongly constrained by the requirement that they be invariant under dualities generated by discrete subgroups, E_n(Z), of simply-laced Lie groups in the E_n series (n<= 8). In particular, expanding the four-supergraviton amplitude at low (...)

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[hal-00663502] Small representations, string instantons, and Fourier modes of Eisenstein series (with an appendix by D. Ciubotaru and P. Trapa)

8 août 2016

This paper concerns some novel features of maximal parabolic Eisenstein series at certain special values of their analytic parameter s. These series arise as coefficients in the R4 and D4R4 interactions in the low energy expansion of scattering amplitudes in maximally supersymmetric string (...)

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[hal-01348147] Une nouvelle approche dans la théorie des entiers friables

3 août 2016

Grâce a une nouvelle approche, dont le point de départ est un calcul de résidu , nous précisons certaines des estimations connues pour la fonction de comptage des entiers friables. Le gain se révèle crucial pour diverses applications, dont certaines concernent des questions fondamentales de la (...)

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[hal-01350056] Amélioration d'une congruence de Glaisher

2 août 2016

Nous prouvons que pour tout nombre premier p ≥ 5, [(2^p-1 - 1)/p] ≡ - 1/2 (Σ^(p-1) (k=1) 2^k/k) (modp^2)

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[hal-01348147] Une nouvelle approche dans la théorie des entiers friables

29 juillet 2016

Grâce a une nouvelle approche, dont le point de départ est un calcul de résidu , nous précisons certaines des estimations connues pour la fonction de comptage des entiers friables. Le gain se révèle crucial pour diverses applications, dont certaines concernent des questions fondamentales de la (...)

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[tel-01201663] Equidistribution problems of squarefree numbers

29 juillet 2016

This thesis concerns a few problems linked with the distribution of squarefree integers in arithmeticprogressions. Such problems are usually phrased in terms of upper bounds for the error term relatedto this distribution.The first, second and fourth chapter focus on the satistical study of the (...)

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[tel-01141624] Formes modulaires p-adiques sur les courbes de Shimura unitaires et compatibilité local-global

29 juillet 2016

Cette thèse s'inscrit dans le cadre du programme de Langlands local p-adique. Soient L une extension finie de Q_p, \rho_L une représentation p-adique de dimension 2 du groupe de Galois Gal(\overlineQ_p/L) de L, lorsque \rho_L provient d'une représentation \rho globale et modulaire (i.e. \rho (...)

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[medihal-01345152] Umberto Zannier - Unlikely Intersections and Pell's equations in polynomials

27 juillet 2016

Unlikely Intersections and Pell's equations in polynomials

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[medihal-01345137] Laurent Fargues - Courbes et fibrés vectoriels en théorie de Hodge p-adique

27 juillet 2016

Courbes et fibrés vectoriels en théorie de Hodge p-adique

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[medihal-01345120] Charles Favre - Application to complex dynamics of the equidistribution of points of small heights

27 juillet 2016

Application to complex dynamics of the equidistribution of points of small heights

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[medihal-01345037] Mahesh Kakde - Congruences in noncommutative Iwasawa theory

27 juillet 2016

Congruences in noncommutative Iwasawa theory

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[medihal-01345030] Kaisa Matomäki - On the distribution of Fourier coefficients of modular forms

27 juillet 2016

On the distribution of Fourier coefficients of modular forms

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[medihal-01345067] Tom Sanders - Roth's theorem on arithmetic progressions

27 juillet 2016

Roth's theorem on arithmetic progressions

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[medihal-01344859] Rachel Pries - The geometry of p-torsion stratifications of the moduli space of curve

27 juillet 2016

The geometry of p-torsion stratifications of the moduli space of curve

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[hal-01292727] A functoriality principle for blocks of p-adic linear groups

27 juillet 2016

Bernstein blocks of complex representations of p-adic reductive groups have been computed in a large amount of examples, in part thanks to the theory of types a la Bushnell and Kutzko. The output of these purely representation-theoretic computations is that many of these blocks are equivalent. (...)

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[tel-01346112] Produit harmonique, sommation de Ramanujan et fonctions zêta d'Arakawa-Kaneko. Mémoire d'Habilitation à Diriger des Recherches.

26 juillet 2016

Ce mémoire s'articule autour des trois principaux thèmes auxquels j'ai consacré mes recherches au cours de ces 15 dernières années, thèmes qui sont assez étroitement reliés entre-eux. Il s'agit du produit harmonique, du procédé de sommation de Ramanujan et de la fonction zêta d'Arakawa-Kaneko qui ont (...)

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[tel-01346112] Produit harmonique, sommation de Ramanujan et fonctions zêta d'Arakawa-Kaneko. Mémoire d'Habilitation à Diriger des Recherches.

26 juillet 2016

Ce mémoire s'articule autour des trois principaux thèmes auxquels j'ai consacré mes recherches au cours de ces 15 dernières années, thèmes qui sont assez étroitement reliés entre-eux. Il s'agit du produit harmonique, du procédé de sommation de Ramanujan et de la fonction zêta d'Arakawa-Kaneko qui ont (...)

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[hal-01111744] Elliptic curves with 2-torsion contained in the 3-torsion field

26 juillet 2016

There is a modular curve X'(6) of level 6 defined over Q whose Q-rational points correspond to j-invariants of elliptic curves E over Q for which Q(E[2]) is a subfield of Q(E[3]). In this note we characterize the j-invariants of elliptic curves with this property by exhibiting an explicit model (...)

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[inria-00462008] Computing quadratic function fields with high 3-rank via cubic field tabulation

26 juillet 2016

We present recent results on the computation of quadratic function fields with high 3-rank. Using a generalization of a method of Belabas on cubic field tabulation and a theorem of Hasse, we compute quadratic function fields with 3-rank $ \geq 1$, of imaginary or unusual discriminant $D$, for a (...)

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[hal-01348576] From a quartic continued fraction in $F_3((T^-1))$ to a transcendental continued fraction in $Q((T^-1))$ through an infinite word over {1,2}

25 juillet 2016

We explicitly describe a noteworthy transcendental continued fraction in the field of power series over Q, having irrationality measure equal to 3. This continued fraction is a generating function of a particular sequence in the set 1, 2. The origin of this sequence, whose study was initiated (...)

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[in2p3-00824664] Des nouvelles solutions au problème des trois corps

20 juillet 2016

Cet article présente les solutions du problème des trois corps découvertes récemment.

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[hal-00601448] Ramification and cleanliness

19 juillet 2016

This article is devoted to studying the ramification of Galois torsors and of $\ell$-adic sheaves in characteristic $p>0$ (with $\ell\not=p$). Let $k$ be a perfect field of characteristic $p>0$, $X$ be a smooth, separated and quasi-compact $k$-scheme, $D$ be a simple normal crossing (...)

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[hal-00001471] Equidistribution des sous-variétés de petite hauteur

12 juillet 2016

In this paper, the equidistribution theorem of Szpiro-Ullmo-Zhang about sequences of small points in an abelian variety is extended to the case of sequences of higher dimensional subvarieties. A quantitative version of this result is also (...)

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[tel-01226270] Sur le spectre des exposants d'approximation diophantienne classiques et pondérés

9 juillet 2016

Pour un n-uplet de nombres réels, vu comme un point de l'espace projectif, on définit pour chaqueindice d entre 0 et n-1 deux exposants d'approximation diophantienne (un ordinaire et un uniforme)qui mesurent l'approximabilité de celui-ci par des sous-espaces rationnels de dimension d dansl'espace (...)

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[tel-01226270] Sur le spectre des exposants d'approximation diophantienne classiques et pondérés

8 juillet 2016

Pour un n-uplet de nombres réels, vu comme un point de l'espace projectif, on définit pour chaqueindice d entre 0 et n-1 deux exposants d'approximation diophantienne (un ordinaire et un uniforme)qui mesurent l'approximabilité de celui-ci par des sous-espaces rationnels de dimension d dansl'espace (...)

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[hal-01341168] Amelioration of Glaisher's Congruence

7 juillet 2016

We prove that if $p\geq 5$ is a prime, then \beginequation* \frac2^p-1-1p\equiv -\frac12\left( \sum_k=1^p-1\frac2^kk% \right)(\bmod p^2) \endequation*

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[tel-00674975] Implémentation matérielle de coprocesseurs haute performance pour la cryptographie asymétrique

7 juillet 2016

Dans cette thèse, je propose des architectures de coprocesseurs haute performance pour implémenter les primitives de cryptographie asymétrique, comme le RSA, les courbes elliptiques ou le couplage. Les coprocesseurs décrits dans cette thèse ont été implémentés dans des FPGA, et (...)

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[hal-01338806] Sur l'exemple d'Euler d'une fonction CMO

4 juillet 2016

Cet article développe et démontre les énoncés donnés dans la note aux Comptes Rendus de l'Académie des Sciences CRAS, 354 (2016), 559-561.

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[hal-01283042] Series Representation of Power Function

1er juillet 2016

This paper presents the way to make expansion for the next form function: $y=x^n, \ \forall(x,n) \in \mathbbN$ to the numerical series. The most widely used methods to solve this problem are Newton's Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are (...)

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[hal-01338806] Sur l'exemple d'Euler d'une fonction CMO

29 juin 2016

Cet article développe et démontre les énoncés donnés dans la note aux Comptes Rendus de l'Académie des Sciences CRAS, 354 (2016), 559-561.

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[hal-01337295] NUMERICAL SEMIGROUPS OF TWO GENERATORS

26 juin 2016

This paper will represent in a simple way some known facts about semigroups especially when the number of minimal generators equals two or in general semigroups with at least two minimal generators. The originality of this exposition is that it is a straight application of a remark written by (...)

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[hal-00630391] Elliptic periods for finite fields

23 juin 2016

We construct two new families of basis for finite field extensions. Basis in the first family, the so-called elliptic basis, are not quite normal basis, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Basis in the second family, the so-called (...)

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[hal-01255343] Iwasawa theory and $F$-analytic Lubin-Tate $(\varphi,\Gamma)$-modules

23 juin 2016

Let $K$ be a finite extension of $\mathbfQ_p$. We use the theory of $(\varphi,\Gamma)$-modules in the Lubin-Tate setting to construct some corestriction-compatible families of classes in the cohomology of $V$, for certain representations $V$ of $\mathrmGal(\overline\mathbfQ_p/K)$. If in (...)

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[hal-00004037] Fonction zêta des hauteurs des variétés toriques non déployées

23 juin 2016

We investigate the anticanonical height zeta function of a (non necessarily split) toric variety defined over a global field of positive characteristic, drawing our inspiration from the method used by Batyrev and Tschinkel to deal with the analogous problem over a number field. By the way, we (...)

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[hal-00268204] Comptage de courbes sur le plan projectif éclaté en trois points alignés

23 juin 2016

We prove a version of Manin's conjecture for the projective plane blown up in three collinear points, the base field being a global field of positive characteristic.

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[hal-00663474] Finite Morphisms to Projective Space and Capacity Theory

23 juin 2016

We study conditions on a commutative ring R which are equivalent to the following requirement; whenever X is a projective scheme over S = Spec(R) of fiber dimension \leq d for some integer d \geq 0, there is a finite morphism from X to P^d_S over S such that the pullbacks of coordinate (...)

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[hal-00843112] Some remarks concerning the Grothendieck Period Conjecture

23 juin 2016

We discuss various results and questions around the Grothendieck period conjecture, which is a counterpart, concerning the de Rham-Betti realization of algebraic varieties over number fields, of the classical conjectures of Hodge and Tate. These results give new evidence towards the conjectures (...)

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[hal-00694117] Fast computation of isomorphisms of hyperelliptic curves and explicit descent

23 juin 2016

We show how to speed up the computation of isomorphisms of hyperelliptic curves by using covariants. We also obtain new theoretical and practical results concerning models of these curves over their field of moduli.

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[hal-01162557] Variétés de Kisin stratifiées et déformations potentiellement Barsotti-Tate

22 juin 2016

Soient F une extension finie non ramifi\'ee de Q_p et rhobar une représentation modulo p irréductible de dimension 2 du groupe de Galois absolu de F. L'objet de ce travail est la détermination de la variété de Kisin qui paramètre les modules de Breuil-Kisin associés à certaines familles de (...)

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[hal-00601448] Ramification and cleanliness

22 juin 2016

This article is devoted to studying the ramification of Galois torsors and of $\ell$-adic sheaves in characteristic $p>0$ (with $\ell\not=p$). Let $k$ be a perfect field of characteristic $p>0$, $X$ be a smooth, separated and quasi-compact $k$-scheme, $D$ be a simple normal crossing (...)

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[hal-00000740] Monogenous algebras. Back to Kronecker.

22 juin 2016

In this note we develop some properties of those algebras (called here locally simple) which can be generated by a single element after, if need be, a faithfullyflat extension. For finite algebras, this is shown to be in fact a property of the geometric fibers. Morphisms between rings of (...)

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[hal-00600371] Integral points of bounded height on toric varieties

22 juin 2016

We establish asymptotic formulas for the number of integral points of bounded height on toric varieties.

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[hal-01102016] Improvements to the general number field sieve for discrete logarithms in prime fields

22 juin 2016

In this paper, we describe many improvements to the number field sieve. Our main contribution consists of a new way to compute individual logarithms with the number field sieve without solving a very large linear system for each logarithm. We show that, with these improvements, the number field (...)

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[hal-01102025] Computing isogenies between elliptic curves over $GF(p^n)$ using Couveignes's algorithm

22 juin 2016

The heart of the improvements of Elkies to Schoof's algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies' approach is well suited for the case where the characteristic of the field is large. Couveignes showed (...)

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[hal-01095359] Efficient computation of pairings on Jacobi quartic elliptic curves

22 juin 2016

This paper proposes the computation of the Tate pairing, Ate pairing and its variations on the special Jacobi quartic elliptic curve Y 2 D dX 4 C Z 4 . We improve the doubling and addition steps in Miller's algorithm to compute the Tate pairing. We use the birational equivalence between Jacobi (...)

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[hal-00765577] Some algorithms for skew polynomials over finite fields

22 juin 2016

In this paper, we study the arithmetics of skew polynomial rings over finite fields, mostly from an algorithmic point of view. We give various algorithms for fast multiplication, division and extended Euclidean division. We give a precise description of quotients of skew polynomial rings by a (...)

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[hal-00315608] Fonctions L d'Artin et nombre de Tamagawa motiviques

22 juin 2016

In the first part of this text, we define motivic Artin L-fonctions via a motivic Euler product, and show that they coincide with the analogous functions introduced by Dhillon and Minac. In the second part, we define under some assumptions a motivic Tamagawa number and show that it specializes (...)

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[hal-01334181] Indefinite theta series and generalized error functions

20 juin 2016

Theta series for lattices with indefinite signature $(n_+,n_-)$ arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case ($n_+=1$), but have remained obscure when $n_+\geq 2$. Using (...)

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[hal-01333682] Multidimensional Heilbronn sets

18 juin 2016

We show in the context of $\mathbbZ^k$-actions that every van der Corput set is a Heilbronn set. Furthermore we establish Diophantine inequalities of the Heilbronn type for generalized polynomials $g$ in particular for sequences $\nu(n)=\lfloor n^c\rfloor+n^k$ with $c>1$ a non-integral real (...)

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[hal-01207158] Forms of differing degrees over number fields

18 juin 2016

Consider a system of polynomials in many variables over the ring of integers of a number field $K$. We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any smooth and geometrically integral variety $X\subseteq (...)

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[hal-01278708] On digital blocks of polynomial values and extractions in the Rudin–Shapiro sequence

17 juin 2016

Let P (x) ∈ Z[x] be an integer-valued polynomial taking only positive values and let d be a fixed positive integer. The aim of this short note is to show, by elementary means, that for any sufficiently large integer N ≥ N_0(P, d) there exists n such that P(n) contains exactly N occurrences of the (...)

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[hal-01320388] Computing cardinalities of Q-curve reductions over finite fields

17 juin 2016

We present a specialized point-counting algorithm for a class of elliptic curves over F_p^2 that includes reductions of quadratic Q-curves modulo inert primes and, more generally, any elliptic curve over F_p^2 with a low-degree isogeny to its Galois conjugate curve. These curves have (...)

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[hal-01332997] On a variant of Schanuel conjecture for the Carlitz exponential

16 juin 2016

We introduce and discuss a variant of Schanuel conjecture in the framework of the Carlitz exponential function over Tate algebras and allied functions. Another purpose of the present paper is to widen the horizons of possible investigations in transcendence and algebraic independence in (...)

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[hal-01332417] SYSTEMS OF SETS OF LENGTHS: TRANSFER KRULL MONOIDS VERSUS WEAKLY KRULL MONOIDS

16 juin 2016

Transfer Krull monoids are monoids which allow a weak transfer homomorphism to a commutative Krull monoid, and hence the system of sets of lengths of a transfer Krull monoid coincides with that of the associated commutative Krull monoid. We unveil a couple of new features of the system of sets (...)

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[hal-01285685] CONJUGACY GROWTH SERIES OF SOME INFINITELY GENERATED GROUPS

15 juin 2016

It is observed that the conjugacy growth series of the infinite fini-tary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed, for other generating sets, for restricted permutational (...)

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[hal-01300328] Valeurs moyennes effectives de fonctions multiplicatives complexes

10 juin 2016

Nous établissons des estimations de valeurs moyennes pour une large classe de fonctions arithmétiques multiplicatives, fournissant ainsi des versions quantitatives essentiellement optimales des évaluations classiques de Wirsing et étendant celles de Halász. Plusieurs applications sont explicitées, (...)

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[hal-01329432] Graded structures and differential operators on nearly holomorphic and quasimodular forms on classical groups

9 juin 2016

We wish to use graded structures [KrVu87], [Vu01] on dffierential operators and quasimodular forms on classical groups and show that these structures provide a tool to construct p-adic measures and p-adic L-functions on the corresponding non-archimedean weight spaces. An approach to (...)

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[hal-01329292] Sur la variation quadratique totale de la suite des parties fractionnaires des quotients d'un nombre réel positif par les nombres entiers naturels consécutifs

9 juin 2016

We give an asymptotic formula for the quadratic total variation of the sequence of fractional parts of the quotients of a positive real number by the consecutive natural numbers.

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[hal-01329206] Sur la variation totale de la suite des parties fractionnaires des quotients d'un nombre réel positif par les nombres entiers naturels consécutifs

9 juin 2016

We give an asymptotic formula for the total variation of the sequence of fractional parts of the quotients of a positive real number by the consecutive natural numbers.

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[hal-01300328] Valeurs moyennes effectives de fonctions multiplicatives complexes

8 juin 2016

Nous établissons des estimations de valeurs moyennes pour une large classe de fonctions arithmétiques multiplicatives, fournissant ainsi des versions quantitatives essentiellement optimales des évaluations classiques de Wirsing et étendant celles de Halász. Plusieurs applications sont explicitées, (...)

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[hal-01227699] Computing Jacobi's $\theta$ in quasi-linear time

7 juin 2016

Jacobi's $\theta$ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of $\theta(z, \tau)$, for $z$, $\tau$ verifying certain conditions, with precision $P$ in $O(M(P) \sqrtP)$ bit operations, where $M(P)$ denotes the number of (...)

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[hal-01320388] Computing cardinalities of Q-curve reductions over finite fields

7 juin 2016

We present a specialized point-counting algorithm for a class of elliptic curves over F_p^2 that includes reductions of quadratic Q-curves modulo inert primes and, more generally, any elliptic curve over F_p^2 with a low-degree isogeny to its Galois conjugate curve. These curves have (...)

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[hal-01295254] Lubin's conjecture for full $p$-adic dynamical systems

6 juin 2016

We give a short proof of a conjecture of Lubin concerning certain families of $p$-adic power series that commute under composition. We prove that if the family is \emphfull (large enough), there exists a Lubin-Tate formal group such that all the power series in the family are endomorphisms of (...)

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[hal-01325350] On the denominators of the Taylor coefficients of G-functions

5 juin 2016

Let $\sum_n=0^\infty a_n z^n\in \overline\mathbb Q$ be a $G$-function, and, for any $n\ge0$, let $\delta_n\ge 1$ denote the least integer such that $\delta_n a_0, \delta_n a_1, ..., \delta_n a_n$ are all algebraic integers. By definition of a $G$-function, there exists some constant $c\ge 1$ (...)

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[tel-01084492] Généralisations du critère d’indépendance linéaire de Nesterenko

31 mai 2016

Cette thèse s'inscrit dans le prolongement du résultat d'Apéry donnant l'irrationalité de ζ (3) et de celui de Ball-Rivoal prouvant qu'il existe une infinité d'entiers impairs en lesquels la fonction zêta de Riemann prend des valeurs irrationnelles. Un outil crucial dans la démonstration de (...)

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[hal-00797625] Uniformisation des variétés de Laumon-Rapoport-Stuhler et conjecture de Drinfeld-Carayol

31 mai 2016

Considérons les variétés de "D-faisceaux elliptiques~" Ell introduites par Laumon, Rapoport et Stuhler, définies sur un corps de fonctions F d'une variable sur un corps fini, pour D une algèbre à division de dimension d^2 sur F. Nous montrons que ces variétés admettent, en une place o de F (...)

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[hal-01320940] Explicit isogenies in quadratic time in any characteristic

31 mai 2016

Consider two ordinary elliptic curves $E,E'$ defined over a finite field $\F_q$, and suppose that there exists an isogeny $\psi$ between $E$ and $E'$. We propose an algorithm that determines $\psi$ from the knowledge of $E$, $E'$ and of its degree $r$, by using the structure of the $ℓ$-torsion (...)

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[hal-00797609] Algèbre T1 : Groupes, Corps et Théorie de Galois

31 mai 2016

Ce livre s'adresse aux étudiants de licence ou master de mathématiques (L3-M1) et à ceux qui préparent le Capes ou l'agrégation. Il traite de la théorie des groupes, de la théorie des corps et d'un de leurs points communs essentiels, la théorie de Galois des extensions finies. Chacune de ces théories (...)

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[hal-01320940] Explicit isogenies in quadratic time in any characteristic

30 mai 2016

Consider two ordinary elliptic curves $E,E'$ defined over a finite field $\F_q$, and suppose that there exists an isogeny $\psi$ between $E$ and $E'$. We propose an algorithm that determines $\psi$ from the knowledge of $E$, $E'$ and of its degree $r$, by using the structure of the $ℓ$-torsion (...)

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[hal-00797622] Sur les réalisations des correspondances de Langlands dans la cohomologie des tours de Drinfeld et de Lubin-Tate

29 mai 2016

Le cadre de cet article est celui du programme de Langlands : on réalise les correspondances de Langlands (du moins pour les supercuspidales) et de Jacquet-Langlands locales dans la cohomologie de certains espaces de modules de groupes $p$-divisibles, à savoir la tour dite de Lubin-Tate et la (...)

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[tel-01237604] Mathematical model of the role and temporal dynamics of protein p53 after drug-induced DNA damage

29 mai 2016

Various molecular pharmacokinetic–pharmacodynamic models have been proposed in the last decades to represent and predict drug effects in anticancer therapies. Most of these models are cell population based models since clearly measurable effects of drugs can be seen on populations of (healthy (...)

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[tel-01178288] P-adic local Langlands correspondence and geometry

29 mai 2016

This thesis concerns the geometry behind the p-adic local Langlands correspondence. We give a formalism of methods of Emerton, which would permit to establish the Fontaine-Mazur conjecture in the general case for unitary groups. Then, we verify that our formalism works well in the case of U(3) (...)

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[hal-00078974] Géométrie des surfaces algébriques et points entiers

28 mai 2016

Let $X$ be a projective normal surface over a number field $K$. Let $H$ be the sum of four properly intersecting ample effective divisors on $X$. We show that any set of $S$-integral points in $X-H$ is not Zariski dense.

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[hal-00482405] Using hyperelliptic curves to find positive polynomials that are not a sum of three squares in R(x, y)

28 mai 2016

This article deals with a quantitative aspect of Hilbert's seventeenth problem: producing a collection of real polynomials in two variables of degree 8 in one variable which are positive but are not a sum of three squares of rational fractions. As explained by Huisman and Mahe, a given monic (...)

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[hal-00374041] Local Fourier transform and epsilon factors

28 mai 2016

Laumon introduced the local Fourier transform for $\ell$-adic Galois representations of local fields, of equal characteristic $p$ different from $\ell$, as a powerful tool to study the Fourier-Deligne transform of $\ell$-adic sheaves over the affine line. In this article, we compute explicitly (...)

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[hal-00374047] Analyse micro-locale l-adique en caracteristique p>0: Le cas d'un trait

28 mai 2016

We develop, for an l-adic etale sheaf on a complete trait of characteristic p>0, the notion of characteristic variety. Our approach, inspired by the microlocal analysis of Kashiwara and Schapira, is a complement to our ramification theory for local fields with general residue fields. We (...)

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[hal-00455582] Igusa integrals and volume asymptotics in analytic and adelic geometry

28 mai 2016

We establish asymptotic formulae for volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying (...)

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[hal-00136041] Abelian varieties with l-adic Galois representation of Mumford's type

28 mai 2016

Nous étudions des variétés abéliennes sur des corps de nombres telles que l'algèbre de Lie de l'image d'une représentation l-adique associée soit géométriquement isomorphe à c + sl(2)^3. On appelle de telles variétés abéliennes des `variétés abéliennes avec représentation l-adique du type de Mumford'. On (...)

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[hal-00136040] On Mumford's families of abelian varieties

28 mai 2016

Mumford a construit des familles de variétés abéliennes de dimension 4 qui sont paramétrées par des variétés de Shimura, mais qui ne sont pas de type PEL. Le but du présent article est d'étudier les familles en question. Pour chaque fibre sur un corps de nombres dans une telle famille, on détermine les (...)

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[hal-00944666] Un calcul d'anneaux de déformations potentiellement Barsotti--Tate

27 mai 2016

Let F be an unramified extension of Qp. The first aim of this work is to develop a purely local method to compute the potentially Barsotti-Tate deformations rings with tame Galois type of irreducible two-dimensional representations of the absolute Galois group of F. We then apply our method in (...)

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[hal-00455574] Heights and measures on analytic spaces. A survey of recent results, and some remarks

27 mai 2016

This paper has two goals. The first is to present the construction, due to the author, of measures on non-archimedean analytic varieties associated to metrized line bundles and some of its applications. We take this opportunity to add remarks, examples and mention related (...)

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[hal-00018574] Produit eulérien motivique et courbes rationnelles sur les variétés toriques

27 mai 2016

We study the asymptotical behaviour of the moduli space of morphisms of given anticanonical degree from a rational curve to a split toric variety, when the degree goes to infinity. We obtain in this case a geometric analogue of Manin's conjecture about rational points of bounded height on (...)

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[hal-00952952] Tracking p-adic precision

27 mai 2016

We present a new method to propagate $p$-adic precision in computations, which also applies to other ultrametric fields. We illustrate it with many examples and give a toy application to the stable computation of the SOMOS 4 (...)

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[hal-00294979] Sur la classification de quelques phi-modules simples

27 mai 2016

This note is an appendix to a preprint by E. Hellmann. We give a complete classification of simple objects of the category of vector spaces D over K = Fpbar((u)) equipped with an endomorphism phi whose image generates D and that are semi-linear with respect to the ring morphism sending u to u^b (...)

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[hal-00455578] Integral points of bounded height on partial equivariant compactifications of vector groups

27 mai 2016

We establish asymptotic formulas for the number of integral points of bounded height on partial equivariant compactifications of vector groups.

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[hal-00449471] La conjecture de Manin géométrique pour une famille de quadriques intrinsèques

27 mai 2016

We prove a version of Manin's conjecture for a certain family of intrinsic quadrics, the base field being a global field of positive characteristic. We also explain how a very slight variation of the method we use allows to establish the conjecture for a certain generalized del Pezzo (...)

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[hal-00368090] Points de vue sur les valeurs aux entiers des formes quadratiques binaires

27 mai 2016

The primary focus of this largely expository paper is to present different points of view on the values assumed by binary quadratic forms, sometimes with irrational coefficients, at integer values. Three different points of view are taken. One is the traditional approximation of real numbers by (...)

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[hal-01293573] A note on certain representations in characteristic p and associated functions

26 mai 2016

We discuss certain representations of GL 2 Fq[T] in equal characteristic and associated vectorial modular forms

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[hal-01321849] Character theory approach to Sato–Tate groups

26 mai 2016

In this article, we propose to use the character theory of compact Lie groups and their orthogonality relations for the study of Frobenius distribution and Sato-Tate groups. The results show the advantages of this new approach in several aspects. With samples of Frobenius ranging in size much (...)

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[hal-01320388] Computing cardinalities of Q-curve reductions over finite fields

25 mai 2016

We present a specialized point-counting algorithm for a class of elliptic curves over F_p^2 that includes reductions of quadratic Q-curves modulo inert primes and, more generally, any elliptic curve over F_p^2 with a low-degree isogeny to its Galois conjugate curve. These curves have (...)

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[hal-01320496] Solving discrete logarithms on a 170-bit MNT curve by pairing reduction

25 mai 2016

Pairing based cryptography is in a dangerous position following the breakthroughs on discrete logarithms computations in finite fields of small characteristic. Remaining instances are built over finite fields of large characteristic and their security relies on the fact the embedding field of (...)

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[hal-01295749] Strong modularity of reducible Galois representations

25 mai 2016

In this paper, we call strongly modular those reducible semi-simple odd mod $l$ Galois representations for which the conclusion of the strongest form of Serre's original modularity conjecture holds. Under the assumption that the Serre weight $k$ satisfies $l>k+1$, we give a precise (...)

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[tel-01247743] Equations diophantiennes et corps cyclotomiques

25 mai 2016

This thesis examines some approaches to address Diophantine equations, specifically we focus on the connection between the Diophantine analysis and the theory of cyclotomic fields.First, we propose a quick introduction to the methods of Diophantine approximation we have used in this research (...)

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[hal-01292805] SUMS OF TWO ${S}$-UNITS VIA FREY-HELLEGOUARCH CURVES

24 mai 2016

In this paper, we develop a new method for finding all perfect powers which can be expressed as the sum of two rational S-units, where S is a finite set of primes. Our approach is based upon the modularity of Galois representations and, for the most part, does not require lower bounds for (...)

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[hal-01196990] On the Conjecture of Lehmer, limit Mahler measure of trinomials and asymptotic expansions

22 mai 2016

Let $n ≥ 2$ be an integer and denote by $\theta_n$ the real root in $(0, 1)$ of the trinomial $G_n(X) = −1 + X + X^n$ . The sequence of Perron numbers $(\theta_n^−1 )_n≥2$ tends to 1. We prove that the Conjecture of Lehmer is true for $\\theta_n^−1 | n ≥ 2\$ by the direct method of Poincar\'e (...)

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[cea-01318414] Multiple D3-instantons and mock modular forms I

21 mai 2016

We study D3-instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In a previous work, consistency of D3-instantons with S-duality was established at first order in the instanton expansion, using the modular properties of the (...)

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[hal-01319255] Multiple D3-instantons and mock modular forms I

20 mai 2016

We study D3-instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In a previous work, consistency of D3-instantons with S-duality was established at first order in the instanton expansion, using the modular properties of the (...)

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[hal-01318564] Central Limit Theorem for probability measures defined by sum-of-digits function in base 2

20 mai 2016

In this paper we prove a central limit theorem for some probability measures defined as asymtotic densities of integer sets defined via sum-of-digit-function. To any integer a we can associate a measure on Z called µa such that, for any d, µa(d) is the asymptotic density of the set of integers n (...)

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[hal-01318944] Hecke algebras and affine flag varieties in characteristic p

20 mai 2016

Let G be a split semi-simple p-adic group and let H be its Iwahori-Hecke algebra with coefficients in the algebraic closure k of the finite field with p elements. Let F be the affine flag variety over k associated with G. We show, in the simply connected simple case, that a torus-equivariant (...)

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[hal-01314014] Calcul de pi par la méthode originale des carrés inscrits

19 mai 2016

Le nombre pi est utilisé dans des domaines variés des mathématiques, de la physique et d’ingénierie. Le défi que représente la détermination de sa valeur a stimulé la créativité et l'enthousiasme de nombreux chercheurs depuis l'antiquité. De nombreuses méthodes plus ou moins complexes permettent de (...)

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[hal-01279192] Local behavior of arithmetical functions with applications to automorphic L-functions

19 mai 2016

We derive a Voronoi-type series approximation for the local weighted mean of an arithmetical function that is associated to Dirichlet series satisfying a functional equation with gamma factors. The series is exploited to study the oscillation frequency with a method of Heath-Brown and Tsang (...)

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[hal-01280194] Sign changes of Fourier coefficients of modular forms of half integral weight, 2

19 mai 2016

In this paper, we investigate the sign changes of Fourier coefficients of half-integral weight Hecke eigenforms and give two quantitative results on the number of sign changes.

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[hal-01278944] The number of coefficients of automorphic L-functions for GL m of same signs

19 mai 2016

Let π be an irreducible unitary cuspidal representation for GL m (A Q), and let L(s, π) be the automorphic L-function attached to π, which has a Dirich-let series expression in the half-plane Re s > 1. When π is self-contragredient, all the coefficients in the Dirichlet series expression are (...)

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[hal-01278937] Distribution of Hecke eigenvalues of newforms in short intervals

19 mai 2016

We explore nonvanishing of Hecke eigenvalues in short arithmetic progressions and their signs in short intervals by further developing B-free number method and studying moments in short intervals. We improve some results of Alkan & Zaharescu [3] and Lau & Wu [20] on the distribution of (...)

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[inria-00211875] Worst-Case Hermite-Korkine-Zolotarev Reduced Lattice Bases

19 mai 2016

The Hermite-Korkine-Zolotarev reduction plays a central role in strong lattice reduction algorithms. By building upon a technique introduced by Ajtai, we show the existence of Hermite-Korkine-Zolotarev reduced bases that are arguably least reduced. We prove that for such bases, Kannan's (...)

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[hal-01278408] On a divisor problem related to the Epstein Zeta-Function, IV

18 mai 2016

In this paper we study several divisor problems related to the Epstein zeta-function. We are able to improve previous results and establish some new results by applying some classical techniques.

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[hal-01278402] On a divisor problem related to the Epstein Zeta-Function, III

18 mai 2016

In this paper we study the mean square of the error term ∆ * k (Q, x) in a divisor problem related to the Epstein zeta-function. An asymptotic formula has been obtained when k = 2.

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[hal-01277992] On a divisor problem related to the Epstein Zeta-Function, II

18 mai 2016

Recently by the theory of modular forms and the Riemann zeta-function, Lü improved the estimates for the error term in a divisor problem related to the Epstein zeta-function established by Sankaranarayanan. In this short note, we are able to further sharpen some results of Sankaranarayanan and (...)

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[tel-01316937] Éléments explicites en théorie algébrique des nombres

18 mai 2016

Ce mémoire présente une synthèse de mes travaux de recherche en théorie algébrique des nombres menés entre 2003 et 2013, seul ou en collaboration. Ils portent principalement sur l'étude de la structure galoisienne de modules associés à des extensions de corps de nombres, sous diverses hypothèses en (...)

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[hal-01316201] Small sumsets in real numbers : a continuous 3k-4 theorem

18 mai 2016

We prove a continuous Freiman's 3k-4 theorem for small sumsets in R by using some ideas from Ruzsa's work on measure of sumsets in R as well as some graphic representation of density functions of sets. We thereby get some structural properties of A, B and A+B when (...)

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[inria-00187880] LLL: a tool for effective diophantine approximation

18 mai 2016

The purpose of this paper is to survey in a unified setting some of the results in diophantine approximation that the LLL algorithm can make effective in an efficient way. We mostly study the problems of finding good rational approximations to vectors of real and p-adic numbers, and of finding (...)

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[inria-00000904] Euclidean minima of totally real fields : Algorithmic determination.

18 mai 2016

This article deals with the determination of the Euclidean minimum $ M(K)$ of a totally real number field $ K$ of degree $ n\geq 2$, using techniques from the geometry of numbers. Our improvements of existing algorithms allow us to compute Euclidean minima for fields of degree $ 2$ to $ 8$ and (...)

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[hal-00091199] Moyennes de certaines fonctions multiplicatives sur les entiers friables, 2

18 mai 2016

Nous évaluons les moyennes indiquées dans le titre sous des hypothèses analytiques concernant la série de Dirichlet associée

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[hal-00136940] Inhomogeneous and Euclidean spectra of number fields with unit rank strictly greater than 1

18 mai 2016

Let K be a number field with unit rank r > 1. In this article we show that the inhomogeneous minimum of K is attained by at least one rational point. In particular, if M(K) is the Euclidean minimum of K, we have . This phenomenon has consequences on the decidability of the Euclidean nature (...)

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[hal-00136941] Euclidean minima of totally real number fields, algorithmic determination

18 mai 2016

Cet article traite de la détermination explicite du minimum euclidien (pour la norme) des corps de nombres totalement réels, voire de la partie supérieure de leur spectre euclidien. L'efficacité de l'algorithme mis en oeuvre permet de travailler jusqu'au degré (...)

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[hal-00107309] LLL on the average

18 mai 2016

Despite their popularity, lattice reduction algorithms remain mysterious in many ways. It has been widely reported that they behave much more nicely than what was expected from the worst-case proved bounds, both in terms of the running time and the output quality. In this article, we (...)

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[hal-01282675] HIGHER INTEGRAL MOMENTS OF AUTOMORPHIC L-FUNCTIONS IN SHORT INTERVALS

17 mai 2016

We consider the higher integral moments for automorphic L-functions in short intervals and give a proof for the conjecture of Conrey et al. under Generalized Riemann Hypothesis for automorphic L-function.

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[hal-01282682] Distribution of values of symmetric power L-functions at the edge of the critical strip

17 mai 2016

We study some problems on the distribution of values of symmetric power L-functions at s = 1 in both aspects of level and weight: bounds of these values, extreme values, Montgomery-Vaughan's conjecture and distribution functions. Our results generalise and/or improve related results of (...)

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[tel-01282669] Valeurs centrales et valeurs au bord de la bande critique de fonctions L automorphe

17 mai 2016

Cette thèse, constitué en trois parties, est consacrée à l’étudie des valeurs spéciales de fonctions L automorphes. La première partie contient un survol rapide de la théorie des formes modulaires et des fonctions L de puissance symétrique associées qui est nécessaire dans la suite. Dans la seconde (...)

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[hal-01316252] A note on generalized radix representations and dynamical systems

16 mai 2016

Akiyama et al. proved an asymptotic formula for the distribution of CNS poly-nomials with fixed constant term. The objective of the present paper is to improve that result by providing an error term too.

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[hal-01316251] Construction of Normal Numbers via Generalized Prime Power Sequences

16 mai 2016

We construct normal numbers in base q by concatenating q-adic expansions of prime powers $\left\lfloor\alpha p^\theta\right\rfloor$ with $\alpha>0$ and $\theta>1$.

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[hal-01316249] Construction of μ -normal sequences

16 mai 2016

In the present paper we extend Champernowne's construction of normal numbers to provide sequences which are generic for a given invariant probability measure, which need not be the maximal one. We present a construction together with estimates and examples for normal numbers with respect to (...)

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[hal-00875091] On the distribution functions of two oscillating sequences

16 mai 2016

We investigate the set of all distribution functions of two special sequences on the unit interval, which involve logarithmic and trigonometric terms. We completely characterise the set of all distribution functions $G(x_n)$ for $(x_n)_n \geq 1 = (\\cos (\alpha n)^n\)_n \geq 1$ and arbitrary (...)

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[hal-01314745] Sign changes of the coefficients of automorphic L-functions

13 mai 2016

This survey gives an account of background and the recent development concerning sign changes of Fourier coefficients of modular forms, which includes the great contributions of other authors. We are attempting to elucidate interesting viewpoints, ideas and methods which, even unspecified, may (...)

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[hal-01315080] Multidimensional Bell inequalities and quantum cryptography. Codes, cryptology, and information security.

12 mai 2016

The laws of quantum physics allow the design of cryptographic protocols for which the security is based on physical principles. The main cryptographic quantum protocols are key distribution schemes, in which two parties generate a shared random secret string. The privacy of the key can be (...)

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[hal-01315075] A qutrit quantum key distribution protocol using Bell inequalities with larger violation capabilities.

12 mai 2016

The Ekert quantum key distribution protocol [1] uses pairs of entangled qubits and performs checks based on a Bell inequality to detect eavesdropping. The 3DEB protocol [2] uses instead pairs of entangled qutrits to achieve better noise resistance than the Ekert protocol. It performs checks (...)

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[hal-01315047] Estimates of ψ,θ for large values of x without the Riemann hypothesis

12 mai 2016

The enlargement of known zero-free regions has enabled us to find better effective estimates for classical number-theoretic functions linked to the distribution of prime numbers. In particular we draw the quintessence of the method of Rosser and Schoenfeld on the upper bounds for the usual (...)

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[hal-01314997] On the splitting of the exact sequence, relating the wild and tame kernels

12 mai 2016

Let k be a number field. For an odd prime p and an integer i>1, the i-th \'etale wild kernel is contained in the second cohomology group of o'_k with coefficients in Zp(i), where o'_k is the ring of p-integers of k. Using Iwasawa theory, we give conditions for this inclusion to split. In (...)

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[hal-01313346] Shifted convolution of cusp-forms with θ-series

10 mai 2016

In this paper we apply simple approachs to improve a recent result due to Luo, concerning a shifted convolution sum involving the Fourier coefficients of cusp forms with those of theta series. In order to explore what should be the best possible bound, a mean square result of this shifted (...)

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[hal-01307789] There are infinitely many twin primes 30n+11 and 30n+13, 30n+17 and 30n+19, 30n+29 and 30n+31

10 mai 2016

We proved that $\liminf\limits_n \rightarrow +\infty(p_n+1-p_n)=2$ where $p_n$ is the $n-th$ prime number. We showed the conditions on which an integer $10X+1$, $10X+3$, $10X+7$ or $10X+9$ can be a prime number. We studied the conditions required to get some twin primes and proved the twin (...)

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[hal-01281342] Sur la répartition du noyau d'un entier

10 mai 2016

We investigate the asymptotic behaviour of the number $N(x,y)$ of those integers $n\leqslant x$ with squarefree kernel $k(n)\leqslant y$.

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[hal-01281305] Mesures quadratiques de la proximité des diviseurs

10 mai 2016

Diverses mesures naturelles de la proximité des diviseurs d'un entier sont introduites et plusieurs conjectures sont résolues.

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[hal-01313347] Fourier coefficients of symmetric power L-functions

10 mai 2016

Let f be a Hecke eigencusp form of even integral weight k or Maass cusp form for the full modular group SL 2 (Z). Denote by λ sym m f (n) the nth normalized coefficient of the Dirichlet expansion of the mth symmetric power L-function associated to f. In this paper, we establish some bounds for (...)

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[hal-01280840] On irregularities of distribution of binary sequences relative to arithmetic progressions, I. (General results)

10 mai 2016

In 1964 K. F. Roth initiated the study of irregularities of distribution of binary sequences relative to arithmetic progressions and since that numerous papers have been written on this subject. In the applications one needs binary sequences which are well distributed relative to arithmetic (...)

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[hal-01280845] On a phenomenon of Turán concerning the summands of partitions

10 mai 2016

Turán [12] proved that for almost all pairs of partitions of an integer, the proportion of common parts is very high, that is greater than 1/ 2 − ε with ε > 0 arbitrarily small. In this paper we prove that this surprising phenomenon persists when we look only at the summands in a fixed (...)

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[hal-01280829] Various approaches for the study of the complexity of some families of pseudorandom subsets

10 mai 2016

Studying randomness in different structures is important from the development of applications and theory. Dartyge, Mosaki and Sárközy (among others) have studied measures of randomness for families of subsets of integers. In this article, we improve results on the complexity of some families (...)

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[hal-01303279] Primes in [kn, (k + 1)n] , for 1⩽. k ⩽. 10

9 mai 2016

In 1919 S. Ramanujan proved [Ram] the Bertrand postulate elegantly with approximations based on the Stirling formula. P. Erdös proved also [Erd] the same result in 1931 by a carefull study of the binomial coeficient (2n n). The Bertrand postulate was first proved by P. Tchebichef in 1850. (...)

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[hal-01312603] Diophantine approximation with improvement of the simultaneous control of the error and of the denominator

9 mai 2016

In this work we proof the following theorem which is, in addition to some other lemmas, our main result: \noindent \textbftheorem. Let$\ X=\left\ \left( x_1\text, % t_1\right) \text, \left( x_2\text, t_2\right) \text, ..., \left( x_n\text, t_n\right) \right\ $ be a finite part of (...)

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[hal-01311299] Lower Bounds for the Height in Galois Extensions

9 mai 2016

We prove essentially sharp lower bounds for the height of an algebraic number in a Galois extension

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[hal-01081309] Note sur les valeurs moyennes criblées de certaines fonctions arithmétiques

8 mai 2016

We provide, in a wide range of the parameters, an estimate for the mean-value over sifted integers of certain arithmetic functions with Dirichlet series analytically close to $1/\zeta_\mathbbK(s)$, where $\mathbbK$ is a number field and $\zeta_\mathbbK$ its Dedekind zeta (...)

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[hal-01281714] Sur l'inégalité de Turán–Kubilius friable

7 mai 2016

Nous établissons une version uniforme, relativement à tous les paramètres, de l’inégalité de Turán-Kubilius friable (i.e.relative aux entiers sans grand facteur premier), comparant la variance empirique d’une fonction arithmétique additive à support friable à celle de son modèle probabiliste. Nous (...)

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[hal-01281724] An Ingham-Müntz type theorem and simultaneous observation problems

7 mai 2016

We establish a theorem combining the estimates of Ingham and Müntz – Szász. Moreover, we allow complex exponents instead of purely imaginary exponents for the Ingham type part or purely real exponents for the Müntz – Szász part. A very special case of this theorem allows us to prove the simultaneous (...)

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[hal-01281732] Fonctions multiplicatives, sommes d'exponentielles, et loi des grands nombres

7 mai 2016

Nous donnons des conditions suffisantes effectives quasi-optimales pour que la majoration de Halberstam–Richert fournisse effectivement l'ordre de grandeur de la valeur moyenne d'une fonction multiplicative positive ou nulle.

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[hal-01067949] On the generic local Langlands correspondence for GSpin groups

5 mai 2016

In the case of split $GSpin$ groups, we prove an equality of $L$-functions between automorphic local $L$-functions defined by the Langlands-Shahidi method and local Artin $L$-functions. Our method of proof is based on previous results of the first author which allow to reduce the problem to (...)

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[hal-01296795] Application de réciprocité de Shafarevich et groupe de Nottingham

4 mai 2016

Soient K le corps local View the MathML source et Kab sa clôture abélienne. On note View the MathML source le groupe des Fq-automorphismes continus σ de Kab tel que σ(K)=K et tel que σ opère sur le corps résiduel View the MathML source comme une puissance entière de l'automorphisme de Frobénius. On (...)

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[hal-01311391] Sums of seventh powers of polynomials over a finite field with 8 elements

4 mai 2016

Let F be a finite field with 8 elements. We study representations of polynomials over F as sums and strict sums of seventh powers.

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[hal-01311367] Sums of seventh powers in the ring of polynomials over the finite field with four elements

4 mai 2016

We study representations of polynomials P in F4[T] as sums P = X17 + . . . + Xs7.

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[hal-01311352] Waring's problem for polynomials in two variables

4 mai 2016

We prove that all polynomials in several variables can be decomposed as the sums of $k$th powers: $P(x_1,...,x_n) = Q_1(x_1,...,x_n)^k+...+ Q_s(x_1,...,x_n)^k$, provided that elements of the base field are themselves sums of $k$th powers. We also give bounds for the number of terms $s$ and the (...)

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[hal-01311343] Sums of $(p^r+1)$-th powers in the polynomial ring $\Bbb F_{p^m}[T]$

4 mai 2016

Let p be an odd prime number and let F be a finite field with p(m) elements. We study representations and strict representations of polynomials M epsilon F[T] by sums of (p(r) + 1)-th powers. A representation M = M-1(k) + ... + M-s(k) of M epsilon F[T] as a sum of k-th powers of polynomials is (...)

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[hal-01311330] On Murata densities

4 mai 2016

In this paper, we set up an abstract theory of Murata densities, well tailored to general arithmetical semigroups. In [On certain densities of sets of primes, Proc. Japan Acad. Ser. A Math. Sci. 56(7) (1980) 351-353; On some fundamental relations among certain asymptotic densities, Math. Rep. (...)

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[hal-01311322] Sums of seventh powers in the polynomial ring $\mathbb{F}_{2^{m}}[T]$

4 mai 2016

Let F be a finite field with even characteristic and q >= 16 elements. We study representations of polynomials P is an element of F [T] as sums P = X-1(7) + ... + X-s(7).

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[hal-01297936] Répartition des fonctions Q-additives le long des carrés de polynômes sur un corps fini

4 mai 2016

Soit F un corps fini à q éléments (q impair), Q F [T ] et f une fonction complètement Q-additive à valeurs entières. L'objet de ce travail est l'étude des sommes d'exponentielles _XF[T]X

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[hal-01272342] Sur les puissances des polynômes sur un corps fini

4 mai 2016

Let F a finite field, Q a given element of F[T] with positive degree and k a positive integer. The goal of this work is to study the Q-automaticity of the set of kth powers of elements of F[T] and to calculate the number of polynomials X ∈ F[T] of degree N such that the sum of digits of Xk in (...)

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[hal-01309134] Are monochromatic Pythagorean triples avoidable?

4 mai 2016

A Pythagorean triple is a triple of positive integers a, b, c ∈ N-0 satisfying a^2 + b^2 = c^2. Is it true that, for any finite coloring of N-0, at least one Pythagorean triple must be monochromatic? In other words, is the Diophantine equation X^2 +Y^2 = Z^2 regular? This problem has been open (...)

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[hal-01311138] Weak Néron models for cubic polynomial maps over a non-Archimedean field

4 mai 2016

The aim of this note is to give an effective criterion to verify whether a cubic polynomial over a non-Archimedean field has a weak N\'eron model or not.

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[tel-01310561] Le théorème de Gauss sur les sommes de 3 carrés, de faisceaux, et composition de Gauss

4 mai 2016

Le théorème de Gauss sur les sommes de 3 carrés relie le nombre de points entiers primitifs sur la sphère de rayon la racine carrée de n au nombre de classes d'un ordre quadratique imaginaire. En 2011, Edixhoven a esquissée une preuve du théorème de Gauss en utilisant une approche de la géométrie (...)

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[hal-00995770] Inverse binomial series and values of Arakawa-Kaneko zeta functions

3 mai 2016

In this article, we present a variety of evaluations of series of polylogarithmic nature. More precisely, we express the special values at positive integers of two classes of zeta functions of Arakawa-Kaneko-type by means of certain inverse binomial series involving harmonic sums which appeared (...)

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[hal-00604393] Le produit harmonique des suites

3 mai 2016

Au moyen d'une transformation binomiale involutive dans l'espace des suites à valeurs complexes, on définit un nouveau produit dénommé " harmonique " en raison de ses remarquables propriétés à l'égard des sommes harmoniques. La transformation d'Euler des séries permet de déduire de ces (...)

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[hal-00495767] A new class of identities involving Cauchy numbers, harmonic numbers and zeta values

3 mai 2016

Improving an old idea of Hermite, we associate to each natural number $k$ a modified zeta function of order $k$. The evaluation of the values of these functions $F_k$ at positive integers reveals a wide class of identities linking Cauchy numbers, harmonic numbers and zeta (...)

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[hal-00499548] Relations entre les sommes d'Euler de poids fixé et les valeurs de zêta

3 mai 2016

Dans cet article, on montre comment déduire d'une remarquable formule sommatoire d'Ohno une relation non-triviale entre les sommes d'Euler de poids fixé $k$ et $\zeta(k)$. In this article, we show how to derive from a deep sum formula of Ohno a non-trivial relation between Euler sums of fixed (...)

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[hal-01202725] Produit harmonique, sommation de Ramanujan et fonctions zêta d'Arakawa-Kaneko

3 mai 2016

Ce mémoire s'articule autour des trois thèmes principaux auxquels j'ai consacré mes recherches au cours de ces 15 dernières années, thèmes qui sont assez étroitement reliés entre-eux comme je m'attacherai à le montrer. Il s'agit du produit harmonique, du procédé de sommation de Ramanujan et de la (...)

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[inria-00395839] Noether's forms for the study of non-composite rational functions and their spectrum

3 mai 2016

In this paper, the spectrum and the decomposability of a multivariate rational function are studied by means of the effective Noether's irreducibility theorem given by Ruppert. With this approach, some new effective results are obtained. In particular, we show that the reduction modulo p of the (...)

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[hal-01226725] Sur le radical kummérien des Zl-extensions

22 avril 2016

Sur la base d'un travail antérieur, nous donnons une description nouvelle du radical iniial attaché au compositum des Zl-extensions d'un corps de nombres en termes de limites projectives pour la norme dans la Zl-extension cyclotomique de ce corps. Le résultat précis que nous obtenons contient, (...)

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[hal-01280172] On Fourier coefficients of modular forms of half integral weight at squarefree integers

21 avril 2016

We show that the Dirichlet series associated to the Fourier coefficients of a half-integral weight Hecke eigenform at squarefree integers extends analytically to a holomorphic function in the half-plane $\re s>\tfrac12$. This exhibits a high fluctuation of the coefficients at squarefree (...)

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[hal-00131641] Points de hauteur bornee sur les varietes de drapeaux en caracteristique finie

20 avril 2016

The aim of this paper is to apply the work of Morris on Eisenstein series over global function fields to the study of the asymptotic behavior of the points of bounded height on a generalized flag variety defined as the quotient of a semi-simple algebraic group by a reduced parabolic subgroup (...)

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[hal-01304309] Ramification in Iwasawa Theory and Splitting Conjectures

19 avril 2016

We make a reciprocity conjecture that extends Iwasawa's analogy of direct limits of class groups along the cyclotomic tower of a totally real number field F to torsion points of Jacobians of curves over finite fields. The extension is to generalized class groups and generalized Jacobians. We (...)

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[hal-01263097] A NOTE ON MULTIPLE ZETA VALUES IN TATE ALGEBRAS

19 avril 2016

In this note, we shall discuss a generalization of Thakur's multiple zeta values and allied objects, in the framework of function fields of positive characteristic and more precisely, of periods in Tate algebras.

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[hal-01300328] Valeurs moyennes effectives de fonctions multiplicatives complexes

19 avril 2016

Nous établissons des estimations de valeurs moyennes pour une large classe de fonctions arithmétiques multiplicatives, fournissant ainsi des versions quantitatives essentiellement optimales des évaluations classiques de Wirsing et étendant celles de Halász. Plusieurs applications sont explicitées, (...)

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[hal-01277623] On shifted Mascheroni Series and hyperharmonic numbers

17 avril 2016

In this article, we study the nature of the forward shifted series σ r = n>r |bn| n−r where r is a positive integer and b n are Bernoulli numbers of the second kind, expressing them in terms of the derivatives ζ (−k) of zeta at the negative integers and Euler's constant γ. These expressions may (...)

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[hal-01302600] On the average distribution of divisors of friable numbers

16 avril 2016

A number is said to be y-friable if it has no prime factor greater than y. In this paper, we prove a central limit theorem on average for the distribution of divisors of y-friable numbers less than x, for all (x, y) satisfying 2 ≤ y ≤ e (log x)/(log log x) 1+ε. This was previously known under the (...)

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[hal-01302604] Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

16 avril 2016

We prove a bound for quintilinear sums of Kloosterman sums, with congruence conditions on the "smooth" summation variables. This generalizes classical work of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms in applications, notably the dispersion method. As (...)

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[hal-01283042] Series Representation of Power Function

16 avril 2016

This paper presents the way to make expansion for the next form function: $y=x^n, \ \forall(x,n) \in \mathbbN$ to the numerical series. The most widely used methods to solve this problem are Newton's Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are (...)

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[hal-01283042] Series Representation of Power Function

16 avril 2016

This paper presents the way to make expansion for the next form function: $y=x^n, \ \forall(x,n) \in \mathbbN$ to the numerical series. The most widely used methods to solve this problem are Newton's Binomial Theorem and Fundamental Theorem of Calculus (that is, derivative and integral are (...)

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[hal-00771342] Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group

15 avril 2016

This paper is devoted to the study of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism groups are cyclic of order coprime to the characteristic of their ground field. We give an explicit and effectively computable description of this (...)

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[hal-01251577] DES APPLICATIONS GÉNÉRATRICES DES NOMBRES PREMIERS ET CINQ PREUVES DE L’HYPOTHÈSE DE RIEMANN

15 avril 2016

I will prove that there exists one application $\psi(\psi^-,\psi^+)$ on $\mathbbR^2$ such that $\mathcalP = \\pm2,\pm3 \ \cup 6\times\mathcalF^-+1 \cup6\times\mathcalF^+-1$ where : $ \mathcal P $ is the set of relatively prime numbers, $\mathcalF^- = \mathbbZ\cap( \psi^+ ( \mathbbZ^*\times (...)

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[hal-01280172] ON FOURIER COEFFICIENTS OF MODULAR FORMS OF HALF INTEGRAL WEIGHT AT SQUAREFREE INTEGERS

14 avril 2016

We show that the Dirichlet series associated to the Fourier coefficients of a half-integral weight Hecke eigenform at squarefree integers extends analytically to a holomorphic function in the half-plane ℜe s > 1 2. This exhibits a high fluctuation of the coefficients at squarefree (...)

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[hal-00584431] Calendriers et fractions continues

30 mars 2016

Cet article explique dans un premier temps l'histoire du calendrier grégorien. Ceci est un prétexte pour expliquer le développement en fractions continues d'un nombre réel et d'en donner les principales propriétés. A la fin de l'article, on introduit l'algorithme de Jacobi-Perron qui donne des (...)

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[hal-00085832] Finitude pour les representations lisses de groupes p-adiques

29 mars 2016

We study basic properties of the category of smooth representations of a p-adic group G with coefficients in any commutative ring R in which p is invertible. Our main purpose is to prove that Hecke algebras are noetherian whenever R is ; a question left open since Bernstein's fundamental work (...)

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[hal-00566314] Circular words and three applications: factors of the Fibonacci word, ${\mathcal F}$-adic numbers, and the sequence $1$, $5$, $16$, $45$, $121$, $320$,\ldots

29 mars 2016

We introduce the notion of \em circular words with a combinatorial constraint derived from the Zeckendorf (Fibonacci) numeration system, and get explicit group structures for these words. As a first application, we give a new result on factors of the Fibonacci word $abaababaabaab\ldots$. (...)

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[hal-00022116] Theorie de Lubin-Tate non-abelienne et representations elliptiques

29 mars 2016

Harris and Taylor proved that the supercuspidal part of the cohomology of the Lubin-Tate tower realizes both the local Langlands and Jacquet-Langlands correspondences, as conjectured by Carayol. Recently, Boyer computed the remaining part of the cohomology and exhibited two defects : first, the (...)

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[hal-00851556] Some exact values of the Harborth constant and its plus-minus weighted analogue

29 mars 2016

The Harborth constant of a finite abelian group is the smallest integer $\ell$ such that each subset of $G$ of cardinality $\ell$ has a subset of cardinality equal to the exponent of the group whose elements sum to the neutral element of the group. The plus-minus weighted analogue of this (...)

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[hal-00835688] Remarks on the plus-minus weighted Davenport constant

29 mars 2016

For $(G,+)$ a finite abelian group the plus-minus weighted Davenport constant, denoted $\mathsfD_\pm(G)$, is the smallest $\ell$ such that each sequence $g_1 \dots g_\ell$ over $G$ has a weighted zero-subsum with weights $+1$ and $-1$, i.e., there is a non-empty subset $I \subset \1,\dots, (...)

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[hal-00911140] Improvements on the accelerated integer GCD algorithm

29 mars 2016

The present paper analyses and presents several improvements to the algorithm for finding the $(a,b)$-pairs of integers used in the $k$-ary reduction of the right-shift $k$-ary integer GCD algorithm. While the worst-case complexity of Weber's ''Accelerated integer GCD algorithm'' is (...)

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[hal-00689464] Some remarks on barycentric-sum problems over cyclic groups

29 mars 2016

We derive some new results on the k-th barycentric Olson constants of abelian groups (mainly cyclic). This quantity, for a finite abelian (additive) group (G,+), is defined as the smallest integer l such that each subset A of G with at least l elements contains a subset with k elements g_1, ... (...)

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[hal-01292805] SUMS OF TWO S-UNITS VIA FREY-HELLEGOUARCH CURVES

25 mars 2016

In this paper, we develop a new method for finding all perfect powers which can be expressed as the sum of two rational S-units, where S is a finite set of primes. Our approach is based upon the modularity of Galois representations and, for the most part, does not require lower bounds for (...)

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[hal-01293564] Vectorial Drinfeld modular forms over Tate algebras

25 mars 2016

In this text, we develop the theory of vectorial modular forms with values in Tate algebras introduced by the first author, in a very special case (dimension two, for a very particular representation of Γ := GL 2 (Fq[θ])). Among several results that we prove here, we determine the complete (...)

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[hal-01293573] A note on certain representations in characteristic p and associated functions

25 mars 2016

We discuss certain representations of GL 2 Fq[T] in equal characteristic and associated vectorial modular forms

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[hal-01293687] On a torsor of paths of an elliptic curve minus a point

25 mars 2016

We are studying some aspects of the action of the Galois groups on torsors of paths on an elliptic curve minus a point. We construct objects whose behaviour is similar to the classical polylogarithms on the projective line minus three (...)

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[hal-01293667] Non-abelian unipotent periods and monodromy of iterated integrals. (2003) 2(1),

25 mars 2016

In this note we are studying the Lie algebras associated to non-abelian unipotent periods on P1ℚ(μn)∖0,μn,∞. Let n be a prime number. We assume that for any m≥1 the numbers Lim+1(ξkn) for 1≤k≤(n−1)/2 are linearly independent over ℚ in ℂ/(2π\ri)m+1ℚ. Let S=k1,⋯,kq be a subset of 1,…,p−1 such that if k∈S, (...)

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[hal-01293611] Cosimplicial objects in algebraic geometry ,

25 mars 2016

Let X be an arc-connected and locally arc-connected topological space and let I be the unit interval. Applying the connected component functor to each fibre of the fibration of the total space map(I, X) over X × X, P(w) = (w(0), w(1)), we get a local system of sets (Poincaré groupoid) over X × X. (...)

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[hal-01293604] On functional equations of p-adic polylogarithms. Bull. Soc. Math. France , 119 (1991)

25 mars 2016

ABSTRACT . — The n-t h order polylogarithm Ln(z) i s defined by th e series E^Ll zk ''t^ n on tn e Q? 611 un it disc. This functio n has multivalued analytic prolon- gation t o C \ 0,1 . The same series ^^ z k f^ defines an analytic p-adic functio n on th e open uni t disc in Cp (a completion o (...)

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[hal-01292754] Milnor K-theory and the graded representation ring

24 mars 2016

Let F be a field, let G = Gal(¯ F /F) be its absolute Galois group, and let R(G, k) be the representation ring of G over a suitable field k. In this preprint we construct a ring homomorphism from the mod 2 Milnor K-theory k * (F) to the graded ring gr R(G, k) associated to Grothendieck's (...)

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[hal-01293244] Central extentions and coverings .Publications Universitat Autonoma de Barcelona. vol. 29 (1985), 145-153.

24 mars 2016

The theory of central extensions has a lot of analogy with the theory of covering spaces. In this paper we show that the category of central extensions of a perfect group and a certain category of covering spaces are equivalent. Then the facts about central extensions will follow from the (...)

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[hal-01292727] A functoriality principle for blocks of p-adic linear groups

24 mars 2016

Bernstein blocks of complex representations of p-adic reductive groups have been computed in a large amount of examples, in part thanks to the theory of types a la Bushnell and Kutzko. The output of these purely representation-theoretic computations is that many of these blocks are equivalent. (...)

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[hal-00371233] On the multiplicative order of $a^n$ modulo $n$

24 mars 2016

Let $n$ be a positive integer and $\alpha_n$ be the arithmetic function which assigns the multiplicative order of $a^n$ modulo $n$ to every integer $a$ coprime to $n$ and vanishes elsewhere. Similarly, let $\beta_n$ assign the projective multiplicative order of $a^n$ modulo $n$ to every integer (...)

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[hal-01291833] The Fourier expansion of $\eta(z)\eta(2z)\eta(3z)/\eta(6z)$

23 mars 2016

We compute the Fourier coefficients of the weight one modular form $\eta(z)\eta(2z)\eta(3z)/\eta(6z)$ in terms of the number of representations of an integer as a sum of two squares. We deduce a relation between this modular form and translates of the modular form (...)

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[hal-00371228] On a problem of Molluzzo concerning Steinhaus triangles in finite cyclic groups

23 mars 2016

Let $X$ be a finite sequence of length $m\geqslant 1$ in $\mathbbZ/n\mathbbZ$. The derived sequence $\partial X$ of $X$ is the sequence of length $m-1$ obtained by pairwise adding consecutive terms of $X$. The collection of iterated derived sequences of $X$, until length $1$ is reached, (...)

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[hal-00823402] On special values of spinor L-functions of Siegel cusp eigenforms of genus 3

22 mars 2016

We compute the special values for the spinor L-function L(s,F12) in the critical strip s=12,...,19, where F12 is the unique (up to a scalar) Siegel cusp form of degree 3 and weight 12, which was constructed by Miyawaki. These values are proportional to the product of Petersson inner products of (...)

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[hal-01160765] Microsolutions of differential operators and values of arithmetic Gevrey series

22 mars 2016

We continue our investigation of E-operators, in particular their connection with G-operators; these differential operators are fundamental in understanding the dio-phantine properties of Siegel's E and G-functions. We study in detail microsolutions (in Kashiwara's sense) of Fuchsian (...)

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