# GDR STN

Site du GDR Structuration de la théorie des nombres

## 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 : Emmanuel Royer, Professeur à l’Université Blaise Pascal, emmanuel.royer@math.univ-bpclermont.fr

## Nouveaux articles en théorie des nombres

### [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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5 décembre 2016

[...]

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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 (...)

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### [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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25 novembre 2016

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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-01345061] Alex Kontorovich - Diophantine problems in thin orbits

27 juillet 2016

Diophantine problems in thin orbits

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### [medihal-01345051] Yuri Manin - Numbers as functions

27 juillet 2016

Numbers as functions

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### [medihal-01345045] Francis Brown - Quantum Field Theory and Arithmetic

27 juillet 2016

Quantum Field Theory and Arithmetic

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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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20 juillet 2016

[...]

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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 présentent des (...)

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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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23 juin 2016

[...]

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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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23 juin 2016

[...]

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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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18 juin 2016

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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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29 mars 2016

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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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