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 : 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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[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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[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 $Q((T^-1))$ 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 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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[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-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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