# GDR STN

Site du GDR Structuration de la théorie des nombres

## Nouveaux articles en théorie des nombres

### [hal-02073665] SOME RESULTS ON THE FLYNN-POONEN-SCHAEFER CONJECTURE

22 mars 2019

For c ∈ Q, consider the quadratic polynomial map ϕ_c (x) = x^2 − c. Flynn, Poonen and Schaefer conjectured in 1997 that no rational cycle of ϕ_c under iteration has length more than 3. Here we discuss this conjecture using arithmetic and combinatorial means, leading to three main results. First, we (...)

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### [hal-00649593] Subset sums in abelian groups

22 mars 2019

Denoting by Sigma(S) the set of subset sums of a subset S of a finite abelian group G, we prove that |Sigma(S)| >= |S|(|S|+2)/4-1 whenever S is symmetric, |G| is odd and Sigma(S) is aperiodic. Up to an additive constant of 2 this result is best possible, and we obtain the stronger (exact (...)

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### [hal-02075857] Rudin–Shapiro sequences along squares

21 mars 2019

We estimate exponential sums of the form Sigma(n <= x) f(n(2)) e(nu n) for a large class of digital functions f and nu is an element of R. We deduce from these estimates the distribution along squares of this class of digital functions which includes the Rudin-Shapiro sequence and some of (...)

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### [hal-02075688] Discrepancy estimates for generalized polynomials

21 mars 2019

We obtain an upper bound for the discrepancy of the sequence ([p(n) a] ss) n= 0 generated by the generalized polynomial [p(x) a] ss, where p(x) is a monic polynomial with real coefficients, a and ss are irrational numbers satisfying certain (...)

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21 mars 2019

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### [hal-00605393] Familles de représentations de de Rham et monodromie p-adique

21 mars 2019

On donne une formalisation de la méthode de Sen pour les représentations $p$-adiques. Comme application de ces techniques, on montre que (1) toute représentation $p$-adique est surconvergente (2) si on se donne un espace $\calX = \mathrmSpm(S)$ qui paramétrise des représentations $p$-adiques $V_x$, (...)

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### [hal-01590183] A program to test the p-rationality of any number field

21 mars 2019

Let K be a number field. We prove that its ray class group modulo p 2 (resp. 8) if p > 2 (resp. p = 2) characterizes its p-rationality. Then we give two short, very fast PARI Programs (§ § 3.1, 3.2) testing if K (defined by an irreducible monic polynomial) is p-rational or not. For quadratic (...)

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### [hal-02059441] Genus theory and ε-conjectures on p-class groups

21 mars 2019

We suspect that the genus part'' of the class number of a number field K may be an obstruction for an easy proof'' of the classical ε-conjecture for p-class groups and, a fortiori, for a proof of the strong ε-conjecture'': #Cl_K (...)

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### [hal-02062503] Odd Collatz Sequence and Binary Representations

21 mars 2019

In this work, we investigate the odd Collatz sequences (odd sequences arising from the 3n + 1 problem) and use binary arithmetic to provide proof of some results in the 3n + 1 problem. Aside from the main result, the paper also provides a new perspective on how to approach the 3n + 1 problem by (...)

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### [hal-02070778] Integer multiplication in time O(n log n)

20 mars 2019

We present an algorithm that computes the product of two n-bit integers in O(n log n) bit operations.

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### [hal-02070816] Polynomial multiplication over finite fields in time O(n log n)

20 mars 2019

Assuming a widely-believed hypothesis concerning the least prime in an arithmetic progression, we show that two n-bit integers can be multiplied in time O(n log n) on a Turing machine with a finite number of tapes; we also show that polynomials of degree less than n over a finite field F_q with (...)

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### [hal-02071479] BOUND ON THE NUMBER OF RATIONAL POINTS ON CURVES ON HIRZEBRUCH SURFACES OVER FINITE FIELDS

20 mars 2019

This paper gives a bound on the number of rational points on an absolutely irreducible curve C lying on a minimal toric surface X. This upper bound improves pre-existing ones if C has large genus. The strategy consists in finding another curve that intersects C with good multiplicity at its (...)

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### [hal-02072156] The Ramificant Determinant

20 mars 2019

We give an introduction to the transalgebraic theory of simply connected log-Riemann surfaces with a finite number of infinite ramification points (transalgebraic curves of genus 0). We define the base vector space of transcendental functions and establish by elementary means some (...)

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### [hal-02061466] Proof that Bombieri's question about the abc-conjecture is Yes

20 mars 2019

In [1] Enrico Bombieri, asked the question "For example, is it always true that c ≤ p/abc p 2 ?." in present paper we prove that the answer is: Yes. 1 The series ∞ n=1 R(n) t n s We introduce the series ∞ n=1 R(n) t n s (1) where R(n) := the multiplicative radical of n, t := a real (...)

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### [hal-02072689] EXPLICIT BOUNDS FOR THE DIOPHANTINE EQUATION A!B! = C!

20 mars 2019

A nontrivial solution of the equation A!B! = C! is a triple of positive integers (A, B, C) with A ≤ B ≤ C − 2. It is conjectured that the only nontrivial solution is (6, 7, 10), and this conjecture has been checked up to C = 10 6. Several estimates on the relative size of the parameters are known, (...)

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### [hal-01131955] A characterization of class groups via sets of lengths

20 mars 2019

Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor. Then every nonunit $a \in H$ can be written as a finite product of irreducible elements. If $a=u_1 \cdot \ldots \cdot u_k$, with irreducibles $u_1, \ldots u_k \in H$, then $k$ is called the length of (...)

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### [hal-01853797] Spectrum of a linear differential equation over a field of formal power series

20 mars 2019

In this paper we associate to a linear differential equation with coefficients in the field of Laurent formal power series a new geometric object, a spectrum in the sense of Berkovich. We will compute this spectrum and show that it contains interesting informations about the (...)

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

19 mars 2019

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

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### [hal-00565895] Twisted moments of automorphic L-functions

19 mars 2019

We study the moments of the symmetric power L-functions of primitive forms at the edge of the critical strip twisted by the square of the value of the standard L-function at the center of the critical strip. We give a precise expansion of the moments as the order goes to (...)

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### [hal-00347529] The number of Hecke eigenvalues of same signs

19 mars 2019

We give the best possible lower bounds in order of magnitude for the number of positive and negative Hecke eigenvalues. This improves upon a recent work of Kohnen, Lau & Shparlinski. Also, we study an analogous problem for short (...)

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19 mars 2019

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19 mars 2019

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19 mars 2019

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### [hal-01869161] Inverse zero-sum problems for certain groups of rank three

18 mars 2019

The inverse problem associated to the Erdős-Ginzburg-Ziv constant and the $\eta$-constant is solved for finite abelian groups of the form $C_2 \oplus C_2 \oplus C_2n$ where $n \ge 2$ is an integer.

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### [hal-01852413] Black holes and higher depth mock modular forms

18 mars 2019

By enforcing invariance under S-duality in type IIB string theory compactified on a Calabi-Yau threefold, we derive modular properties of the gene rating function of BPS degeneracies of D4-D2-D0 black holes in type IIA string theory compactified on the same space. Mathematically, these BPS (...)

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### [hal-01848021] On path partitions of the divisor graph

18 mars 2019

It is known that the longest simple path in the divisor graph that uses integers ≤ N is of length N/ log N. We study the partitions of 1, 2,. .. , N into a minimal number of paths of the divisor graph, and we show that in such a partition, the longest path can have length asymptotically (...)

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### [hal-01592317] An asymptotically tight bound for the Davenport constant

18 mars 2019

We prove that for every integer r => 1 the Davenport constant D(C^r_n) is asymptotic to rn when n tends to infinity. An extension of this theorem is also provided.

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### [hal-00252031] On Jacobi Sums in $\mathbb Q(\zeta_p)$

18 mars 2019

We study the p-adic behavior of Jacobi Sums for $\mathbb Q(\zeta_p)$ and link this study to the p-Sylow subgroup of the ideal class group of $\mathbb Q(\zeta_pà^+$

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### [hal-02069722] Coleman Map in Coleman Families

16 mars 2019

In this paper, we construct a two-variable Coleman map for a given $p$-adic family of eigen cuspforms with a fixed non-zero slope (Coleman family). A Coleman map is a machinary which transforms a hypothetical $p$-adic family of zeta elements to a $p$-adic $L$-function. The result obtained in (...)

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### [hal-01715832] Deterministic factoring with oracles

16 mars 2019

We revisit the problem of integer factorization with number-theoretic oracles, including a well-known problem: can we factor an integer N unconditionally, in deterministic polynomial time, given the value of the Euler totient $ϕ(N)$? We show that this can be done, under certain size conditions (...)

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### [hal-00947148] On Jacobi Sums in Q(zeta_p)

15 mars 2019

We study the arithmetic of Jacobi Sums in Q(\zeta_p) and we show that they play a somehow parallel role "in the minus part" as to Units "in the plus part" of the Class Field Theory of Cyclotomic fields.

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### [hal-00947151] Cyclotomic Units and Class Groups in Z_p-extensions of real abelian number fields

15 mars 2019

For a real abelian number field F and for a prime p we study the relation between the p-parts of the class groups and of the quotients of global units modulo cyclotomic units along the cyclotomic Z_p-extension of F. Assuming Greenberg's conjecture about the van- ishing of the λ-invariant of the (...)

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### [hal-00947158] Fitting Ideals

15 mars 2019

This document is a summary of two talks given at the School on Arithmetic Geometry held in Guwahati, India, from September 22nd − 29th on Fitting ideals and their applications to Wiles' proof of Iwasawa Conjecture for totally real (...)

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### [hal-02067885] Hash functions from superspecial genus-2 curves using Richelot isogenies

15 mars 2019

Last year Takashima proposed a version of Charles, Goren and Lauter's hash function using Richelot isogenies, starting from a genus-2 curve that allows for all subsequent arithmetic to be performed over a quadratic finite field Fp2. In a very recent paper Flynn and Ti point out that Takashima's (...)

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15 mars 2019

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### [hal-01918849] AN ABELIAN ANALOGUE OF SCHANUEL'S CONJECTURE AND APPLICATIONS

14 mars 2019

In this article we study an abelian analogue of Schanuel's conjecture. This conjecture falls in the realm of the generalised period conjecture of Y. André. As shown by C. Bertolin, the generalised period conjecture includes Schanuel's conjecture as a special case. Extending methods of Bertolin, (...)

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### [hal-01766202] La constante de Manin et le degré modulaire d'une courbe elliptique

14 mars 2019

On donne une méthode de calcul de la courbe de Weil forte d'une classe d'isogénie de courbes elliptiques sur Q et de la constante de Manin en utilisant les symboles modulaires de Pollack-Stevens.

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### [hal-01919518] Modulo $p$ representations of reductive $p$-adic groups: functorial properties

14 mars 2019

Let $F$ be a local field with residue characteristic $p$, let $C$ be an algebraically closed field of characteristic $p$, and let $\mathbfG$ be a connected reductive $F$-group. In a previous paper, Florian Herzig and the authors classified irreducible admissible $C$-representations of (...)

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14 mars 2019

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### [hal-01758986] Volume function and Mahler measure of exact polynomials

14 mars 2019

We study a class of 2-variable polynomials called exact polynomials which contains $A$-polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the (...)

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### [hal-01819644] Morphisms of 1-motives Defined by Line Bundles

13 mars 2019

Let S be a normal base scheme. The aim of this paper is to study the line bundles on 1-motives defined over S. We first compute a dévissage of the Picard group of a 1-motive M according to the weight filtration of M. This dévissage allows us to associate, to each line bundle L on M , a linear (...)

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### [hal-01272983] Moments of a Thue-Morse generating function

13 mars 2019

We study the moments of even order of the generating function of the Thue-Morse sequence and present several conjectures related to these moments.

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13 mars 2019

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13 mars 2019

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### [hal-00125109] Irrationalite d'au moins un des neuf nombres zeta(5),zeta(7),...,zeta(21)

12 mars 2019

We prove that there is at least one irrationnal among the nine numbers zeta(5), zeta(7),..., zeta(21).

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### [hal-00125108] La fonction Zeta de Riemann prend une infinite de valeurs irrationnelles aux entiers impairs

12 mars 2019

We provide a lower bound for the dimension of the vector space spanned by 1 and by the values of the Riemann Zeta function at the first odd integers. As a consequence, the Zeta function takes infinitely many irrational values at odd (...)

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### [hal-02064687] Relative Aspects in Representation Theory, Langlands Functoriality and Automorphic Forms

12 mars 2019

This volume presents a panorama of the diverse activities organized by V. Heiermann and D. Prasad in Marseille at the CIRM for the Chaire Morlet event during the first semester of 2016. It assembles together expository articles on topics which previously could only be found in research papers. (...)

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### [hal-01978093] Local triviality for G-torsors

12 mars 2019

Let C → Spec(R) be a relative proper flat curve over an henselian base. Let G be a reductive C-group scheme. Under mild technical assumptions, we show that a G-torsor over C which is trivial on the closed fiber of C is locally trivial for the Zariski (...)

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### [hal-00138566] Fonction zeta et matrices aléatoires

9 mars 2019

In this survey, we explain the introduction of random matrix theory in the study of Riemann ζ function and L functions.

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### [hal-02059441] Genus theory and ε-conjectures on p-class groups

9 mars 2019

We suspect that the "genus part" of the class number of a number field K is an obstruction for an "easy proof" of a "strong ε-conjecture" for p-class groups: # (Cℓ_K ⊗ Z_p) ≪_(d,p,ε) (√ D_K)^ε for all K totally real of degree d. We analyze the weight of genus theory in (...)

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### [hal-01845011] Modular invariants for genus 3 hyperelliptic curves

8 mars 2019

In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary oc-tics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume (...)

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### [hal-01705641] A note on the size of the set A² + A

8 mars 2019

Let F(x,y,z)=xy+z. We consider some properties of expansion of the polynomial F in different settings, namely in the integers and in prime fields. The main results concern the question of covering 0,1,…,N (resp. Fp) by A2+A with some thin sets (...)

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### [hal-01067333] The Legendre determinant form for Drinfeld modules in arbitrary rank

8 mars 2019

For each positive integer $r$, we construct a nowhere-vanishing, single-cuspidal Drinfeld modular form for $\GL_r(\FF_q[\theta])$, necessarily of least possible weight, via determinants using rigid analytic trivializations of the universal Drinfeld module of rank $r$ and deformations of (...)

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### [hal-01483629] Ergodic measures on spaces of infinite matrices over non-Archimedean locally compact fields

7 mars 2019

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

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### [ujm-02060809] A note on the set $\boldsymbol{A(A+A)}$

7 mars 2019

Let $p$ a large enough prime number. When $A$ is a subset of $\mathbbF_p\smallsetminus\0\$ of cardinality $|A|> (p+1)/3$, then an application of Cauchy-Davenport Theorem gives $\mathbbF_p\smallsetminus\0\\subset A(A+A)$. In this note, we improve on this and we show that if $|A|\ge 0.3051 p$ (...)

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### [ujm-02060805] On the density of sumsets and product sets

7 mars 2019

In this paper some links between the density of a set of integers and the density of its sumset, product set and set of subset sums are presented.

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### [hal-01389696] Factorization of bivariate sparse polynomials

7 mars 2019

We prove a function field analogue of a conjecture of Schinzel on the factorization of univariate polynomials over the rationals. We derive from it a finiteness theorem for the irreducible factorizations of the bivariate Laurent polynomials in families with fixed set of complex coefficients and (...)

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### [hal-02060277] Monochromatic sums of squares

7 mars 2019

For any integer let s(K) be the smallest integer such that in any colouring of the set of squares of the integers in K colours every large enough integer can be written as a sum of no more than s(K) squares, all of the same colour. A problem proposed by Sarkozy asks for optimal bounds for s(K) (...)

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6 mars 2019

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### [hal-01063511] Tower of algebraic function fields with maximal Hasse-Witt invariant and tensor rank of multiplication in any extension of $\mathbb{F}_2$ and $\mathbb{F}_3$

5 mars 2019

Up until now, it was recognized that a detailed study of the p-rank in towers of function fields is relevant for their applications in coding theory and cryptography. In particular, it appears that having a large p-rank may be a barrier for a tower to lead to competitive bounds for the (...)

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### [hal-02057370] Trace of products in finite fields

5 mars 2019

Let p be a prime number and let q = p(r). If C and D are large subsets of F-q(*) we study the trace of products cd with c is an element of C and d is an element of D and show that it is well distributed in F-p. We give an optimal condition (up to an absolute constant factor) on the size of the (...)

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### [hal-01063511] Tower of algebraic function fields with maximal Hasse-Witt invariant and tensor rank of multiplication in any extension of $\mathbb{F}_2$ and $\mathbb{F}_3$

5 mars 2019

Up until now, it was recognized that a detailed study of the p-rank in towers of function fields is relevant for their applications in coding theory and cryptography. In particular, it appears that having a large p-rank may be a barrier for a tower to lead to competitive bounds for the (...)

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### [hal-00318546] Théorie des Codes : compression, cryptage, correction

5 mars 2019

La transmission d'information sous forme numérique doit répondre à des impératifs de sécurité, d'efficacité et d'intégrité. Les techniques de codage que l'on utilise pour y parvenir reposent sur un socle théorique commun issu de l'algèbre linéaire, des probabilités, de l'algorithmique et de la (...)

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### [hal-01278962] Non-vanishing and sign changes of Hecke eigenvalues for Siegel cusp forms of genus two (with an appendix)

5 mars 2019

In this paper, we show that half of non-zero coefficients of the spinor zeta function of a Siegel cusp form of genus 2 are positive and half are negative. We also prove results concerning the non-vanishing in short intervals and strong cancellation among the coefficients evaluated at powers of (...)

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### [hal-01845023] Lower Bounds on Lattice Enumeration with Extreme Pruning

4 mars 2019

At Eurocrypt '10, Gama, Nguyen and Regev introduced lattice enumeration with extreme pruning: this algorithm is implemented in state-of-the-art lattice reduction software and used in challenge records. They showed that extreme pruning provided an exponential speed-up over full enumeration. (...)

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### [hal-01061897] Resonances and convex co-compact congruence subgroups of PSL2(Z)

4 mars 2019

This papers deals with congruence subgroups of convex cocompact subgroups of PSL2(Z). We examine the behaviour of the resonance spectrum when the congruence parameter q goes to infinity: we show a lower bound for the counting function in discs and an upper bound in vertical strips. These (...)

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### [hal-02053125] Rotation number of contracted rotations

2 mars 2019

Let 0 < lambda< 1. We consider the one-parameter family of circle lambda-affine contractions f(delta): x is an element of [0,1] -> lambda x + delta mod 1, where 0 <= delta < 1. Let rho be the rotation number of the map f(delta). We will give some numerical relations between the (...)

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### [hal-02052124] The Lagrange spectrum of some square-tiled surfaces

1er mars 2019

Lagrange spectra have been defined for closed submanifolds of the moduli space of translation surfaces which are invariant under the action of SL(2, R). We consider the closed orbit generated by a specific covering of degree 7 of the standard torus, which is an element of the stratum H(2). We (...)

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### [hal-01965547] ON THE MALLE CONJECTURE AND THE GRUNWALD PROBLEM

1er mars 2019

We contribute to the Malle conjecture on the number N (K, G, y) of finite Galois extensions E of some number field K of finite group G and of discriminant of norm |N K/Q (d E)| ≤ y. We prove the lower bound part of the conjecture for every group G and every number field K containing a certain (...)

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### [hal-01008774] Local zero estimates and effective division in rings of algebraic power series

1er mars 2019

We give a necessary condition for algebraicity of finite modules over the ring of formal power series. This condition is given in terms of local zero estimates. In fact, we show that this condition is also sufficient when the module is a ring with some additional properties. To prove this (...)

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### [tel-01832063] Iwasawa algebras for p-adic Lie groups and Galois groups

1er mars 2019

A key tool in p-adic representation theory is the Iwasawa algebra, originally constructed by Iwasawa in 1960's to study the class groups of number fields. Since then, it appeared in varied settings such as Lazard's work on p-adic Lie groups and Fontaine's work on local Galois representations. (...)

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### [hal-02045588] On the Reducibility and the Lenticular Sets of Zeroes of Almost Newman Lacunary Polynomials

28 février 2019

The class B of lacunary polynomials f(x) := -1 + x + x^n + x^m_1 + x^m_2 + ... + x^m_s, where s >= 0, m_1 - n >= n - 1, m_q+1 - m_q >= n - 1 for 1 <= q < s, n >= 3 is studied. A polynomial having its coefficients in 0, 1 except its constant coefficient equal to -1 is called an almost (...)

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### [hal-01161856] Nearby slopes and boundedness for ℓ-adic sheaves in positive characteristic

27 février 2019

The goal of this paper is to motivate a boundedness conjecture on nearby slopes of ℓ-adic sheaves in positive characteristic, and to prove it for smooth curves. For a constructible ℓ-adic sheaf, we prove the finiteness of the set of nearby slopes associated to a given morphism. For the constant (...)

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### [hal-01995881] Gröbner bases over Tate algebras

26 février 2019

Tate algebras are fundamental objects in the context of analytic geometry over the p-adics. Roughly speaking, they play the same role as polynomial algebras play in classical algebraic geometry. In the present article, we develop the formalism of Gröbner bases for Tate algebras. We prove an (...)

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### [hal-01759083] Some formulae which match with the prime counting function infinitely often

26 février 2019

Recently K. Gaitanas gave a formula which matches with the counting prime function for an infinite set of values of its argument. In this note, we give a construction of an infinite number of such formulae.

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### [hal-00947165] A criterion for Greenberg's Conjecture

26 février 2019

We give a criterion for the vanishing of the Iwasawa λ invariants of totally real number fields K based on the class number of K(ζp) by evaluating the p-adic L functions at s = −1.

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### [hal-01131955] A characterization of class groups via sets of lengths

26 février 2019

Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor. Then every nonunit $a \in H$ can be written as a finite product of irreducible elements. If $a=u_1 \cdot \ldots \cdot u_k$, with irreducibles $u_1, \ldots u_k \in H$, then $k$ is called the length of (...)

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### [hal-01568564] Méthode de Mahler en caractéristique non nulle : un analogue du Théorème de Ku. Nishioka

26 février 2019

En 1990, Ku. Nishioka [21] démontre un théorème fondamental pour la méthode de Mahler, qui constitue l'analogue du théorème de Siegel-Shidlovskii pour les fonctions mahlériennes. Le but de cet article est d'établir une version du théorème de Ku. Nishioka qui soit également valable pour des systèmes (...)

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### [hal-01852496] Regular extensions and algebraic relations between values of Mahler functions in positive characteristic

26 février 2019

Let K be a function field of characteristic p > 0. We recently established the analogue of a theorem of Ku. Nishioka for linear Mahler systems defined over K(z). This paper is dedicated to proving the following refinement of this theorem. Let f 1 (z),. .. fn(z) be d-Mahler functions such (...)

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### [hal-01272296] Generalized Hausdorff dimensions of sets of real numbers with zero entropy expansion

25 février 2019

The complexity function of an infinite word w on a finite alphabet A is the sequence counting, for each non-negative integer n, the number of words of length n on the alphabet A that are factors of the infinite word w. Let f be a given function with subexponential growth. The goal of this work (...)

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### [hal-01272331] Family Complexity and VC-Dimension

25 février 2019

In 2003 Ahlswede, Khachatrian, Mauduit and Sárközy introduced the notion of family complexity of binary sequences, and in 2006 Ahlswede, Mauduit and Sárközy extended this definition to sequences of k symbols. Since that several further related papers have been published on this subject. In this (...)

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### [hal-02024884] Une nouvelle démonstration du théorème d'André sur les E-fonctions au sens large

24 février 2019

Nous donnons une nouvelle preuve d'un théorème d'André (2014) affirmant que toute relation polynomiale sur Qbar entre des valeurs d'une famille de E-fonctions au sens large (f_1,..., f_n) provient d'une relation polynomiale sur Qbar(z) entre les f_i(z). Pour cela, nous prouvons un théorème de (...)

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### [tel-01922312] Calculs explicites en th?orie d'Iwasawa

23 février 2019

Dans le premier chapitre de cette th?se on rappelle l'?nonc? ainsi que des ?quivalents de la conjecture de Leopoldt puis l'on donne un algorithme permettant de v?rifier cette conjecture pour un corps de nombre et premier donn?s. Pour la suite on suppose cette conjecture vraie pour le premier p (...)

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### [hal-00138566] Fonction zeta et matrices aléatoires

23 février 2019

In this survey, we explain the introduction of random matrix theory in the study of Riemann ζ function and L functions.

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### [hal-02043515] New Bounds on Dark Energy Induced Fifth Forces

23 février 2019

We consider the gravitational Wilsonian effective action at low energy when all the particles of the standard model have decoupled. When the $\cal R^2$ terms dominate, the theory is equivalent to a scalar-tensor theory with the universal coupling $\beta=1/\sqrt 6$ to matter for which we present (...)

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### [hal-00904678] A dynamical point of view on the set of B-free integers

15 novembre 2017

We extend the study of the square-free flow, recently introduced by Sarnak, to the more general context of B-free integers, that is to say integers with no factor in a given family B of pairwise relatively prime integers, the sum of whose reciprocals is finite. Relying on dynamical arguments, (...)

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### [hal-01070531] The Chowla and the Sarnak conjectures from ergodic theory point of view

15 novembre 2017

We rephrase the conditions from the Chowla and the Sarnak conjectures in abstract setting, that is, for sequences of numbers in -1,0,1, and introduce several natural generalizations. We study the relationships between these properties and other notions from topological dynamics and ergodic (...)

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### [hal-00574623] Averaging along Uniform Random Integers

15 novembre 2017

Motivated by giving a meaning to ''The probability that a random integer has initial digit d'', we define a URI-set as a random set E of natural integers such that each n>0 belongs to E with probability 1/n, independently of other integers. This enables us to introduce two notions of (...)

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

15 novembre 2017

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

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### [hal-00767319] Points rationnels de la fonction Gamma d'Euler

14 novembre 2017

We use a method, first developed for the Riemann zeta-function by Masser in ["Rational values of the Riemann zeta function", Journ. Num. Th. 131 (2011), 2037-2046], to prove a new zero estimate for polynomials in z and 1/Gamma(z). This allows us to prove that, for all n>=2, there (...)

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### [hal-01632280] On Veech's proof of Sarnak's theorem on the Möbius flow

14 novembre 2017

We present Veech's proof of Sarnak's theorem on the Möbius flow which say that there is a unique admissible measure on the Möbius flow. As a consequence, we obtain that Sarnak's conjecture is equivalent to Chowla conjecture with the help of Tao's logarithmic Theorem which assert that the (...)

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### [hal-01634444] $R^2$ Dark Energy in the Laboratory

14 novembre 2017

We analyse the role, on large cosmological scales and laboratory experiments, of the leading curvature squared contributions to the low energy effective action of gravity. We argue for a natural relationship $c_0\lambda^2\simeq 1$ at low-energy between the $\cal R^2$ coefficients $c_0$ of the (...)

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### [hal-01625653] A conjecture which implies that there are infinitely many primes of the form n!+1

13 novembre 2017

Let f(6)=720, and let f(n+1)=f(n)! for every integer n \geq 6. For an integer n \geq 6, let \Lambda_n denote the following statement: if a system S \subseteq x_i!=x_j: 1 \leq i < j \leq n \cup x_i \cdot x_j=x_j+1: 1 \leq i < j \leq n-1 has at most finitely many solutions in integers (...)

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### [hal-01176039] Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals

13 novembre 2017

We show that Sarnak's conjecture on M\"obius disjointness holds in every uniquely ergodic model of a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial $P\in\R[x]$ with irrational leading coefficient and for each multiplicative function (...)

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### [hal-01630363] Equations with powers of singular moduli

13 novembre 2017

We treat two different equations involving powers of singular moduli. On the one hand, we show that, with two possible (explicitly specified) exceptions, two distinct singular moduli j(τ), j(τ ′) such that the numbers 1, j(τ) m and j(τ ′) n are linearly dependent over Q for some positive integers m, (...)

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### [hal-00786176] A converse to linear independence criteria, valid almost everywhere

11 novembre 2017

We prove a weighted analogue of the Khintchine-Groshev Theorem, where the distance to the nearest integer is replaced by the absolute value. This is subsequently applied to proving the optimality of several linear independence criteria over the field of rational (...)

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### [ujm-01115751] The p-Adic Golden Section

11 novembre 2017

The aim of this text is to calculate the p-adic valuation des termes de la suite de Fibonacci

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### [hal-00863138] Counting Primes in Residue Classes

11 novembre 2017

We explain how the Meissel-Lehmer-Lagarias-Miller-Odlyzko method for computing π(x), the number of primes up to x, can be used for computing efficiently π(x,k,l), the number of primes congruent to l modulo k up to x. As an application, we computed the number of prime numbers of the form 4n±1 less (...)

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### [hal-00863445] Propriétés dʼindépendance algébrique de valeurs de séries de Hecke-Mahler

11 novembre 2017

Nous caractérisons complètement et explicitement les relations de dépendance algébrique sur les nombres algébriques entre valeurs complexes de séries de Hecke-Mahler prises en des points algébriques du groupe multiplicatif. Notre résultat contient des résultats antérieurs de Loxton et van der Poorten, (...)

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### [ujm-01214263] Note on a Result of Chung on Weil Type Sums

10 novembre 2017

Following previous works of Chung we are interested in Vinogradov’s type inequalities for some multivariate character sums. Using Johnsen’s bound on a complete unidimensional character sum we obtain a sharper result in several ranges of (...)

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### [hal-01066478] On certain generating functions in positive characteristic

10 novembre 2017

We present new methods for the study of a class of generating functions introduced by the second author which carry some formal similarities with the Hurwitz zeta function. We prove functional identities which establish an explicit connection with certain deformations of the Carlitz logarithm (...)

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### [hal-00863149] Testing the Congruence Conjecture for Rubin-Stark Elements

10 novembre 2017

The 'Congruence Conjecture' was developed by the second author in a previous paper. It provides a conjectural explicit reciprocity law for a certain element associated to an abelian extension of a totally real number field whose existence is predicted by earlier conjectures of Rubin and Stark. (...)

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### [tel-01628170] Arithmétique et D-modules

9 novembre 2017

Ce texte est un survol de mes travaux de recherche comprenant deux parties indépendantes. Dans la première nous présentons les résultats obtenus en collaboration avec T. Abe. On se place sur une courbe propre et lisse $X$ sur un corps fini de caractéristique $p$. Le résultat principal est une formule (...)

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### [hal-00932851] Théorie des Codes : compression, cryptage, correction, 2e edition

9 novembre 2017

La transmission d'information sous forme numérique doit répondre à des impératifs de sécurité, d'efficacité et d'intégrité. Les techniques de codage que l'on utilise pour y parvenir reposent sur un socle théorique commun issu de l'algèbre linéaire, des probabilités, de l'algorithmique et de la (...)

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

8 novembre 2017

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

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### [hal-01463665] Periodic balanced binary triangles

7 novembre 2017

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

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### [hal-00806923] A generalisation of Miller's algorithm and applications to pairing computations on abelian varieties

6 novembre 2017

In this paper, we use the theory of theta functions to generalize to all abelian varieties the usual Miller's algorithm to compute a function associated to a principal divisor. We also explain how to use the Frobenius morphism on abelian varieties defined over a finite field in order to shorten (...)

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### [hal-01629760] Numerical stability of Euclidean algorithm over ultrametric fields

6 novembre 2017

Nous étudions le problème de la stabilité du calcul des résultants et sous-résultants des polynômes définis sur des anneaux de valuation discrète complets (e.g. ℤ p ou k où k est un corps). Nous démontrons que les algorithmes de type Euclide sont très instables en moyenne et, dans de nombreux cas, nous (...)

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### [hal-00834222] Poisson structures and star products on quasimodular forms

6 novembre 2017

We construct and classify all Poisson structures on quasimodular forms that extend the one coming from the first Rankin-Cohen bracket on the modular forms. We use them to build formal deformations on the algebra of quasimodular (...)

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### [hal-00905887] Multiplicate inverse forms of terminating hypergeometric series

6 novembre 2017

The multiplicate form of Gould--Hsu's inverse series relations enables to investigate the dual relations of the Chu--Vandermonde--Gauß's, the Pfaff--Saalschütz's summation theorems and the binomial convolution formula due to Hagen and Rothe. Several identitity and reciprocal relations are thus (...)

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### [hal-00905902] A note on Pr\"{u}fer--like coding and counting forests of uniform hypertrees

6 novembre 2017

This note presents an encoding and a decoding algorithms for a forest of (la- belled) rooted uniform hypertrees and hypercycles in linear time, by using as few as n − 2 integers in the range [1, n]. It is a simple extension of the classical Prüfer code for (labelled) rooted trees to an encoding (...)

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### [hal-00009425] Diophantus' 20th Problem and Fermat's Last Theorem for n=4: Formalization of Fermat's Proofs in the Coq Proof Assistant

6 novembre 2017

We present the proof of Diophantus' 20th problem (book VI of Diophantus' Arithmetica), which consists in wondering if there exist right triangles whose sides may be measured as integers and whose surface may be a square. This problem was negatively solved by Fermat in the 17th century, who used (...)

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### [hal-01254293] ON THE COMPUTATIONAL COMPLEXITY OF ALGEBRAIC NUMBERS: THE HARTMANIS–STEARNS PROBLEM REVISITED

5 novembre 2017

— We consider the complexity of integer base expansions of algebraic irrational numbers from a computational point of view. We show that the Hartmanis–Stearns problem can be solved in a satisfactory way for the class of multistack machines. In this direction, our main result is that the base-b (...)

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### [hal-01627846] Multiple D3-instantons and mock modular forms II

2 novembre 2017

We analyze the modular properties of D3-brane instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In Part I, we found a necessary condition for the existence of an isometric action of S-duality on this moduli space: the (...)

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### [hal-01389696] Factorization of bivariate sparse polynomials

2 novembre 2017

We prove a function field analogue of a conjecture of Schinzel on the factorization of univariate polynomials over the rationals. We derive from it a finiteness theorem for the irreducible factorizations of the bivariate Laurent polynomials in families with fixed set of complex coefficients and (...)

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### [hal-01625653] A conjecture which implies that there are infinitely many primes of the form n!+1

31 octobre 2017

Let f(6)=720, and let f(n+1)=f(n)! for every integer n \geq 6. For an integer n \geq 6, let \Lambda_n denote the following statement: if a system S \subseteq x_i!=x_j: 1 \leq i < j \leq n \cup x_i \cdot x_j=x_j+1: 1 \leq i < j \leq n-1 has at most finitely many solutions in integers (...)

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### [hal-01626429] Julia Robinson's Numbers

31 octobre 2017

We partially answer to a question of Vidaux and Videla by constructing an infinite family of rings of algebraic integers of totally real subfields of Q whose Julia Robinson's Number is distinct from 4 and +∞. Moreover the set of the Julia Robinson's Number that we construct is (...)

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### [hal-01161783] Decoding of Repeated-Root Cyclic Codes up to New Bounds on Their Minimum Distance

27 octobre 2017

The well-known approach of Bose, Ray-Chaudhuri and Hocquenghem and its generalization by Hartmann and Tzeng are lower bounds on the minimum distance of simple-root cyclic codes. We generalize these two bounds to the case of repeated-root cyclic codes and present a syndrome-based burst error (...)

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### [hal-01625166] On the Explicit Torsion Anomalous Conjecture

27 octobre 2017

The Torsion Anomalous Conjecture states that an irreducible variety $V$ embedded in a semi-abelian variety contains only finitely many maximal $V$-torsion anomalous varieties. In this paper we consider an irreducible variety embedded in a product of elliptic curves. Our main result provides a (...)

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### [hal-01625161] On torsion anomalous intersections

27 octobre 2017

A deep conjecture on torsion anomalous varieties states that if $V$ is a weak-transverse variety in an abelian variety, then the complement $V^ta$ of all $V$-torsion anomalous varieties is open and dense in $V$. We prove some cases of this conjecture. We show that the $V$-torsion anomalous (...)

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### [hal-00637093] Some remarks concerning the rank of mapping tori and ascending HNN-extensions of abelian groups.

27 octobre 2017

Let A be a matrix in GL_d(Z) of infinite order. In a recent paper, G. Levitt and V. Metaftsis prove that for any sufficiently large n the matrix A^n is not conjugated to a companion matrix. We first prove a local version of this theorem. Then, we give an effective statement, using linear form (...)

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### [hal-01625058] On the Torsion Anomalous Conjecture in CM abelian varieties

27 octobre 2017

The Torsion Anomalous Conjecture (TAC) states that a subvariety V of an abelian variety A has only finitely many maximal torsion anomalous subvarieties. In this work we prove, with an effective method, some cases of the TAC when the ambient variety A has CM, generalising our previous results in (...)

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### [tel-01624238] Contributions to the Langlands program

26 octobre 2017

This thesis deals with two problems within the Langlands program. For the first problem, in the situation of $\GL_2$ and a non-minuscule cocharacter, we provide a counter-example (under some natural assumptions) to the Rapoport-Zink conjecture, communicated to us by Laurent Fargues.The second (...)

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### [hal-00632997] Computation of the Euclidean minimum of algebraic number fields

26 octobre 2017

We present an algorithm to compute the Euclidean minimum of an algebraic number field, which is a generalization of the algorithm restricted to the totally real case described by Cerri. With a practical implementation, we obtain unknown values of the Euclidean minima of algebraic number fields (...)

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### [tel-00765252] Questions d'Euclidianité

26 octobre 2017

Nous étudions l'euclidianité des corps de nombres pour la norme et quelques unes de ses généralisations. Nous donnons en particulier un algorithme qui calcule le minimum euclidien d'un corps de nombres de signature quelconque. Cela nous permet de prouver que de nombreux corps sont euclidiens ou non (...)

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### [hal-00634643] Examples of norm-Euclidean ideal classes

25 octobre 2017

In 1978, Lenstra defined the notion of Euclidean ideal class. Using a slight modification of an algorithm computing the Euclidean minimum, we give new examples of number fields with norm-Euclidean ideal classes. Extending the results of Cioffari, we also establish the complete list of pure (...)

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### [hal-01258906] The Euclidean algorithm in quintic and septic cyclic fields

25 octobre 2017

Conditionally on the Generalized Riemann Hypothesis (GRH), we prove the following results: (1) a cyclic number field of degree $5$ is norm-Euclidean if and only if $\Delta=11^4,31^4,41^4$; (2) a cyclic number field of degree $7$ is norm-Euclidean if and only if $\Delta=29^6,43^6$; (3) there are (...)

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### [hal-01622008] Unramified 2-extensions of totally imaginary number fields and 2-adic analytic groups

25 octobre 2017

— Let K be a totally imaginary number field. Denote by G ur K (2) the Galois group of the maximal unramified pro-2 extension of K. By comparing cup-products in étale cohomology of SpecO K and cohomology of uniform pro-2 groups, we obtain situations where G ur K (2) has no non-trivial uniform (...)

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### [hal-01622014] Analytic lie extensions of number fields with cyclic fixed points and tame ramification

25 octobre 2017

— Let p be a prime number and K an algebraic number field. What is the arithmetic structure of Galois extensions L/K having p-adic analytic Galois group Γ = Gal(L/K)? The celebrated Tame Fontaine-Mazur conjecture predicts that such extensions are either deeply ramified (at some prime dividing p) (...)

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### [hal-01623558] Tchebotarev theorems for function fields

25 octobre 2017

We prove Tchebotarev type theorems for function field extensions over various base fields: number fields, finite fields, p-adic fields, PAC fields, etc. The Tchebotarev conclusion - existence of appropriate cyclic residue extensions - also compares to the Hilbert specialization property. It is (...)

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### [lirmm-00870376] Empirical optimization of divisor arithmetic on hyperelliptic curves over $\mathbf{F}_{2^m}$

25 octobre 2017

A significant amount of effort has been devoted to improving divisor arithmetic on low-genus hyperelliptic curves via explicit versions of generic algorithms. Moderate and high genus curves also arise in cryptographic applications, for example, via the Weil descent attack on the elliptic curve (...)

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### [lirmm-01104898] On the maximal weight of (p,q)-ary chain partitions with bounded parts

25 octobre 2017

A (p,q) -ary chain is a special type of chain partition of integers with parts of the form paqb for some fixed integers p and q. In this note, we are interested in the maximal weight of such partitions when their parts are distinct and cannot exceed a given bound m. Characterizing the cases (...)

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### [hal-00804810] An algorithm computing non solvable spectral radii of $p$-adic differential equations

24 octobre 2017

We obtain an algorithm computing explicitly the values of the non solvable spectral radii of convergence of the solutions of a differential module over a point of type 2, 3 or 4 of the Berkovich affine line.

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### [hal-00008577] Orbitwise countings in H(2) and quasimodular forms

24 octobre 2017

We prove formulae for the countings by orbit of square-tiled surfaces of genus two with one singularity. These formulae were conjectured by Hubert & Lelièvre. We show that these countings admit quasimodular forms as generating (...)

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### [hal-00482447] A Generalization of Siegel's Theorem and Hall's Conjecture

24 octobre 2017

Consider an elliptic curve, defined over the rational numbers, and embedded in projective space. The rational points on the curve are viewed as integer vectors with coprime coordinates. What can be said about a rational point if a bound is placed upon the number of prime factors dividing a (...)

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### [hal-00138566] Fonction zeta et matrices aléatoires

24 octobre 2017

In this survey, we explain the introduction of random matrix theory in the study of Riemann ζ function and L functions.

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### [hal-00328183] Test vectors for trilinear forms : the case of two principal series

24 octobre 2017

Let F be a finite extension of Qp and G be GL(2,F). When V is the tensor product of three admissible, irreducible, finite dimensional representations of G, the space of G-invariant linear forms has dimension at most one. When a non zero linear form exists, one wants to find an element of V (...)

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23 octobre 2017

[...]

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23 octobre 2017

[...]

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### [hal-00797877] Iwasawa modules with extra structure and p-modular representations of GL2

23 octobre 2017

Let F be a nite extension of Qp. We associate, to certain smooth p-modular representations \pi of GL2(F), a module S(\pi) on the mod p Iwasawa of the standard Iwahori subgroup I of GL2(F). When F is unrami ed, we obtain a module on a suitable formally smooth Fq-algebra, endowed with an action (...)

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### [hal-00907410] Théorème de Chebotarev effectif

23 octobre 2017

Let K be a number field, and L be a finite normal extension of K with Galois group G. It is known that the number of Frobenius automorphisms corresponding to prime ideals, whose norms are less than x, is equivalent to the logarithmic integral as x tends to infinity, and these automorphisms are (...)

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### [hal-01257924] Very strong approximation for certain algebraic varieties

23 octobre 2017

Let F be a global field. In this work, we show that the Brauer-Manin condition on adelic points for subvarieties of a torus T over F cuts out exactly the rational points, if either F is a function field or, if F is the field of rational numbers and T is split. As an application, we prove a (...)

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### [hal-00801781] Surjectivity of Galois representations associated with quadratic Q-curves

23 octobre 2017

We prove in this paper an uniform surjectivity result for Galois representations associated with non-CM $\mathbbQ$-curves over imaginary quadratic fields, using various tools for the proof, such as Mazur's method, isogeny theorems, Runge's method and analytic estimates of sums of (...)

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### [hal-01620848] Efficient Optimal Ate Pairing at 128-bit Security Level

23 octobre 2017

Following the emergence of Kim and Barbulescu's new number field sieve (exTNFS) algorithm at CRYPTO'16 [21] for solving discrete logarithm problem (DLP) over the finite field; pairing-based cryptography researchers are intrigued to find new parameters that confirm standard security levels (...)

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### [hal-01620765] Finite beta-expansions with negative bases

21 octobre 2017

The finiteness property is an important arithmetical property of beta-expansions. We exhibit classes of Pisot numbers $\beta$ having the negative finiteness property, that is the set of finite $(-\beta)$-expansions is equal to $\mathbbZ[\beta^-1]$. For a class of numbers including the (...)

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### [hal-01620763] On the regularity of the generalised golden ratio function

21 octobre 2017

Given a finite set of real numbers $A$, the generalised golden ratio is the unique real number $\mathcalG(A) > 1$ for which we only have trivial unique expansions in smaller bases, and have non-trivial unique expansions in larger bases. We show that $\mathcalG(A)$ varies continuously with (...)

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### [hal-01620410] Fundamental units for a family of totally real cubic orders and the diophantine equation u(u + a)(u + 2a) = v(v + 1)

20 octobre 2017

We determine a system of fundamental units of the totally real cubic orders Z[alpha], where alpha is a root of X-3 - 4a(2)X + 2, 1 <= a epsilon Z, which arise in our study here of the equation E-a : v(v + 1) = u(u + a)(u + 2a). For potentially infinitely many values of a, we reduce the (...)

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### [hal-01221814] The μ-ordinary Hasse invariant of unitary Shimura varieties

20 octobre 2017

We construct a generalization of the Hasse invariant for any Shimura variety of PEL type A over a prime of good reduction, whose vanishing locus is the open and dense \mu-ordinary locus.

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### [tel-01619918] Benjamini-Schramm convergence of locally symmetric spaces

19 octobre 2017

The main theme of this work is the study of geometry and topology of locally symmetric spaces Gamma\ X as ther volume Vol(\Gamma\ X) tends to infinity. Our first main result concerns the Benjamini-Schramm convergence for arithmetic hyperbolic 2 or 3-manifolds. A sequence of locally symmetric (...)

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### [hal-01619235] Is Euler's constant a value of an arithmetic special function?

19 octobre 2017

Euler's constant γ is one of the mathematical constants with the most different analytic representations, probably on par with π. Yet, none of these representations proves that γ is a value of an E-function, a G-function or an M-function at an algebraic point. In fact, it is plausible that no such (...)

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### [hal-01619460] Statistics for biquadratic covers of the projective line over finite fields

19 octobre 2017

We study the distribution of the traces of the Frobenius endomorphism of genus $g$ curves which are quartic non-cyclic covers of $\mathbbP^1_\mathbbF_q$, as the curve varies in an irreducible component of the moduli space. We show that for $q$ fixed, the limiting distribution of the trace of (...)

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### [hal-01619346] Bad reduction of genus $3$ curves with complex multiplication

19 octobre 2017

Let $C$ be a smooth, absolutely irreducible genus-$3$ curve over a number field $M$. Suppose that the Jacobian of $C$ has complex multiplication by a sextic CM-field $K$. Suppose further that $K$ contains no imaginary quadratic subfield. We give a bound on the primes $\mathfrakp$ of $M$ such (...)

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### [hal-01619330] Twists of non-hyperelliptic curves of genus 3

19 octobre 2017

In this paper we explicitly compute equations for the twists of all the smooth plane quartic curves defined over a number field k. Since the plane quartic curves are non-hyperelliptic curves of genus 3 we can apply the method developed by the author in a previous article. The starting point is (...)

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18 octobre 2017

[...]

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18 octobre 2017

[...]

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### [hal-01618799] Construction of Brauer-Severi Varieties

18 octobre 2017

In this paper we give an algorithm for computing equations of Brauer-Severi varieties over perfect fields of characteristic 0. As an example we show the equations of all Brauer-Severi surfaces defined over $\mathbbQ$.

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### [hal-01618783] The Picard Group of Brauer-Severi Varieties

18 octobre 2017

In this note we provide explicit generators of the Picard groups of cyclic Brauer-Severi varieties defined over the base field. In particular, for all Brauer-Severi surfaces. To produce these generators we use the Twisting Theory for smooth plane (...)

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### [hal-01618759] On twists of smooth plane curves

18 octobre 2017

Given a smooth curve defined over a field $k$ that admits a non-singular plane model over $\overlinek$, a fixed separable closure of $k$, it does not necessarily have a non-singular plane model defined over the field $k$. We determine under which conditions this happens and we show an example (...)

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### [hal-01618727] A note on the stratification by automorphisms of smooth plane curves of genus 6

18 octobre 2017

In this note, we give a so-called representative classification for the strata by automorphism group of smooth $\bark$-plane curves of genus $6$, where $\bark$ is a fixed separable closure of a field $k$ of characteristic $p = 0$ or $p > 13$. We start with a classification already obtained (...)

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### [hal-01618637] Computing twists of hyperelliptic curves

18 octobre 2017

We give an efficient algorithm to compute equations of twists of hyperelliptic curves of arbitrary genus over any separable field (of characteristic different from 2), and we explicitly describe some interesting examples.

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### [hal-01618616] A bound on the primes of bad reduction for CM curves of genus 3

18 octobre 2017

We give bounds on the primes of geometric bad reduction for curves of genus three of primitive CM type in terms of the CM orders. In the case of genus one, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction (...)

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### [hal-01174473] Principe de Mazur pour U(1,d)

17 octobre 2017

Le principe de Mazur fournit des conditions simples pour qu'une F l-représentation irréductible non ramifiée provenant d'une forme modulaire de niveau Γ 0 (N p) provienne aussi d'une forme de niveau Γ 0 (N). L'objectif de ce travail est de proposer une généralisation de ce principe en dimension (...)

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### [hal-01614087] A common approach to Brocard's problem, the problem of the infinitude of primes of the form n^2+1, and the twin prime problem

16 octobre 2017

Let f(3)=4, and let f(n+1)=f(n)! for every integer n \geq 3. For an integer n \geq 3, let \Phi_n denote the following statement: if a system S \subseteq x_i!=x_i+1: 1 \leq i \leq n-1 \cup x_i \cdot x_j=x_j+1: 1 \leq i \leq j \leq n-1 has at most finitely many solutions in integers x_1,...,x_n (...)

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### [hal-00806518] The Cohen-Lenstra heuristics, moments and $p^j$-ranks of some groups

16 octobre 2017

This article deals with the coherence of the model given by the Cohen-Lenstra heuristic philosophy for class groups and also for their generalizations to Tate-Shafarevich groups. More precisely, our first goal is to extend a previous result due to E. Fouvry and J. Klüners which proves that a (...)

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### [hal-00868110] Conditional expanding bounds for two-variables functions over prime fields

16 octobre 2017

In this paper we provide in $\bFp$ expanding lower bounds for two variables functions $f(x,y)$ in connection with the product set or the sumset. The sum-product problem has been hugely studied in the recent past. A typical result in $\bFp^*$ is the existenceness of $\Delta(\alpha)>0$ such (...)

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### [hal-00868107] A Structure result for bricks in Heisenberg groups

16 octobre 2017

We show that for a sufficiently big \textitbrick $B$ of the $(2n+1)$-dimensional Heisenberg group $H_n$ over the finite field $\mathbbF_p$, the product set $B\cdot B$ contains at least $|B|/p$ many cosets of some non trivial subgroup of (...)

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### [hal-00867368] Distribution of residues in approximate subgroups of $\mathbb{F}_p^*$

16 octobre 2017

We extend a result due to Bourgain on the uniform distribution of residues by proving that subsets of the type $f(I)\cdot H$ is equidistributed (as $p$ tends to infinity) where $f$ is a polynomial, $I$ is an interval of $\Fp$ and $H$ is an approximate subgroup of $\mathbbF_p^*$ with size larger (...)

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### [hal-01053353] q-Ehrhart polynomials of Gorenstein polytopes,\\Bernoulli umbra and related Dirichlet series

16 octobre 2017

This article considers some q-analogues of classical results concerning the Ehrhart polynomials of Gorenstein polytopes, namely properties of their q-Ehrhart polynomial with respect to a good linear form. Another theme is a specific linear form Ψ (involving Carlitz' q-analogues of Bernoulli (...)

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### [hal-01287140] Algebraic independence of $G$-functions and congruences "à la Lucas".

16 octobre 2017

We develop a new method for proving algebraic independence of $G$-functions. Our approach rests on the following observation: $G$-functions do not always come with a single linear differential equation, but also sometimes with an infinite family of linear difference equations associated with (...)

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### [hal-00666154] Small values of the Euler function and the Riemann hypothesis

16 octobre 2017

Let $\vfi$ be Euler's function, $\ga$ be Euler's constant and $N_k$ be the product of the first $k$ primes. In this article, we consider the function $c(n) =(n/\vfi(n)-e^\ga\log\log n)\sqrt\log n$. Under Riemann's hypothesis, it is proved that $c(N_k)$ is bounded and explicit bounds are given (...)

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### [hal-00873552] Arithmetic properties of Apéry-like numbers

16 octobre 2017

We provide lower bounds for p-adic valuations of multisums of factorial ratios which satisfy an Apéry-like recurrence relation: these include Apéry, Domb, Franel numbers, the numbers of abelian squares over a finite alphabet, and constant terms of powers of certain Laurent polynomials. In (...)

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### [hal-00947516] Multiple zeta-functions associated with linear recurrence sequences and the vectorial sum formula.

16 octobre 2017

We prove the holomorphic continuation of certain multi-variable multiple zeta-functions whose coefficients satisfy a suitable recurrence condition. Actually we introduce more general vectorial zeta-functions, and prove their holomorphic continuation. Moreover we show a vectorial sum formula (...)

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### [hal-00816198] Hodge theory of the middle convolution

13 octobre 2017

We compute the behaviour of Hodge data by tensor product with a unitary rank-one local system and middle convolution by a Kummer unitary rank-one local system for an irreducible variation of polarized complex Hodge structure on a punctured complex affine line. We give applications of these (...)

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### [hal-00863022] Numerical Verification of the Brumer-Stark Conjecture

13 octobre 2017

We study the Brumer-Stark conjecture computationally in the simplest situation in which it is unproven: an extension K/k with k quadratic, Gal(K/k) isomorphic to ℤ/4ℤ, the class group of K non trivial, K/ℚ non Galois. We verify the conjecture in 379 such cases and study the problem of whether the (...)

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### [hal-00864745] On Dwork's p-adic formal congruences theorem and hypergeometric mirror maps

13 octobre 2017

Using Dwork's theory, we prove a broad generalisation of his famous p-adic formal congruences theorem. This enables us to prove certain p-adic congruences for the generalized hypergeometric series with rational parameters; in particular, they hold for any prime number p and not only for almost (...)

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### [hal-00756399] Hardy-Littlewood series and even continued fractions

12 octobre 2017

For any $s\in (1/2,1]$, the series$F_s(x)=\sum_n=1^\infty e^i\pi n^2 x/n^s$ converges almost everywhere on $[-1,1]$ by a result of Hardy-Littlewood, but not everywhere. However, there does not yet exist an intrinsic description of the set of convergence for $F_s$. In this paper, we define in (...)

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### [hal-00860302] Multiple zeta values, Padé approximation and Vasilyev's conjecture

12 octobre 2017

Sorokin gave in 1996 a new proof that pi is transcendental. It is based on a simultaneous Padé approximation problem involving certain multiple polylogarithms, which evaluated at the point 1 are multiple zeta values equal to powers of pi. In this paper we construct a Padé approximation problem of (...)

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

12 octobre 2017

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

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### [hal-01010964] Arithmetic theory of E-operators

12 octobre 2017

In [Séries Gevrey de type arithmétique I Théorémes de pureté et de dualité, Annals of Math. 151 (2000), 705--740], André has introduced E-operators, a class of differential operators intimately related to E-functions, and constructed local bases of solutions for these operators. In this paper we (...)

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### [hal-00101376] Séries hypergéométriques multiples et polyzêtas

12 octobre 2017

We describe a theoretical and effective algorithm which enables us to prove that rather general hypergeometric series and integrals can be decomposed as linear combinations of multiple zeta values, with rational coefficients.

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### [hal-00101377] Phénomènes de symétrie dans des formes linéaires en polyzêtas

12 octobre 2017

We give two generalizations, in arbitrary depth, of the symmetry phenomenon used by Ball-Rivoal to prove that infinitely many values of Riemann $\zeta$ function at odd integers are irrational. These generalizations concern multiple series of hypergeometric type, which can be written as linear (...)

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

12 octobre 2017

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

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### [hal-01615120] A realization theorem for sets of lengths in numerical monoids

12 octobre 2017

We show that for every finite nonempty subset of N ≥2 there are a numerical monoid H and a squarefree element a ∈ H whose set of lengths L(a) is equal to L.

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### [hal-00403683] Characterizing algebraic curves with infinitely many integral points

11 octobre 2017

A classical theorem of Siegel asserts that the set of S-integral points of an algebraic curve C over a number field is finite unless C has genus 0 and at most two points at infinity. In this paper we give necessary and sufficient conditions for C to have infinitely many S-integral (...)

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### [hal-01245428] Rational approximation to values of G-functions, and their expansions in integer bases

11 octobre 2017

Building upon previous works of André and Chudnovsky, we prove a general result concerning the approximations of values at rational points a/b of any G-function F with rational Taylor coefficients by fractions of the form n/(B ·b^m), where the integer B is fixed. As a corollary, we show that if F (...)

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### [hal-00686683] Structure interne des représentations modulo p de SL2(Qp)

11 octobre 2017

Soit p >3 un nombre premier. À partir de travaux antérieurs dus aux deux auteurs, nous déterminons la filtration par le SL2(Zp)-socle des \bar Fp-représentations lisses irréductibles de SL2(Qp)

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### [hal-00856932] Polynomial configurations in the primes

10 octobre 2017

The Bergelson-Leibman theorem states that if P_1, ..., P_k are polynomials with integer coefficients, then any subset of the integers of positive upper density contains a polynomial configuration x+P_1(m), ..., x+P_k(m), where x,m are integers. Various generalizations of this theorem are known. (...)

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### [hal-00803373] A NEW LOWER BOUND FOR HERMITE'S CONSTANT FOR SYMPLECTIC LATTICES

10 octobre 2017

In this paper we give an improved lower bound on Hermite's constant δ2g for symplectic lattices in even dimensions (g = 2n) by applying a mean-value argument from the geometry of numbers to a subset of symmetric lattices. We obtain only a slight improvement. However, we believe that the method (...)

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### [hal-00793402] Global Jacquet-Langlands correspondence for division algebras in characteristic p

10 octobre 2017

We prove a full global Jacquet-Langlands correspondence between GL(n) and division algebras over global fields of non zero characteristic. If $D$ is a central division algebra of dimension $n^2$ over a global field $F$ of non zero characteristic, we prove that there exists an injective map from (...)

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### [hal-01614076] Metrical results on the distribution of fractional parts of powers of real numbers

10 octobre 2017

Denote by · the fractional part. We establish several new metrical results on the distribution properties of the sequence (x n ) n≥1. Many of them are presented in a more general framework, in which the sequence of functions (x → x n) n≥1 is replaced by a sequence (fn) n≥1 , under some growth and (...)

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### [hal-00782907] On geodesics of phyllotaxis

10 octobre 2017

Seeds of sunflowers are often modelled by the map $n\longmapsto \varphi_\theta(n)=\sqrtne^2i\pi n\theta$ leading to a roughly uniform repartition with two consecutive seeds separated by the divergence angle $2\pi\theta$ for $\theta$ the golden ratio. We associate to an arbitrary real divergence (...)

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### [hal-00931612] Construction of some perfect integral lattices with minimum 4

10 octobre 2017

We construct several families of perfect sublattices with minimum $4$ of $\mathbb Z^d$. In particular, the number of $d-$dimensional perfect integral lattices with minimum $4$ grows faster than $d^k$ for every exponent $k$.

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### [lirmm-00344945] Asymptotic Behavior of the Number of Solutions for Non-Archimedean Diophantine Approximations with Restricted Denominators

10 octobre 2017

We consider metric results for the asymptotic behavior of the number of solutions of Diophantine approximation inequalities with restricted denominators for Laurent formal power series with coefficients in a finite field. We consider, in particular, approximations by rational functions whose (...)

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### [hal-01613530] Improved Complexity Bounds for Counting Points on Hyperelliptic Curves

9 octobre 2017

We present a probabilistic Las Vegas algorithm for computing the local zeta function of a hyperelliptic curve of genus $g$ defined over $\mathbbF_q$. It is based on the approaches by Schoof and Pila combined with a modeling of the $\ell$-torsion by structured polynomial systems. Our main result (...)

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### [lirmm-00484731] Fast Ideal Cubing in Quadratic Number and Function Fields

9 octobre 2017

We present algorithms for computing the cube of an ideal in an imaginary quadratic number field or function field. In addition to a version that computes a non-reduced output, we present a variation based on Shanks' NUCOMP algorithm that computes a reduced output and keeps the sizes of the (...)

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### [hal-01592317] On Davenport constant

6 octobre 2017

We prove that for every fixed integer r => 1 the Davenport constant D(C^r_n) is asymptotic to rn when n tends to infinity.

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### [hal-00804859] Frobenius structure for rank one $p-$adic differential equations

6 octobre 2017

According to a criterion of B. Chiarellotto and G. Christol [Compositio Math. 100 (1996), no. 1, 77-99; MR1377409 (97b:14021)], a solvable rank one p-adic differential operator d/dx−g, with g=∑ni=1a−ixi, has a Frobenius structure if and only if a−1 is p-integral. Using natural estimates on tensor (...)

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### [hal-00947520] Height zeta functions on generalized projective toric varieties.

6 octobre 2017

In this paper we study the analytic properties of height zeta functions associated to generalized projective toric varieties. As an application, we obtain asymptotic expansions of the counting functions of rational points of generalized projective toric varieties provided with a large class of (...)

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### [hal-00947525] k-configurations of points in discrete self-similar sets.

6 octobre 2017

This paper uses the method of zeta functions to study k-con gurations and distinct volumes of k-simplices determined by k-tuples of points of a discrete fractal set F for which the similarity transformations pairwise commute. Under certain reasonable hypotheses on F, we fi nd nontrivial lower (...)

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### [hal-00365948] On perfection relations in lattices

6 octobre 2017

We explore a few special cases of the following problem in the theory of Euclidean lattices: suppose that the s invariant of lattice exceeds its perfection rank; what are the linear relations induced on the symmetric square?

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### [inria-00632118] Computing low-degree isogenies in genus 2 with the Dolgachev-Lehavi method

6 octobre 2017

Let ell be a prime, and H a curve of genus 2 over a field k of characteristic not 2 or ell. If S is a maximal Weil-isotropic subgroup of Jac(H)[ell], then Jac(H)/S is isomorphic to the Jacobian of some (possibly reducible) curve X. We investigate the Dolgachev--Lehavi method for constructing (...)

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6 octobre 2017

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29 août 2017

[...]

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29 août 2017

[...]

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### [hal-00466426] Twisting the Stern sequence

29 août 2017

We describe a twisted version of the Stern sequence and study a few of its properties.

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### [hal-00292997] On exponentials of exponential generating series

29 août 2017

Identifying the algebra of exponential generating series with the shuffle algebra of formal power series, one can define an exponential map $\mathopexp_!:X\mathbb K\longrightarrow 1+X\mathbb K$ for the associated Lie group formed by exponential generating series with constant coefficient $1$ (...)

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### [hal-00149241] Arithmetic properties related to the shuffle-product

29 août 2017

Properties of the shuffle product suggest the definition of a quadratic form with domain and values in formal power series over a field of characteristic 2. This quadratic form preserves rational (respectively algebraic) power series and its restriction to the affine subspace of series with (...)

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29 août 2017

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

26 juin 2017

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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20 juin 2017

[...]

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20 juin 2017

[...]

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20 juin 2017

[...]

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### [hal-01542564] Choosing and generating parameters for pairing implementation on BN curves

19 juin 2017

Because pairings have many applications, many hardware and software pairing implementations can be found in the literature. However, the parameters generally used have been invalidated by the recent results on the discrete logarithm problem over pairing friendly elliptic curves (Kim and (...)

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19 juin 2017

[...]

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19 juin 2017

[...]

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15 juin 2017

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### [hal-00353156] Geometric study of the beta-integers for a Perron number and mathematical quasicrystals

29 mai 2017

We investigate in a geometrical way the point sets of ~$\rb$~ obtained by the ~$\beta$-numeration that are the ~$\beta$-integers ~$\zb_\beta \subset \zb[\beta]$~ where ~$\beta$~ is a Perron number. We show that there exist two canonical cut-and-project schemes associated with the (...)

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### [hal-00020490] Modular representations of p-adic groups

29 mai 2017

I will survey some results in the theory of modular representations of a reductive $p$-adic group, in positive characteristic $\ell \neq p$ and $\ell=p$.

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### [hal-00956365] Stabilisation de la formule des traces tordue IV: transfert spectral archimédien

29 mai 2017

It is one of a series of papers whose goal is to stabilize the twisted trace formula. We consider here a "twisted space" over the real field. We prove in this twisted situation the results obtained by Arthur in his Selecta's paper. That is the existence of transfer of tempered (...)

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### [hal-00407221] Words and Transcendence

29 mai 2017

Is it possible to distinguish algebraic from transcendental real numbers by considering the $b$-ary expansion in some base $b\ge2$? In 1950, É. Borel suggested that the answer is no and that for any real irrational algebraic number $x$ and for any base $g\ge2$, the $g$-ary expansion of $x$ (...)

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### [hal-00488098] Fonctions PN sur une infinité d'extensions de $\mathbb{F}_p$, $p$ impair

29 mai 2017

Let $p$ be an odd prime number. We prove that for $m\equiv1\mod p$, $x^m$ is perfectly nonlinear over $\mathbbF_p^n$ for infinitely many $n$ if and only if $m$ is of the form $p^l+1$, $l\in\mathbbN$. First, we study singularities of $f(x,y)=\frac(x+1)^m-x^m-(y+1)^m+y^mx-y$ and we use Bezout (...)

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

24 mai 2017

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

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

23 mai 2017

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

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### [hal-00338806] F_p-représentations semi-stables

23 mai 2017

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

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### [hal-01522391] A proof of Syracuse conjecture

17 mai 2017

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

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

17 mai 2017

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

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### [hal-01283042] Series Representation of Power Function

17 mai 2017

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

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

9 mai 2017

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

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

9 mai 2017

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

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

6 mai 2017

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

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

28 avril 2017

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

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

27 avril 2017

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

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### [hal-01177065] Fonctions complètement multiplicatives de somme nulle

27 avril 2017

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

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

24 avril 2017

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

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

24 avril 2017

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

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

21 avril 2017

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

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

18 avril 2017

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

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

15 avril 2017

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

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

14 avril 2017

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

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7 avril 2017

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### [hal-01355926] Short addition sequences for theta functions

24 août 2016

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

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

24 août 2016

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

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

18 août 2016

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

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

18 août 2016

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

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### [hal-00823745] Computing class polynomials for abelian surfaces

18 août 2016

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

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

17 août 2016

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

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

17 août 2016

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

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

17 août 2016

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

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

17 août 2016

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

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

17 août 2016

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

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

17 août 2016

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

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

17 août 2016

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

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

17 août 2016

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

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

17 août 2016

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

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

10 août 2016

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

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

9 août 2016

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

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

9 août 2016

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

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

9 août 2016

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

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

8 août 2016

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

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

8 août 2016

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

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

8 août 2016

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

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

8 août 2016

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

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

3 août 2016

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

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

2 août 2016

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

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

29 juillet 2016

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

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### [tel-01201663] Equidistribution problems of squarefree numbers

29 juillet 2016

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

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

29 juillet 2016

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

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

27 juillet 2016

Unlikely Intersections and Pell's equations in polynomials

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

27 juillet 2016

Courbes et fibrés vectoriels en théorie de Hodge p-adique

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

27 juillet 2016

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

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### [medihal-01345061] Alex Kontorovich - Diophantine problems in thin orbits

27 juillet 2016

Diophantine problems in thin orbits

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

27 juillet 2016

Numbers as functions

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

27 juillet 2016

Quantum Field Theory and Arithmetic

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

27 juillet 2016

Congruences in noncommutative Iwasawa theory

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

27 juillet 2016

On the distribution of Fourier coefficients of modular forms

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

27 juillet 2016

Roth's theorem on arithmetic progressions

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

27 juillet 2016

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

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

27 juillet 2016

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

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

26 juillet 2016

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

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

26 juillet 2016

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

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

26 juillet 2016

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

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

26 juillet 2016

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

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

25 juillet 2016

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

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20 juillet 2016

[...]

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

20 juillet 2016

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

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### [hal-00601448] Ramification and cleanliness

19 juillet 2016

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

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

12 juillet 2016

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

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

9 juillet 2016

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

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

8 juillet 2016

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

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### [hal-01341168] Amelioration of Glaisher's Congruence

7 juillet 2016

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

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

7 juillet 2016

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

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

4 juillet 2016

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

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### [hal-01283042] Series Representation of Power Function

1er juillet 2016

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

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

29 juin 2016

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

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### [hal-01337295] NUMERICAL SEMIGROUPS OF TWO GENERATORS

26 juin 2016

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

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23 juin 2016

[...]

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### [hal-00630391] Elliptic periods for finite fields

23 juin 2016

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

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23 juin 2016

[...]

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

23 juin 2016

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

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

23 juin 2016

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

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

23 juin 2016

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

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

23 juin 2016

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

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

23 juin 2016

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

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

23 juin 2016

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

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

22 juin 2016

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

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### [hal-00601448] Ramification and cleanliness

22 juin 2016

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

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

22 juin 2016

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

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

22 juin 2016

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

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

22 juin 2016

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

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

22 juin 2016

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

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

22 juin 2016

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

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

22 juin 2016

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

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

22 juin 2016

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

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

20 juin 2016

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

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### [hal-01333682] Multidimensional Heilbronn sets

18 juin 2016

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

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18 juin 2016

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### [hal-01280172] ON FOURIER COEFFICIENTS OF MODULAR FORMS OF HALF INTEGRAL WEIGHT AT SQUAREFREE INTEGERS

14 avril 2016

We show that the Dirichlet series associated to the Fourier coefficients of a half-integral weight Hecke eigenform at squarefree integers extends analytically to a holomorphic function in the half-plane ℜe s > 1 2. This exhibits a high fluctuation of the coefficients at squarefree (...)

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### [hal-00584431] Calendriers et fractions continues

30 mars 2016

Cet article explique dans un premier temps l'histoire du calendrier grégorien. Ceci est un prétexte pour expliquer le développement en fractions continues d'un nombre réel et d'en donner les principales propriétés. A la fin de l'article, on introduit l'algorithme de Jacobi-Perron qui donne des (...)

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### [hal-00085832] Finitude pour les representations lisses de groupes p-adiques

29 mars 2016

We study basic properties of the category of smooth representations of a p-adic group G with coefficients in any commutative ring R in which p is invertible. Our main purpose is to prove that Hecke algebras are noetherian whenever R is ; a question left open since Bernstein's fundamental work (...)

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### [hal-00566314] Circular words and three applications: factors of the Fibonacci word, ${\mathcal F}$-adic numbers, and the sequence $1$, $5$, $16$, $45$, $121$, $320$,\ldots

29 mars 2016

We introduce the notion of \em circular words with a combinatorial constraint derived from the Zeckendorf (Fibonacci) numeration system, and get explicit group structures for these words. As a first application, we give a new result on factors of the Fibonacci word $abaababaabaab\ldots$. (...)

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### [hal-00022116] Theorie de Lubin-Tate non-abelienne et representations elliptiques

29 mars 2016

Harris and Taylor proved that the supercuspidal part of the cohomology of the Lubin-Tate tower realizes both the local Langlands and Jacquet-Langlands correspondences, as conjectured by Carayol. Recently, Boyer computed the remaining part of the cohomology and exhibited two defects : first, the (...)

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### [hal-00851556] Some exact values of the Harborth constant and its plus-minus weighted analogue

29 mars 2016

The Harborth constant of a finite abelian group is the smallest integer $\ell$ such that each subset of $G$ of cardinality $\ell$ has a subset of cardinality equal to the exponent of the group whose elements sum to the neutral element of the group. The plus-minus weighted analogue of this (...)

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29 mars 2016

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### [hal-00823402] On special values of spinor L-functions of Siegel cusp eigenforms of genus 3

22 mars 2016

We compute the special values for the spinor L-function L(s,F12) in the critical strip s=12,...,19, where F12 is the unique (up to a scalar) Siegel cusp form of degree 3 and weight 12, which was constructed by Miyawaki. These values are proportional to the product of Petersson inner products of (...)

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### [hal-01160765] Microsolutions of differential operators and values of arithmetic Gevrey series

22 mars 2016

We continue our investigation of E-operators, in particular their connection with G-operators; these differential operators are fundamental in understanding the dio-phantine properties of Siegel's E and G-functions. We study in detail microsolutions (in Kashiwara's sense) of Fuchsian (...)

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