# GDR STN

Site du GDR Structuration de la théorie des nombres

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

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

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

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

## Nouveaux articles en théorie des nombres

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

20 janvier 2017

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

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

20 janvier 2017

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

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

19 janvier 2017

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

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

17 janvier 2017

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

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

17 janvier 2017

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

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

17 janvier 2017

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

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

17 janvier 2017

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

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

17 janvier 2017

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

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

16 janvier 2017

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

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

12 janvier 2017

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

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

12 janvier 2017

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

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

11 janvier 2017

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

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

10 janvier 2017

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

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### [hal-01426832] Certification of modular Galois representations

9 janvier 2017

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

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

6 janvier 2017

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

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

6 janvier 2017

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

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### [hal-00694113] A faster pseudo-primality test

6 janvier 2017

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

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### [hal-00133407] Mesures de Mahler et équidistribution logarithmique

6 janvier 2017

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

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

6 janvier 2017

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

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### [hal-00373511] Galois invariant smoothness basis

6 janvier 2017

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

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

6 janvier 2017

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

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

6 janvier 2017

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

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6 janvier 2017

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

6 janvier 2017

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

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### [hal-00630394] Galois invariant smoothness basis

6 janvier 2017

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

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

6 janvier 2017

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

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

6 janvier 2017

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

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

6 janvier 2017

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

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

5 janvier 2017

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

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

5 janvier 2017

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

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### [inria-00386309] Modular Curves of Composite Level

5 janvier 2017

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

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

5 janvier 2017

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

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

5 janvier 2017

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

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

5 janvier 2017

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

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

5 janvier 2017

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

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

5 janvier 2017

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

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5 janvier 2017

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

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

5 janvier 2017

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

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5 janvier 2017

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

5 janvier 2017

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

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

5 janvier 2017

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

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

5 janvier 2017

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

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### [hal-00085619] Sur la complexité des nombres algébriques

5 janvier 2017

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

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5 janvier 2017

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

5 janvier 2017

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

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

5 janvier 2017

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

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

5 janvier 2017

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

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

5 janvier 2017

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

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### [hal-00880994] New identities relating wild Goppa codes

5 janvier 2017

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

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

5 janvier 2017

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

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

5 janvier 2017

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

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5 janvier 2017

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

5 janvier 2017

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

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

4 janvier 2017

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

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

4 janvier 2017

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

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

4 janvier 2017

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

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

4 janvier 2017

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

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

4 janvier 2017

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

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

4 janvier 2017

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

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

4 janvier 2017

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

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4 janvier 2017

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4 janvier 2017

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### [hal-01419948] Conjecture de Syracuse : avancées inédites

4 janvier 2017

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

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

4 janvier 2017

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

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

27 décembre 2016

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

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

27 décembre 2016

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

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

27 décembre 2016

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

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

23 décembre 2016

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

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

22 décembre 2016

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

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

21 décembre 2016

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

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

21 décembre 2016

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

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

21 décembre 2016

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

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

21 décembre 2016

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

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

21 décembre 2016

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

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

21 décembre 2016

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

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### [hal-01420485] Discrete Logarithms

21 décembre 2016

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

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

20 décembre 2016

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

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

14 décembre 2016

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

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

13 décembre 2016

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

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

13 décembre 2016

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

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### [inria-00246115] Computing Hilbert Class Polynomials

13 décembre 2016

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

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

13 décembre 2016

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

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

13 décembre 2016

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

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

12 décembre 2016

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

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

8 décembre 2016

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

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

6 décembre 2016

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

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

6 décembre 2016

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

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

6 décembre 2016

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

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

6 décembre 2016

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

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

5 décembre 2016

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

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5 décembre 2016

[...]

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### [hal-01407300] Note sur la conjecture de Greenberg

2 décembre 2016

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

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

1er décembre 2016

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

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

1er décembre 2016

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

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

30 novembre 2016

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

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

30 novembre 2016

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

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

30 novembre 2016

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

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

30 novembre 2016

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

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

29 novembre 2016

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

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

29 novembre 2016

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

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25 novembre 2016

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

24 août 2016

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

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

24 août 2016

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

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

18 août 2016

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

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

18 août 2016

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

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

18 août 2016

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

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

17 août 2016

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

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

17 août 2016

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

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

17 août 2016

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

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

17 août 2016

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

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

17 août 2016

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

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

17 août 2016

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

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

17 août 2016

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

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

17 août 2016

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

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

17 août 2016

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

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

10 août 2016

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

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

9 août 2016

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

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

9 août 2016

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

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

9 août 2016

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

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

8 août 2016

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

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

8 août 2016

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

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

8 août 2016

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

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

8 août 2016

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

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

3 août 2016

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

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

2 août 2016

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

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

29 juillet 2016

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

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

29 juillet 2016

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

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

29 juillet 2016

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

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

27 juillet 2016

Unlikely Intersections and Pell's equations in polynomials

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

27 juillet 2016

Courbes et fibrés vectoriels en théorie de Hodge p-adique

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

27 juillet 2016

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

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

27 juillet 2016

Diophantine problems in thin orbits

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

27 juillet 2016

Numbers as functions

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

27 juillet 2016

Quantum Field Theory and Arithmetic

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

27 juillet 2016

Congruences in noncommutative Iwasawa theory

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

27 juillet 2016

On the distribution of Fourier coefficients of modular forms

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

27 juillet 2016

Roth's theorem on arithmetic progressions

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

27 juillet 2016

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

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

27 juillet 2016

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

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

26 juillet 2016

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

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

26 juillet 2016

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

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

26 juillet 2016

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

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

26 juillet 2016

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

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

25 juillet 2016

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

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20 juillet 2016

[...]

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

20 juillet 2016

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

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### [hal-00601448] Ramification and cleanliness

19 juillet 2016

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

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

12 juillet 2016

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

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

9 juillet 2016

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

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

8 juillet 2016

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

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

7 juillet 2016

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

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

7 juillet 2016

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

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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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