# GDR STN

Site du GDR Structuration de la théorie des nombres

## Nouveaux articles en théorie des nombres

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

14 août 2017

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

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

10 août 2017

In this paper described numerical expansion of natural-valued power function $x^n$, in point $x=x_0$ where $n, \ x_0$ - natural numbers. Applying numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared (...)

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

9 août 2017

In this paper described numerical expansion of natural-valued power function $x^n$, in point $x=x_0$ where $n, \ x_0$ - natural numbers. Applying numerical methods, that is calculus of finite differences, namely, discrete case of Binomial expansion is reached. Received results were compared (...)

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### [hal-01568373] Breaking DLP in $GF(p^5)$ using 3-dimensional sieving

3 août 2017

We report on a discrete logarithm computation in $GF(p^5)$ for a 20-decimal digit prime, using the number field sieve algorithm (NFS-DL), and a relation collection phase over degree-two polynomials, instead of the more classical degree-one (...)

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### [hal-01569570] Sur la comparaison entre les minima et les pentes

1er août 2017

On étudie la comparaison entre les minima et les pentes successifs d'un fibré vectoriel hermitien sur une courbe arithmétique et démontre un encadrement uniform de leurs différences.

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### [hal-01571121] On special values of standard L-functions of Siegel cusp eigenforms of genus 3

1er août 2017

We explicitly compute the special values of the standard $L$-function $L(s, F_12, \mathrmSt)$ at the critical points $s\in \lbrace -8, -6, -4, -2, 0, 1, 3, 5, 7, 9\rbrace$, where $F_12$ is the unique (up to a scalar) Siegel cusp form of degree $3$ and weight $12$, which was constructed by (...)

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### [hal-01571117] Explicit Hecke series for symplectic group of genus 4

1er août 2017

Shimura conjectured the rationality of the generating series for Hecke operators for the symplectic group of genus $n$. This conjecture was proved by Andrianov for arbitrary genus $n$, but the explicit expression was out of reach for genus higher than 3. For genus $n=4$, we explicitly compute (...)

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### [hal-01571090] Spherical Image of Local Hecke Series of Genus Four

1er août 2017

Получена явная формула образа при сферическом отображении дроби многочленов, предложенная Шимурой в 1963 г. для производящих рядов Гекке рода 4. Для вычислений мы использовали формулы Андрианова для сферического отображения Сатаке, которые подробно описаны в нашей предыдущей (...)

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### [hal-01571086] Spherical image of local Hecke series of genus four

1er août 2017

For the spherical image of a polynomial fraction, we obtain the explicit formula conjectured by Shimura in 1963 for generating Hecke series in the particular case of genus 4. As in our previous work, we use formulas due to Andrianov for the Satake spherical (...)

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### [hal-01571077] Explicit Shimura’s conjecture for Sp3 on a computer

1er août 2017

We compute by a different method the generating series in Shimura's conjecture for Sp3, proved by Andrianov in 1967. We develop formulas for the Satake spherical maps for Spn and GLn.

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### [hal-01570933] Exceptional values of E-functions at algebraic points

1er août 2017

E-functions are entire functions with algebraic Taylor coefficients satisfying certain arithmetic conditions, and which are also solutions of linear differential equations with rational functions coefficients. They were introduced by Siegel in 1929 to generalize Diophantine properties of the (...)

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

29 juillet 2017

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

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### [hal-01568068] A Disturbing Combination of Geometrical and Modular Rotations in the World of Arithmetic

28 juillet 2017

In previous papers [1] [2], we have examined two types of 'rotations' in associated Stirling numbers of first and second kind at any order r, respectively : 1. Geometrical rotations, which helped to compact all those Stirling numbers in an arithmeti-cal triangle's structure. 2. Modular (...)

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

26 juillet 2017

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-01562678] On some Generalizations of Wilson's Theorem and How to Characterize Carmichael Numbers by Stirling Numbers

25 juillet 2017

This work uses a combinatorial identity involving Stirling's numbers of the second kind to prove and generalize Wilson's theorem in an original manner. Then, again using this relation , we show new characterizations of Carmichael's (...)

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

25 juillet 2017

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-01565953] SURCONVERGENCE DE LA MONODROMIE p-ADIQUE DES FAMILLES UNIVERSELLES DE VARIÉTÉS ABÉLIENNES ORDINAIRES

24 juillet 2017

Nous montrons que la tour d’Igusa des variétés de Siegel (de niveau ≥ 3 premier à p) sur Q_p est surconvergente, ie qu'elle se prolonge au-dessus d'un voisinage strict du lieu ordinaire. On construit la représentation de monodromie associée en utilisant un morphisme syntomique à coefficients dans des (...)

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### [hal-00122369] Theoreme de Dobrowolski-Laurent pour les extensions abeliennes sur une courbe elliptique a multiplication complexe

20 juillet 2017

Let E/K be an elliptic curve with complex multiplication and let $K^ab$ be the Abelian closure of $K$. We prove in this article that there exists a constant $c(E/K)$ such that : for all point $P\in E(\barK)-E_tors$, we have $\hath(P)\geq\fracc(E/K)D(\frac\log \log 5D\log 2D)^13,$ where (...)

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### [hal-01562677] Two Types of Rotations in Associated Stirling Numbers

19 juillet 2017

In a previous article [1], we have found that associated Stirling numbers of first and second kind can be compacted, at any order and using a linear transformation, in a structure of arithmetical triangle. We show then a strong link between the congruence of such numbers and this common (...)

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### [hal-01560947] REPRÉSENTATIONS GALOISIENNES DIÉDRALES ET FORMES À MULTIPLICATION COMPLEXE

19 juillet 2017

Pour une représentation galoisienne diédrale en caractéristique ℓ on établit (sous certaines hypothèses) l'existence d'une newform à multiplication complexe, dont on contrôle le poids, le niveau et le caractère, telle que la représentation ℓ-adique associée est congrue modulo ℓ à celle de départ. Given a (...)

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### [hal-01300328] Moyennes effectives de fonctions multiplicatives complexes

18 juillet 2017

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

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### [hal-01562597] A new result about prime numbers: lim n→+∞ n/(p(n) − n(ln n + ln ln n − 1)) = +∞

17 juillet 2017

In this short paper we propose a new result about prime numbers: lim n→+∞ n/(p(n) − n(ln n + ln ln n − 1)) = +∞ .

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### [tel-01559293] Propriétés algébriques et analytiques de certaines suites indexées par les nombres premiers

13 juillet 2017

Dans la première partie de cette thèse, on s'intéresse à la suite NX(p) [mod p] où X est un schéma séparé réduit de type fini sur Z,et pour tout p premier, NX(p) est le nombre de Fp-points de la réduction modulo p de X.Sous certaines hypothèses sur la géométrie de X, on donne une condition simple pour (...)

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### [hal-01560672] A conjecture about prime numbers assuming the Riemann hypothesis

12 juillet 2017

In this paper we propose a conjecture about prime numbers. Based on the result of Pierre Dusart stating that the n th prime number is smaller than n(ln n+ln ln n−0.9484) for n ≥ 39017 we propose that the n th prime number is smaller than n(ln n + ln ln n − 1 +) when n → (...)

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### [hal-01558507] Semi-simple groups that are quasi-split over a tamely-ramified extension

10 juillet 2017

Let K be a discretly henselian field whose residue field is separably closed. Answering a question raised by G. Prasad, we show that a semisimple K– group G is quasi-split if and only if it quasi–splits after a finite tamely ramified extension of (...)

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### [hal-01555728] A classification of R-Fuchsian subgroups of Picard modular groups

7 juillet 2017

Given an imaginary quadratic extension K of Q, we classify the maximal nonelementary subgroups of the Picard modular group PU(1, 2; O_K) preserving a totally real totally geodesic plane in the complex hyperbolic plane H^2_C . We prove that these maximal R-Fuchsian subgroups are arithmetic, and (...)

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### [hal-00452827] On Manin's conjecture for a family of Châtelet surfaces

7 juillet 2017

The Manin conjecture is established for Châtelet surfaces over Q arising as minimal proper smooth models of the surface Y^2+Z^2=f(X) where f is a totally reducible polynomial of degree 3 without repeated roots. These surfaces do not satisfy weak (...)

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### [hal-00512870] On generalized Frame-Stewart numbers

7 juillet 2017

For the multi-peg Tower of Hanoi problem with $k \geqslant 4$ pegs, so far the best solution is obtained by the Stewart's algorithm based on the the following recurrence relation: $\mathrmS_k(n)=\min_1 \leqslant t \leqslant n \left\2 \cdot \mathrmS_k(n-t) + \mathrmS_k-1(t)\right\$, (...)

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### [tel-01557497] Correspondances de Simpson p-adique et modulo pⁿ

6 juillet 2017

Cette thèse est consacrée à deux variantes arithmétiques de la correspondance de Simpson. Dans la première partie, on compare la correspondance de Simpson p-adique à un analogue p-adique de la correspondance de Narasimhan et Seshadri pour les courbes sur les corps p-adiques dû à Deninger et Werner. (...)

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### [hal-00949618] Balanced simplices

6 juillet 2017

An additive cellular automaton is a linear map on the set of infinite multidimensional arrays of elements in a finite cyclic group $\mathbbZ/m\mathbbZ$. In this paper, we consider simplices appearing in the orbits generated from arithmetic arrays by additive cellular automata. We prove that (...)

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### [hal-00837647] Modular Schur numbers

6 juillet 2017

For any positive integers $l$ and $m$, a set of integers is said to be (weakly) $l$-sum-free modulo $m$ if it contains no (pairwise distinct) elements $x_1,x_2,\ldots,x_l,y$ satisfying the congruence $x_1+\ldots+x_l\equiv y\bmodm$. It is proved that, for any positive integers $k$ and $l$, there (...)

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### [hal-00981841] Möbius function of semigroup posets through Hilbert series

6 juillet 2017

In this paper, we investigate the Möbius function $\mu_\mathcalS$ associated to a (locally finite) poset arising from a semigroup $\mathcalS$ of $\mathbbZ^m$. We introduce and develop a new approach to study $\mu_\mathcalS$ by using the Hilbert series of $\mathcalS$. The latter enables us to (...)

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### [tel-01289444] Points de hauteur bornée sur les hypersurfaces des variétés toriques

5 juillet 2017

Depuis les 50 dernières années, de nombreux progrès ont été faits dans la compréhension du comportement asymptotique du nombre de points rationnels de hauteur bornée sur les variétés algébriques. Des conjectures précises ont été avancées par Baryrev, Manin et Peyre quant à la formule asymptotique attendue (...)

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### [tel-01552854] p-adic admissible measures attached to Siegel modular forms of arbitrary genus

3 juillet 2017

L'auteur n'a pas fourni de résumé en anglais.

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

3 juillet 2017

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-01550708] Quotient and product sets of thin subsets of the positive integers

29 juin 2017

We study the cardinalities of A/A and AA for thin subsets A of the set of the first n positive integers. In particular, we consider the typical size of these quantities for random sets A of zero density and compare them with the sizes of A/A and AA for subsets of the shifted primes and the set (...)

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### [hal-01550584] Rationality of the zeta function of the subgroups of abelian p-groups

29 juin 2017

Given a finite abelian p-group F, we prove an efficient recursive formula for sigma(alpha)(F) = Sigma(H <= F) vertical bar H vertical bar(alpha) where H ranges over the subgroups of F. We infer from this formula that the p-component of the corresponding zeta-function on groups of p-rank (...)

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### [hal-01550555] Explicit averages of non-negative multiplicative functions: going beyond the main term

29 juin 2017

We produce an explicit formula for averages of the type Sigma(d <= D) (g star 1)(d) / d, where star is the Dirichlet convolution and g a function that vanishes at infinity (more precise conditions are needed, a typical example of an acceptable function is g(m) - mu(m) / m). This formula (...)

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### [hal-01044876] Thue equations and CM-fields

29 juin 2017

We obtain a polynomial type upper bound for the size of the integral solutions of Thue equations $F(X,Y) = b$ defined over a totally real number field $K$, assuming that $F(X,1)$ has a root $\alpha$ such that $K(\alpha)$ is a CM-field. Furthermore, we give an algorithm for the computation of (...)

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### [hal-01272915] Répartition simultanée de S(n) et S(n+1) dans les progressions arithmétiques

29 juin 2017

Si q≥2 est un nombre entier, on désigne par Sq(n) la somme des chiffres en base q du nombre entier naturel n et par vq(n) sa valuation q-adique. L’objectif de cet article est d’étudier des sommes d’exponentielles de la forme ∑n≤xexp(2iπ(lmSq(n)+km′Sq(n+1)+θn)) afin d’en déduire certaines propriétés (...)

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28 juin 2017

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### [hal-01522391] Arithmétique cantus, graphe de Syracuse, et preuve de la conjecture de Collatz

28 juin 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-01546656] CONJECTURE DE GREENBERG

27 juin 2017

Let k be a totally real number field and let k_∞ be its cyclotomic Z_p-extension for p totally split in k. This text completes our article entitled: Approche p-adique de la conjecture de Greenberg pour les corps totalement r\'eels'' (Annales Mathématiques Blaise Pascal 2017), by means of (...)

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27 juin 2017

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7 avril 2017

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### [hal-01503560] Twists of non-hyperelliptic curves

7 avril 2017

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

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

6 avril 2017

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

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### [hal-01502378] Sufficient conditions for the dirichlet property

5 avril 2017

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

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

5 avril 2017

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

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### [tel-01501685] Blocs des chiffres des nombres premiers

4 avril 2017

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

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

4 avril 2017

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

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4 avril 2017

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

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3 avril 2017

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

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

3 avril 2017

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

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

3 avril 2017

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

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

31 mars 2017

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

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

29 mars 2017

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

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

24 mars 2017

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

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

24 mars 2017

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

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

23 mars 2017

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

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

23 mars 2017

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

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

23 mars 2017

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

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

22 mars 2017

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

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

21 mars 2017

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

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20 mars 2017

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

21 octobre 2016

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

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

21 octobre 2016

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

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21 octobre 2016

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21 octobre 2016

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### [inria-00539742] Topological properties of Rauzy fractals

21 octobre 2016

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

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

21 octobre 2016

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

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

21 octobre 2016

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

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

20 octobre 2016

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

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

20 octobre 2016

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

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

20 octobre 2016

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

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

20 octobre 2016

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

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

20 octobre 2016

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

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

20 octobre 2016

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

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

20 octobre 2016

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

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

20 octobre 2016

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

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

20 octobre 2016

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

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

19 octobre 2016

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

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### [hal-01382899] PRINCIPAL QUARTIC EXTENSIONS AND ELLIPTIC CURVES

19 octobre 2016

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

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

18 octobre 2016

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

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

18 octobre 2016

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

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### [hal-01383161] Le théorème des nombres premiers

18 octobre 2016

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

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

17 octobre 2016

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

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

17 octobre 2016

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

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### [hal-01382385] Counting Co-Cyclic Lattices

17 octobre 2016

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

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### [hal-01230858] Groupes de Galois et nombres automatiques

13 octobre 2016

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

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

13 octobre 2016

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

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

13 octobre 2016

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

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### [hal-01379484] Le grand théorème de Fermat

13 octobre 2016

grand théorème de Fermat

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

12 octobre 2016

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

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

12 octobre 2016

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

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

12 octobre 2016

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

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

12 octobre 2016

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

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

12 octobre 2016

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

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

12 octobre 2016

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

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12 octobre 2016

[...]

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### [hal-00672824] Sekiguchi-Suwa theory revisited

12 octobre 2016

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

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12 octobre 2016

[...]

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12 octobre 2016

[...]

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

11 octobre 2016

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

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

11 octobre 2016

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

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

11 octobre 2016

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

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### [tel-01378003] Algorithmes pour les anneaux finis

11 octobre 2016

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

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### [tel-01378003] Algorithmes pour les anneaux finis

10 octobre 2016

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

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

10 octobre 2016

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

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

7 octobre 2016

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

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

4 octobre 2016

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

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

30 septembre 2016

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

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### [inria-00000406] Increasing Self-Described Sequences

30 septembre 2016

[...]

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

30 septembre 2016

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

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

29 septembre 2016

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

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

29 septembre 2016

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

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

29 septembre 2016

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

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### [hal-00013042] Hauteur asymptotique des points de Heegner

28 septembre 2016

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

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

28 septembre 2016

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

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

28 septembre 2016

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

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

28 septembre 2016

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

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

28 septembre 2016

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

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

28 septembre 2016

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

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### [halshs-00119444] Matrimonial Ring Structures

28 septembre 2016

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

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

28 septembre 2016

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

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

28 septembre 2016

We obtain a decidability result for the Davenport constant.

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

28 septembre 2016

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

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

28 septembre 2016

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

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### [hal-00694296] Geometric and differential Galois theories

28 septembre 2016

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

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

28 septembre 2016

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

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

28 septembre 2016

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

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

26 septembre 2016

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

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

23 septembre 2016

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

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

23 septembre 2016

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

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

22 septembre 2016

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

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

22 septembre 2016

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

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

19 septembre 2016

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

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

16 septembre 2016

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

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

16 septembre 2016

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

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

16 septembre 2016

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

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### [hal-00448031] Pairing the Volcano

16 septembre 2016

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

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

16 septembre 2016

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

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

16 septembre 2016

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

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

16 septembre 2016

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

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### [inria-00544503] Algorithmic Number Theory. 9th. International Symposium, ANTS-IX. Nancy, France, July 2010. Proceedings

12 septembre 2016

[...]

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

12 septembre 2016

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

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

9 septembre 2016

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

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

9 septembre 2016

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

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### [hal-01361276] Normes cyclotomiques naïves et unités logarithmiques

7 septembre 2016

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

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

7 septembre 2016

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

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### [hal-01357338] LA CONJECTURE DU FACTEUR DIRECT

7 septembre 2016

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

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

3 septembre 2016

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

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

2 septembre 2016

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

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

2 septembre 2016

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

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### [hal-00687289] Classes logarithmiques et capitulation

1er septembre 2016

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

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

1er septembre 2016

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

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

31 août 2016

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

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

31 août 2016

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

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

30 août 2016

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

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

27 août 2016

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

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

26 août 2016

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

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

26 août 2016

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

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26 août 2016

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

17 juin 2016

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

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

17 juin 2016

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

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

16 juin 2016

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

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

16 juin 2016

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

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

15 juin 2016

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

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

10 juin 2016

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

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

9 juin 2016

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

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

9 juin 2016

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

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

9 juin 2016

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

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

8 juin 2016

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

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### [hal-01227699] Computing Jacobi's $\theta$ in quasi-linear time

7 juin 2016

Jacobi's $\theta$ function has numerous applications in mathematics and computer science; a naive algorithm allows the computation of $\theta(z, \tau)$, for $z$, $\tau$ verifying certain conditions, with precision $P$ in $O(M(P) \sqrtP)$ bit operations, where $M(P)$ denotes the number of (...)

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

7 juin 2016

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

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### [hal-01295254] Lubin's conjecture for full $p$-adic dynamical systems

6 juin 2016

We give a short proof of a conjecture of Lubin concerning certain families of $p$-adic power series that commute under composition. We prove that if the family is \emphfull (large enough), there exists a Lubin-Tate formal group such that all the power series in the family are endomorphisms of (...)

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### [hal-01325350] On the denominators of the Taylor coefficients of G-functions

5 juin 2016

Let $\sum_n=0^\infty a_n z^n\in \overline\mathbb Q$ be a $G$-function, and, for any $n\ge0$, let $\delta_n\ge 1$ denote the least integer such that $\delta_n a_0, \delta_n a_1, ..., \delta_n a_n$ are all algebraic integers. By definition of a $G$-function, there exists some constant $c\ge 1$ (...)

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### [tel-01084492] Généralisations du critère d’indépendance linéaire de Nesterenko

31 mai 2016

Cette thèse s'inscrit dans le prolongement du résultat d'Apéry donnant l'irrationalité de ζ (3) et de celui de Ball-Rivoal prouvant qu'il existe une infinité d'entiers impairs en lesquels la fonction zêta de Riemann prend des valeurs irrationnelles. Un outil crucial dans la démonstration de (...)

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### [hal-00797625] Uniformisation des variétés de Laumon-Rapoport-Stuhler et conjecture de Drinfeld-Carayol

31 mai 2016

Considérons les variétés de "D-faisceaux elliptiques~" Ell introduites par Laumon, Rapoport et Stuhler, définies sur un corps de fonctions F d'une variable sur un corps fini, pour D une algèbre à division de dimension d^2 sur F. Nous montrons que ces variétés admettent, en une place o de F (...)

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### [hal-01320940] Explicit isogenies in quadratic time in any characteristic

31 mai 2016

Consider two ordinary elliptic curves $E,E'$ defined over a finite field $\F_q$, and suppose that there exists an isogeny $\psi$ between $E$ and $E'$. We propose an algorithm that determines $\psi$ from the knowledge of $E$, $E'$ and of its degree $r$, by using the structure of the $ℓ$-torsion (...)

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### [hal-00797609] Algèbre T1 : Groupes, Corps et Théorie de Galois

31 mai 2016

Ce livre s'adresse aux étudiants de licence ou master de mathématiques (L3-M1) et à ceux qui préparent le Capes ou l'agrégation. Il traite de la théorie des groupes, de la théorie des corps et d'un de leurs points communs essentiels, la théorie de Galois des extensions finies. Chacune de ces théories (...)

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### [hal-01320940] Explicit isogenies in quadratic time in any characteristic

30 mai 2016

Consider two ordinary elliptic curves $E,E'$ defined over a finite field $\F_q$, and suppose that there exists an isogeny $\psi$ between $E$ and $E'$. We propose an algorithm that determines $\psi$ from the knowledge of $E$, $E'$ and of its degree $r$, by using the structure of the $ℓ$-torsion (...)

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### [hal-00797622] Sur les réalisations des correspondances de Langlands dans la cohomologie des tours de Drinfeld et de Lubin-Tate

29 mai 2016

Le cadre de cet article est celui du programme de Langlands : on réalise les correspondances de Langlands (du moins pour les supercuspidales) et de Jacquet-Langlands locales dans la cohomologie de certains espaces de modules de groupes $p$-divisibles, à savoir la tour dite de Lubin-Tate et la (...)

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### [tel-01237604] Mathematical model of the role and temporal dynamics of protein p53 after drug-induced DNA damage

29 mai 2016

Various molecular pharmacokinetic–pharmacodynamic models have been proposed in the last decades to represent and predict drug effects in anticancer therapies. Most of these models are cell population based models since clearly measurable effects of drugs can be seen on populations of (healthy (...)

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### [tel-01178288] P-adic local Langlands correspondence and geometry

29 mai 2016

This thesis concerns the geometry behind the p-adic local Langlands correspondence. We give a formalism of methods of Emerton, which would permit to establish the Fontaine-Mazur conjecture in the general case for unitary groups. Then, we verify that our formalism works well in the case of U(3) (...)

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### [hal-00078974] Géométrie des surfaces algébriques et points entiers

28 mai 2016

Let $X$ be a projective normal surface over a number field $K$. Let $H$ be the sum of four properly intersecting ample effective divisors on $X$. We show that any set of $S$-integral points in $X-H$ is not Zariski dense.

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### [hal-00482405] Using hyperelliptic curves to find positive polynomials that are not a sum of three squares in R(x, y)

28 mai 2016

This article deals with a quantitative aspect of Hilbert's seventeenth problem: producing a collection of real polynomials in two variables of degree 8 in one variable which are positive but are not a sum of three squares of rational fractions. As explained by Huisman and Mahe, a given monic (...)

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### [hal-00374041] Local Fourier transform and epsilon factors

28 mai 2016

Laumon introduced the local Fourier transform for $\ell$-adic Galois representations of local fields, of equal characteristic $p$ different from $\ell$, as a powerful tool to study the Fourier-Deligne transform of $\ell$-adic sheaves over the affine line. In this article, we compute explicitly (...)

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### [hal-00374047] Analyse micro-locale l-adique en caracteristique p>0: Le cas d'un trait

28 mai 2016

We develop, for an l-adic etale sheaf on a complete trait of characteristic p>0, the notion of characteristic variety. Our approach, inspired by the microlocal analysis of Kashiwara and Schapira, is a complement to our ramification theory for local fields with general residue fields. We (...)

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### [hal-00455582] Igusa integrals and volume asymptotics in analytic and adelic geometry

28 mai 2016

We establish asymptotic formulae for volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying (...)

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### [hal-00136041] Abelian varieties with l-adic Galois representation of Mumford's type

28 mai 2016

Nous étudions des variétés abéliennes sur des corps de nombres telles que l'algèbre de Lie de l'image d'une représentation l-adique associée soit géométriquement isomorphe à c + sl(2)^3. On appelle de telles variétés abéliennes des `variétés abéliennes avec représentation l-adique du type de Mumford'. On (...)

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### [hal-00136040] On Mumford's families of abelian varieties

28 mai 2016

Mumford a construit des familles de variétés abéliennes de dimension 4 qui sont paramétrées par des variétés de Shimura, mais qui ne sont pas de type PEL. Le but du présent article est d'étudier les familles en question. Pour chaque fibre sur un corps de nombres dans une telle famille, on détermine les (...)

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### [hal-00944666] Un calcul d'anneaux de déformations potentiellement Barsotti--Tate

27 mai 2016

Let F be an unramified extension of Qp. The first aim of this work is to develop a purely local method to compute the potentially Barsotti-Tate deformations rings with tame Galois type of irreducible two-dimensional representations of the absolute Galois group of F. We then apply our method in (...)

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### [hal-00455574] Heights and measures on analytic spaces. A survey of recent results, and some remarks

27 mai 2016

This paper has two goals. The first is to present the construction, due to the author, of measures on non-archimedean analytic varieties associated to metrized line bundles and some of its applications. We take this opportunity to add remarks, examples and mention related (...)

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### [hal-00018574] Produit eulérien motivique et courbes rationnelles sur les variétés toriques

27 mai 2016

We study the asymptotical behaviour of the moduli space of morphisms of given anticanonical degree from a rational curve to a split toric variety, when the degree goes to infinity. We obtain in this case a geometric analogue of Manin's conjecture about rational points of bounded height on (...)

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27 mai 2016

We present a new method to propagate $p$-adic precision in computations, which also applies to other ultrametric fields. We illustrate it with many examples and give a toy application to the stable computation of the SOMOS 4 (...)

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### [hal-00294979] Sur la classification de quelques phi-modules simples

27 mai 2016

This note is an appendix to a preprint by E. Hellmann. We give a complete classification of simple objects of the category of vector spaces D over K = Fpbar((u)) equipped with an endomorphism phi whose image generates D and that are semi-linear with respect to the ring morphism sending u to u^b (...)

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27 mai 2016

[...]

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### [hal-00455578] Integral points of bounded height on partial equivariant compactifications of vector groups

27 mai 2016

We establish asymptotic formulas for the number of integral points of bounded height on partial equivariant compactifications of vector groups.

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### [hal-00449471] La conjecture de Manin géométrique pour une famille de quadriques intrinsèques

27 mai 2016

We prove a version of Manin's conjecture for a certain family of intrinsic quadrics, the base field being a global field of positive characteristic. We also explain how a very slight variation of the method we use allows to establish the conjecture for a certain generalized del Pezzo (...)

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27 mai 2016

[...]

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### [hal-00368090] Points de vue sur les valeurs aux entiers des formes quadratiques binaires

27 mai 2016

The primary focus of this largely expository paper is to present different points of view on the values assumed by binary quadratic forms, sometimes with irrational coefficients, at integer values. Three different points of view are taken. One is the traditional approximation of real numbers by (...)

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

26 mai 2016

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

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### [hal-01321849] Character theory approach to Sato–Tate groups

26 mai 2016

In this article, we propose to use the character theory of compact Lie groups and their orthogonality relations for the study of Frobenius distribution and Sato-Tate groups. The results show the advantages of this new approach in several aspects. With samples of Frobenius ranging in size much (...)

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

25 mai 2016

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

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

25 mai 2016

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

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### [hal-01295749] Strong modularity of reducible Galois representations

25 mai 2016

In this paper, we call strongly modular those reducible semi-simple odd mod $l$ Galois representations for which the conclusion of the strongest form of Serre's original modularity conjecture holds. Under the assumption that the Serre weight $k$ satisfies $l>k+1$, we give a precise (...)

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### [tel-01247743] Equations diophantiennes et corps cyclotomiques

25 mai 2016

This thesis examines some approaches to address Diophantine equations, specifically we focus on the connection between the Diophantine analysis and the theory of cyclotomic fields.First, we propose a quick introduction to the methods of Diophantine approximation we have used in this research (...)

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### [hal-01292805] SUMS OF TWO ${S}$-UNITS VIA FREY-HELLEGOUARCH CURVES

24 mai 2016

In this paper, we develop a new method for finding all perfect powers which can be expressed as the sum of two rational S-units, where S is a finite set of primes. Our approach is based upon the modularity of Galois representations and, for the most part, does not require lower bounds for (...)

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### [hal-01196990] On the Conjecture of Lehmer, limit Mahler measure of trinomials and asymptotic expansions

22 mai 2016

Let $n ≥ 2$ be an integer and denote by $\theta_n$ the real root in $(0, 1)$ of the trinomial $G_n(X) = −1 + X + X^n$ . The sequence of Perron numbers $(\theta_n^−1 )_n≥2$ tends to 1. We prove that the Conjecture of Lehmer is true for $\\theta_n^−1 | n ≥ 2\$ by the direct method of Poincar\'e (...)

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### [cea-01318414] Multiple D3-instantons and mock modular forms I

21 mai 2016

We study D3-instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In a previous work, consistency of D3-instantons with S-duality was established at first order in the instanton expansion, using the modular properties of the (...)

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### [hal-01319255] Multiple D3-instantons and mock modular forms I

20 mai 2016

We study D3-instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In a previous work, consistency of D3-instantons with S-duality was established at first order in the instanton expansion, using the modular properties of the (...)

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### [hal-01318564] Central Limit Theorem for probability measures defined by sum-of-digits function in base 2

20 mai 2016

In this paper we prove a central limit theorem for some probability measures defined as asymtotic densities of integer sets defined via sum-of-digit-function. To any integer a we can associate a measure on Z called µa such that, for any d, µa(d) is the asymptotic density of the set of integers n (...)

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### [hal-01318944] Hecke algebras and affine flag varieties in characteristic p

20 mai 2016

Let G be a split semi-simple p-adic group and let H be its Iwahori-Hecke algebra with coefficients in the algebraic closure k of the finite field with p elements. Let F be the affine flag variety over k associated with G. We show, in the simply connected simple case, that a torus-equivariant (...)

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### [hal-01314014] Calcul de pi par la méthode originale des carrés inscrits

19 mai 2016

Le nombre pi est utilisé dans des domaines variés des mathématiques, de la physique et d’ingénierie. Le défi que représente la détermination de sa valeur a stimulé la créativité et l'enthousiasme de nombreux chercheurs depuis l'antiquité. De nombreuses méthodes plus ou moins complexes permettent de (...)

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### [hal-01279192] Local behavior of arithmetical functions with applications to automorphic L-functions

19 mai 2016

We derive a Voronoi-type series approximation for the local weighted mean of an arithmetical function that is associated to Dirichlet series satisfying a functional equation with gamma factors. The series is exploited to study the oscillation frequency with a method of Heath-Brown and Tsang (...)

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### [hal-01280194] Sign changes of Fourier coefficients of modular forms of half integral weight, 2

19 mai 2016

In this paper, we investigate the sign changes of Fourier coefficients of half-integral weight Hecke eigenforms and give two quantitative results on the number of sign changes.

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### [hal-01278944] The number of coefficients of automorphic L-functions for GL m of same signs

19 mai 2016

Let π be an irreducible unitary cuspidal representation for GL m (A Q), and let L(s, π) be the automorphic L-function attached to π, which has a Dirich-let series expression in the half-plane Re s > 1. When π is self-contragredient, all the coefficients in the Dirichlet series expression are (...)

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### [hal-01278937] Distribution of Hecke eigenvalues of newforms in short intervals

19 mai 2016

We explore nonvanishing of Hecke eigenvalues in short arithmetic progressions and their signs in short intervals by further developing B-free number method and studying moments in short intervals. We improve some results of Alkan & Zaharescu [3] and Lau & Wu [20] on the distribution of (...)

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### [inria-00211875] Worst-Case Hermite-Korkine-Zolotarev Reduced Lattice Bases

19 mai 2016

The Hermite-Korkine-Zolotarev reduction plays a central role in strong lattice reduction algorithms. By building upon a technique introduced by Ajtai, we show the existence of Hermite-Korkine-Zolotarev reduced bases that are arguably least reduced. We prove that for such bases, Kannan's (...)

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### [hal-01278408] On a divisor problem related to the Epstein Zeta-Function, IV

18 mai 2016

In this paper we study several divisor problems related to the Epstein zeta-function. We are able to improve previous results and establish some new results by applying some classical techniques.

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### [hal-01278402] On a divisor problem related to the Epstein Zeta-Function, III

18 mai 2016

In this paper we study the mean square of the error term ∆ * k (Q, x) in a divisor problem related to the Epstein zeta-function. An asymptotic formula has been obtained when k = 2.

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### [hal-01277992] On a divisor problem related to the Epstein Zeta-Function, II

18 mai 2016

Recently by the theory of modular forms and the Riemann zeta-function, Lü improved the estimates for the error term in a divisor problem related to the Epstein zeta-function established by Sankaranarayanan. In this short note, we are able to further sharpen some results of Sankaranarayanan and (...)

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### [tel-01316937] Éléments explicites en théorie algébrique des nombres

18 mai 2016

Ce mémoire présente une synthèse de mes travaux de recherche en théorie algébrique des nombres menés entre 2003 et 2013, seul ou en collaboration. Ils portent principalement sur l'étude de la structure galoisienne de modules associés à des extensions de corps de nombres, sous diverses hypothèses en (...)

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### [hal-01316201] Small sumsets in real numbers : a continuous 3k-4 theorem

18 mai 2016

We prove a continuous Freiman's 3k-4 theorem for small sumsets in R by using some ideas from Ruzsa's work on measure of sumsets in R as well as some graphic representation of density functions of sets. We thereby get some structural properties of A, B and A+B when (...)

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18 mai 2016

[...]

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### [inria-00187880] LLL: a tool for effective diophantine approximation

18 mai 2016

The purpose of this paper is to survey in a unified setting some of the results in diophantine approximation that the LLL algorithm can make effective in an efficient way. We mostly study the problems of finding good rational approximations to vectors of real and p-adic numbers, and of finding (...)

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18 mai 2016

[...]

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### [inria-00000904] Euclidean minima of totally real fields : Algorithmic determination.

18 mai 2016

This article deals with the determination of the Euclidean minimum $M(K)$ of a totally real number field $K$ of degree $n\geq 2$, using techniques from the geometry of numbers. Our improvements of existing algorithms allow us to compute Euclidean minima for fields of degree $2$ to $8$ and (...)

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18 mai 2016

[...]

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### [hal-00091199] Moyennes de certaines fonctions multiplicatives sur les entiers friables, 2

18 mai 2016

Nous évaluons les moyennes indiquées dans le titre sous des hypothèses analytiques concernant la série de Dirichlet associée

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### [hal-00136940] Inhomogeneous and Euclidean spectra of number fields with unit rank strictly greater than 1

18 mai 2016

Let K be a number field with unit rank r > 1. In this article we show that the inhomogeneous minimum of K is attained by at least one rational point. In particular, if M(K) is the Euclidean minimum of K, we have . This phenomenon has consequences on the decidability of the Euclidean nature (...)

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### [hal-00136941] Euclidean minima of totally real number fields, algorithmic determination

18 mai 2016

Cet article traite de la détermination explicite du minimum euclidien (pour la norme) des corps de nombres totalement réels, voire de la partie supérieure de leur spectre euclidien. L'efficacité de l'algorithme mis en oeuvre permet de travailler jusqu'au degré (...)

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### [hal-00107309] LLL on the average

18 mai 2016

Despite their popularity, lattice reduction algorithms remain mysterious in many ways. It has been widely reported that they behave much more nicely than what was expected from the worst-case proved bounds, both in terms of the running time and the output quality. In this article, we (...)

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### [hal-01282675] HIGHER INTEGRAL MOMENTS OF AUTOMORPHIC L-FUNCTIONS IN SHORT INTERVALS

17 mai 2016

We consider the higher integral moments for automorphic L-functions in short intervals and give a proof for the conjecture of Conrey et al. under Generalized Riemann Hypothesis for automorphic L-function.

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### [hal-01282682] Distribution of values of symmetric power L-functions at the edge of the critical strip

17 mai 2016

We study some problems on the distribution of values of symmetric power L-functions at s = 1 in both aspects of level and weight: bounds of these values, extreme values, Montgomery-Vaughan's conjecture and distribution functions. Our results generalise and/or improve related results of (...)

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### [tel-01282669] Valeurs centrales et valeurs au bord de la bande critique de fonctions L automorphe

17 mai 2016

Cette thèse, constitué en trois parties, est consacrée à l’étudie des valeurs spéciales de fonctions L automorphes. La première partie contient un survol rapide de la théorie des formes modulaires et des fonctions L de puissance symétrique associées qui est nécessaire dans la suite. Dans la seconde (...)

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### [hal-01316252] A note on generalized radix representations and dynamical systems

16 mai 2016

Akiyama et al. proved an asymptotic formula for the distribution of CNS poly-nomials with fixed constant term. The objective of the present paper is to improve that result by providing an error term too.

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### [hal-01316251] Construction of Normal Numbers via Generalized Prime Power Sequences

16 mai 2016

We construct normal numbers in base q by concatenating q-adic expansions of prime powers $\left\lfloor\alpha p^\theta\right\rfloor$ with $\alpha>0$ and $\theta>1$.

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### [hal-01316249] Construction of μ -normal sequences

16 mai 2016

In the present paper we extend Champernowne's construction of normal numbers to provide sequences which are generic for a given invariant probability measure, which need not be the maximal one. We present a construction together with estimates and examples for normal numbers with respect to (...)

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### [hal-00875091] On the distribution functions of two oscillating sequences

16 mai 2016

We investigate the set of all distribution functions of two special sequences on the unit interval, which involve logarithmic and trigonometric terms. We completely characterise the set of all distribution functions $G(x_n)$ for $(x_n)_n \geq 1 = (\\cos (\alpha n)^n\)_n \geq 1$ and arbitrary (...)

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### [hal-01314745] Sign changes of the coefficients of automorphic L-functions

13 mai 2016

This survey gives an account of background and the recent development concerning sign changes of Fourier coefficients of modular forms, which includes the great contributions of other authors. We are attempting to elucidate interesting viewpoints, ideas and methods which, even unspecified, may (...)

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12 mai 2016

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### [hal-01315080] Multidimensional Bell inequalities and quantum cryptography. Codes, cryptology, and information security.

12 mai 2016

The laws of quantum physics allow the design of cryptographic protocols for which the security is based on physical principles. The main cryptographic quantum protocols are key distribution schemes, in which two parties generate a shared random secret string. The privacy of the key can be (...)

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### [hal-01315075] A qutrit quantum key distribution protocol using Bell inequalities with larger violation capabilities.

12 mai 2016

The Ekert quantum key distribution protocol [1] uses pairs of entangled qubits and performs checks based on a Bell inequality to detect eavesdropping. The 3DEB protocol [2] uses instead pairs of entangled qutrits to achieve better noise resistance than the Ekert protocol. It performs checks (...)

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12 mai 2016

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### [hal-01315047] Estimates of ψ,θ for large values of x without the Riemann hypothesis

12 mai 2016

The enlargement of known zero-free regions has enabled us to find better effective estimates for classical number-theoretic functions linked to the distribution of prime numbers. In particular we draw the quintessence of the method of Rosser and Schoenfeld on the upper bounds for the usual (...)

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### [hal-01314997] On the splitting of the exact sequence, relating the wild and tame kernels

12 mai 2016

Let k be a number field. For an odd prime p and an integer i>1, the i-th \'etale wild kernel is contained in the second cohomology group of o'_k with coefficients in Zp(i), where o'_k is the ring of p-integers of k. Using Iwasawa theory, we give conditions for this inclusion to split. In (...)

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12 mai 2016

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10 mai 2016

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### [hal-01313346] Shifted convolution of cusp-forms with θ-series

10 mai 2016

In this paper we apply simple approachs to improve a recent result due to Luo, concerning a shifted convolution sum involving the Fourier coefficients of cusp forms with those of theta series. In order to explore what should be the best possible bound, a mean square result of this shifted (...)

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### [hal-01307789] There are infinitely many twin primes 30n+11 and 30n+13, 30n+17 and 30n+19, 30n+29 and 30n+31

10 mai 2016

We proved that $\liminf\limits_n \rightarrow +\infty(p_n+1-p_n)=2$ where $p_n$ is the $n-th$ prime number. We showed the conditions on which an integer $10X+1$, $10X+3$, $10X+7$ or $10X+9$ can be a prime number. We studied the conditions required to get some twin primes and proved the twin (...)

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10 mai 2016

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### [hal-01281342] Sur la répartition du noyau d'un entier

10 mai 2016

We investigate the asymptotic behaviour of the number $N(x,y)$ of those integers $n\leqslant x$ with squarefree kernel $k(n)\leqslant y$.

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### [hal-01281305] Mesures quadratiques de la proximité des diviseurs

10 mai 2016

Diverses mesures naturelles de la proximité des diviseurs d'un entier sont introduites et plusieurs conjectures sont résolues.

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### [hal-01313347] Fourier coefficients of symmetric power L-functions

10 mai 2016

Let f be a Hecke eigencusp form of even integral weight k or Maass cusp form for the full modular group SL 2 (Z). Denote by λ sym m f (n) the nth normalized coefficient of the Dirichlet expansion of the mth symmetric power L-function associated to f. In this paper, we establish some bounds for (...)

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### [hal-01280840] On irregularities of distribution of binary sequences relative to arithmetic progressions, I. (General results)

10 mai 2016

In 1964 K. F. Roth initiated the study of irregularities of distribution of binary sequences relative to arithmetic progressions and since that numerous papers have been written on this subject. In the applications one needs binary sequences which are well distributed relative to arithmetic (...)

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### [hal-01280845] On a phenomenon of Turán concerning the summands of partitions

10 mai 2016

Turán [12] proved that for almost all pairs of partitions of an integer, the proportion of common parts is very high, that is greater than 1/ 2 − ε with ε > 0 arbitrarily small. In this paper we prove that this surprising phenomenon persists when we look only at the summands in a fixed (...)

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### [hal-01280829] Various approaches for the study of the complexity of some families of pseudorandom subsets

10 mai 2016

Studying randomness in different structures is important from the development of applications and theory. Dartyge, Mosaki and Sárközy (among others) have studied measures of randomness for families of subsets of integers. In this article, we improve results on the complexity of some families (...)

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### [hal-01303279] Primes in [kn, (k + 1)n] , for 1⩽. k ⩽. 10

9 mai 2016

In 1919 S. Ramanujan proved [Ram] the Bertrand postulate elegantly with approximations based on the Stirling formula. P. Erdös proved also [Erd] the same result in 1931 by a carefull study of the binomial coeficient (2n n). The Bertrand postulate was first proved by P. Tchebichef in 1850. (...)

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### [hal-01312603] Diophantine approximation with improvement of the simultaneous control of the error and of the denominator

9 mai 2016

In this work we proof the following theorem which is, in addition to some other lemmas, our main result: \noindent \textbftheorem. Let$\ X=\left\ \left( x_1\text, % t_1\right) \text, \left( x_2\text, t_2\right) \text, ..., \left( x_n\text, t_n\right) \right\$ be a finite part of (...)

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### [hal-01311299] Lower Bounds for the Height in Galois Extensions

9 mai 2016

We prove essentially sharp lower bounds for the height of an algebraic number in a Galois extension

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### [hal-01081309] Note sur les valeurs moyennes criblées de certaines fonctions arithmétiques

8 mai 2016

We provide, in a wide range of the parameters, an estimate for the mean-value over sifted integers of certain arithmetic functions with Dirichlet series analytically close to $1/\zeta_\mathbbK(s)$, where $\mathbbK$ is a number field and $\zeta_\mathbbK$ its Dedekind zeta (...)

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7 mai 2016

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### [hal-01281714] Sur l'inégalité de Turán–Kubilius friable

7 mai 2016

Nous établissons une version uniforme, relativement à tous les paramètres, de l’inégalité de Turán-Kubilius friable (i.e.relative aux entiers sans grand facteur premier), comparant la variance empirique d’une fonction arithmétique additive à support friable à celle de son modèle probabiliste. Nous (...)

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### [hal-01281724] An Ingham-Müntz type theorem and simultaneous observation problems

7 mai 2016

We establish a theorem combining the estimates of Ingham and Müntz – Szász. Moreover, we allow complex exponents instead of purely imaginary exponents for the Ingham type part or purely real exponents for the Müntz – Szász part. A very special case of this theorem allows us to prove the simultaneous (...)

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7 mai 2016

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### [hal-01281732] Fonctions multiplicatives, sommes d'exponentielles, et loi des grands nombres

7 mai 2016

Nous donnons des conditions suffisantes effectives quasi-optimales pour que la majoration de Halberstam–Richert fournisse effectivement l'ordre de grandeur de la valeur moyenne d'une fonction multiplicative positive ou nulle.

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5 mai 2016

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### [hal-01067949] On the generic local Langlands correspondence for GSpin groups

5 mai 2016

In the case of split $GSpin$ groups, we prove an equality of $L$-functions between automorphic local $L$-functions defined by the Langlands-Shahidi method and local Artin $L$-functions. Our method of proof is based on previous results of the first author which allow to reduce the problem to (...)

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### [hal-01296795] Application de réciprocité de Shafarevich et groupe de Nottingham

4 mai 2016

Soient K le corps local View the MathML source et Kab sa clôture abélienne. On note View the MathML source le groupe des Fq-automorphismes continus σ de Kab tel que σ(K)=K et tel que σ opère sur le corps résiduel View the MathML source comme une puissance entière de l'automorphisme de Frobénius. On (...)

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### [hal-01311391] Sums of seventh powers of polynomials over a finite field with 8 elements

4 mai 2016

Let F be a finite field with 8 elements. We study representations of polynomials over F as sums and strict sums of seventh powers.

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### [hal-01311367] Sums of seventh powers in the ring of polynomials over the finite field with four elements

4 mai 2016

We study representations of polynomials P in F4[T] as sums P = X17 + . . . + Xs7.

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### [hal-01311352] Waring's problem for polynomials in two variables

4 mai 2016

We prove that all polynomials in several variables can be decomposed as the sums of $k$th powers: $P(x_1,...,x_n) = Q_1(x_1,...,x_n)^k+...+ Q_s(x_1,...,x_n)^k$, provided that elements of the base field are themselves sums of $k$th powers. We also give bounds for the number of terms $s$ and the (...)

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### [hal-01311343] Sums of $(p^r+1)$-th powers in the polynomial ring $\Bbb F_{p^m}[T]$

4 mai 2016

Let p be an odd prime number and let F be a finite field with p(m) elements. We study representations and strict representations of polynomials M epsilon F[T] by sums of (p(r) + 1)-th powers. A representation M = M-1(k) + ... + M-s(k) of M epsilon F[T] as a sum of k-th powers of polynomials is (...)

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### [hal-01311330] On Murata densities

4 mai 2016

In this paper, we set up an abstract theory of Murata densities, well tailored to general arithmetical semigroups. In [On certain densities of sets of primes, Proc. Japan Acad. Ser. A Math. Sci. 56(7) (1980) 351-353; On some fundamental relations among certain asymptotic densities, Math. Rep. (...)

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### [hal-01311322] Sums of seventh powers in the polynomial ring $\mathbb{F}_{2^{m}}[T]$

4 mai 2016

Let F be a finite field with even characteristic and q >= 16 elements. We study representations of polynomials P is an element of F [T] as sums P = X-1(7) + ... + X-s(7).

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### [hal-01297936] Répartition des fonctions Q-additives le long des carrés de polynômes sur un corps fini

4 mai 2016

Soit F un corps fini à q éléments (q impair), Q F [T ] et f une fonction complètement Q-additive à valeurs entières. L'objet de ce travail est l'étude des sommes d'exponentielles _XF[T]X

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### [hal-01272342] Sur les puissances des polynômes sur un corps fini

4 mai 2016

Let F a finite field, Q a given element of F[T] with positive degree and k a positive integer. The goal of this work is to study the Q-automaticity of the set of kth powers of elements of F[T] and to calculate the number of polynomials X ∈ F[T] of degree N such that the sum of digits of Xk in (...)

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### [hal-01309134] Are monochromatic Pythagorean triples avoidable?

4 mai 2016

A Pythagorean triple is a triple of positive integers a, b, c ∈ N-0 satisfying a^2 + b^2 = c^2. Is it true that, for any finite coloring of N-0, at least one Pythagorean triple must be monochromatic? In other words, is the Diophantine equation X^2 +Y^2 = Z^2 regular? This problem has been open (...)

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### [hal-01311138] Weak Néron models for cubic polynomial maps over a non-Archimedean field

4 mai 2016

The aim of this note is to give an effective criterion to verify whether a cubic polynomial over a non-Archimedean field has a weak N\'eron model or not.

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### [tel-01310561] Le théorème de Gauss sur les sommes de 3 carrés, de faisceaux, et composition de Gauss

4 mai 2016

Le théorème de Gauss sur les sommes de 3 carrés relie le nombre de points entiers primitifs sur la sphère de rayon la racine carrée de n au nombre de classes d'un ordre quadratique imaginaire. En 2011, Edixhoven a esquissée une preuve du théorème de Gauss en utilisant une approche de la géométrie (...)

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### [hal-00995770] Inverse binomial series and values of Arakawa-Kaneko zeta functions

3 mai 2016

In this article, we present a variety of evaluations of series of polylogarithmic nature. More precisely, we express the special values at positive integers of two classes of zeta functions of Arakawa-Kaneko-type by means of certain inverse binomial series involving harmonic sums which appeared (...)

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### [hal-00604393] Le produit harmonique des suites

3 mai 2016

Au moyen d'une transformation binomiale involutive dans l'espace des suites à valeurs complexes, on définit un nouveau produit dénommé " harmonique " en raison de ses remarquables propriétés à l'égard des sommes harmoniques. La transformation d'Euler des séries permet de déduire de ces (...)

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### [hal-00495767] A new class of identities involving Cauchy numbers, harmonic numbers and zeta values

3 mai 2016

Improving an old idea of Hermite, we associate to each natural number $k$ a modified zeta function of order $k$. The evaluation of the values of these functions $F_k$ at positive integers reveals a wide class of identities linking Cauchy numbers, harmonic numbers and zeta (...)

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### [hal-00499548] Relations entre les sommes d'Euler de poids fixé et les valeurs de zêta

3 mai 2016

Dans cet article, on montre comment déduire d'une remarquable formule sommatoire d'Ohno une relation non-triviale entre les sommes d'Euler de poids fixé $k$ et $\zeta(k)$. In this article, we show how to derive from a deep sum formula of Ohno a non-trivial relation between Euler sums of fixed (...)

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### [hal-01202725] Produit harmonique, sommation de Ramanujan et fonctions zêta d'Arakawa-Kaneko

3 mai 2016

Ce mémoire s'articule autour des trois thèmes principaux auxquels j'ai consacré mes recherches au cours de ces 15 dernières années, thèmes qui sont assez étroitement reliés entre-eux comme je m'attacherai à le montrer. Il s'agit du produit harmonique, du procédé de sommation de Ramanujan et de la (...)

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### [inria-00395839] Noether's forms for the study of non-composite rational functions and their spectrum

3 mai 2016

In this paper, the spectrum and the decomposability of a multivariate rational function are studied by means of the effective Noether's irreducibility theorem given by Ruppert. With this approach, some new effective results are obtained. In particular, we show that the reduction modulo p of the (...)

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### [hal-01226725] Sur le radical kummérien des Zl-extensions

22 avril 2016

Sur la base d'un travail antérieur, nous donnons une description nouvelle du radical iniial attaché au compositum des Zl-extensions d'un corps de nombres en termes de limites projectives pour la norme dans la Zl-extension cyclotomique de ce corps. Le résultat précis que nous obtenons contient, (...)

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### [hal-01280172] On Fourier coefficients of modular forms of half integral weight at squarefree integers

21 avril 2016

We show that the Dirichlet series associated to the Fourier coefficients of a half-integral weight Hecke eigenform at squarefree integers extends analytically to a holomorphic function in the half-plane $\re s>\tfrac12$. This exhibits a high fluctuation of the coefficients at squarefree (...)

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### [hal-00131641] Points de hauteur bornee sur les varietes de drapeaux en caracteristique finie

20 avril 2016

The aim of this paper is to apply the work of Morris on Eisenstein series over global function fields to the study of the asymptotic behavior of the points of bounded height on a generalized flag variety defined as the quotient of a semi-simple algebraic group by a reduced parabolic subgroup (...)

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### [hal-01304309] Ramification in Iwasawa Theory and Splitting Conjectures

19 avril 2016

We make a reciprocity conjecture that extends Iwasawa's analogy of direct limits of class groups along the cyclotomic tower of a totally real number field F to torsion points of Jacobians of curves over finite fields. The extension is to generalized class groups and generalized Jacobians. We (...)

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### [hal-01263097] A NOTE ON MULTIPLE ZETA VALUES IN TATE ALGEBRAS

19 avril 2016

In this note, we shall discuss a generalization of Thakur's multiple zeta values and allied objects, in the framework of function fields of positive characteristic and more precisely, of periods in Tate algebras.

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

19 avril 2016

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

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### [hal-01277623] On shifted Mascheroni Series and hyperharmonic numbers

17 avril 2016

In this article, we study the nature of the forward shifted series σ r = n>r |bn| n−r where r is a positive integer and b n are Bernoulli numbers of the second kind, expressing them in terms of the derivatives ζ (−k) of zeta at the negative integers and Euler's constant γ. These expressions may (...)

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### [hal-01302600] On the average distribution of divisors of friable numbers

16 avril 2016

A number is said to be y-friable if it has no prime factor greater than y. In this paper, we prove a central limit theorem on average for the distribution of divisors of y-friable numbers less than x, for all (x, y) satisfying 2 ≤ y ≤ e (log x)/(log log x) 1+ε. This was previously known under the (...)

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### [hal-01302604] Sums of Kloosterman sums in arithmetic progressions, and the error term in the dispersion method

16 avril 2016

We prove a bound for quintilinear sums of Kloosterman sums, with congruence conditions on the "smooth" summation variables. This generalizes classical work of Deshouillers and Iwaniec, and is key to obtaining power-saving error terms in applications, notably the dispersion method. As (...)

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

16 avril 2016

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

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

16 avril 2016

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

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### [hal-00771342] Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group

15 avril 2016

This paper is devoted to the study of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism groups are cyclic of order coprime to the characteristic of their ground field. We give an explicit and effectively computable description of this (...)

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15 avril 2016

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### [hal-00911140] Improvements on the accelerated integer GCD algorithm

29 mars 2016

The present paper analyses and presents several improvements to the algorithm for finding the $(a,b)$-pairs of integers used in the $k$-ary reduction of the right-shift $k$-ary integer GCD algorithm. While the worst-case complexity of Weber's ''Accelerated integer GCD algorithm'' is (...)

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### [hal-00689464] Some remarks on barycentric-sum problems over cyclic groups

29 mars 2016

We derive some new results on the k-th barycentric Olson constants of abelian groups (mainly cyclic). This quantity, for a finite abelian (additive) group (G,+), is defined as the smallest integer l such that each subset A of G with at least l elements contains a subset with k elements g_1, ... (...)

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### [hal-01292805] SUMS OF TWO S-UNITS VIA FREY-HELLEGOUARCH CURVES

25 mars 2016

In this paper, we develop a new method for finding all perfect powers which can be expressed as the sum of two rational S-units, where S is a finite set of primes. Our approach is based upon the modularity of Galois representations and, for the most part, does not require lower bounds for (...)

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### [hal-01293564] Vectorial Drinfeld modular forms over Tate algebras

25 mars 2016

In this text, we develop the theory of vectorial modular forms with values in Tate algebras introduced by the first author, in a very special case (dimension two, for a very particular representation of Γ := GL 2 (Fq[θ])). Among several results that we prove here, we determine the complete (...)

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

25 mars 2016

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

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### [hal-01293687] On a torsor of paths of an elliptic curve minus a point

25 mars 2016

We are studying some aspects of the action of the Galois groups on torsors of paths on an elliptic curve minus a point. We construct objects whose behaviour is similar to the classical polylogarithms on the projective line minus three (...)

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### [hal-01293667] Non-abelian unipotent periods and monodromy of iterated integrals. (2003) 2(1),

25 mars 2016

In this note we are studying the Lie algebras associated to non-abelian unipotent periods on P1ℚ(μn)∖0,μn,∞. Let n be a prime number. We assume that for any m≥1 the numbers Lim+1(ξkn) for 1≤k≤(n−1)/2 are linearly independent over ℚ in ℂ/(2π\ri)m+1ℚ. Let S=k1,⋯,kq be a subset of 1,…,p−1 such that if k∈S, (...)

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25 mars 2016

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25 mars 2016

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25 mars 2016

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### [hal-01293611] Cosimplicial objects in algebraic geometry ,

25 mars 2016

Let X be an arc-connected and locally arc-connected topological space and let I be the unit interval. Applying the connected component functor to each fibre of the fibration of the total space map(I, X) over X × X, P(w) = (w(0), w(1)), we get a local system of sets (Poincaré groupoid) over X × X. (...)

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### [hal-01293604] On functional equations of p-adic polylogarithms. Bull. Soc. Math. France , 119 (1991)

25 mars 2016

ABSTRACT . — The n-t h order polylogarithm Ln(z) i s defined by th e series E^Ll zk ''t^ n on tn e Q? 611 un it disc. This functio n has multivalued analytic prolon- gation t o C \ 0,1 . The same series ^^ z k f^ defines an analytic p-adic functio n on th e open uni t disc in Cp (a completion o (...)

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25 mars 2016

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24 mars 2016

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24 mars 2016

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24 mars 2016

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24 mars 2016

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### [hal-01292754] Milnor K-theory and the graded representation ring

24 mars 2016

Let F be a field, let G = Gal(¯ F /F) be its absolute Galois group, and let R(G, k) be the representation ring of G over a suitable field k. In this preprint we construct a ring homomorphism from the mod 2 Milnor K-theory k * (F) to the graded ring gr R(G, k) associated to Grothendieck's (...)

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### [hal-01293244] Central extentions and coverings .Publications Universitat Autonoma de Barcelona. vol. 29 (1985), 145-153.

24 mars 2016

The theory of central extensions has a lot of analogy with the theory of covering spaces. In this paper we show that the category of central extensions of a perfect group and a certain category of covering spaces are equivalent. Then the facts about central extensions will follow from the (...)

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24 mars 2016

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24 mars 2016

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

24 mars 2016

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

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### [hal-00371233] On the multiplicative order of $a^n$ modulo $n$

24 mars 2016

Let $n$ be a positive integer and $\alpha_n$ be the arithmetic function which assigns the multiplicative order of $a^n$ modulo $n$ to every integer $a$ coprime to $n$ and vanishes elsewhere. Similarly, let $\beta_n$ assign the projective multiplicative order of $a^n$ modulo $n$ to every integer (...)

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### [hal-01291833] The Fourier expansion of $\eta(z)\eta(2z)\eta(3z)/\eta(6z)$

23 mars 2016

We compute the Fourier coefficients of the weight one modular form $\eta(z)\eta(2z)\eta(3z)/\eta(6z)$ in terms of the number of representations of an integer as a sum of two squares. We deduce a relation between this modular form and translates of the modular form (...)

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### [hal-00371228] On a problem of Molluzzo concerning Steinhaus triangles in finite cyclic groups

23 mars 2016

Let $X$ be a finite sequence of length $m\geqslant 1$ in $\mathbbZ/n\mathbbZ$. The derived sequence $\partial X$ of $X$ is the sequence of length $m-1$ obtained by pairwise adding consecutive terms of $X$. The collection of iterated derived sequences of $X$, until length $1$ is reached, (...)

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

22 mars 2016

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

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

22 mars 2016

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

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