# GDR STN

Site du GDR Structuration de la théorie des nombres

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

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

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

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

## Nouveaux articles en théorie des nombres

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

25 mars 2016

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

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

25 mars 2016

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

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

24 mars 2016

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

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

24 mars 2016

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

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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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