Emmanuel Royer

EVALUATING CONVOLUTION SUMS OF THE DIVISOR FUNCTION BY QUASIMODULAR FORMS

We provide a systematic method to compute arithmetic sums including some previously computed by Alaca, Besge, Cheng, Glaisher, Huard, Lahiri, Lemire, Melfi, Ou, Ramanujan, Spearman and Williams. Our method is based on quasimodular forms. This extension of modular forms has been constructed by Kaneko and Zagier.

Statistics for low-lying zeros of symmetric power L-functions in the level aspect

Abstract We study one-level and two-level densities for low-lying zeros of symmetric power L-functions in the level aspect. This allows us to completely determine the symmetry types of some families of symmetric power L-functions with prescribed sign of functional equation. We also compute the moments of one-level density and exhibit mock-Gaussian behavior discovered by Hughes & Rudnick.

Orbitwise Countings in H(2) and Quasimodular Forms

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

Interprétation combinatoire des moments négatifs des valeurs de fonctions L au bord de la bande critique

Resume On donne une interpretation combinatoire des moments negatifs de la valeur au bord de la bande critique de fonctions L de formes modulaires de GL(2) et GL(3). On en deduit des renseignements sur la taille de ces nombres.

SIGN CHANGES IN SHORT INTERVALS OF COEFFICIENTS OF SPINOR ZETA FUNCTION OF A SIEGEL CUSP FORM OF GENUS 2

In this paper, we establish a Voronoi formula for the spinor zeta function of a Siegel cusp form of genus 2. We deduce from this formula quantitative results on the number of its positive (respectively, negative) coefficients in some short intervals.

Sign of Fourier coefficients of modular forms of half integral weight

We establish lower bounds for (i) the numbers of positive and negative terms and (ii) the number of sign changes in the sequence of Fourier coefficients at squarefree integers of a half-integral weight modular Hecke eigenform.

Poisson structures and star products on quasimodular forms

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

Fonction ζ et matrices aléatoires

Since Euler and Riemann, various links have been established between the behaviour of prime numbers and the analytical properties of the Riemann ζ function. The zeroes of ζ have a great importance that justifies their fine study. Since seventies, a rich tool has appeared for the study of zeroes: the statistical spectral properties of the unitary matrices that are a model for ζ . The aim of this survey is to explain the link between unitary matrices and ζ , and its extension to the L-functions.

Kloosterman paths of prime powers moduli

Emmanuel Kowalski and William Sawin proved, using a deep independence result of Kloosterman sheaves, that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums S(a,b0;p)/p^{1/2} converge in the sense of finite distributions to a specific random Fourier series, as a varies over (Z/pZ)^*, b0 is fixed in (Z/pz)* and p tends to infinity among the odd prime numbers. This article considers the case of S(a,b0;p^n)/p^{n/2}, as a varies over (Z/p^nZ)^*, b0 is fixed in (Z/p^nZ)^*, p tends to infinity among the odd prime numbers and n>=2 is a fixed integer. A convergence in law in the Banach space of complex-valued continuous function on [0,1] is also established, as (a,b) varies over (Z/p^nZ)*.(Z/p^nZ)*, p tends to infinity among the odd prime numbers and n>=2 is a fixed integer. This is the analogue of the result obtained by Emmanuel Kowalski and William Sawin in the prime moduli case.

THE AMPLIFICATION METHOD IN THE GL(3) HECKE ALGEBRA

This article contains all of the technical ingredients required to implement an effective, explicit and unconditional amplifier in the context of GL(3) automorphic forms. In particular, several coset decomposition computations in the GL(3) Hecke algebra are explicitly done.