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

We study one-level and two-level densities for low lying zeros of symmetric power L-functions in the level aspect. It 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.Comment: 45 page

Rankin-Cohen brackets on quasimodular forms

We give the algebra of quasimodular forms a collection of Rankin-Cohen operators. These operators extend those defined by Cohen on modular forms and, as for modular forms, the first of them provide a Lie structure on quasimodular forms. They also satisfy a ``Leibniz rule'' for the usual derivation. Rankin-Cohen operators are useful for proving arithmetic identities. In particular we give an interpretation of the Chazy equation and explain why such an equation has to exist.Comment: 17 page

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 & Leli\`{e}vre. We show that these countings admit quasimodular forms as generating functions.Comment: 22 pages, 6 figure

Special values of symmetric power LL-functions and Hecke eigenvalues

We compute the moments of L-functions of symmetric powers of modular forms at the edge of the critical strip, twisted by the central value of the L-functions of modular forms. We show that, in the case of even powers, it is equivalent to twist by the value at the edge of the critical strip of the symmetric square L-functions. We deduce information on the size of symmetric power L-functions at the edge of the critical strip under conditions. In a second part, we study the distribution of small and large Hecke eigenvalues. We deduce information on the simultaneous extremality conditions on the values of L-functions of symmetric powers of modular forms at the edge of the critical strip.Comment: 42 pages Previously circulated under the title "Central values and values at the edge of the critical strip of symmetric power L-functions and Hecke eigenvalues

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

International audienceWe study one-level and two-level densities for low lying zeros of symmetric power L-functions in the level aspect. It 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

Taille des valeurs de fonctions L de carrés symétriquez au bord de la bande critique

On prouve l'existence, pour tout poids $k$ et tout niveau $N$ sans facteur carré et sans petit facteur premier, de formes primitives $f_+$ et $f_-$ de poids $k$ et de niveau $N$ telles que $$L(1,{\rm sym}^2f_+)\gg_{k}[\log\log(3N)]^{3} \quad{\rm et}\quad L(1,{\rm sym}^2f_-)\ll_{k}[\log\log(3N)]^{-1}.$$ L'existence de ces formes est déduite d'une étude minutieuse des moments de $L(1,{\rm sym}^2f)$. Cette étude permet aussi de tra{\^\i}ter le cas des niveaux sans facteur carré mais avec petits facteurs premiers. On en déduit un contre--exemple à l'équivalence entre moyennes harmonique et naturelle

Lower order terms for the one-level densities of symmetric power LL-functions in the level aspect

International audienceIn a previous paper, the authors determined, among other things, the main terms for the one-level densities for low-lying zeros of symmetric power L-functions in the level aspect. In this paper, the lower order terms of these one-level densities are found. The combinatorial difficulties, which should arise in such context, are drastically reduced thanks to Chebyshev polynomials, which are the characters of the irreducible representations of SU(2).