arXiv subjects
Publications and source records attributed to Emmanuel Royer.
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.
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.
We provide a systematic method to compute arithmetic sums including some previously computed by Alaca, 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 andZagier.
We prove formulae for the countings by orbit of square-tiled surfaces of genus two with one singularity. These formulae were conjectured by Hubert & Lelièvre. We show that these countings admit quasimodular forms as generating functions.