Non annulation des fonctions $L$ des formes modulaires de Hilbert en le point central

TL;DR

This paper extends non-vanishing results of L-functions at the central point from classical modular forms over Q to forms over arbitrary totally real fields, using advanced analytic and automorphic techniques.

Contribution

It generalizes the non-vanishing results of L-functions at the central point to totally real fields, employing Jacquet-Langlands theory and adelization.

Findings

Unconditional density of non-vanishing L-functions matches the case over Q.

Generalized Petersson formula for totally real fields.

Controlled old forms contribution in the non-vanishing analysis.

Abstract

Birch and Swinnerton-Dyer conjecture allows for sharp estimates on the rank of certain abelian varieties defined over $ \Q$. in the case of the jacobian of the modular curves, this problem is equivalent to the estimation of the order of vanishing at 1/2 of $L$-functions of classical modular forms, and was treated, without assuming the Riemann hypothesis, by Kowalski, Michel and VanderKam. The purpose of this paper is to extend this approach in the case of an arbitrary totally real field, which necessitates an appeal of Jacquet-Langlands' theory and the adelization of the problem. To show that the $L$-function (resp. its derivative) of a positive density of forms does not vanish at 1/2, we follow Selberg's method of mollified moments (Iwaniec, Sarnak, Kowalski, Michel and VanderKam among others applied it successfully in the case of classical modular forms). We generalize the Petersson formula, and use it to estimate the first two harmonic moments, this then allows us to match the same unconditional densities as the ones proved over $\Q$ by Kowalski, Michel and VanderKam. In this setting, there is an additional term, coming from old forms, to control. Finally we convert our estimates for the harmonic moments into ones for the natural moments.