Non annulation des fonctions L automorphes au point central

TL;DR

This paper investigates the non-vanishing of automorphic L-functions at the central point for primitive forms with prime power levels, extending previous results from prime and squarefree levels to more general levels.

Contribution

It generalizes non-vanishing results of automorphic L-functions to primitive forms with prime power levels, broadening understanding of level arithmetic influence.

Findings

Proves positive proportion of non-vanishing for prime power level forms

Extends previous prime and squarefree level results

Provides new insights into level structure effects

Abstract

The question about modular forms have recently received a lot of attention; concerning the non-vanishing of automorphic L-functions Michel, Kowalski and Vanderkam proved (among others results) that there's positive proportion of non-vanishing of primitives forms at the critical point. This result was proved by these authors in the prime level case; on the othe hand, Iwaniec, Luo and Sarnak showed the same result for the squarefree level case. In order to understand the influence of the arithmetic shape of level forms in their vanishing, this paper studies a generalisation to the primitives forms with prime powers level.