Linnik-type problems for automorphic L-functions; Problemes de type Linnik pour les fonctions L de formes automorphes

TL;DR

This thesis investigates the distribution and sign changes of automorphic L-function coefficients for GL(m), providing new inequalities, estimates, and generalizations of classical theorems for higher rank groups.

Contribution

It introduces a novel Hecke-type inequality for automorphic L-functions and generalizes Selberg's normal density theorem to GL(m).

Findings

Established bounds on the first negative coefficient of automorphic L-functions.

Proved a Hecke-type inequality showing coefficients on prime powers cannot all be small.

Generalized Selberg's normal density theorem for GL(m).

Abstract

The thesis gave a fine study on the distribution of the coefficients of automorphic L-functions for GL(m) with m>1. In particular we have treated two types of problems: change of signs of these coefficients (when they are real) and their decompensation on the prime numbers. Chapter 2 is devoted to the results known concerning the first problem in the case of classical modular forms and to explain our motivation of research. In Chapter 3, we treated the case of primitive Maass forms and obtained an estimate of the subconvexity-type for the least integer n such that the n-th coefficient is negative. Chapter 4 is the main part of the thesis. For the coefficients of the L-function associated with an irreducible unitary cuspidal representation for GL(m) with m>1, we established an elegant inequality of Hecke type, which shows that these coefficients on the first m powers of an arbitrary unramified prime number cannot be small simultaneously. As an application, we obtained a polynomial type estimate for the least integer n such that the n-th coefficient is negative. In Chapters 5 and 6, we studied the long and short summations of coefficients of L-functions for GL(m) on the prime numbers to test their decompensation, respectively. In particular we gave a perfect generalization of Selberg's normal density theorem when m>1.