Spectre automorphe des vari\'et\'es hyperboliques et applications topologiques

TL;DR

This work explores the automorphic spectrum of hyperbolic varieties, proving Selberg-type theorems for eigenvalues and introducing a novel cohomology lifting method, bridging spectral theory and differential geometry.

Contribution

It establishes new spectral bounds for hyperbolic manifolds using representation theory and introduces a novel cohomology lifting technique.

Findings

Proved Selberg-type theorems for eigenvalues of the Laplacian.

Developed a new cohomology lifting method.

Connected automorphic forms with differential geometry applications.

Abstract

This book is made of two parts. The first is concerned with the differential form spectrum of congruence hyperbolic manifolds. We prove Selberg type theorems on the first eigenvalue of the laplacian on differential forms. The method of proof is representation theoritic, we hope the different chapters may as well serve as an introduction to the modern theory of automorphic forms and its application to spectral questions. The second part of the book is of a more differential geometric flavor, a new kind of lifting of cohomology classes is proved.