Automorphic Forms and Arithmetic

Introduction

This proposal is at the interface of number theory and automorphic forms and aims at a cross-fertilization of both areas. It focuses on explicit arithmetic problems that can be solved using the full force of the automorphic machinery. Conversely, it develops the theory of automorphic forms in higher rank using number theoretic methods.

Core Conjectures

At the core are three sets of deep and longstanding open conjectures in higher rank cases:

1. The joint equidistribution conjectures of Michel and Venkatesh concerning the distribution of pairs of shifted CM points.
2. The “beyond endoscopy” program of Langlands of identifying functional lifts by a comparison of different trace formulae.
3. The density and optimal lifting conjectures of Sarnak concerning the behaviour of exceptional eigenvalues and their arithmetic consequences.

Key Objects

The key objects are certain Shimura varieties associated with abelian varieties, higher rank Kloosterman sums, families of L-functions, trace formulae, and Hecke eigenvalues.

Objectives

By an interdisciplinary combination of analytic number theory and automorphic forms, the proposal aims at definitive progress and solutions to these three conjectures, each of which has been open for at least 15 years.