Strong local-global phenomena for Galois and automorphic representations

TL;DR

This paper surveys known and conjectured relationships between local and global properties of Galois and automorphic representations, highlighting recent work on comparing degree 2 Artin and automorphic representations at various places.

Contribution

It presents a survey of existing results and introduces new joint work on comparing degree 2 Artin and automorphic representations with potential discrepancies at infinite sets of places.

Findings

Comparison of degree 2 Artin and automorphic representations at various places

Identification of conditions under which local data determines global objects

Insights into the limitations of local-global principles in automorphic forms

Abstract

Many results are known regarding how much local information is required to determine a global object, such as a modular form, or a Galois or automorphic representation. We survey some things that are known and expected, and then explain recent joint work with Dinakar Ramakrishnan about comparing degree 2 Artin and automorphic representations which a priori may not correspond at certain infinite sets of places.