Motives and Automorphic Representations

TL;DR

This paper investigates the deep connections between algebraic geometry and automorphic representation theory, aiming to construct universal groups that unify these mathematical frameworks and elucidate their fundamental duality.

Contribution

It introduces conjectural constructions of universal groups linking algebraic geometry and automorphic representations, advancing understanding of their underlying duality.

Findings

Proposes explicit conjectural models for universal groups.

Explores the duality between geometric and spectral objects.

Lays groundwork for future unification of theories.

Abstract

This paper explores relations between two separate worlds. They are the algebraic geometry of Alexander Grothendieck and the automorphic representation theory of Robert Langlands. The relation between them would be a very broad example of the fundamental duality between geometric objects and spectral objects that permeates much of modern mathematics. Our goal is to introduce explicit conjectural constructions of the universal groups that govern much of these theories, and to explore the relations between them.