Geometrization of Trace Formulas

TL;DR

This paper initiates the development of geometric methods for analyzing trace formulas over function fields and proposes a conjectural framework for complex curves using geometric Langlands correspondence.

Contribution

It introduces the first steps toward geometric analysis of trace formulas over function fields and suggests a new conjectural framework for complex curves.

Findings

Proposed geometric approach to trace formulas over function fields.

Suggested a conjectural geometric trace formula framework for complex curves.

Connected geometric trace formulas with the categorical geometric Langlands correspondence.

Abstract

Following our joint work arXiv:1003.4578 with Robert Langlands, we make the first steps toward developing geometric methods for analyzing trace formulas in the case of the function field of a curve defined over a finite field. We also suggest a conjectural framework of geometric trace formulas for curves defined over the complex field, which exploits the categorical version of the geometric Langlands correspondence.