Automorphic functions on moduli spaces of bundles on curves over local fields: a survey

TL;DR

This survey reviews recent advances and conjectures in the theory of automorphic functions on moduli spaces of bundles over curves defined over local fields, highlighting analogies with classical automorphic theory.

Contribution

It compiles and discusses recent results and conjectures on automorphic functions in the context of moduli spaces over local fields, connecting to joint works of Kazhdan and collaborators.

Findings

Recent results on automorphic functions on moduli spaces

Conjectures for constructing analogs of classical automorphic theory

Connections between local field settings and moduli space geometry

Abstract

This paper is the written version of D.Kazhdan's plenary talk at ICM 2022. It is dedicated to an exposition of recent results and (mostly) conjectures attempting to construct an analog of the theory of automorphic functions on moduli spaces of bundles on curves over local fields (both archimedian and non-archimedian). The talk is based on joint works of D.Kazhdan with A.Braverman, P.Etingof, E.Frenkel and A.Polishchuk.