Unramified two-dimensional Langlands correspondence

TL;DR

This paper develops an unramified Langlands correspondence for two-dimensional local fields, constructing categorical analogues of representations and establishing noncommutative reciprocity laws linking local and global structures.

Contribution

It introduces a categorical framework for the unramified Langlands correspondence in two dimensions and proves reciprocity laws for central extensions over surfaces.

Findings

Construction of a categorical analogue of principal series representations

Proof of noncommutative reciprocity laws for arithmetic and projective surfaces

Establishment of connections between local and global central extensions

Abstract

In this paper we describe the unramified Langlands correspondence for two-dimensional local fields, we construct a categorical analogue of the unramified principal series representations and study its properties. The main tool for this description is the construction of a central extension. For this (and other) central extension we prove noncommutative reciprocity laws (i.e. the splitting of the central extensions over some subgroups) for arithmetic surfaces and projective surfaces over a finite field. These reciprocity laws connect central extensions which are constructed locally and globally.