Noncommutative reciprocity laws on algebraic surfaces: a case of tame ramification

TL;DR

This paper establishes non-commutative reciprocity laws on algebraic surfaces over perfect fields, linking central extensions of groups to tame ramification in the context of two-dimensional local fields.

Contribution

It proves non-commutative reciprocity laws on algebraic surfaces and connects local central extensions to tame ramification in the two-dimensional local Langlands correspondence.

Findings

Proved reciprocity laws for algebraic surfaces over perfect fields.

Constructed local central extensions related to tame ramification.

Linked global group extensions to local tame ramification cases.

Abstract

We prove non-commutative reciprocity laws on an algebraic surface defined over a perfect field. These reciprocity laws claim the splittings of some central extensions of globally constructed groups over some subgroups constructed by points or projective curves on a surface. For a two-dimensional local field with a finite last residue field the constructed local central extension is isomorphic to a central extension which comes from the case of tame ramification of the Abelian two-dimensional local Langlands correspondence suggested by M. Kapranov.