Two-Dimensional Local-Global Class Field Theory in Positive Characteristic

TL;DR

This paper extends class field theory for higher local fields to local-global fields on algebraic surfaces over finite fields, utilizing higher tame symbols and Witt vector methods to define reciprocity maps.

Contribution

It introduces an extension of Parshin's techniques to algebraic surfaces, broadening the scope of class field theory in positive characteristic.

Findings

Developed reciprocity maps for local-global fields on surfaces

Extended higher tame symbol methods to new geometric contexts

Enhanced understanding of ramification in algebraic surface fields

Abstract

Using the higher tame symbol and Kawada and Satake's Witt vector method, A. N. Parshin developed class field theory for higher local fields, defining reciprocity maps separately for the tamely ramified and wildly ramified cases. We extend this technique to the case of local-global fields associated to points and curves on an algebraic surface over a finite field.