Arithmetic Central Extensions and Reciprocity Laws for Arithmetic Surface

TL;DR

This paper establishes three types of reciprocity laws for arithmetic surfaces using central extensions, adelic complexes, and intersection theory, advancing the understanding of arithmetic surface reciprocity.

Contribution

It introduces new arithmetic central extensions and applies adelic cohomology and intersection theory to prove reciprocity laws on arithmetic surfaces.

Findings

Constructed $K_2$-type central extensions around points and vertical curves.

Proved reciprocity laws using adelic complexes and intersection theory.

Developed a unified framework within arithmetic central extensions.

Abstract

Three types of reciprocity laws for arithmetic surfaces are established. For these around a point or along a vertical curve, we first construct $K_2$ type central extensions, then introduce reciprocity symbols, and finally prove the law as an application of Parshin-Beilinson's theory of adelic complex. For reciprocity law along a horizontal curve, we first introduce a new type of arithmetic central extensions, then apply our arithmetic adelic cohomology theory and arithmetic intersection theory to prove the related reciprocity law. All this can be interpreted within the framework of arithmetic central extensions. We add an appendix to deal with some basic structures of such extensions.