Tame class field theory over local fields

TL;DR

This paper develops a generalized tame class field theory over local fields for schemes with smooth compactifications, establishing a reciprocity homomorphism and analyzing its kernel, cokernel, and finiteness properties.

Contribution

It introduces a continuous reciprocity homomorphism for tame class groups over local fields and extends higher dimensional unramified class field theory results.

Findings

Constructed a reciprocity homomorphism from tame class group to abelian tame etale fundamental group.

Described prime-to-p parts of the kernel and cokernel of the homomorphism.

Proved a finiteness theorem for the geometric part of the abelian tame etale fundamental group.

Abstract

For a quasi-projective scheme $X$ admitting a smooth compactification over a local field of residue characteristic $p > 0$, we construct a continuous reciprocity homomorphism from a tame class group to the abelian tame etale fundamental group of $X$. We describe the prime-to-$p$ parts of its kernel and cokernel. This generalizes the higher dimensional unramified class field theory over local fields by Jannsen-Saito and Forre. We also prove a finiteness theorem for the geometric part of the abelian tame etale fundamental group, generalizing the results of Grothendieck and Yoshida for the unramified fundamental group.