Torsion in abelian fundamental group and its application

TL;DR

This paper proves the finiteness of the torsion subgroup of the abelian fundamental group for certain varieties over local fields and explores related algebraic structures, leading to advances in class field theory.

Contribution

It establishes the finiteness of the torsion subgroup of the abelian fundamental group and analyzes the structure of $SK_1(X)$ for varieties over local fields, applying these results to class field theory.

Findings

Finiteness of the torsion subgroup of the abelian fundamental group.

Structural insights into $SK_1(X)$ for regular projective varieties.

Application to class field theory for curves over local fields.

Abstract

We prove that the torsion subgroup of the abelian fundamental group is finite for a regular geometrically integral projective variety over a local field. We also study the structure of $SK_1(X)$ for a regular projective variety $X$ over a local field. As an application, we get class field theory for regular projective curves over local fields.