Torsion in abelian fundamental group and its application

TL;DR

The paper proves finiteness of the torsion in the abelianized étale fundamental group $\pi^{\mathrm{ab}}_1(X)$ for regular geometrically integral projective varieties over local fields of positive characteristic, and analyzes the $p$-primary and prime-to-$p$ components via a two-step strategy. It develops a structure theorem for $V(X/k)$ in terms of $SK_1(X)$, and uses alterations to extend finiteness results from smooth cases to regular (not necessarily smooth) varieties. These results feed into unramified class field theory for regular curves, establishing dualities and injectivity of reciprocity maps both in the prime-to-$p$ and $p$-primary settings. The work also provides a detailed duality framework for curves through trace maps and logarithmic de Rham–Walo (or related) cohomology, connecting local reciprocity with global SK$_1$-to-$\pi^{\mathrm{ab}}_1$ maps. Overall, the paper extends geometric class field theory to regular projective varieties over local fields in positive characteristic, with concrete consequences for curves and SK$_1$-theory.

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.

Theorems & Definitions (48)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Corollary 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 3.1