Finiteness of local torsion for abelian t-modules

TL;DR

This paper proves that for abelian A-modules over local fields derived from a function field, the torsion submodule of rational points is always finite, extending understanding of torsion structures in positive characteristic.

Contribution

It establishes the finiteness of the local torsion submodule for abelian A-modules over local fields, a new result in the theory of Anderson modules.

Findings

The torsion submodule M(L_v)_{tors} is finite.

The result applies to abelian A-modules over local fields.

Supports the broader understanding of torsion in positive characteristic.

Abstract

Let $C/\mathbb{F}_q$ be a regular projective curve, $\infty \in C$ a closed point, $A := \Gamma(C - \{\infty\}, \mathcal{O}_C)$, and $K := K(C)$ the fraction field of $A$. Consider a finite extension $L/K$, a place $v$ of $L$, and an abelian $A$-module $M$ (in the sense of Anderson) over $L_v$. We prove that the $L_v$-rational torsion submodule $M(L_v)_{\mathrm{tors}}$ of $M$ is a finite $A$-module.