Non-openness of v-adic Galois representation for A-motives

TL;DR

This paper investigates the openness of the Galois representation image associated with A-motives, revealing that in rank one cases, the image is open precisely when the virtual dimension is coprime to the ground field's characteristic.

Contribution

It clarifies the conditions under which the Galois image is open for rank one A-motives, extending understanding beyond Drinfeld-modules.

Findings

Galois image is open iff virtual dimension is coprime to characteristic

Results extend known cases for Drinfeld-modules to A-motives

Provides criteria for openness in rank one A-motives

Abstract

Drinfeld-modules and $A$-motives are the function field analogous of elliptic curves and abelian varieties. For the latter one can construct the $l$-adic Galois representation and can ask if its image is open. For Drinfeld-modules this question was answered by Pink and his coauthors. We clarify the rank one case for $A$-motives and show that the image of Galois is open if and only if the virtual dimension is prime to the characteristic of the ground field.