Congruences of Galois representations attached to effective $A$-motives over global function fields

TL;DR

This paper establishes a criterion for when congruent $rak{p}$-adic Galois representations from effective $A$-motives over global function fields are isomorphic up to semi-simplification, extending number field analogs and exploring non-existence results.

Contribution

It provides a new criterion for congruences of $A$-motivic Galois representations over function fields, paralleling number field results and addressing non-existence conjectures.

Findings

Criterion for isomorphism of congruent $A$-motivic Galois representations

Extension of Ozeki-Taguchi's criterion to function fields

Non-existence of certain strongly semi-stable effective $A$-motives

Abstract

This article investigates congruences of $\mathfrak{p}$-adic representations arising from effective $A$-motives defined over a global function field $K$. We give a criterion for two congruent $\mathfrak{p}$-adic representations coming from strongly semi-stable effective $A$-motives to be isomorphic up to semi-simplification when restricted to decomposition groups of suitable places of $K$. This is a function field analog of Ozeki-Taguchi's criterion for $\ell$-adic representations of number fields. Motivated by a non-existence conjecture on abelian varieties over number fields stated by Rasmussen and Tamagawa, we also show that there exist no strongly semi-stable effective $A$-motives with some constrained.