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

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 $\mathrm{\ell }$-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.