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

This paper investigates congruences of [Formula: see text]-adic representations arising from effective [Formula: see text]-motives defined over a global function field [Formula: see text]. We give a criterion for two congruent [Formula: see text]-adic representations coming from strongly semi-stable effective [Formula: see text]-motives to be isomorphic up to semi-simplification when restricted to decomposition groups of suitable places of [Formula: see text]. This is a function field analog of Ozeki–Taguchi’s criterion for [Formula: see text]-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 [Formula: see text]-motives with some constrained.