The notion of $A$-motives  is a higher-dimensional generalization of Drinfeld modules and $\mathfrak{p}$-adic  representations attached to them are one of the most important objects in function field arithmetic.

In this talk, we give a criterion  for two congruent $\mathfrak{p}$-adic representations  arising from $A$-motives defined over a global function field to be isomorphic up to semi-simplification when they are restricted to a decomposition group, in terms of Hodge-Pink theory.

This is a function field analogue of the criterion for $\ell$-adic representations given by Ozeki and  Taguchi.