First we need a criterion for potentially good reduction in terms of Galois representation. The following is Theorem 2 from Serre & Tate’s paper Good Reduction of Abelian Varieties:
The abelian variety has potential good reduction at a place of if and only if the image by of the inertia group is finite.
Now the main result we aim to prove is the following (Corollary 1):
Suppose that the residue field is finite of characteristic , and that, for some , the image of in is abelian. Then has potential good reduction at .
First we can complete WLOG (since the Tate module of is isomorphic to that of where is base change of to the completion of at so if has potentially good reduction at so is by the above criterion). Local class field theory tells us that the image of inertia is actually a quotient of the group of units of . But structure theorem of unit groups of local fields tells us that is the product of a finite group and a pro- group. Thus the image intersect the pro- group trivially. Hence the image inject into , which is finite.