Frobenius Traces for Rank-2 Drinfeld Modules, Higher-Dimensional Galois Representations, and a Strong Multiplicity One Theorem in Positive Characteristic

In this paper, we prove that if the Frobenius traces agree at all but finitely many places, then two $l$-adic Galois representations, associated to rank-$2$ non-CM Drinfeld modules of generic characteristic, are isomorphic. As a generalization, we show that the “Frobenius trace equality at all but finitely many places forces isomorphism” between two Galois representations over a local field of positive characteristic holds under an absolute irreducibility assumption. Moreover, we formulate and prove a function field analogue of strong multiplicity one property for semisimple Galois representations over a local field of positive characteristic.