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

TL;DR

This paper establishes that Frobenius trace agreement at almost all places implies isomorphism of certain Galois representations in positive characteristic, generalizing multiplicity one results for function fields.

Contribution

It proves a Frobenius trace-based isomorphism criterion for rank-2 Drinfeld modules and Galois representations, extending strong multiplicity one principles in positive characteristic.

Findings

Frobenius trace equality at all but finitely many places implies Galois representation isomorphism.

Generalization to Galois representations over local fields of positive characteristic.

Formulation and proof of a function field analogue of strong multiplicity one property.

Abstract

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.