A Converse Theorem for Epsilon Factors

Abstract

We prove the following theorem: Let F be a nonarchimedean local field of characteristic zero and K a quadratic extension of F. Let S be the set of characters of K* trivial on F*. Let χ1 and χ2 be two characters of K* such that χ1∣F*=χ2∣F*≠1. Let ψ be a nontrivial additive character of F and ψK=ψ∘trK/F. If ε(χ1λ, ψK)=ε(χ2λ, ψK) for all λ∈S then χ1 and χ2 agree on all units in the ring of integers in K and on all elements of trace zero. If, in addition, the conductor of χ1∣F* is not zero then χ1=χ2.