A converse theorem for epsilon factors

We prove the following theorem: Let F be a nonarchimedean local field of ch aracteristic zero and K a quadratic extension of F. Let S be the set of cha racters of K* trivial on F*. Let chi (1) and chi (2) be two characters of K * such that chi (1 \) (F*) = chi (2 \F*) not equal 1. Let psi be a nontrivi al additive character of F and psi (K) = psi tr (K/F). If epsilon(chi (1)la mbda, psi (K)) = epsilon(chi (2)lambda, psi (K)) for all lambda is an eleme nt of S then chi (1) and chi (2) agree on all units in the ring of integers in K and on all elements of trace zero. If, in addition, the conductor of chi (1 \F*) is not zero then chi (1) = chi (2). (C) 2001 Academic Press.