Twisted zeta functions of quaternion orders

Abstract

Given an abelian Galois extension K/F of number fields, a quaternion algebra A over F that is ramified at all infinite primes, and a character χ of the Galois group of K over F, we consider the twist of the zeta function of A by the character χ. We show that such twisted zeta functions provide a factorization of the zeta function of A(K)=A⊗FK. Also, the quotient of the zeta function for A(K) by the zeta function for A is entire when K/F is Galois of odd degree, not necessarily abelian. Examples show that the hypothesis of odd degree here may be relaxed, but not eliminated. So the analogue of the Aramata–Brauer holomorphy theorem in this situation requires some restriction. A key step in the proof of this analogue for K/F Galois of odd degree consists in showing that if a finite prime of K contains a ramified prime for A over F, then it is ramified for A(K) over K.