Zeta functions and ideal classes of quaternion orders

Abstract

Inspired by Stark's analytic proof of class number finiteness of a ring of integers in an algebraic number field, we give a new proof of the finiteness of the number of classes of ideals in a maximal order of a totally definite quaternion algebra over a totally real number field. Our proof makes use of Epstein zeta function properties. This approach leads to alternative proofs of Eichler's mass formula and even parity of the number of ramified primes in the quaternion algebra.