ZETA FUNCTIONS AND IDEAL CLASSES OF QUATERNION ORDERS

摘要

Inspired by Starks analytic proof of the finiteness of the class number 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 quaternion division algebra over a totally real number field. Previous proofs of this well-known result have used adeles or geometry of numbers, while our proof uses the classical analytic theory of zeta functions. We also note that this approach leads to alternative proofs of Eichlers mass formula and the even parity of the number of ramified primes in the quaternion algebra.