Supersingular abelian surfaces and Eichler’s class number formula

Abstract

Let F be a totally real field with ring of integers OF , and D be a totally definite quaternion algebra over F . A well-known formula established by Eichler and then extended by Korner computes the class number of any OF -order in D. In this paper we generalize the Eichler class number formula so that it works for arbitrary Z-orders in D. Our motivation is to count the isomorphism classes of supersingular abelian surfaces in a simple isogeny class over a finite prime field Fp. We give explicit formulas for the number of these isomorphism classes for all primes p.