Supersingular abelian surfaces and Eichler class number formula

Abstract

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

29 pages, 3 numerical tables, shortened revised version with same results, Sections 7-9 of v2 are removed