On superspecial abelian surfaces over finite fields II

Abstract

Extending the results of the current authors [Doc. Math., 21 (2016), 1607–1643] and [Asian J. Math. to appear, arXiv:1404.2978], we calculated explicitly the number of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field of odd degree over the prime field $\mathbb{F}_p$. A key step was to reduce the calculation to the prime field case, and we calculated the number of isomorphism classes in each isogeny class through a concrete lattice description. In the present paper we treat the even degree case by a different method. We first translate the problem by Galois cohomology into a seemingly unrelated problem of computing conjugacy classes of elements of finite order in arithmetic subgroups, which is of independent interest. We then explain how to calculate the number of these classes for the arithmetic subgroups concerned, and complete the computation in the case of rank two. This complements our earlier results and completes the explicit calculation of superspecial abelian surfaces over finite fields.