On superspecial abelian surfaces over finite fields II

TL;DR

This paper completes the explicit enumeration of superspecial abelian surfaces over finite fields by extending previous methods to even degree extensions using Galois cohomology and arithmetic subgroup analysis.

Contribution

It introduces a new approach for even degree cases by translating the problem into conjugacy class computations in arithmetic groups, completing prior work on odd degree cases.

Findings

Explicit count of superspecial abelian surfaces over finite fields of even degree

Development of a Galois cohomology approach for conjugacy class enumeration

Complete classification of superspecial abelian surfaces over all finite fields

Abstract

Extending the results of [Asian J. Math. 2019], in [Doc. Math. \textbf{21}, 2016] we calculated explicitly the number of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field of \textit{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 \textit{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 tocalculate 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.