On superspecial abelian surfaces over finite fields III

TL;DR

This paper completes the enumeration of superspecial abelian surfaces over finite fields of even degree, including primes 2, 3, and 5, by addressing ramification issues and classifying lattices over quaternion Bass orders.

Contribution

It extends previous work by including the remaining primes and classifying lattices over local quaternion Bass orders, completing the enumeration in the even degree case.

Findings

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

Classification of lattices over local quaternion Bass orders.

Addresses ramification issues at primes 2, 3, and 5.

Abstract

In the paper [On superspecial abelian surfaces over finite fields II. J. Math. Soc. Japan, 72(1):303--331, 2020], Tse-Chung Yang and the first two current authors computed explicitly the number $\lvert \mathrm{SSp}_2(\mathbb{F}_q)\rvert$ of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field $\mathbb{F}_q$ of even degree over the prime field $\mathbb{F}_p$. There it was assumed that certain commutative $\mathbb{Z}_p$-orders satisfy an \'etale condition that excludes the primes $p=2, 3, 5$. We treat these remaining primes in the present paper, where the computations are more involved because of the ramifications. This completes the calculation of $\lvert \mathrm{SSp}_2(\mathbb{F}_q)\rvert$ in the even degree case. The odd degree case was previous treated by Tse-Chung Yang and the first two current authors in [On superspecial abelian surfaces over finite fields. Doc. Math., 21:1607--1643, 2016]. Along the proof of our main theorem, we give the classification of lattices over local quaternion Bass orders, which is a new input to our previous works.