On superspecial abelian surfaces over finite fields III

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 $|{\mathrm{SSp}}_2({\mathbb{F}}_q)|$ 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 étale 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 $|{\mathrm{SSp}}_2({\mathbb{F}}_q)|$ 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.

21 pages