On Superspecial abelian surfaces over finite fields III

Abstract

In the paper (J Math Soc Jpn 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 ramification. 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 (Doc Math 21:1607–1643, 2016). To complete the proof of our main theorem, we give a classification of lattices over local quaternion Bass orders, which is a new input to our previous works.