2021/02/02 by Xue, Jiangwei, Yu, Chia-Fu, Zheng, Yuqiang
#11G10 #11R52 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2102.01296
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 | SSp2(\mathbbFq)| of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field \mathbbFq of even degree over the prime field \mathbbFp. There it was assumed that certain commutative ℤp-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 | SSp2(\mathbbFq)| 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.