Asian Journal of Mathematics

Volume 23 (2019)

Number 4

Supersingular abelian surfaces and Eichler’s class number formula

Pages: 651 – 680

DOI: https://dx.doi.org/10.4310/AJM.2019.v23.n4.a6

Authors

Jiangwei Xue (Collaborative Innovation Centre of Mathematics, School of Mathematics and Statistics, Wuhan University, Luojiashan, Wuhan, Hubei, China)

Tse-Chung Yang (Institute of Mathematics, Academia Sinica, Taipei, Taiwan)

Chia-Fu Yu (Institute of Mathematics, Academia Sinica and NCTS, Taipei, Taiwan)

Abstract

In [Ann. Sci. École Norm. Sup. (4), 1969], Waterhouse classified simple abelian varieties over a prime field $\mathbb{F}_p$ in terms of lattices, except for the isogeny class that corresponds to the conjugacy class of Weil numbers $\pm \sqrt{p}$. He gave a description only for those with maximal endomorphism rings in this isogeny class, and suggested to apply Eichler’s trace formula to compute the number of them. The main result of this paper gives an explicit formula for the number of isomorphism classes in this isogeny class, generalizing the classical formula for supersingular elliptic curves by Eichler and Deuring. To achieve this, we give a self-contained treatment of Eichler’s trace formula for an arbitrary $\mathbb{Z}$-order in any totally definite quaternion algebra.

Keywords

supersingular abelian surfaces, class number formula, Brandt matrices, trace formula

2010 Mathematics Subject Classification

11G10, 11R52

Received 14 November 2016

Accepted 23 May 2018

Published 7 January 2020