Supersingular abelian surfaces and Eichler class number formula

Xue, Jiangwei; Yang, Tse-Chung; Yu, Chia-Fu

doi:10.4310/AJM.2019.v23.n4.a6

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Supersingular abelian surfaces and Eichler class number formula

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Yu, Chia-Fu
Max Planck Institute for Mathematics, Max Planck Society;

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https://dx.doi.org/10.4310/AJM.2019.v23.n4.a6
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https://doi.org/10.48550/arXiv.1404.2978
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Citation

Xue, J., Yang, T.-C., & Yu, C.-F. (2019). Supersingular abelian surfaces and Eichler class number formula. Asian Journal of Mathematics, 23(4), 651-680. doi:10.4310/AJM.2019.v23.n4.a6.

Cite as: https://hdl.handle.net/21.11116/0000-0009-7198-C

Abstract

In [Ann. Sci. École Norm. Sup. (4), 1969], Waterhouse classified simple abelian varieties over a prime field Fp in terms of lattices, except for the isogeny class that corresponds to the conjugacy class of Weil numbers ±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 Z-order in any totally definite quaternion algebra.