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January, 2020 On superspecial abelian surfaces over finite fields II
Jiangwei XUE, Tse-Chung YANG, Chia-Fu YU
J. Math. Soc. Japan 72(1): 303-331 (January, 2020). DOI: 10.2969/jmsj/81438143

Abstract

Extending the results of the current authors [Doc. Math., 21 (2016), 1607–1643] and [Asian J. Math. to appear, arXiv:1404.2978], we calculated explicitly the number of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field of odd degree over the prime field $\mathbb{F}_p$. A key step was to reduce the calculation to the prime field case, and we calculated the number of isomorphism classes in each isogeny class through a concrete lattice description. In the present paper we treat the even degree case by a different method. We first translate the problem by Galois cohomology into a seemingly unrelated problem of computing conjugacy classes of elements of finite order in arithmetic subgroups, which is of independent interest. We then explain how to calculate the number of these classes for the arithmetic subgroups concerned, and complete the computation in the case of rank two. This complements our earlier results and completes the explicit calculation of superspecial abelian surfaces over finite fields.

Funding Statement

The first author is partially supported by the Natural Science Foundation of China grant #11601395. The second and third authors are partially supported by the MoST grants 104-2115-M-001-001MY3, 104-2811-M-001-066, 105-2811-M-001-108 and 107-2115-M-001-001-MY2.

Citation

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Jiangwei XUE. Tse-Chung YANG. Chia-Fu YU. "On superspecial abelian surfaces over finite fields II." J. Math. Soc. Japan 72 (1) 303 - 331, January, 2020. https://doi.org/10.2969/jmsj/81438143

Information

Received: 9 October 2018; Published: January, 2020
First available in Project Euclid: 25 November 2019

zbMATH: 1385.11042
MathSciNet: MR4055096
Digital Object Identifier: 10.2969/jmsj/81438143

Subjects:
Primary: 11R52
Secondary: 11G10

Rights: Copyright © 2020 Mathematical Society of Japan

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Vol.72 • No. 1 • January, 2020
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