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arXiv · 1404.2978

Supersingular abelian surfaces and Eichler class number formula

Jiangwei Xue·Tse-Chung Yang·Chia-Fu Yu
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Abstract

Let $F$ be a totally real field with ring of integers $O_F$, and $D$ be a totally definite quaternion algebra over $F$. A well-known formula established by Eichler and then extended by Körner computes the class number of any $O_F$-order in $D$. In this paper we generalize the Eichler class number formula so that it works for arbitrary $\mathbb{Z}$-orders in $D$. The motivation is to count the isomorphism classes of supersingular abelian surfaces in a simple isogeny class over a prime finite field $\mathbb{F}_p$. We give explicit formulas for the number of these isomorphism classes for all primes $p$.

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Jiangwei Xue, Tse-Chung Yang, Chia-Fu Yu. 2015-05-08. Supersingular abelian surfaces and Eichler class number formula. https://arxiv.org/abs/1404.2978

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