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July 2022 Supersingular loci of low dimensions and parahoric subgroups
Tomoyoshi Ibukiyama
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Osaka J. Math. 59(3): 703-726 (July 2022).

Abstract

The theory of polarized supersingular abelian varieties $(A,\lambda)$ is often essentially related to the theory of quaternion hermitian lattices. In this paper, we add one more such relation by giving an adelic description of supersingular loci of low dimensions. Katsura, Li and Oort have shown that the supersingular locus in the moduli of principally polarized abelian varieties is not irreducible and that the number of its irreducible components is equal to the class number of certain maximal quaternion hermitian lattices. Now the locus has certain natural algebraic subsets consisting of $A$ with fixed \textit{$a$-numbers} that are defined as the dimensions of embeddings from $\alpha_p$ to $A$. For low dimensional cases when $\dim A\leq 3$, we describe configuration of these subsets in the locus by intersection properties of some adelic subgroups of quaternion hermitian groups corresponding to parahoric subgroups locally at characteristic $p$. In particular, we describe which components each superspecial point lies on. This is proved by using certain liftability property of isogenies of abelian varieties, where the isogenies are interpreted to cosets of $GL_n$ and of parahoric subgroups of the quaternion hermitian groups acting on quaternion hermitian matrices.

Funding Statement

This work was supported by JSPS KAKENHI Grant Numbers JP19K03424 and JP20H00115.

Acknowledgments

The author would like to express his sincere gratitude to Professor Chia-Fu Yu for answering a lot of his questions on known facts on geometry and for suggestion of references. He also thanks Academia Sinica for their kind hospitality during he was preparing the paper. He also thanks Professor Frans Oort for a lot of discussions and advices in an early stage when he started this project a long time ago. He also thanks the anonymous referee for the careful reading and fruitful suggestions.

Citation

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Tomoyoshi Ibukiyama. "Supersingular loci of low dimensions and parahoric subgroups." Osaka J. Math. 59 (3) 703 - 726, July 2022.

Information

Received: 18 January 2021; Revised: 11 June 2021; Published: July 2022
First available in Project Euclid: 23 June 2022

Subjects:
Primary: 14K10
Secondary: 11E41 , 11G10 , 20G25

Rights: Copyright © 2022 Osaka University and Osaka City University, Departments of Mathematics

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Vol.59 • No. 3 • July 2022
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