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October, 2020 Principal polarizations of supersingular abelian surfaces
Tomoyoshi IBUKIYAMA
J. Math. Soc. Japan 72(4): 1161-1180 (October, 2020). DOI: 10.2969/jmsj/82528252

Abstract

We consider supersingular abelian surfaces $A$ over a field $k$ of characteristic $p$ which are not superspecial. For any such fixed $A$, we give an explicit formula of numbers of principal polarizations $\lambda$ of $A$ up to isomorphisms over the algebraic closure of $k$. We also determine all the automorphism groups of $(A, \lambda)$ over algebraically closed field explicitly for every prime $p$. When $p \geq 5$, any automorphism group of $(A,\lambda)$ is either $\mathbb{Z}/2\mathbb{Z} = \{\pm 1\}$ or $\mathbb{Z}/10\mathbb{Z}$. When $p=2$ or 3, it is a little more complicated but explicitly given. The number of principal polarizations having such automorphism groups is counted exactly. In particular, for any odd prime $p$, we prove that the automorphism group of any generic $(A, \lambda)$ is $\{\pm 1\}$. This is a part of a conjecture by Oort that the automorphism group of any generic principally polarized supersingular abelian variety should be $\{\pm 1\}$. On the other hand, we prove that the conjecture is false for $p=2$ in case of dimension two by showing that the automorphism group of any $(A, \lambda)$ (with $\dim A = 2$) is never equal to $\{\pm 1\}$.

Funding Statement

This work was supported by JSPS KAKENHI JP25247001 and JP19K03424.

Citation

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Tomoyoshi IBUKIYAMA. "Principal polarizations of supersingular abelian surfaces." J. Math. Soc. Japan 72 (4) 1161 - 1180, October, 2020. https://doi.org/10.2969/jmsj/82528252

Information

Received: 2 May 2019; Published: October, 2020
First available in Project Euclid: 25 March 2020

MathSciNet: MR4165927
Digital Object Identifier: 10.2969/jmsj/82528252

Subjects:
Primary: 14K15
Secondary: 11R52, 11R58, 14K10

Rights: Copyright © 2020 Mathematical Society of Japan

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Vol.72 • No. 4 • October, 2020
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