Principal polarizations of supersingular abelian surfaces
type: Item
instance of: scholarly article
title: Principal polarizations of supersingular abelian surfaces
zbMATH Open document ID: 1471.14091
author: Tomoyoshi Ibukiyama
published in: Journal of the Mathematical Society of Japan
publication date: 2021-01-04T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jmsj/1585123222
review text: Let \(A\) be a (fixed) supersingular abelian surface over a field \(k\) of characteristic \(p>0\), which is not superspecial, and let \(\overline{k}\) be an algebraic closure of \(k.\) The main problem that was dealt with in the paper under review is on the number of isomorphism classes of principal polarizations of any fixed such an abelian surface \(A\). (Recall that it was known in [the author et al., Compos. Math. 57, 127--152 (1986; Zbl 0589.14028)] that the number of principal polarizations of \(E^n\) (\(n \geq 2\)) for a supersingular elliptic curve \(E\) is equal to the class number of quaternion hermitian lattices of rank \(n\) that belong to the principal genus, and explicit such values were obtained for \(n=2\) and for \(n=3\) in other papers.) In this regard, the first main result of this paper is an explicit formula for the numbers of principal polarizations of \(A\) up to \(\overline{k}\)-isomorphism. More precisely, the number of principal polarizations of \(A\) up to \(\Aut(A)\) was obtained depending on two factors: i) the condition whether the tangent of the embedding \(\iota : \alpha_p \rightarrow \alpha_p^2\), which gives an isomorphism \(A \cong E^2 / \iota (\alpha_p )\) lies outside \(\mathbb{F}_{p^4}\) (called Case I) or is an element of \(\mathbb{F}_{p^4} \setminus \mathbb{F}_{p^2}\) (called Case II), and ii) of course, the value of the characteristic \(p>0\). (For a concrete description, see Theorem 1.1.) Another interesting result that the author gave was on the description of the automorphism group \(\Aut(A,\lambda)\) of a principally polarized abelian surface \((A, \lambda)\). In Case I, the cardinality of the group \(\Aut(A, \lambda)\) (with the unique principal polarization \(\lambda\) on \(A\)) is equal to \(32\) if \(p=2,\) and equals \(2\) (that is, \(\Aut(A, \lambda) = \{ \pm 1 \})\) for any principal polarization \(\lambda\) on \(A\) if \(p \geq 3\). Similarly, in Case II, the cardinality of the group \(\Aut(A, \lambda)\) (with the…
reviewed by: WonTae Hwang
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 11R52
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 7290145
zbMATH Keywords: abelian variety
zbMATH Keywords: supersingular
zbMATH Keywords: polarization
zbMATH Keywords: quaternion
MaRDI profile type: Publication
OpenAlex ID: W3013481336
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DOI: 10.2969/JMSJ/82528252
description: scientific article; zbMATH DE number 7290145