Superspecial abelian varieties over finite prime fields
type: Item
instance of: scholarly article
title: Superspecial abelian varieties over finite prime fields
zbMATH Open document ID: 1255.11031
author: Chia-Fu Yu
published in: Journal of Pure and Applied Algebra
publication date: 2012-10-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1004.0120
reviewed by: Ernst-Ulrich Gekeler
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14K10
zbMATH DE Number: 6094128
zbMATH Keywords: superspecial abelian variety
zbMATH Keywords: supersingular abelian variety
zbMATH Keywords: class numbers
MaRDI profile type: Publication
OpenAlex ID: W2963429311
arXiv ID: 1004.0120
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DOI: 10.1016/J.JPAA.2011.12.007
review text: An abelian variety defined over a field \(K\) of characteristic \(p\) is called superspecial if it is isomorphic over the algebraic closure \(\overline{K}\) to a product of supersingular elliptic curves. Fix a supersingular elliptic curve \(E_0/{\mathbb F}_p\) (with an extra condition if \(p = 2\) or \(3\)).NEWLINENEWLINEIn the paper under consideration, the number of \(g\)-dimensional superspecial abelian varieties \(A\) over \({\mathbb F}_p\) is determined such that there is an \({\mathbb F}_p\)-isogeny from \(E_0^g\) to \(A\). The result, Theorem 1.1, relates the number in question with the class number of the imaginary quadratic field \({\mathbb Q}(\sqrt{-p})\), and thereby generalizes a classical result of Deuring for the case \(g=1\),NEWLINENEWLINEThe proof is by algebraic means and uses the module theory of a certain not necessarily maximal order in a semisimple commutative \({\mathbb Q}\)-algebra.NEWLINENEWLINEBesides the proof, the article presents a condensed but highly valuable discussion of similar problems: relations with weighted/unweighted class numbers, geometric and arithmetic mass formulas, supersingular instead of superspecial abelian varieties, with or without polarizations, Drinfeld modules, etc.
description: scientific article; zbMATH DE number 6094128