On superspecial abelian surfaces over finite fields
type: Item
instance of: scholarly article
title: On superspecial abelian surfaces over finite fields
zbMATH Open document ID: 1385.11042
author: Tse-Chung Yang
author: Chia-Fu Yu
author: Jiang Wei Xue
published in: Documenta Mathematica
publication date: 2017-01-16T00:00:00Z
full work available at URL: 41.html
review text: Let \(q = p^a\) be a prime power. An abelian variety over a field \(k\) is called superspecial if it is isomorphic to a product of supersingular elliptic curves over an algebraic closure of \(k\) and superspecial if it is isogenous to a product of supersingular elliptic curves over an algebraic closure of \(k\). The authors denote the set of isomorphism classes of \(d\)-dimensional superspecial abelian varieties over \(\mathbb{F}_q\) by \(\mathrm{Sp}_d(\mathbb{F}_q)\). The purpose of this article is to determine \(\mathrm{Sp}_d(\mathbb{F}_q)\) for \(d=1\) and \(d=2\) and \(a\) odd. Furthermore, it is proved that for \(q, q'\) powers of \(p\) of the same exponent parity, there is a natural bijection \(\mathrm{Sp}_d(\mathbb{F}_q) \cong \mathrm{Sp}_d(\mathbb{F}_{q'})\) preserving isogeny classes. This allows to reduce the case \(a\) odd to \(q = p\) using Galois cohomology; Proposition 5.1 only works for \(q = p\). \[ \mathrm{Sp}_d(\mathbb{F}_p) \cong \mathrm{H}^1_{\text{ét}}(\mathbb{F}_p, \mathrm{Aut}(X_0 \otimes \overline{\mathbb{F}}_p)), \quad d > 1, \] where \(X_0\) is a fixed \(d\)-dimensional supersingular abelian variety over \(\mathbb{F}_p\). The method of computing the cardinality of \(\mathrm{Sp}_d(\mathbb{F}_p)\) is as follows: One divides the set of isomorphism classes of superspecial abelian varieties into isogeny classes and classifies them. Isogeny classes of supersingular elliptic curves are in bijection with supersingular Weil \(p\)-numbers by Honda-Tate theory (these have been studied by Deuring, see [\textit{W. C. Waterhouse}, Ann. Sci. Éc. Norm. Supér. (4) 2, 521--560 (1969; Zbl 0188.53001), Section 4], so isogeny classes of superspecial abelian varieties are in bijection with supersingular multiple Weil \(p\)-numbers. Proposition 4.4. Denote the class number of \(\mathbb{Q}(\sqrt{m})\) by \(h(\sqrt{m})\). (1) If \(a\) is odd, \[ |\mathrm{Sp}_1(\mathbb{F}_q)| = \begin{cases} 3, & p=2, \\ 4, & p=3, \\ h(\sqrt{-p}), &p \equiv 1 \pmod{4},\\ \big(3-(\fr…
reviewed by: Timo Keller
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11G10
zbMATH DE Number: 6674986
zbMATH Keywords: supersingular abelian surfaces
zbMATH Keywords: class number formula
zbMATH Keywords: Galois cohomology
describes a project that uses: Magma
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6674986