Moduli of supersingular abelian varieties
type: Item
instance of: scholarly article
title: Moduli of supersingular abelian varieties
zbMATH Open document ID: 0920.14021
author: Kezheng Li
author: Frans Oort
published in: Lecture Notes in Mathematics
publication date: 1998-02-10T00:00:00Z
review text: This book contains a systematic study of the moduli spaces of supersingular abelian varieties. An abelian variety \(X\) of dimension \(g\) defined over a field \(K\) of characteristic \(p\) is said to be supersingular if and only if \(X\otimes\overline K\sim E^g\) where \(E\) is a supersingular elliptic curve over \(\overline K\), \(\overline K\) is an algebraically closed field containing \(K\) and \(\sim\) means isogeny equivalence. Let \({\mathcal A}_{g,d} \otimes \mathbb{F}_p\) be the moduli space of abelian varieties of dimension \(g\) with a polarization of degree \(d^2\) over a field of characteristic \(p\). The subset, \(S_{g,d}\), of \({\mathcal A}_{g,d} \otimes\mathbb{F}_p\) representing supersingular polarized abelian varieties is a closed algebraic subset. The main results proved in the book are: (a) The dimension of \(S_{g,1}\) is \([g^2/4]\) (b) The number of irreducible components of \(S_{g,1}\) is: \[ H_g(p,1)\text{ if }g \text{ is odd};\quad H_q(1,p) \text{ if }g\text{ is even}, \] \(H_g(p,1)\) (resp., \(H_g(1,p))\) being the class number of the principal genus (resp. the non-principal genus) defined by \textit{K. Hashimoto} and \textit{T. Ibukiyama} [J. Fac. Sci., Univ. Tokyo, Sect. IA 27, 549-601 (1980; Zbl 0452.10029) and 28, 695-699 (1981; Zbl 0493.10030)]. The main ingredient in the description of the moduli spaces \(s_g\) of supersingular abelian varieties is the study of polarized flat type quotients: A flat type quotient is a sequence of isogenies of abelian varieties: \[ E^g=X_{g-1} @>\rho_{g-1}>> \cdots @>\rho_i>> X_i/ \alpha_p^i\simeq X_{i-1} @>\rho_{i-1}>> \cdots @>\rho_1>> X_0 \] where \(E\) is a supersingular elliptic curve and \(\alpha_p\) the simple finite group scheme. A polarized flag type quotient with respect to a polarization \(\eta\) of degree \(d^2p^{g(g-1)}\) is a family of polarized abelian varieties \(\{(X_i\eta_i)\), \(0\leq i\leq g-1\}\) and isogenies \(\rho_i:X_i\to X_{i-1}\) compatible with the polarizations such that \…
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14D22
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 11-02
zbMATH DE Number: 1116599
zbMATH Keywords: moduli spaces of supersingular abelian varieties
zbMATH Keywords: non-principal genus
zbMATH Keywords: flag type quotients
zbMATH Keywords: polarized flag type quotient
reviewed by: J. M. Muñoz Porras
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0095931
OpenAlex ID: W131393994
DOI: 10.1007/BFB0095931
description: scientific article; zbMATH DE number 1116599