Lifting abelian varieties with additional structures
type: Item
instance of: scholarly article
title: Lifting abelian varieties with additional structures
zbMATH Open document ID: 1051.14052
author: Chia-Fu Yu
published in: Mathematische Zeitschrift
publication date: 2004-06-17T00:00:00Z
review text: The author investigates the moduli space of polarized \({\mathcal O}_B\) abelian varieties with a determinant condition. The main result stated as Theorem 4.5 is that when \(p\) is unramified in \(B\) and greater than 2, the moduli space is a complete intersection relative to the mixed characteristic base in the sense of \textit{A. Grothendieck} [Eléments de Geométrie Algébrique IV, Publ. Math., Inst. Hautes Étud. Sci. 32, 361 (1967; Zbl 0153.22301), Def. 19.3.6]. The author's main theorem proves a conjecture of \textit{M. Rapoport} and \textit{Th. Zink} [Period spaces for \(p\)-divisible groups, Ann. Math. Stud. 141 (1996; Zbl 0873.14039)] for the case of parahoric level structures which \textit{U. Görtz} [Adv. Math. 176, 89--115 (2003; Zbl 1051.14027)] has verified for the case of symplectic groups. Section 1 contains the definition of the moduli spaces under consideration with a determinant condition following [\textit{R. Kottwitz}, J. Am. Math. Soc. 5, 373--444 (1992; Zbl 0796.14014)]. Section 2 reviews the classical classification of Albert of division algebras with a positive involution. Section 3 contains the classification of Dieudonné modules attached to polarized \({\mathcal O}_B\) abelian varieties in terms of normal forms. In section 4, the author uses Grothendieck-Messing's deformation theory [\textit{W. Messing}, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lect. Notes Math. 264 (1972; Zbl 0243.14013)] and Fontaine's theory [\textit{J.-M. Fontaine}, Groupes p-divisibles sur les corps locaux, Astérisque 47--48 (1977; Zbl 0377.14009)] to investigate the local moduli spaces. The author concludes stating and proving theorem 4.5 relying on Proposition 4.3 and lemma 4.4 contained in this section.
reviewed by: Isidro Nieto Baños
Mathematics Subject Classification ID: 14K05
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 2075738
zbMATH Keywords: algebraic theory
zbMATH Keywords: Abelian varieties of dimension \(> 1\)
zbMATH Keywords: polarized abelian varieties
zbMATH Keywords: moduli space
zbMATH Keywords: parahoric level structures
MaRDI profile type: Publication
full work available at URL: s002090100350
OpenAlex ID: W2072184314
DOI: 10.1007/S002090100350
description: scientific article; zbMATH DE number 2075738