Canonical subgroups and p-adic vanishing cycles for abelian varieties
type: Item
instance of: scholarly article
title: Canonical subgroups and p-adic vanishing cycles for abelian varieties
zbMATH Open document ID: 1062.14057
author: Ahmed Abbes
author: Abdellah Mokrane
published in: Publications Mathématiques
publication date: 2004-09-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104204
review text: Let \(k\) be an algebraically closed field of characteristic \(p>0\), \(W=W(k)\) its ring of Witt vectors and \(\sigma\) the endormorphism of Frobenius on \(k\) and \(W\). Let \(A\) be an ordinary \(k\)-abelian variety of dimension \(g\) and \(\mathfrak M\) the moduli space of formal deformations of \(A\) over \(W\)-artinian local algebras with residue field \(k\). It follows from Serre-Tate theory [\textit{N. Katz}, in Surfaces algébriques, Lect. Notes Math. 868, 138--202 (1981; Zbl 0477.14007)] that there exists a canonical isomorphism of \(W\)-formal schemes \(\mathfrak M\overset{\cong}\longrightarrow\text{Hom}_{\mathbb Z_p}(T_pA(k)\otimes T_p\hat{A}(k),\hat{\mathbb G}_m)\), where \(\hat{A}\) is the dual abelian variety of \(A\) and \(T_p\) denotes the \(p\)-adic Tate module. \textit{B. Dwork} showed in [appendix of \textit{P. Deligne}, \textit{L. Illusie}, in: Surfaces algébriques, Lect. Notes Math. 868, 80--137 (1981; Zbl 0537.14012)] that the structure of the formal toric group over \(\mathfrak M\) is imposed by a given \(W\)-morphism \(\Phi:\mathfrak M\to\mathfrak M^{(\sigma)}\) that lifts the Frobenius morphism. In particular, the structure of the formal group of Serre-Tate is completely determined by the canonical lifting of Frobenius \(\Phi_{\text{can}}:\mathfrak M\to\mathfrak M^{(\sigma)}\). In this case Dwork conjectured that the canonical lifting is overconvergent (the problem of the excellent lifting). More precisely, let \(\mathcal X\) be a formal \(W\)-scheme topologically of finite type and \(\mathcal A\to\mathcal X\) a formal abelian scheme such that \(\mathcal A_k\to\mathcal X_k\) is a family of ordinary abelian varieties. Let \({}_p\mathcal A\) be the kernel of multiplication by \(p\), \({}_p\mathcal A^{\text{ét}}\) the greatest étale quotient and \({}_p\mathcal A^0\) the kernel of the natural morphism \({}_p\mathcal A\to{}_p\mathcal A^{\text{ét}}\). We suppose given a cartesian diagram \[ \begin{tikzcd} \mathcal A/{}_p\mathcal A^0\ar[r,"\Theta"]…
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 14F30
zbMATH DE Number: 2103609
zbMATH Keywords: vanishing cycles
zbMATH Keywords: abelian varieties
reviewed by: Amílcar Pacheco
MaRDI profile type: Publication
OpenAlex ID: W2013902307
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DOI: 10.1007/S10240-004-0022-X
description: scientific article; zbMATH DE number 2103609