Split reductions of simple abelian varieties
type: Item
instance of: scholarly article
title: Split reductions of simple abelian varieties
zbMATH Open document ID: 1180.14046
published in: Mathematical Research Letters
publication date: 2009-09-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/0806.4421
review text: \textit{V. Kumar Murty} and \textit{V. M. Patankar} [Int. Math. Res. Not. 2008, Article ID rnn033, 27 p. (2008; Zbl 1152.14043)] study the splitting behavior of abelian varieties over number field, and advance the following Conjecture. Let \(X/K\) be an absolutely simple abelian variety over a number field. The set of primes of \(K\) where \(X\) split has positive density if and only if \(\mathrm{End}_{\overline{K}}(X)\) is noncommutative. Theorem A. Let \(X/K\) be an absolutely simple abelian variety over a number field. Suppose that either (i) \(\mathrm{End}_{\overline{K}}(X) \otimes {\mathbb{Q}} \cong F\) a totally real field, and \(\dim X/[F:{\mathbb{Q}}]\) is odd; or (ii) \(\mathrm{End}_{\overline{K}}(X) \otimes {\mathbb{Q}} \cong E\) a totally imaginary field, and the action of \(E\) on \(X\) is not special. Then for almost every prime \(p,\) \(X_{p}\) is absolutely simple. Theorem B. Suppose \(X/K\) is an absolutely simple abelian variety over a number field, and that \(\mathrm{End}_{\overline{K}}(X)\) is noncommutative. (i) For \(p\) in a set of positive density, \(X_{p}\) absolutely reducible. (ii) Suppose \(\mathrm{End}_{\overline{K}}(X) \otimes {\mathbb{Q}}\) is an indefinite quaternion algebra over a totally real field \(F,\) and that \(\dim X/2[F:{\mathbb{Q}}]\) is odd. For \(p\) in a set of positive density, \(X_{p}\) is geometrically isogenous to the self-product of an absolutely simple abelian variety.
reviewed by: V. V. Chueshev
Mathematics Subject Classification ID: 14K02
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 11G25
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 5608638
zbMATH Keywords: an absolutely simple abelian varieties over number field
zbMATH Keywords: a totally real field
zbMATH Keywords: a totally imaginary field
zbMATH Keywords: indefinite quaternion algebra over a totally real field
author: Jeffrey D. Achter
MaRDI profile type: Publication
arXiv ID: 0806.4421
DOI: 10.4310/MRL.2009.V16.N2.A1
Recommended article: Explicit bounds for split reductions of simple abelian varieties
Recommended article: Splitting of Abelian Varieties
Recommended article: Split abelian surfaces over finite fields and reductions of genus-2 curves
Recommended article: Exceptional splitting of reductions of abelian surfaces
Recommended article: Splitting of abelian varieties
Recommended article: Q5703066
Recommended article: The splitting of reductions of an abelian variety
Recommended article: On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field
Recommended article: Abelian Varieties with Extra Twist, Cusp Forms, and Elliptic Curves Over Imaginary Quadratic Fields
Recommended article: Isogeny classes and endomorphism algebras of abelian varieties over finite fields
description: scientific article