The splitting of reductions of an abelian variety
type: Item
instance of: scholarly article
zbMATH Open document ID: 1318.14040
author: David Zywina
publication date: 2014-11-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1111.0624
Mathematics Subject Classification ID: 14K02
Mathematics Subject Classification ID: 14K05
Mathematics Subject Classification ID: 11G10
zbMATH DE Number: 6365884
zbMATH Keywords: abelian varieties
zbMATH Keywords: good reductions
MaRDI profile type: Publication
OpenAlex ID: W2962762360
arXiv ID: 1111.0624
DOI: 10.1093/IMRN/RNT113
published in: IMRN. International Mathematics Research Notices
title: The splitting of reductions of an abelian variety
review text: Let \(A\) be an absolutely simple abelian variety defined over a number field \(K\) such that all \(\ell\)-adic monodromy groups associated to \(A\) over \(K\) are connected. Consider the integer \(m:= [\text{End}(A)\otimes_{\mathbb{Z}}\mathbb{Q}: E]^{{1\over 2}}\), where \(E\) is the center of the division algebra \(\text{End}(A)\otimes_{\mathbb{Z}}\mathbb{Q}\).NEWLINENEWLINE The main result of the paper is the following theorem: For all places \(v\) of \(K\) away from a set of density \(0\), \(A_v\) is isogenous to \(B^m\) for some abelian variety \(B\) over \(\mathbb{F}_v\). Under the assumption that the Mumford-Tate conjecture holds for \(A\), the abelian variety \(B\) is absolutely simple. This implies the following conjecture due to Murty-Patankar: Let \(A\) be as above and let \({\mathcal V}\) be the set of finite places \(v\) of \(K\) for which \(A\) has good reduction and \(A_v\) over \(\mathbb{F}_v\) is simple. Then, after possibly replacing \(K\) by a finite extension, \({\mathcal V}\) has density \(1\) if and only if \(\text{End}(A_{\overline K})\) is commutative. Under the same assumptions, the paper also describes the Galois extensions of \(\mathbb{Q}\) generated by the Weil numbers of \(A_v\) for most \(v\).
reviewed by: Herbert Lange
description: scientific article; zbMATH DE number 6365884