On the image of Galois \(l\)-adic representations for abelian varieties of type III
type: Item
instance of: scholarly article
title: On the image of Galois \(l\)-adic representations for abelian varieties of type III
zbMATH Open document ID: 1202.14042
author: Grzegorz Banaszak
author: Wojciech Gajda
author: Piotr Krasoń
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2010-09-07T00:00:00Z
review text: Let \(F\) be a number field with absolute Galois group \(G_{F}\). Let \(A\) be a simple abelian variety defined over \(F\). If (i) \(\mathcal{R}:=\text{End}_{\bar{F} }(A)=\)End\(_{F}(A)\), (ii) \(\mathcal{R} \otimes_{\mathbb{Z}}\mathbb{\mathbb{Q}}\) is of type III, (iii) \(G_{l} ^{\text{alg}}\), the Zariski closure of the image of the \(l\)-adic Galois representation \(\rho_{l}\) in \(\text{GL}_{V_{l}(A)}/\mathbb{Q}_{l}\), is a connected algebraic group, and (iv)\ \(\dim A=2h[E:\mathbb{Q}]\) where \(h\) is odd and \(E\) is the center of \(\mathcal{R}\otimes_{\mathbb{Z}}\mathbb{\mathbb{Q}}\); then \(A\) is said to be of class \(\mathcal{B}\). The work under review is a study of the image of \(\rho_{l}\) when \(A\) is an abelian variety of class \(\mathcal{B}\). After an embedding \(F\hookrightarrow\mathbb{C}\) we can define a cocharacter \(\mu_{\infty}:\mathbb{G}_{m}(\mathbb{C})\to \text{GL}_{2g}(\mathbb{C})\). The smallest algebraic subgroup of \(\text{GL}_{2g}\) defined over \(\mathbb{Q}\) such that its geometric points include the image of \(\mu_{\infty}\) is called the Mumford-Tate group -- its base change to \(\mathbb{Q}_{l}\) is denoted MT\((A)_{\mathbb{Q}_{l}}\). One always has \((G_{l}^{\text{alg}})^{\circ}\subset\)MT\((A)_{\mathbb{Q} _{l}}\) and a well-known conjecture of Mumford and Tate states that these two groups are equal. Here, it is shown that the Mumford-Tate conjecture holds whenever \(A\) is of class \(\mathcal{B}\); more precisely we have \(G_{l}^{\text{alg}}=\text{MT}(A)_{\mathbb{Q}_{l}}\) in this case since the first group is connected by assumption. It is a conjecture of Lang that for large \(l\) the group \(\rho_{l}(G_{F})\) contains all of the homotheties in \(\text{GL}_{T_{l}(A)}(\mathbb{Z}_{l})\). This is proven in the case that \(A\) is of class \(\mathcal{B}\), in fact it follows immediately from the result above. For any profinite group \(G\), let \(G^{\prime}\) be the closure of the commutator subgroup of \(G\). Then for large \(l\) we …
reviewed by: Alan Koch
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 17B45
zbMATH DE Number: 5780337
zbMATH Keywords: Abelian variety
zbMATH Keywords: \(l\)-adic representations
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1277298644
OpenAlex ID: W2066501987
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DOI: 10.2748/TMJ/1277298644
description: scientific article; zbMATH DE number 5780337