On Galois representations for abelian varieties with complex and real multiplications.
type: Item
instance of: scholarly article
title: On Galois representations for abelian varieties with complex and real multiplications.
zbMATH Open document ID: 1056.11034
DOI: 10.1016/S0022-314X(02)00121-X
author: Piotr Krasoń
author: Grzegorz Banaszak
author: Wojciech Gajda
published in: Journal of Number Theory
publication date: 2003-06-25T00:00:00Z
review text: This paper studies the image of \(\ell\)-adic representations coming from the Tate module of an abelian variety. Let \(A\) be a simple abelian variety of nondegerate CM-type defined over a number field \(F\) with \(\text{End}(A)\otimes\mathbb{Q}= E\subset F\), and let \(\phi:T_{E'}\to T_E\) be the homomorphism of tori defined in [\textit{K. A. Ribet}, Mém. Soc. Math. Fr., Nouv. Sér. 2, 75--94 (1980; Zbl 0452.14009)], where \(E'\) denotes the reflex field of \(E\). The authors determine the image of \(\overline{\rho_\ell}: G_F\to \text{GL}_{2g}(\mathbb{F}_\ell)\) explicitly under the assumption that the kernel of the map \(\phi\) is connected. When \(\text{End}(A)\otimes\mathbb{Q}\) is a totally real field of degree \(e\) with \(g/e\) odd, they determine the commutator subgroup of \(\overline{\rho_\ell}(G_F)\) under some conditions on \(\ell\). As a consequence they show the validity of the Mumford-Tate conjecture for such \(A\).
reviewed by: Fumio Hazama
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 11R34
zbMATH DE Number: 1934760
zbMATH Keywords: Galois representation
zbMATH Keywords: Abelian variety
zbMATH Keywords: Complex multiplication
zbMATH Keywords: Real multiplication
Wikidata QID: Q114122662
MaRDI profile type: Publication
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full work available at URL: s0022-314x(02)00121-x
OpenAlex ID: W2066751578
description: scientific article; zbMATH DE number 1934760