Abelian varieties, \(\ell\)-adic representations, and \(\ell\)-independence
type: Item
instance of: scholarly article
title: Abelian varieties, \(\ell\)-adic representations, and \(\ell\)-independence
zbMATH Open document ID: 0867.14019
DOI: 10.1007/BF01444508
author: Michael Larsen
author: Richard Pink
published in: Mathematische Annalen
publication date: 1997-08-10T00:00:00Z
full work available at URL: https://eudml.org/doc/165344
review text: Let \(G=\text{Gal} (\overline{K}/K)\) be the Galois group of the extension of a global field \(K\) by a separable closure \(\overline{K}\). For any abelian variety \(A\) of dimension \(g\) over \(K\), the Galois group \(G\) acts on the group \(A[\ell^n]\) of \(\ell^n\)-torsion points of \(A\) where \(\ell\) is a prime number different from \(\text{char} (K)\). Therefore, \(G\) acts continuously on the vector space \(V_\ell=(\displaystyle\varprojlim_n A[\ell^n]) \otimes_{\mathbb{Z}_\ell}, \mathbb{Q}_\ell\cong \mathbb{Q}_\ell^{2g}\), and the corresponding system \(\rho_\ell: \Aut(V_\ell)\), \(\ell\neq\text{char} (K)\), of representations is strictly compatible in the sense of Serre. Let \(G_\ell\) denote the image of \(G\) under \(\rho_\ell\) in \(\text{GL} (2g,\mathbb{Q}_\ell)\), \(G_\ell^0\) the identity component of \(G_\ell\), and \(\rho_\ell^{\text{alg}}\) the induced representation of \(G_\ell^0\) and \(V_\ell\). The present paper is devoted to a closer study of the identity component \(G_\ell^0\), and that mainly under the aspect of searching for more evidence for the following well-known conjecture: There exists a connected reductive group \(H\) over \(\mathbb{Q}\), together with a faithful representation \(\rho\) of \(H\) on a \(\mathbb{Q}\)-vector space \(V\) auch that for sufficiently big prime numbers \(\ell\neq \text{char}(K)\) there is an isomorphism of representations \((G_\ell^0, V_\ell, \rho_\ell^{\text{alg}})= (H,V,\rho) \otimes_\mathbb{Q} \mathbb{Q}_\ell\). Building upon the results obtained in two earlier papers [cf. \textit{M. Larsen} and \textit{R. Pink}, Invent. Math. 102, No. 2, 377-398 (1990; Zbl 0687.22004) and ibid. 107, No. 3, 603-636 (1992; Zbl 0778.11036)] the authors prove that \(G_\ell^0\) is unramified for \(\ell\gg 0\), and that the conjecture is true, at least for primes \(\ell\) that form a set of Dirichlet density one, under additional assumptions on \(\text{End}_{\overline{K}} (A)\), \(g=\dim A\) and the splitting field of the rep…
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 20C11
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 14K10
Mathematics Subject Classification ID: 11G35
zbMATH DE Number: 799560
zbMATH Keywords: \(\ell\)-adic representations
zbMATH Keywords: Serre conjecture
zbMATH Keywords: torsion points
zbMATH Keywords: abelian variety
zbMATH Keywords: Mumford-Tate conjecture
zbMATH Keywords: Tate conjecture
reviewed by: Werner Kleinert
MaRDI profile type: Publication
cites work: Q3941610
cites work: On the l-adic representations attached to simple abelian varieties of type IV
cites work: ℓ-Adic and λ-Adic Representations Associated to Abelian Varieties Defined Over Number Fields
cites work: Hodge cycles, motives, and Shimura varieties
cites work: Finiteness theorems for abelian varieties over number fields.
cites work: Determining representations from invariant dimensions
cites work: On \(\ell\)-independence of algebraic monodromy groups in compatible systems of representations
cites work: Q5597385
cites work: On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne
cites work: Q5580133
cites work: Q5615218
cites work: ISOGENIES OF ABELIAN VARIETIES OVER FIELDS OF FINITE CHARACTERISTIC
cites work: A REMARK ON ENDOMORPHISMS OF ABELIAN VARIETIES OVER FUNCTION FIELDS OF FINITE CHARACTERISTIC
cites work: Abelian varieties, \(\ell\)-adic representations and Lie algebras. Rank independence on \(\ell\)
cites work: Abelian varieties having a reduction of \(K\)3 type
OpenAlex ID: W2912558695
description: scientific article; zbMATH DE number 799560