Torsion of abelian varieties over GL(2)-extensions of number fields
type: Item
instance of: scholarly article
title: Torsion of abelian varieties over GL(2)-extensions of number fields
zbMATH Open document ID: 0661.14032
DOI: 10.1007/BF01443356
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164577
review text: Let K be a number field and A an abelian variety over K (which we assume for the moment to be absolutely simple). Suppose that L/K is a Galois extension such that Gal(L/K) is an \(\ell\)-adic Lie group with Lie algebra \({\mathfrak g}\). Consider the following question: Fixing the (isomorphism class of the) Lie algebra \({\mathfrak g}\), which abelian varieties A can possible have an infinite number of torsion points rational over the field L? For example, if \({\mathfrak g}\) is \({\mathbb{Q}}_{\ell}\) then from work of Serre [see also \textit{K. Wingberg}, Math. Ann. 279, 9-24 (1987; Zbl 0657.14024)] one knows that A must be of CM-type, and indeed the same conclusion holds if A has an infinite number of torsion points rational over the maximal abelian extension field of K. In the present paper, the author treats the ``next'' case, which is of interest not only because of the role it plays in the above general question but also because of its relationship to classical modular forms. He considers the case where \({\mathfrak g}\) is the Lie algebra either of \(GL_ 2({\mathbb{Z}}_ p)\) or of the units in a quaternion algebra over \({\mathbb{Z}}_ p\). He proves that, in these cases, if A has an infinite number of torsion points rational over L, then End(A)\(\otimes {\mathbb{Q}}\) contains a number field of degree equal to the dimension of A. In fact he assumes something a bit weaker than absolute irreducibility of A to obtain this conclusion: he needs only that A is irreducible over the field L.
Mathematics Subject Classification ID: 14K05
zbMATH DE Number: 4079613
zbMATH Keywords: infinite number of torsion points
zbMATH Keywords: rationality
author: Yu. G. Zarhin
MaRDI profile type: Publication
cites work: Ramified torsion points on curves
cites work: Finiteness theorems for abelian varieties over number fields.
cites work: A remark on the rational points of abelian varieties with values in cyclotomic $Z_p$-extensions
cites work: ELLIPTIC CURVES OF INFINITE RANK OVER Γ-EXTENSIONS
cites work: CYCLOTOMIC FIELDS AND MODULAR CURVES
cites work: Rational points of Abelian varieties with values in towers of number fields
cites work: Q4090463
cites work: Galois Action on Division Points of Abelian Varieties with Real Multiplications
cites work: Galois properties of points of finite order of elliptic curves
cites work: Q5625265
cites work: Endomorphisms and torsion of Abelian varieties
cites work: A finiteness theorem for unpolarized Abelian varieties over number fields with prescribed places of bad reduction
cites work: ABELIAN VARIETIES,l-ADIC REPRESENTATIONS AND SL2
cites work: On the rational points of abelian varieties over \({\mathbb{Z}}_ p\)- extensions of number fields
Recommended article: TORSION AND ENDOMORPHISMS OF ABELIAN VARIETIES OVER INFINITE EXTENSIONS OF NUMBER FIELDS
Recommended article: Q3363052
Recommended article: ℓ-Adic and λ-Adic Representations Associated to Abelian Varieties Defined Over Number Fields
Recommended article: Torsion of abelian varieties, Weil classes and cyclotomic extensions
Recommended article: Abelian varieties having a reduction of \(K\)3 type
Recommended article: Abelian varieties without homotheties
Recommended article: On the torsion points of Drinfeld modules in abelian extensions
Recommended article: Which quaternion algebras act on a modular abelian variety?
Recommended article: Abelian varieties over number fields, tame ramification and big Galois image
Recommended article: A remark on the modularity of abelian varieties of \(GL_2\)-type over \(\mathbb{Q}\)
description: scientific article; zbMATH DE number 4079613