Torsion of abelian varieties over GL(2)-extensions of number fields
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.
scientific article; zbMATH DE number 4079613
scientific article