ℓ-Adic and λ-Adic Representations Associated to Abelian Varieties Defined Over Number Fields

Abstract

0. Introduction. Let A be an abelian variety over a number field K of dimension n -1, and let G = Gal (KIK). For each prime number (, let Vi(A) be the Qe-adic Tate module of A. One general problem is to study the image of the e-adic representation Vr(A) of G. Let ge be the e-adic Lie algebra of the image of this f-adic representation. Fix an embedding cr: K c-* C, and denote by A(C) the abelian variety A x ,,, C. In [12], Mumford and Tate conjectured that ge = m 0Q Qe, where mn is the Lie algebra of the Mumford-Tate group MT(A(C)) (cf. [9], Section 3). As is now well known, in general g is contained in m 0Q Qe (cf. [9], [16]). Various results toward this general problem have been obtained for some classes of abelian varieties (cf. [22], Section 4). More recently, some important general results on e-adic representations attached to abelian varieties have been established. Especially, one has the following results: (i) the rank of g is independent of C (Serre [24], Zarhin [34]) (ii) geis algebraic and contains the homotheties (Bogomolov [1]) (iii) g is reductive and End,,(Ve(A)) = EndK(A) 0,, Qe (Faltings [10]). These make the determination of g possible for some other classes of abelian varieties. For example, when d = dim A is odd and EndT(A) = Z, J-P. Serre has proved that ge = M 0Q Qe sp(2d, Qe) 3 Qe id., where id. is the 2d x 2d identity matrix (cf. [25]). The purpose of this article is to extend Serre's method to some other cases. As a result, ge can be determined for the following types of simple abelian varieties: