On the image of Galois $l$-adic representations for abelian varieties of type III

Abstract

In this paper we investigate the image of the l-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert classification. We compute the image explicitly and verify the classical conjectures of Mumford-Tate, Hodge, Lang and Tate, for a large family of abelian varieties of type I and II. In addition, for this family, we prove an analogue of the open image theorem of Serre.be the prime ideal of E[X, Y ] corresponding to the point X t ∈ GL r (E). Let J = t∈T J t .

Families of Motives and the Mumford–Tate Conjecture

“…In particular, for abelian varieties X with End(X) = Z satisfying some numerical conditions on dim(X), Pink proves that the HC, TC and MTC are true for all powers of X; this improves earlier results of Serre [82], [83] and Tankeev [94]. For some other types of endomorphism algebras, analogous results have been obtained for instance in [9] and [10]. See [55], Section 2, for an application of Pink's results in the context of motives of K3 type.…”

Explicit bounds for split reductions of simple abelian varieties

“…In this section, recent work by Banaszak, Gajda and Krasoń [2] is used to characterize the splitting behavior of reductions of abelian varieties of type III. We make use of Katz's analysis of orthogonal groups over finite fields [7], which was carried out in the service of an irreducibility statement somewhat like Lemma 2.5.…”

“…Each of these properties is preserved by any finite extension; and field extensions satisfying (i), (ii) and (iii) are explicitly calculated in [ The groups H X/K, are calculated for abelian varieties of type III in [2]. Suppose X/K is a polarized simple abelian variety such that End K (X) ⊗ Q is a definite quaternion algebra.…”

“…Moreover, a preprint of [1] elicited two further questions. W. Gajda shared a preprint of his work with Banaszak and Krasoń on the MumfordTate conjecture for abelian varieties of type III [2], and inquired whether it might be used to refine the results of [1] in that case. V.K.…”

Split reductions of simple abelian varieties

“…Quite recently, Banaszak et al have extended the methods of [2] to abelian varieties of type III, which allows an extension of Theorem 5.4 to the case of definite quaternion algebras. Also, Zywina points out that sieve methods (e.g., those of [28]) can be used to make the density one statements in Section 4 more explicit.…”