Some cases of the Mumford--Tate conjecture and Shimura varieties

Abstract

We prove the Mumford--Tate conjecture for those abelian varieties over number fields whose extensions to C have attached adjoint Shimura varieties that are products of simple, adjoint Shimura varieties of certain Shimura types. In particular, we prove the conjecture for the orthogonal case (i.e., for the $B_n$ and $D_n^R$ Shimura types). As a main tool, we construct embeddings of Shimura varieties (whose adjoints are) of prescribed abelian type into unitary Shimura varieties of PEL type. These constructions implicitly classify the adjoints of Shimura varieties of PEL type.

65 pages. Final version, to appear in Indiana Univ. Math. J

References in corpus (2)

• Very simple 2-adic representations and hyperelliptic jacobians
• Surjectivity Criteria for p-adic Representations, Part I

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• CM-lifts for Isogeny Classes of Shimura F-crystals over Finite Fields
• On the Picard number of K3 surfaces over number fields
• An effective open image theorem for abelian varieties
• Cycles in the de Rham cohomology of abelian varieties over number fields
• A note on images of Galois representations (with an application to a result of Litt)