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Families of motives and the Mumford-Tate conjecture. (English) Zbl 1401.14048

This is an expository article that explains results and techniques related to the Mumford-Tate conjecture. After the author explains some historical background of the conjecture, he introduces in Section 2 Mumford-Tate groups and their \(\ell\)-adic counterparts \(G_{\ell}^0\), and discusses the Mumford-Tate conjecture. In Section 3 he introduces pure motives in the sense of Andrè, as well as motivic Galois groups, and discuss how this leads to a motivic version of the conjecture. In the final section, Section 4, he discusses how working with families of varieties provides us with additional tools that in some cases lead to very nontrivial results. He concludes the introductory section with the following impressive sentence: “To the author’s mind, the fact that conjectures about algebraic cycles have a direct relation to representation theory of reductive groups is one of the wonders of the subject.”

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14F20 Étale and other Grothendieck topologies and (co)homologies
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