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Deligne's idea was that Shimura varieties should be understood as moduli space of motives(with extra structures). lot's of Shimura varieties of abelian type can be understood as moduli space of abelian motives and any way essentially you can study all of them in this way by using Deligne formalism about the relation between two isogenous Shimura datum.

but there are Shimura varieties that are not of abelian type. My question is if there is any such Shimura variety that has a "nice" moduli description?

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  • $\begingroup$ I think none is known. $\endgroup$
    – user166831
    Commented Mar 19, 2021 at 21:04
  • $\begingroup$ There are hopeful indications that it may be possible to realize Shimura varieties of type E6 and E7 as moduli varieties. In the paper below, the authors construct compatible systems of l-adic representations that appear in the cohomology of algebraic varieties and have (for all l) algebraic monodromy groups equal to the exceptional groups of type E6. Boxer, George; Calegari, Frank; Emerton, Matthew; Levin, Brandon; Madapusi Pera, Keerthi; Patrikis, Stefan Compatible systems of Galois representations associated to the exceptional group E6. Forum Math. Sigma 7 (2019), Paper No. e4, 29 pp. $\endgroup$
    – user166831
    Commented Mar 19, 2021 at 21:20

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