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It seems to me that Hilbert modular varieties (forms) are generalization from Q to totally real fields. While Siegel modular varieties (forms) are generalization from 1 dimensional to higher dimensional abelian varieties. But they should both be some kind of Shimura variety (automorphic forms), right?

According to Milne's note of Shimura varieties, Siegel modular varieties are Shimura varieties coming from the Shimura datum (G, X) where G is the symplectic similitude group of a symplectic space (V, \phi). So what is the corresponding Shimura datum for the Hilbert modular variety? Or am I asking a wrong question?

Also, in the definition of Shimura varieties G(Q)\G(A_f)X/K, why we only consider Q and its adele group? why not general number fields and their adeles? (again, maybe a wrong question)

Thank you.

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2 Answers 2

up vote 4 down vote accepted

Both of your questions have the same answer: if $K$ is an arbitrary number field, it is not necessary to consider separately reductive groups over $K$, because if $G_{/K}$ is a reductive group, then $Res_{K/\mathbb{Q}} G$ is a reductive group over Q. Here $Res_{K/\mathbb{Q}}$ denotes Weil restriction.

In particular, if K is a totally real field, then the reductive group whose Shimura variety is the corresponding Hilbert modular variety is $Res_{K/\mathbb{Q}} \operatorname{GL}_2$.

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The Shimura datum for a Hilbert modular variety is the pair (G,X) where G is the group over Q obtained from GL(2) over a totally real number field F by (Weil) restriction of scalars and X is a product of copies of C minus the real axis indexed by the real embeddings of F.

In the definition of Shimura varieties we work only over Q by convention: we could work over number fields, but this gives nothing new (one can always take Weil restriction). However, in some cases, e.g., Hilbert modular varieties, it seems (to me to be) more natural to work over a number field other than Q.

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