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5
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1answer
408 views

different Shimura data with common underlying group?

A pure Shimura datum is of the form $(G,X)$ with $G$ a connected reductive $\mathbb{Q}$-group, and $X$ a homogeneous space under $G(\mathbb{R})$, subject to Deligne's conditions in terms of Hodge ...
5
votes
1answer
690 views

Generalizing Eichler-Shimura to higher dimension, again

This question is related to Intuition behind the Eichler-Shimura relation? and L-functions and higher-dimensional Eichler-Shimura relation Answering the first question above, Matt Emerton gives a ...
30
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3answers
2k views

Are there motives which do not, or should not, show up in the cohomology of any Shimura variety?

Let $F$ be a real quadratic field and let $E/F$ be an elliptic curve with conductor 1 (i.e. with good reduction everywhere; these things can and do exist) (perhaps also I should assume E has no CM, ...
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3answers
1k views

Constructing coherent sheaves on Shimura varieties.

Let me first run through the setting of my question in an example I understand well; that of modular curves. If $Y_1(N)$ denotes the usual modular curve over the complexes, the quotient of the upper ...
6
votes
2answers
804 views

modularity of algebraic varieties

Hello, Are there any examples of varieties which are not Shimura varieties or abelian varieties and whose L-functions have been shown to be a product of automorphic L-functions? Thanks. N
3
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1answer
425 views

What is the image of complex conjugation under Siegel Galois representations?

Let $G$ be the reductive group $\operatorname{GSp}_{4}$. Let $\pi$ be a smooth admissible cuspidal representation of $\operatorname{GSp}_{4}(\mathbb{A}^{(\infty)})$ of dominant weight. Assume, for ...