The shimura-varieties tag has no usage guidance.

**11**

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**1**answer

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### Which Shimura varieties are known to be automorphic?

This seems like something that should be well-known, but as an outsider to the field, I'm having trouble locating precise statements.
Hasse-Weil zeta functions of Shimura varieties should be ...

**4**

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**0**answers

532 views

### base change and Langlands' combinatorial exercise

Hi,
Is it correct that Langlands' combinatorial exercise (as he terms it in his paper
"Shimura varieties and the Selberg trace formula") is to establish base change identities between orbital ...

**5**

votes

**1**answer

553 views

### Is the Galois x Hecke action on cohomology of Shimura varieties semi-simple?

Given a reductive group $G/\mathbf Q$ (+ additional data), and a compact open subgroup $K\subset G(\mathbf A^\infty)$, there is a standard construction that produces a Shimura variety $S$ and if we ...

**3**

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**0**answers

474 views

### choice of local system in Deligne's construction of $l$-adic Galois representations

Hello,
Deligne famously constructed $l$-adic representations of $G_\mathbf Q = Gal(\overline{\mathbf Q}/\mathbf Q)$ starting form cusp modular forms of weight $k$ by looking inside the cohomology ...

**4**

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**0**answers

178 views

### vanishing of automorphic bundles

Let $S _K = S_K(G,X)$ be a Shimura variety of dimension $n$. Let $\xi$ be a (finite-dimensional) representation of $G$, which gives rise (by a construction of Harris) to an automorphic bundle $V(\xi)$ ...

**1**

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**0**answers

356 views

### higher direct image, Shimura varieties of PEL-type and representations

Let $M=M(G,X) = (M_K)_K$ be a Shimura variety of PEL-type associated to datum $(G,X)$. Let $A$ be the universal abelian scheme over $M(G,X)$ and $a: A \rightarrow M$.
Now, using notations from ...

**5**

votes

**1**answer

636 views

### Are all Shimura Varieties Special Subvarieties of the Siegel modular Variety?

Given a Shimura variety $S$, is it possible to imbed $S$ as a special Subvariety
of the Siegel modular variety $A_{g,N}$, for some $g$ and level $N$? I expect that the answer is yes, essentially since ...

**6**

votes

**1**answer

451 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

**1**answer

797 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 ...

**31**

votes

**3**answers

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, ...

**14**

votes

**3**answers

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

**2**answers

845 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**

votes

**1**answer

448 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 ...