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452 views

### Langlands program beyond CM fields?

I apologize since this is a quite vague question. And I am personally at an expert in these fields at all.
It seems to me that there are two main directions of the Langlands program, namely, ...

**8**

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371 views

### A frustrating cohomology class on the moduli of abelian surfaces

Here's a very frustrating question that I have been stuck on for some time. I believe that my question could fit in a general framework of what happens when you restrict $L^2$-cohomology classes on a ...

**6**

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160 views

### drinfeld shtukas over higher dimensional spaces

Everytime I encounter Drinfeld Shtukas, the definition begins with vector bundles over a curve $X$ over a finite field. My question is: why the restriction to curves? Is there any interest or results ...

**6**

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194 views

### Is there an integral pairing between quaternionic Hecke algebras and cusp forms?

Let $F$ be a totally real number field with integers $\mathcal{O}_F$ and $B$ a quaternion algebra over $F$ split at exactly one infinity place.Fix $n\geq 1$ and like in the special case $F=\mathbb{Q}, ...

**6**

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831 views

### How do you get algebraic models for modular/shimura curves?

I've got a few questions related to a paper by Lei Yang - "Exotic Arithmetic Structure on the First Hurwitz Triplet" http://arxiv.org/pdf/1209.1783v3.pdf
We know that there are exactly three Hurwitz ...

**5**

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113 views

### Effect of Hecke transform on the Mumford-Tate group

Let $Sh_{K}(G,X)$ be a Shimura variety and $Z\subset Sh_{K}(G,X)$ be a special subvariety. $Z$ is given by a Shimura sub-datum $(H,Y)$ with $H\subset G$ an algebraic subgroup which I call the $Mumford-...

**5**

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719 views

### Motivic Galois group and Shimura varieties

Hi,
Suppose that one has a Shimura variety $Sh(G,X)$ where $(G,X)$ is the corresponding Shimura datum and suppose that it can be interpreted as a moduli space of motives (e.g. PEL type Shimura ...

**4**

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109 views

### Uniqueness of cohomological holomorphic discrete series representation

In Claus Sorenson's PhD thesis, he proves a theorem about level lifting of paramodular forms whose associated automorphic representation has component $\pi_{\infty}$ that is the "cohomological ...

**4**

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152 views

### Shimura varieties and Maximal conditions

Working with Shimura varieties, I have been convinced to call them (or the families giving rise to them especially in $A_{g}$) somehow the "maximal" families. The motivation of this, has been for ...

**4**

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145 views

### Asymptotics of arithmetic Fuchsian groups and Shimura curves.

I'm interested in what is known/expected about some families of arithmetic Fuchsian groups. Here is the simplest family that I'm interested in: Let $E = Z[\omega]$, where $\omega = e^{2 \pi i / 3}$. ...

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262 views

### false elliptic curves and principal polarizations

Hi,
Let $\Delta$ be a quaternion algebra over $\mathbf Q$ and let $\mathcal O_\Delta$ be a maximal order in $\Delta$.
Recall that a false elliptic curve over a field $K$ is a pair $(A/K,i)$ ...

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546 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 ...

**4**

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180 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 $V(\xi)$ ...

**3**

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119 views

### Classification of compact Shimura curves

Is there a classification that determines all isomorphism classes of compact Shimura curves at least Shimura curves in $A_g$? I did not find this in the literature and appreciate any helpful ...

**3**

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150 views

### Ampleness on the P^1 bundle over Siegel threefold

I am looking at the Shimura variety for $\mathrm{GSp}_4(\mathbb Q)$, with hyperspecial level structure at $p$. Let $X$ denote the special fiber over $\mathbb F_p$. For simplicity, let us pretend ...

**3**

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107 views

### Newton point and Newton polygon stratifications

Let $k$ be a field of characteristic $p>0$, with absolute Galois group $\Gamma$. Let $Y$ be a Shimura variety of PEL type, defined over $k$, with associated reductive (connected) quasisplit group $...

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337 views

### Cyclotomic fields and singular moduli

Let $\mu$ be the roots of unity and $S$ be the image under the modular $j$-function of all imaginary quadratic $\tau$. Then what is $\mathbb{Q}(\mu)\cap\mathbb{Q}(S)$?

**3**

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500 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 ...

**2**

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215 views

### algebraic representation over $\mathbb{C}$

In reading the Harris-Taylor book, I encounter expressions like "Let $\xi$ be an algebraic representation of $G$ over $\mathbb{C}$". What does this mean? Here $G$ is a reductive group over $\mathbb{Q}$...

**2**

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181 views

### Morphism of Shimura varieties and differential equations

Is there a way of constructing a morphism between Shimura varieties using differential equations? Maybe, this looks like a completely ridiculous question, so I think that I should explain the context ...

**2**

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243 views

### Bruhat Tits buiding to visualize closed points of affine flag varieties?

In his survey "affine springer fibers and affine Deligne-Lusztig varieties", Goertz gives us a tutorial session on how to use Bruhat Tits buildings to visualize subsets of affine Grassmannians or of ...

**2**

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231 views

### Is the Picard number bounded by $b_2$ in positive characteristic?

We know that for a smooth projective variety $X$ over an algebraically closed field of characteristic 0 (for example $k=\mathbb{C}$), $\rho(X)\leq b_2(X)$. What about in positive characteristic? Is ...

**2**

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369 views

### what are the possible CM-fields of PEL type shimura varieties ?

In the paper "Travaux de Shimura" section 6, Deligne had defined a PEL- type shimura variety, for the following datum $(F,E,D,\psi)$, with $F$ a totally real cubic field, and $E$ a imaginary ...

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123 views

### Removing finitely many points from a Shimura curve

Let $X$ be a compact Shimura curve. If we remove finitely many points from this curve, do we neccessarily get a "non-compact Shimura curve"? I have some reasons to believe that the answer is negative, ...

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330 views

### Different approaches to Shimura varieties

I have just started taking a look at some introductory papers about Shimura varieties, after a friend of mine suggested me to do so. They seem to be a sort of very interesting and many-sided topic, ...

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196 views

### Siegel domains and the Baily-Borel compactification of $\mathcal{A}_2$

Consider the connected, almost simple, algebraic group $Sp_4$ over $\mathbb{Q}$ (embedded canonically in $GL_4$). For the following facts, I refer the reader to Murnaghan, Linear Algebraic Groups, ...

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88 views

### Ampleness of the Canonical Bundle for Siegel Modular Varieties

Background
Throughout I only work with varieties over $\mathbb{C}$.
For $p$ a prime number, Let $Y(p)$ denote the modular curve parametrizing elliptic curves together with full $p$-torsion structure,...

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180 views

### Existence of a point on the Shimura variety of PEL-type correponding to a specific abelian variety

I have been puzzle by the following question for a while. Suppose that we have an a Shimura variety $Sh(G,h_0)$ given by some datatum $(L, V, \psi, h_0)$ such as in Section 4.9 of "Travaux de Shimura"...

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214 views

### symplectic representations: when could the center act trivially?

I'm considering a problem about symplectic representation of real reductive group, which fits more or less into the setting of symplectic representations discussed in Milne's survey ''Shimura ...

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364 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 Milne'...