The shimura-varieties tag has no wiki summary.

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### Quotient of the hyperbolic plane with respect to commutator group of $\pi_1(\Sigma_g)$

Let $\Sigma_g$ be a Riemann surface of genus $g\geq 2$ and $G=\pi_1(\Sigma_g)$.
Let $\pi\colon \mathbb{H}\to \Sigma_g$ be the universal covering map. What kind of surface is $\mathbb{H}/[G,G]$?
...

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208 views

### The special point count on Shimura varieties

This, in view of the analogies between CM points on Shimura curves and torsion points on elliptic curves, is a sequel to an earlier question I had asked: The torsion point count in higher dimension .
...

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207 views

### Is the ordinary locus affine?

Let $p$ be a prime number and let $Y$ over $\mathbb F_p$ be a Siegel modular variety, with minimal compactification $X$. It is well known that $X^{\operatorname{ord}}$, the ordinary locus of $X$ is ...

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137 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}, ...

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### surjective homomorphism with compact kernel (Milne's note on Shimura varieties)

I'm reading Milne's Introduction to Shimura varieties (http://www.jmilne.org/math/xnotes/svi.pdf) and there is something I don't get.
Let $G$ be a connected semisimple algebraic group $G$ over ...

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### 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 ...

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136 views

### On Universal Abelian surfaces over a Shimura curve.

Let ${\cal O}, {\cal O}'$ be two order in ${\mathrm M}_2({\Bbb R})$ that are sets of all $2 \times 2$ matrices over real number ${\Bbb R}$. Assume that we have the relation ${\cal O}' = a{\cal ...

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### Shimura varieties of type C

Are there Shimura Varieties of Hodge-type which are not of PEL-type? I'd like to assume that the derived group is of type C.

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### higher dimensional analogues of the Manin-Drinfeld theorem

The Manin-Drinfeld theorem asserts that a divisor on the compact modular curve $X_0(N)$ which is supported on the cusps is torsion.
Equivalently, if $Y_0(N)$ is the open modular curve, the mixed ...

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### 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 ...

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### 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, ...

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354 views

### Connected cycles of Shimura curves in $A_{g}$ not contained in larger Shimura subvarieties

Is there always a finite family of Shimura curves $(C_{i})$ in $A_{g}$ the moduli space of principally polarized abelian varieties of dimension $g(\geq 2)$, such that the union $\cup C_{i}$ is ...

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260 views

### isogeny and congruence subgroup

Let $G_1$ and $G_1$ be two semisimple algebraic groups defined over $\mathbb{Q}$, suppose we have a surjective homomorphism $f: G_1\to G_2$, with finite kernel contained in the center of $G_1$.
By ...

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### 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 ...

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### Moduli space of motives vs moduli space of varieties

A (projective) abelian variety $A$ over the complex numbers is determined by $H^1(A,\mathbb{Z})$ together with its Hodge structure and polarization. This miracle means that one can parametrise ...

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### 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 ...

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293 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 ...

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### Shimura surfaces that do not contain a Shimura curve

Let $S$ be a Shimura surface i.e. a Shimura variety with $dimS=2$. Does $S$ necessarily contain a Shimura curve? I know that probably the answer is No, but do not have an explicit example. What is the ...

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135 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 ...

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### monodromy of Gauss-Manin over a Shimura variety

This is probably a difficult question. I would like to understand some particular cases and get some references. The rough question is the following:
Let $X$ be a PEL Shimura variety and $\pi: ...

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### Automorphisms of Generic Abelian Varieties

Automorphism groups of elliptic curves are very well understood. Of course, every elliptic curve has the automorphism $[-1]$ of order $2$. If we are over a (algebraically closed) field, this is the ...

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171 views

### Are period domains ever contractible

Which simply-connected period domains are contractible?
Examples. Siegel upper-half space? Poincare upper-half plane? Universal cover of a Shimura variety?
Are these contractible?

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### 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 ...

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### 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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### p-rank stratification in unitary Shimura variety

Let $K$ be a quadratic extension of $\mathbb Q$ and let $p \neq 2$ be a prime that is inert in $K$. Let $X$ be the Shimura variety associated to the unitary group $\operatorname{U}(2,1)$ over $K$ ...

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### 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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### 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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### 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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### Non emptyness of ordinary locus for PEL type Shimura varieties

We let $B$ be a simple algebra over $\mathbb Q$, with the usual notations for PEL type Shimura varieties. In his paper "Ordinariness in good reductions of Shimura varieties of PEL-type" (available ...

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### State of the art for integral models of PEL type Shimura varieties with deep level structure

The theory of PEL type Shimura varieties is nowadays well developed, but it is not easy to be updated with the latest results. Here I am particularly interested in integrals models. Let me describe ...

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### Imaginary quadratic field contained in Hecke orbit field?

Let $\tau$ in the upper half plane lie in an imaginary quadratic field $K$.
Then is $K \subset \mathbb{Q}(\{j(g \tau) \ | \ g \in GL_2^+(\mathbb{Q}) \})$?
(here $j$ is the modular $j$-function, and ...

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### How do you find the genus of a Fuchsian group derived from a quaternion algebra?

Let $G$ be a Fuchsian group with normalizer $N(G)$ inside $PSL(2,13)$
Due to the Hurwitz formula, it suffices to find a presentation of $G$ of the form:
$$\langle ...

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### 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 ...

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### Adelic formulations of complex multiplication and modular curves

In modular curves and modular forms, there is an adelic formulation, in which smaller open subgroups of some adelic group relate to higher level structure. As we know, higher level structure ...

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### semisimplicity of automorphic Galois representations

Is it known that the Galois representation constructed by Harris and Taylor in their book is semisimple? I can't see this proven in the book, but on the other hand, everywhere else the representation ...

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### 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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### Zograf's bound on the index of a modular curve for Shimura curves

I've been reading Voight's paper on Shimura curves and it prompted the following question; see http://www.cems.uvm.edu/~voight/articles/shimura-clay-proceedings-071707.pdf for which notes I'm talking ...

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301 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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### Mumford-Tate groups of abelian varieties with potentially good reduction everywhere

Let $A$ be an abelian variety defined over a number field, and let $MT(A)$ be its Mumford-Tate group. It is a conjecture of Morita that if $MT(A)$ is anisotropic-mod-center (that is, it has no ...

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### 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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### 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 ...

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### A route towards understanding Shimura varieties?

I'm in the embarrassing situation that I want to ask a question that
was already asked, but (for complicated reasons) never answered. I'd
like to try with a blank slate.
Shimura varieties show ...

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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 $\mathbb{Q}(\mu)\cap\mathbb{Q}(S)$?

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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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $V(\xi)$ ...

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### 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 ...