# Questions tagged [shimura-varieties]

The shimura-varieties tag has no usage guidance.

**1**

vote

**1**answer

275 views

### Points of infinite level modular curve

Let us consider the anticanonical adic tower of modular curves which is described in Peter Scholze's paper "On torsion in the cohomology of locally symmetric varieties", and let us call $\mathcal{X}_{\...

**2**

votes

**1**answer

102 views

### Involution on false elliptic curve

Let B be a indefinite quaternion algebra with discriminant $d>1$, maximal order $\mathcal{O}$ and standard involution $'$, then there exists $t\in{B}$ such that $t^2=-d$ and a new involution on B ...

**3**

votes

**1**answer

213 views

### Possible groups appearing in a Shimura datum

Let $\mathbb{S}:=\text{Res}_{\mathbb{C}/\mathbb{R}} \mathbb{G}_{m}$ be the Deligne torus. My question is the following: is there a sort of classification of real reductive algebraic groups $G$ for ...

**3**

votes

**1**answer

107 views

### What role do $(\mathfrak{g},K)$- modules play in the construction of automorphic vector bundles

Looking at the connection between modular forms as sections and automorphic representations it is to me somewhat clear why automorphic representations are (demanded to be) admissible $G(\mathbb{A}^\...

**5**

votes

**0**answers

193 views

### What is the difference between Kisin's and Vasiu's work on models of Shimura varieties?

My research is related to integral model of Shimura varieties. I realized there are two approaches building models for varieties of pre-abelian type and abelian type. I want to know what their ...

**2**

votes

**0**answers

80 views

### Compactification of symmetric spaces

Let $G = GL_n$. In the literature, I see that the symmetric space associated to $G$ is of the form $G(\mathbb{R})/K_\infty$ with $K_\infty = O(n) Z(\mathbb{R})^\circ = O(n) \mathbb{R}^\times_{+}$. ...

**3**

votes

**2**answers

327 views

### What is wrong with this modification of the definition of Shimura datum?

The definition of a "connected Shimura datum" (as in Milne's notes) is a pair $(G, X)$, where $G$ is a reductive algebraic group and $X$ is a $G(\mathbb{R})$-conjugacy class of morphisms
$$
x: \mathbb{...

**2**

votes

**0**answers

149 views

### Orbits under an algebraic group inside a Shimura variety

Let $(G,X)$ be a Shimura datum, $K$ a compact open subgroup of $G(\mathbb A_f)$. Let $H\subset G$ be a $\mathbb Q$-subgroup. Choose a point $x\in X$. What can be said about the orbit $H(\mathbb R)\...

**11**

votes

**0**answers

267 views

### Mysterious “raison d'être” of filtrations of congruence subgroups

I wonder for long why congruence subgroups seem to arise so naturally in certain filtrations. Everything below is on a local field $F_p$.
Filtration for $GL_n$. Casselman and later Jacquet, Piatetski-...

**2**

votes

**0**answers

85 views

### Relation between a conjecture of Pink and semi-abelian varieties

A conjecture of Pink says that in a mixed Shimura variety, every Hodge-generic point is Galois generic (conjecture 6.8 of "A Combination of the Conjectures of Mordell-Lang and Andr\'e-Oort). One can ...

**24**

votes

**0**answers

605 views

### A modern perspective on the relationship between Drinfeld modules and shtukas

Shtukas were defined by Drinfeld as a generalization of Drinfeld modules. While the relationship between the definitions of Drinfeld modules and shtukas is not obvious, one does have a natural ...

**5**

votes

**0**answers

135 views

### On toroidal compactifications of Hilbert Kuga-Sato varieties

Let $F$ be a totally real field of degree d. There are Hilbert modular varieties over $\mathbb{Q}$ that paramatrize abelian varieties of dimension d with an action of $\mathcal{O}_F$ the ring of ...

**5**

votes

**0**answers

276 views

### Is there a concrete way to show the existence of canonical model for non-modular Shimura curves?

I am trying to read Carayol's article on the construction of Galois representations associated to Hilbert modular forms (http://archive.numdam.org/article/CM_1986__59_2_151_0.pdf). The main geometric ...

**4**

votes

**0**answers

144 views

### On the class of Shimura data of Hodge type that cover a given Shimura datum of abelian type

A Shimura datum $(G,X)$ is of Hodge type if there exists an injective morphism of Shimura data $(G,X) \hookrightarrow (\mathrm{GSp}_{2g}, \mathfrak{H}^{\pm})$, for some integer $g$.
Let $(G',X')$ be ...

**1**

vote

**1**answer

182 views

### supersingular Abelian scheme

By a supersingular Abelian scheme, I mean an Abelian scheme which is fibrewise a supersingular Abelian variety, i.e. isogenous to a product of supersingular elliptic curves (F. Oort, Subvarieties of ...

**7**

votes

**0**answers

286 views

### Galois representations in cohomology of quaternion Shimura varieties

Let $F$ be a totally real field, and $E \subseteq F$ a subfield. Choose a quaternion algebra $B$ over $F$ satisfying the following condition:
there is a distinguished infinite place $\tau$ of $E$ ...

**5**

votes

**1**answer

445 views

### What is the maximal order of the automorphism group of a given Shimura variety?

Background:
Given an elliptic curve $E$, it seems that $max(ord(Aut(E)))$ over the prime 2 is 24, and $(max(ord(Aut(E)))$ over the prime 3 is 12.
The endomorphism algebra of an elliptic curve over $...

**4**

votes

**1**answer

221 views

### Albanese of Siegel modular variety $\mathcal{A}_2$

Jacobian varieties of Shimura curves are very interesting objects. For one thing they provide a geometric relation between elliptic curves and modular forms of weight 2 (say we are over $\mathbb{Q}$). ...

**0**

votes

**1**answer

143 views

### Units in indefinite quaternionic algebra

This is the opposite to my last question case.
Let $F$ be a totally real number field, $R$ is a quaternion algebra over $F$ unramified in at least one infinite place of $F$. Let $\mathcal{O}⊂R$ be an ...

**10**

votes

**1**answer

417 views

### Does every Shimura variety contain a generic point defined over a number field?

This question is related to my previous question, to which I got a partial answer.
Consider the cyclotomic field $L={{\mathbb{Q}}}(\zeta_8)={{\mathbb{Q}}}(\sqrt{2},i)$, where $\zeta_8$ is a primitive ...

**11**

votes

**2**answers

469 views

### Abelian variety with prescribed endomorphism ring

Consider the cyclotomic field $L={{\mathbb{Q}}}(\zeta_8)={{\mathbb{Q}}}(\sqrt{2},i)$, where $\zeta_8$ is a primitive 8-th root of unity. Let $\Lambda={{\mathbb{Z}}}[\zeta_8]$ denote the ring of ...

**1**

vote

**1**answer

83 views

### Tangent spaces of an indecomposable family of abelian varieties (parametrized by a Hodge type Shimura variety)

Let $G$ be a $\mathbb{Q}$-subgroup of $\mathrm{GSp}_{2g}$, reductive and defines a Shimura subdatum of $(\mathrm{GSp}_{2g},\mathfrak{H}_g)$. Let $V$ be the natural representation of $\mathrm{GSp}_{2g}$...

**1**

vote

**0**answers

234 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, ...

**6**

votes

**0**answers

221 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 ...

**3**

votes

**1**answer

277 views

### p-adic modular forms, Hecke algebra, deformation theory and modular curves.

Let $h^{ord}(N,\mathcal{O})$ be the $p$-ordinary Hecke algebra, and $\mathfrak{m}$ be a maximal ideal of the semi local ring $h^{ord}(N,\mathcal{O})$ corresponding to a residual representation $\bar{\...

**2**

votes

**0**answers

224 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**

votes

**1**answer

432 views

### Confusion about a result on Shimura and Teichmüller curves

It is shown by M. Moeller (M. Moeller, Shimura- and Teichmüller curves) that there are only 2 Shimura and Teichmüller curves in the moduli space of curves $M_g$, namely, the ones given by $y^4=x(x-1)(...

**6**

votes

**2**answers

469 views

### Reflex fields of Shimura varieties

I am currently learning the theory of Shimura varieties. Out of curiosity, is it known which number fields can occur as reflex fields? More precisely, can one find, for any number field, a positive ...

**1**

vote

**1**answer

191 views

### Is an Isomorphism from an Abelian variety to a Shimura variety always defined over a solvable extension?

Are there counterexamples to the following:
Given two varieties $A$, $\tilde{A}$, both defined over $\mathbb{Q}$, one of which, say $A$, is a Shimura variety. Then, every isomorphism defined over $\...

**9**

votes

**1**answer

188 views

### Matsushima-Murakami Isomorphism for $L^2$-cohomology

Let $\mathbf{G}$ be a reductive connected linear algebraic group over a totally real global number field, say $\mathbb{Q}$. Let $\mathbb{A}=\mathbb{R}\times\mathbb{A}_f$ be the ring of rational adele.
...

**2**

votes

**0**answers

215 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 ...

**3**

votes

**1**answer

419 views

### Paper of Boutot-Carayol in `Courbes modulaires et courbes de Shimura'

I am trying to obtain a copy of the following
J.-F. Boutot and H. Carayol, Uniformisation p-adique des courbes de Shimura: les
théorèmes de Čerednik et de Drinfel'd , Astérisque No. 196-197 (1991)...

**4**

votes

**0**answers

140 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 ...

**3**

votes

**0**answers

162 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 ...

**4**

votes

**1**answer

198 views

### Subgroups of $Sp_{2g}$ giving rise to Shimura data

Consider the Shimura datum $(GSp_{2g},\mathcal{H}_g)$. Let $G$ be a reductive $\mathbb{Q}$-subgroup of $Sp_{2g}$. I want to know under what condition there exists a point $x\in\mathcal{H}_g$ such that ...

**3**

votes

**0**answers

338 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 ...

**8**

votes

**1**answer

478 views

### Do all 0-dimensional Shimura Varieties show up (as CM points) in $\mathcal{A}_g$?

Question: Let $S$ be a 0-dimensional Shimura variety. Does $S$ necessarily admit a morphism (in the category of Shimura varieties) to $\mathcal{A}_g$ for some $g\geq 1$? Here $\mathcal{A}_g$ is the ...

**12**

votes

**1**answer

888 views

### Concrete Examples of Shimura Surfaces

First a disclaimer: I am at best a part-time arithmetic geometer, so please accept my apologies when I am too naive or get something wrong.
From time to time I have tried to learn something about ...

**1**

vote

**1**answer

389 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 .
...

**3**

votes

**1**answer

309 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 ...

**6**

votes

**0**answers

277 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}, ...

**2**

votes

**1**answer

295 views

### 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 $\...

**1**

vote

**0**answers

110 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,...

**3**

votes

**1**answer

202 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 O}a^{-1}...

**1**

vote

**1**answer

411 views

### 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.

**10**

votes

**1**answer

359 views

### 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 ...

**5**

votes

**0**answers

139 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-...

**9**

votes

**0**answers

545 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, ...

**5**

votes

**1**answer

460 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 ...

**3**

votes

**2**answers

353 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 ...