# Questions tagged [shimura-varieties]

The shimura-varieties tag has no usage guidance.

107
questions

**6**

votes

**1**answer

245 views

### On the moduli stack of abelian varieties without polarization

(I am especially interested in abelian surfaces and characteristic 0).
How bad is the moduli stack of abelian varieties (with no polarization or level structure)? Is it an Artin stack? DM (Deligne-...

**3**

votes

**1**answer

141 views

### Quotienting $G(\mathbb{Q})_{+}$ by $G^{\text{sc}}(\mathbb{Q})$ and inner forms

Let $G/\mathbb{Q}$ be a connected reductive group, let $G^{\text{ad}}$ be the adjoint group, let $G^{\text{der}}$ be the derived group and let $\rho\colon G^{\text{sc}} \to G^{\text{der}}$ be the ...

**1**

vote

**0**answers

103 views

### Closed embeddings of Shimura varieties

In Deligne's "Travaux de Shimura" the author proves in Proposition 1.15 (notation from the paper):
For an inclusion of Shimura data $(G^1,h^1)\hookrightarrow(G^2,h^2)$ and any open compact $K^1\...

**4**

votes

**1**answer

150 views

### Definition field of weight homomorphism and moduli interpretation of Shimura varieties

In "Canonical models of Shimura curves" by J.S. Milne (avaliable at https://www.jmilne.org/math/articles/2003a.pdf), he explains the definition of quaternion Shimura curve, and explains the modern ...

**1**

vote

**0**answers

95 views

### Picard rank of Shimura varieties

Let $D$ be a bounded symmetric domain and $\Gamma\subset Aut(D)$ an arithmetic subgroup of automorphisms of $D$. Write $X=\Gamma\backslash D$, it is a connected Shimura variety. Suppose that $X$ is ...

**4**

votes

**0**answers

140 views

### Dependence of X in definition of Shimura variety

(Disclaimer: this question is related to this question, but is different enough that it warrants (in my opinion) a separate question)
Let $G$ be a connected reductive group over $\mathbb{Q}$. To $G$ ...

**11**

votes

**0**answers

543 views

### What is known at $\ell = p$ about realizing Jacquet-Langlands & local Langlands as the cohomology of Lubin-Tate space with level structure?

Background:
(Mostly my paraphrased interpretation of the introduction of Strauch's Deformation spaces of one-dimensional formal
modules and their cohomology, with additional details from Carayol's ...

**1**

vote

**1**answer

175 views

### Hodge variation

I am reading Milne's online book of Shimura Varieties https://www.jmilne.org/math/xnotes/svi.pdf, I confused by a Definition of Hodge variation. On page 29, it was said something is called Hodge ...

**16**

votes

**0**answers

483 views

### Shimura varieties and connected components

Let $G$ be a connected reductive algebraic group over $\mathbf{Q}$. I've seen two slightly different definitions in the literature of the Shimura variety of level $U$, for $U \subseteq G(\mathbf{A}_{\...

**4**

votes

**0**answers

185 views

### Higher dimensional generalization of an identity between traces of Hecke operators and number of elliptic curves over finite fields?

In http://www.math.ubc.ca/~behrend/ladic.pdf, the author uses his generalization of Lefschetz trace formula to smooth algebraic stacks to prove an interesting identity (Proposition 6.4.11.):
$\sum_{k}...

**14**

votes

**3**answers

1k views

### Tower of moduli spaces in Scholze's theory

My question is related to another one I read here in Overflow. I am reading Scholze's papers about moduli spaces of $p$-divisible groups and elliptic curves, and I am very interested in the formal ...

**1**

vote

**0**answers

101 views

### Tate module is canonically isomorphic to a $\mathbb Z_p$-lattice on Shimura Variety of Hodge Type

Let $(G, X)$ be a Shimura datum of Hodge type. Suppose that $K \le G(\mathbb A_f)$ is such a compact open subgroup that its $p$th component $K_p = \mathcal G(\mathbb Z_p)$ is a hyperspecial subgroup ...

**4**

votes

**0**answers

280 views

### What is the analogy between the moduli of shtukas and Shimura varieties?

I have heard that moduli spaces of shtukas are supposed to be the analogue of Shimura varieties in the setting of function fields. Could someone more knowledgeable about these objects explain how this ...

**3**

votes

**0**answers

271 views

### Shimura varieties of Hodge type

I am trying to understand the theory of integral model of Shimura variety of Hodge type, like for example in Kisin's paper "Integral models for Shimura varieties of abelian type".
I understand that ...

**1**

vote

**1**answer

343 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

**2**answers

175 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

241 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

127 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}^\...

**6**

votes

**0**answers

227 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

89 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

359 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

158 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

287 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

112 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 ...

**25**

votes

**0**answers

886 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

180 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

301 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

179 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

202 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

333 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

518 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

253 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

158 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

460 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

522 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

85 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

238 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

238 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

307 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

227 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

451 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)(...

**7**

votes

**2**answers

609 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

202 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

223 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

224 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

467 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

149 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

182 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

202 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 ...

**4**

votes

**0**answers

380 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 ...