Questions tagged [shimura-varieties]

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6
votes
1answer
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
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1answer
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
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0answers
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
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1answer
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 ...
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0answers
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
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0answers
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
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0answers
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
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1answer
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
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0answers
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
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0answers
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
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3answers
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
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0answers
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
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0answers
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
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0answers
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
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1answer
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
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2answers
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
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1answer
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
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1answer
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
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0answers
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
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0answers
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
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2answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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
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0answers
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 ...
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1answer
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
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0answers
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
1answer
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
1answer
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
1answer
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
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1answer
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
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2answers
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
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1answer
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 $\mathrm{GSp}_{2g}$...
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0answers
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
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0answers
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
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1answer
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
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0answers
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
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1answer
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
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2answers
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
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1answer
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
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1answer
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
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0answers
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
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1answer
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
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0answers
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
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0answers
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
1answer
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
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0answers
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 ...