Questions tagged [shimura-varieties]
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157 questions
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Moduli interpretation of Hirzebruch-Zagier divisors
In their famous 1976 paper, Hirzebruch and Zagier define certain divisors $T_N$ on the Hilbert modular surface corresponding to the group $\text{SL}_2(\mathcal{O}_F)$ for $F=\mathbb{Q}(\sqrt{p})$. ...
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Moduli-space interpretation of a morphism of unitary Shimura varieties
Let $G$ be the quasi-split unitary similitude group $GU(2, 1)$, for some choice of imaginary quadratic field $E$; and let $T = GU(1)$ be the torus $Res_{E/Q}\mathbf{G}_m$. Then there's a morphism $\...
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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 ...
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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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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 ...
- 3,539
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131
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versal deformation ring of a p-divisible group with some tensors
I'm trying to read Kisin's paper about the Integral model of Shimura varieties. In section five he discusses versal deformation ring of a p-divisible group. Assume that $K$ is a number field with ...
- 1,093
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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}_{\...
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Is the cohomology of Hilbert modular surfaces spanned by special cycles?
We consider the Hilbert modular surface $X$ that parametrize abelian surface with real multiplication by $\mathcal{O}_F$ and $\mathfrak{a}$-polarization, where $F$ is a real quadratic field with ...
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2
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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 ...
- 14.2k
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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}_{\...
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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 ...
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248
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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 ...
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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 ...
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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 $G$ ...
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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 ...
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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 ...
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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{...
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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 ...
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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}...
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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 ...
- 14.2k
4
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1
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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 ...
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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 ...
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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 ...
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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 $...
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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 ...
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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}^\...
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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 ...
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Constructing coherent sheaves on Shimura varieties.
Let me first run through the setting of my question in an example I understand well; that of modular curves. If $Y_1(N)$ denotes the usual modular curve over the complexes, the quotient of the upper ...
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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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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 $\mathbb{Q}$). ...
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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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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-...
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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 ...
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2
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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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1
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393
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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.
...
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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 ...
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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 ...
user19475
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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 ...
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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)\...
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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 $E$ ...
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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 $K_\infty = O(n) Z(\mathbb{R})^\circ = O(n) \mathbb{R}^\times_{+}$. ...
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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 $\...
- 1,805
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169
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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 ...
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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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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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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 ...
- 3,183
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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{\...
- 1,066
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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)...
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1
answer
496
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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)(...
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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 ...
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