Questions tagged [shimura-varieties]
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70
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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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740
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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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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 ...
- 3,519
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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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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 $\...
- 36.1k
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Is there a classification of non-simple Jacobians?
An abelian variety in the interior of the Torelli locus is non-decomposable, but it could possibly be non-simple (i.e. isogenous to a product of abelian varieties with lower dimension). For certain ...
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9
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647
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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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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 ...
- 10k
7
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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 $E$ ...
- 36.1k
6
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301
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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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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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257
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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 ...
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336
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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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A local model of a Shimura variety and a local Shimura variety
I have a question about the book on p-adic geometry by Scholze and Weinstein.
There are two ‘local theories of Shimura varieties’ written in it.
The one is a local model of a Shimura variety. This is ...
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Heuristics for the very little torsion in the cohomology of Shimura variety
Consider the following statement which is a part of Conjecture 1.3 in the paper titled "The asymptotic growth of torsion homology for arithmetic groups" authored by N. Bergeron and 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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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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272
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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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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 ...
- 2,802
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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 ...
- 2,419
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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 ...
- 2,181
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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 $Mumford-...
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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 $(A/K,i)$ ...
- 2,802
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827
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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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681
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Base change and Langlands' combinatorial exercise
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 integrals of ...
- 2,802
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658
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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 ...
- 2,802
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176
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Several L-functions but one Galois representation: How to choose
Let $\mathbf{G}$ be a reductive group which enjoys all the nice properties a reducive group can dream of. Fix $(\mathbf{G},X)$ a Shimura datum associated with it and assume that if $K\leq\mathbf{G} $ ...
- 1,105
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The coarse moduli schemes of the "Shimura stacks" are the canonical models of the corresponding Shimura varieties
Let $F$ be a number field, $B$ a central simple algebra over $F$, $*$ a positive involution on $B$ which fixes $F$, and
$O_B$ a maximal $O_F$-order of $B$ which is stable under $*$.
Assume that $(B, *)...
- 1,352
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164
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The notion of border for (complex and non-archimedean) analytic spaces and schemes
Is a manifold with corner an analytic space (just show that $\left[0, +\infty \right)^{n}$ is an analytic space, which seems obvious but maybe I'm wrong...) EDIT: as noted in the comments some complex ...
- 1,105
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Explicit toroidal compactification of Hilbert modular varieties
Hirzebruch's construction of toroidal compactification of Hilbert modular surfaces is explicit, namely one can explicitly choose rational polyhedral cone decomposition in a sort of optimal way using ...
- 1,014
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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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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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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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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 ...
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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 ...
- 2,181
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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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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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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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Redefining connected Shimura datum
Firstly, let us fix a semisimple reductive linear algebriac group $G$ over $\mathbb{Q}$.
I am interested in seeing if I can bring the definition of connected Shimura datum (which is defined using some ...
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Does the construction of arithmetic toroidal compactification of $A_{g}$ depend on semistable reduction theorem?
If there is a good theory of arithmetic toroidal compactification over $\mathbb{Z}_{p}$ of the Siegel modular variety with deep enough level structure, then it seems like semistable reduction theorem ...
- 1,014
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544
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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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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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444
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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 ...
- 1,072
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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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350
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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 $\mathbb{Q}(\mu)\cap\mathbb{Q}(S)$?
- 1,895
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Representability of moduli problem of elliptic curves with complex multiplication
I'd like to know whether the moduli problem for elliptic curves with complex multiplication by a fixed imaginary quadratic number field $K$ (and with suitable level structure to be picked) is ...
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166
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Real structure(s) of a Shimura curve ("complex conjugation" of abelian surfaces)
For a complete lattice $L \subseteq \mathbb{C}^2$ let $A_L$ denote the complex abelian algebraic surface that is isomorphic (as a complex manifold) to the complex torus ${\mathbb{C}^2}/{L}$ (this ...
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Is there any work on the intersection loci of the universal theta divisor with torsion sections?
Let $Y$ be a Siegel modular variety of some non-stacky level and genus $g$, carrying over it a universal principally polarized family of dimension-$g$ abelian varieties $A\to Y$. Inside $A$, with fine ...
- 1,964
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The group of the modular automorphisms of the Shimura curves
Let $B$ be a rational indefinite division quaternion algebra, $(X,G)$ the Shimura datum associated with $B$ (i.e., $X$ is the upper half plane and $G(R) = (B \otimes_\mathbb{Q} R)^*$ for a ring $R/\...
- 1,352
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Shintani's unpublished paper on automorphic forms
I'm trying to find Shintani's preprint:
Shintani T., On automorphic forms on unitary groups of order 3, unpublished, 1979.
It seems to be impossible to find, even though several authors quote it. I ...
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