Questions tagged [shimura-varieties]

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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 ...
Fra's user avatar
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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 ...
DGrimm's user avatar
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Embeddings of unitary groups over $\mathbb{Q}$

$\DeclareMathOperator\GU{GU}$$\DeclareMathOperator\GL{GL}$I'm a bit confused by the following situation: suppose we have an Hermitian vector space $V=K^3$ of matrix $$ J=\begin{pmatrix}& & \...
Fra's user avatar
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5 votes
1 answer
384 views

Cohomology of Shimura varieties before and after completion at some prime

Let $(G,X)$ be a Shimura datum with reflex field $E\subset \mathbb C$. For any neat open compact subgroup $K \subset G(\mathbb A_f)$, let $\mathrm{Sh}_K$ denote the associated Shimura variety. It is a ...
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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 ...
xir's user avatar
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2 votes
1 answer
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Understanding an example of abelian-type Shimura varieties

I'd like some help understanding the idea of abelian-type Shimura varieties. In paricular, I understand an abelian-type Shimura datum $(G,X)$ generally parameterizes non-rational Hodge structures ...
xir's user avatar
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17 votes
3 answers
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Are some congruence subgroups better than others?

When I first started studying modular forms, I was told that we can consider any congruence subgroup $\Gamma\subset\operatorname{SL}_2(\mathbb{Z})$ as a level, but very soon the book/lecturer begins ...
Coherent Sheaf's user avatar
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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/\...
k.j.'s user avatar
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3 votes
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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 ...
Coherent Sheaf's user avatar
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Is Krämer's local model for ramified unitary groups isomorphic to the blow-up of Pappas' flat model at the singular point?

I am reading the following two papers: Pappas, On the arithmetic moduli schemes of PEL Shimura varieties, 1999 (it seems to be difficult to find online nowadays - only a .ps file remains available), ...
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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 ...
Fra's user avatar
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4 votes
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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} $ ...
Marsault Chabat's user avatar
6 votes
1 answer
481 views

Definition of locally symmetric space of reductive groups

This might seems like a bit of philosophical question and so maybe if I keep reading a bit more, I might get my answer. But, I ask nonetheless. In my attempt to study Shimura varieties, I came across ...
Coherent Sheaf's user avatar
2 votes
1 answer
138 views

Reference for isomorphism between parabolic and cuspidal cohomology of the Siegel variety

I'm asking for a reference where I can find proof of isomorphism $$H^{3}_{\text{cusp}}(Y(U),F_{\lambda})\simeq H^{3}_{\text{par}}(Y(U),F_{\lambda}),$$ where $Y(U)$ is the level $U$ shimura variety of $...
Marsault Chabat's user avatar
6 votes
1 answer
738 views

B. W. Jordan's thesis on arithmetic of Shimura curves

I'm looking for Bruce W. Jordan's thesis: On the diophantine arithmetic of Shimura curves. Thesis, Harvard University, 1981. I could not find the pdf at the following site. https://www.math.harvard....
k.j.'s user avatar
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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, *)...
k.j.'s user avatar
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1 vote
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Are the irreducible components appearing in the resolution of singularities of a Hilbert modular surface defined over $\mathbb{Q}$?

It seems to me that this is claimed in van der Geer's "Hilbert modular surfaces" on p. 245 at the beginning of XI.2 (without justification). My current state of belief/knowledge: The ...
dfn's user avatar
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1 answer
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Why do we ask Shimura datum to have Hodge weight $(-1,1),(0,0),(1,-1)$?

Why do we ask Shimura datum to have Hodge weight $(-1,1),(0,0),(1,-1)$? I know it's related to the decomposition of a complex Lie algebra $\frak{g}_{\mathbb{C}}=\frak{t}\oplus\frak{p}^{+} \oplus \frak ...
Marsault Chabat's user avatar
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183 views

Why does Deligne's construction of the Galois representation attached to the new cuspidal forms require that the Kuga-Sato manifold be regular?

The origin of this question is related to the construction of Galois representations of Deligne attached to $f$ a new cuspidal form (of weight $k\geq 2$). To do this, we consider the fiber product $k$-...
Marsault Chabat's user avatar
4 votes
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164 views

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 ...
Marsault Chabat's user avatar
1 vote
0 answers
143 views

Moduli interpretation for integral models of PEL Shimura variety at parahoric level?

Kottwitz has built canonical integral models for a large family of PEL Shimura varieties, associated to a certain reductive group $G$ over $\mathbb Q$, when the structure level has the form $K = K_pK^...
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5 votes
1 answer
327 views

modularity lifting theorems for non-compact unitary groups

I am reading David Geraghty's paper, 'Modularity lifting theorems for ordinary Galois representations'(https://link.springer.com/article/10.1007/s00208-018-1742-4) and I have a related question, which,...
Zhan's user avatar
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3 votes
1 answer
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Description of a Shimura variety

Let $(G, X)$ be a Shimura datum and let $U \subseteq G(\mathbb A_f)$ be an open compact subgroup. By the general theory of Shimura varieties, we get a corresponding algebraic variety $Y(U)$ defined ...
dekimashita's user avatar
7 votes
1 answer
804 views

Why are Shimura varieties the "right" objects?

So this is probably blasphemist to ask and I've resisted asking this for a while. Essentially my question is why are locally symmetric spaces/Shimura varieties the "right" object to study ...
curious math guy's user avatar
5 votes
0 answers
385 views

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 ...
Takahiro Matsuda's user avatar
3 votes
0 answers
177 views

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 ...
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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 ...
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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. ...
random123's user avatar
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21 votes
1 answer
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Cohomology of Shimura varieties and coherent sheaves on the stack of Langlands parameters

In Zhu's Coherent sheaves on the stack of Langlands parameters theorem 4.7.1 relates the cohomology of the moduli stack of shtukas to global sections of a certain sheaf on the stack of global ...
Anton Hilado's user avatar
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9 votes
0 answers
155 views

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 ...
Ryan Keast's user avatar
1 vote
0 answers
255 views

Shimura varieties which are not of abelian type but has a good modular description

Deligne's idea was that Shimura varieties should be understood as moduli space of motives(with extra structures). lot's of Shimura varieties of abelian type can be understood as moduli space of ...
ali's user avatar
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3 votes
1 answer
249 views

complement of "good reduction" points in p-adic shimura varieties

assume that $X$ is Siegel Shimura variety defined over $\mathbb{Z}_p$, you can take its p-adic formal completion $\mathfrak{X}$,and than take it's adic generic fiber $\mathcal{X}$ and get an adic ...
ali's user avatar
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10 votes
0 answers
165 views

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 $\...
David Loeffler's user avatar
6 votes
0 answers
301 views

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 $F=\mathbb{Q}(\sqrt{p})$. ...
xir's user avatar
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8 votes
1 answer
502 views

Artin reciprocity via Shimura varieties

The point of Shimura varieties, as far as I've understood it, is that for a given Shimura datum $(G,D)$, there exist models, by which I mean that for congruence subgroups $\Gamma$ there exists a ...
curious math guy's user avatar
5 votes
0 answers
137 views

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 ...
user330928's user avatar
2 votes
0 answers
127 views

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 ...
ali's user avatar
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1 vote
1 answer
373 views

About the type of a polarization of an abelian variety

The following is a question I posted about a week ago on Maths stackexchange there, but it didn't bring any discussion nor comment. For this reason I am posting it here also. Let $X$ be an abelian ...
Suzet's user avatar
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6 votes
1 answer
610 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-...
Asvin's user avatar
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4 votes
1 answer
211 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 ...
Pol van Hoften's user avatar
4 votes
1 answer
248 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 ...
sawdada's user avatar
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1 vote
0 answers
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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 ...
Chris's user avatar
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5 votes
0 answers
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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 $G$ ...
rleigh's user avatar
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11 votes
0 answers
724 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 ...
Catherine Ray's user avatar
1 vote
1 answer
234 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 ...
Qirui Li's user avatar
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16 votes
0 answers
740 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}_{\...
David Loeffler's user avatar
4 votes
0 answers
204 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}...
sawdada's user avatar
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16 votes
3 answers
2k 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 ...
A. Walker's user avatar
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0 answers
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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 ...
Jędrzej Garnek's user avatar
4 votes
0 answers
363 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 ...
Kim's user avatar
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