Questions tagged [shimura-varieties]
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157 questions
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Galois group of shimura varieties with different level structure
Let $(G,X)$ be a shimura data, and $K$ an open compact neat subgroup of $G(\mathbb A_f)$. Suppose $K'\subset K$ is open and normal, then, I see in many references that the finite etale cover $Sh(G,X)_{...
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Central isogeny, Shimura varieties and exceptional cases
For a simple complex Lie algebra $\mathfrak g$, its weight lattice is not equal to the root lattice (i.e. the center of its simply connected form is a non-trivial finite group) iff $\mathfrak g$ is of ...
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Descend local system to the canonical model of Shimura varieties
Suppose $(G,X)$ is a Shimura data, and $E$ be its reflex field. In page 33 of this paper, it constructs an etale local system on the canonical model $\mathrm{Sh}(G,X)_{K,E}$ (variety over $E$) for any ...
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What do we do when $G$ doesn't have a Shimura variety?
Let $G$ be a reductive group. If one can associate to $G$ a Shimura datum $(G,X)$, then the étale cohomology of the associated Shimura variety $\operatorname{Sh}(G,X)$ is a strong tool for the ...
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"Reflex field" for $\mathbb H/\Gamma$ for $\Gamma$ non-congruence
Suppose $\Gamma$ is a non-congruence arithmetic subgroup of $PGL_2(\mathbb Z)$, and $\mathbb H$ is the upper half plane of $\mathbb C$. Then by Belyi's theorem we know $\mathbb H/\Gamma$ is an ...
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Is $\mathcal H/\Gamma$ defined over a number field when $\Gamma$ is not congruent?
Let $\mathcal H$ denote the upper half plane of complex numbers, and $\Gamma$ an arithmetic subgroup of $\operatorname{SL}_2(\mathbb Z )$ (not necessarily congruent). I wonder whether $\mathcal H/\...
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When is a vector bundle on a Shimura variety an automorphic vector bundle?
Let $(G, X)$ be a Shimura datum, let $K \subset G(\mathbb{A}_f)$ be an open compact subgroup, and denote by $\text{Sh}_K(G,X)$ the Shimura variety whose complex points are given by $G(\mathbb{Q})\...
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Reference request for unitary Shimura varieties
Let $K$ be an imaginary quadratic field and let $X$ be the Shimura variety associated to the unitary group $U(m,n)$ over $K$ (after a suitable choice of PEL datum).
Is there a reference that explains ...
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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 ...
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Identification of different components of Hilbert modular surface?
I'm wondering whether the different components of the Hilbert modular surface can be (naturally?) identified with each other, or if they're at least abstractly isomorphic. (I'd also be interested in ...
- 2,054
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Why automorphism group of a Hermitian symmetric domain has trivial center?
Definition: A Hermitian symmetric domain is a Hermitian manifold that is connected, homogeneous, has a symmetry at some point (by homogenity hence every point), and has negative curvature.
I want to ...
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Divisors on product abelian fourfolds
Given a principally polarized abelian surface $A$ with CM of signature $(1,1)$ by an imaginary quadratic number field $K$, I am interested in studying the Néron-Severi group $\text{NS}(A\times A)$. ...
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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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Notion of "Hodge bundle" for abelian type Shimura varieties
For a Siegel type Shimura datum $(\text{GSp}_{2g}, \mathcal{H}^{\pm})$ and level $K$, we construct the Shimura variety $S_{g,K} := \text{Sh}_K(\text{GSp}_{2g},\mathcal{H}^{\pm})$. We have a universal ...
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Fundamental set for families of abelian varieties
I'm considering the universal family of principally polarized g-dimensional abelian varieties with level N-structure. Let's briefly recall the usual construction as a quotient of $\mathbb{C}^g \times \...
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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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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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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 ...
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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....
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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 ...
- 2,054
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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/\...
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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}& & \...
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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 ...
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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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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 $...
- 1,270
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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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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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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} $ ...
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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,...
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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 ...
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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, *)...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $k$-...
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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 ...
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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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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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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 ...
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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 ...
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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-...
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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 ...
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Are there motives which do not, or should not, show up in the cohomology of any Shimura variety?
Let $F$ be a real quadratic field and let $E/F$ be an elliptic curve with conductor 1 (i.e. with good reduction everywhere; these things can and do exist) (perhaps also I should assume E has no CM, ...
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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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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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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 ...
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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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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 ...
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