Questions tagged [shimura-varieties]
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144
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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, ...
29
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0
answers
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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 ...
24
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2
answers
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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 ...
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 ...
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3
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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 ...
16
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3
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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 ...
16
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0
answers
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}_{\...
15
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3
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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 ...
15
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1
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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 ...
13
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1
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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 ...
12
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1
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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 ...
11
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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 ...
11
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1
answer
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higher dimensional analogues of the Manin-Drinfeld theorem
The Manin-Drinfeld theorem asserts that a divisor on the compact modular curve $X_0(N)$ which is supported on the cusps is torsion.
Equivalently, if $Y_0(N)$ is the open modular curve, the mixed ...
11
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1
answer
757
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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 ...
11
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0
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724
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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 ...
11
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0
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314
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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-...
10
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1
answer
558
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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 ...
10
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1
answer
368
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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.
...
10
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1
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328
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Shimura surfaces that do not contain a Shimura curve
Let $S$ be a Shimura surface i.e. a Shimura variety with $dimS=2$. Does $S$ necessarily contain a Shimura curve? I know that probably the answer is No, but do not have an explicit example. What is the ...
10
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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 $\...
9
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Adelic formulations of complex multiplication and modular curves
In modular curves and modular forms, there is an adelic formulation, in which smaller open subgroups of some adelic group relate to higher level structure. As we know, higher level structure ...
9
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1
answer
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Automorphisms of Generic Abelian Varieties
Automorphism groups of elliptic curves are very well understood. Of course, every elliptic curve has the automorphism $[-1]$ of order $2$. If we are over a (algebraically closed) field, this is the ...
9
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0
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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 ...
9
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0
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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, ...
8
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2
answers
1k
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modularity of algebraic varieties
Hello,
Are there any examples of varieties which are not Shimura varieties or abelian varieties
and whose L-functions have been shown to be a product of automorphic L-functions?
Thanks.
N
8
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1
answer
503
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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 ...
8
votes
1
answer
966
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Is the Galois x Hecke action on cohomology of Shimura varieties semi-simple?
Given a reductive group $G/\mathbf Q$ (+ additional data), and a compact open subgroup $K\subset G(\mathbf A^\infty)$, there is a standard construction that produces a Shimura variety $S$ and if we ...
8
votes
2
answers
620
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Non emptyness of ordinary locus for PEL type Shimura varieties
We let $B$ be a simple algebra over $\mathbb Q$, with the usual notations for PEL type Shimura varieties. In his paper "Ordinariness in good reductions of Shimura varieties of PEL-type" (available ...
8
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1
answer
726
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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 ...
8
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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 ...
7
votes
2
answers
508
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Zograf's bound on the index of a modular curve for Shimura curves
I've been reading Voight's paper on Shimura curves and it prompted the following question; see http://www.cems.uvm.edu/~voight/articles/shimura-clay-proceedings-071707.pdf for which notes I'm talking ...
7
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2
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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 ...
7
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State of the art for integral models of PEL type Shimura varieties with deep level structure
The theory of PEL type Shimura varieties is nowadays well developed, but it is not easy to be updated with the latest results. Here I am particularly interested in integrals models. Let me describe ...
7
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1
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356
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p-rank stratification in unitary Shimura variety
Let $K$ be a quadratic extension of $\mathbb Q$ and let $p \neq 2$ be a prime that is inert in $K$. Let $X$ be the Shimura variety associated to the unitary group $\operatorname{U}(2,1)$ over $K$ (...
7
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1
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808
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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 ...
7
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430
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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 $E$ ...
6
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1
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612
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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-...
6
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1
answer
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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....
6
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1
answer
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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 ...
6
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1
answer
681
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different Shimura data with common underlying group?
A pure Shimura datum is of the form $(G,X)$ with $G$ a connected reductive $\mathbb{Q}$-group, and $X$ a homogeneous space under $G(\mathbb{R})$, subject to Deligne's conditions in terms of Hodge ...
6
votes
1
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459
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monodromy of Gauss-Manin over a Shimura variety
This is probably a difficult question. I would like to understand some particular cases and get some references. The rough question is the following:
Let $X$ be a PEL Shimura variety and $\pi: \...
6
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0
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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 $F=\mathbb{Q}(\sqrt{p})$. ...
6
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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 ...
6
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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 ...
6
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0
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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}, ...
5
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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,...
5
votes
1
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488
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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 ...
5
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1
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387
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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 ...
5
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Generalizing Eichler-Shimura to higher dimension, again
This question is related to
Intuition behind the Eichler-Shimura relation?
and
L-functions and higher-dimensional Eichler-Shimura relation
Answering the first question above, Matt Emerton gives a ...
5
votes
1
answer
585
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What is the image of complex conjugation under Siegel Galois representations?
Let $G$ be the reductive group $\operatorname{GSp}_{4}$. Let $\pi$ be a smooth admissible cuspidal representation of $\operatorname{GSp}_{4}(\mathbb{A}^{(\infty)})$ of dominant weight. Assume, for ...