# Questions tagged [shimura-varieties]

The shimura-varieties tag has no usage guidance.

144
questions

2
votes

0
answers

197
views

### 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 ...

2
votes

0
answers

166
views

### 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 ...

0
votes

0
answers

69
views

### 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}& & \...

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 ...

2
votes

0
answers

75
views

### 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 ...

2
votes

1
answer

198
views

### 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 ...

17
votes

3
answers

2k
views

### 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 ...

2
votes

0
answers

244
views

### 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/\...

3
votes

0
answers

146
views

### 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 ...

1
vote

1
answer

126
views

### 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),
...

2
votes

0
answers

189
views

### 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 ...

4
votes

0
answers

175
views

### 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} $ ...

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 ...

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 $...

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....

4
votes

0
answers

202
views

### 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
vote

0
answers

91
views

### 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 ...

2
votes

1
answer

196
views

### 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 ...

1
vote

0
answers

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 $k$-...

4
votes

0
answers

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 ...

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^...

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,...

3
votes

1
answer

355
views

### 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 ...

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 ...

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 ...

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 ...

4
votes

0
answers

134
views

### 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 ...

5
votes

0
answers

189
views

### 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. ...

21
votes

1
answer

2k
views

### 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 ...

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 ...

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 ...

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 ...

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 $\...

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 $F=\mathbb{Q}(\sqrt{p})$. ...

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 ...

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 ...

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 ...

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 ...

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-...

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 ...

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 ...

1
vote

0
answers

152
views

### 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 ...

5
votes

0
answers

165
views

### 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 $G$ ...

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 ...

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 ...

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}_{\...

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}...

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 ...

1
vote

0
answers

133
views

### 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 ...

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 ...