All Questions
28
questions
2
votes
1
answer
226
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 ...
- 1,994
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 ...
- 821
2
votes
0
answers
249
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/\...
- 1,352
3
votes
0
answers
152
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 ...
- 821
6
votes
1
answer
767
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....
- 1,352
4
votes
0
answers
209
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, *)...
- 1,352
3
votes
0
answers
181
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 ...
- 1,014
1
vote
0
answers
270
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 ...
- 1,053
3
votes
1
answer
261
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 ...
- 1,053
6
votes
0
answers
330
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 $F=\mathbb{Q}(\sqrt{p})$. ...
- 1,994
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 ...
- 1,053
6
votes
1
answer
651
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-...
- 7,716
4
votes
1
answer
254
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 ...
- 6,188
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}...
- 6,188
1
vote
0
answers
138
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 ...
- 411