All Questions
16 questions
6
votes
1
answer
305
views
Definition of modular curve associated to $\Gamma(N)$
For a positive integer $N$, we define $$\Gamma(N)=\big \{ \begin{bmatrix} a & b \newline c & d\end{bmatrix}\in \operatorname{SL}_2(\mathbb{Z}): \begin{bmatrix} a & b \newline c & d\end{...
- 821
3
votes
0
answers
96
views
Discrete subgroups of $\text{Sp}_{4}(\mathbb{Q})$ parameterizing polarized Abelian surfaces plus torsion data
I want to start by considering a familiar congruence subgroup of the integral symplectic group $\text{Sp}_{4}(\mathbb{Z})$. For a positive integer $N$, let $\Gamma _{0}^{(2)}(N) \subset \text{Sp} _{4}(...
- 1,701
4
votes
0
answers
136
views
Eichler Shimura in higher genera
Let $\mathcal{M}$ denote the moduli stack of smooth elliptic curves over $\mathbb{C}$. There is a local system, $\mathcal{H}$, on $\mathcal{M}$ with fibre $H^{1}(E,\mathbb{C})$ at the $\mathbb{C}$-...
user108998
17
votes
1
answer
745
views
Special fiber of $X(p)$ in characteristic $p$
Let $p \geq 5$ be a prime. Let $Y(p)$ be the fine moduli space representing elliptic curves + basis of the $p$-torsion over $\mathbb{Q}_p$ and let $Y_0(p)$ be the fine moduli space representing ...
- 1,109
7
votes
1
answer
957
views
Modular curve X(2)
Let $\mathfrak{M}(2)$ be the algberaic stack over $\mathbb{Z}[1/2]$ which classifies the elliptic curves with the two level structure and let $X(2)$ be the coarse moduli space of $\mathfrak{M}(2)$ ($X(...
- 1,066
1
vote
1
answer
426
views
Complex plane mod lattice to elliptic curve correspondence generalization
If we observe the correspondence
$$\mathbb{C}/\Lambda \rightarrow E: Y^{2} = X^{3} - \frac{g_{2}(\Lambda)}{4}X - \frac{g_{3}(\Lambda)}{4},$$
we see the relationship between weight 4 and weight 6 ...
- 1,441
3
votes
1
answer
372
views
Special fibre of the modular curve $X(N)$
Let $N$ be an integer $\geq 3$ and $X(N)\rightarrow \mathrm{Spec } \mathbb{Z}[1/N]$ is the projective smooth modular curve defined in Deligne-Rappoport. Is there an exemple of $N$ for which the ...
- 1,066
5
votes
2
answers
674
views
Why is the supersingular locus the zero locus of a modular form?
This question is related to my other question here: Examples of subspaces singled out by modular forms.
Here I am wondering if there is a philosophical explanation about why the supersingular locus ...
- 231
6
votes
1
answer
815
views
Is there an elliptic surface over $Y(1)$?
Actually I have a few related questions.
Here, by $Y(1)$ I mean the affine $j$-line $\text{SL}_2(\mathbb{Z})\backslash\mathcal{H}$.
I know $Y(1)$ is only a coarse moduli space, so there isn't a ...
- 10.7k
1
vote
1
answer
226
views
coarse moduli space $X(2)$
Let $\mathfrak{M}(2)$ be the algebraic stack over $\mathbb{Z}[1/2]$ which classifies the elliptic curves with the $\Gamma(2)$ level structure and let $M(2)$ be its coarse moduli space. Is there an ...
- 1,066
5
votes
1
answer
752
views
Some help in digesting a paragraph in the introduction of Deligne/Rapoport's "Les Schemas de Modules de Courbes Elliptique"
http://www.springerlink.com/content/04x54gr171v556m4/fulltext.pdf
On page 149 (DeRa-7), in the middle of the page, I can translate the middle paragraph that starts "3. La surface de Riemann ..." as ...
- 10.7k
3
votes
1
answer
229
views
On Universal Abelian surfaces over a Shimura curve.
Let ${\cal O}, {\cal O}'$ be two order in ${\mathrm M}_2({\Bbb R})$ that are sets of all $2 \times 2$ matrices over real number ${\Bbb R}$. Assume that we have the relation ${\cal O}' = a{\cal O}a^{-1}...
- 181
5
votes
1
answer
412
views
Modular curve parametrizing two cyclic subgroups of an elliptic curve
The aim of this question is to better understand the following moduli space/modular curve, for which I propose (temporarily) the name $Y_0(M,N)$. We define $Y_0(M,N)$ as the moduli space parametrizing ...
3
votes
0
answers
154
views
Examples of subspaces singled out by modular forms
I am wondering what subspaces of modular varieties defined as the zero locus of modular forms have been studied in the literature.
To be more clear let me explain the example I have in mind.
Let $N\...
- 231
1
vote
1
answer
174
views
reference request for the finiteness of cuspidal subgroup of $X_0(N)$?
I've seen stated offhand in many sources that the cuspidal subgroup of the Jacobian of $X_0(N)$ is finite.
Do they mean that the subgroup of the jacobian generated by $\mathbb{Q}$-rational cusps is ...
- 10.7k
2
votes
0
answers
227
views
Possible slope for a modular form
Let $f\colon \mathbb{H}_g \to \mathbb{C}$ be a Siegel modular form of weight $k$ with respect to $\Gamma_g$.
Then, $f$ admits a Fourier expansion $f(Z) = \sum_T a(T) \exp(i\pi \mathop{tr} TZ)$, where $...
