All Questions
22
questions
2
votes
0
answers
129
views
Isom-functor for generalized elliptic curves is representable
I am studying Deligne-Rapoport's 'Les Schémas de Modules de Courbes Elliptiques'. The following excerpt is from the proof of Theorem 2.5, Chapter III, page DeRa-61,
(page DeRa-61) (*) For $C_i$, ...
- 21
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
6
votes
1
answer
284
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
1
answer
352
views
Overconvergent modular forms and the level at $p$
I am a little bit confused about the basic theory of overconvergent modular forms, so here is a question that I think will be straightforward for those who know the theory but would help me a lot.
The ...
- 229
6
votes
1
answer
296
views
An explicit equation of the canonical morphism $X_1(N) \to X_0(N)$
I know there are some research about explicit equations for affine models in $\mathbb{A}^2$ of many modular curves over $\mathbb{Q}$, for example of $X_i(N), X(N)$ (where $i = 0, 1, 2$) for small $N$.
...
- 1,352
2
votes
0
answers
340
views
A computation of the rank of the Jacobian of a hyperelliptic curve over a number field using MAGMA
In this paper,
the authors says that, in order to show the rank of a Jacobian over $\mathbb{Q}$ is 0, they use the L function.
In the section 3.3, the authors compute the rank of the Jacobian of $X_1(...
- 1,352
2
votes
0
answers
270
views
Moduli interpretation and Ogg's notation for the cusps on modular curves
In Ogg's paper "rational points on certain elliptic modular curves", the author says, using Ogg's notation for cusps,
that for fiexed $d$, if $(y, N) = d$, then for any $x$ satisfying $(x, y,...
- 1,352
2
votes
1
answer
213
views
Global section of vertical differential 1 forms on universal elliptic curve
Let $B$ be a modular curve (of some level) over a number field $K$ (here, we implicitly assume that $K$ is large enough to make sense the phrase "$B$ is a $K$-variety"). Let $E\to B$ the ...
- 1,408
16
votes
0
answers
268
views
Why should an abelian variety with few places of bad reduction and a lot of endomorphisms not have many points?
In the paper "Points of Order 13 on Elliptic Curves" by Mazur-Tate, they say in the introduction:
It seemed ... that if such an abelian variety $J$, which has bad reduction at only one ...
- 7,716
4
votes
1
answer
334
views
Weight 3 modular form associated to singular abelian surfaces?
Given an extremal K3 surface $S$ over $\mathbb{Q}$ (i.e. a K3 surface with maximal Picard rank) there is a 2-dimensional Galois representation on the transcendental lattice $T(S)$, and an associated ...
- 1,701
2
votes
0
answers
150
views
Moduli interpretation of the integral anticanonical tower
This question is related to my reading of On torsion in the cohomology of locally symmetric varieties by Scholze.
In chapter $3$, using the theory of canonical subgroup, he produces Frobenius maps ...
- 149
4
votes
0
answers
231
views
Carayol's "ramified Eichler-Shimura relation" and its applications
In his paper "Sur la mauvaise reduction des courbes de Shimura" from '86 H. Carayol shows the following congruence relation:
Let $M$ be the tower of Shimura curves over a totally real $F$, associated ...
- 469
3
votes
1
answer
370
views
An explicit correspondence for reductions of modular curves $Y(N)$
Let $Y(N)$ be the modular curve associated with the principal congruence subgroup $\Gamma(N) \subset \mathrm{SL}(2, \mathbb{Z})$ of level $N \in \mathbb{N}$. It is well known that this curve has a ...
- 1,813
14
votes
0
answers
905
views
Relation between Igusa tower and $p$-adic modular forms
As the title suggests, my question is devoted to understand (and maybe get some good references) the relation between the Igusa tower for a modular curves and $p$, or maybe $T$-adic modular forms. I ...
- 445
4
votes
0
answers
403
views
quasi-finite group schemes
The following is what Mazur wrote on page 91 of his paper, Modular curves and the Eisenstein ideal, published in Publ. IHES in 1977, DOI: 10.1007/BF02684339 (freely available at eudml):
Let $m$ be an ...
- 335