8
votes
0answers
179 views

congruences of level 1 and level p modular forms

I've been carrying out some experiments on the computer and I noticed the following congruence phenomenon: fixing a prime $p$, it seems that any modular form over $SL_2(\mathbb{Z})$ and of weight $k ...
6
votes
1answer
611 views

Is Gouvêa-Mazur's “Infinite Fern” a fractal?

[EDIT]: Following Qiaochu Yuan's comment, it is better to clarify that I do not know what the right definition of a fractal in the following question should be. But a nice answer might contain such a ...
9
votes
4answers
720 views

Concrete examples of noncongruence, arithmetic subgroups of SL(2,R)

A subgroup of $SL_2(\mathbb{R})$ is called arithmetic if it is commensurable with $SL_2(\mathbb{Z})$. An arithmetic subgroup is called congruence if it contains a subgroup of type $\Gamma(N)$ for ...
2
votes
2answers
359 views

branch points of modular parametrization of an elliptic curve

Let $E$ be an elliptic curve over a number field K. Then there is a morphism $\phi:X_0(n) \to E$. Consider composition $f:X_0(n)\to \mathbf{P}^1_K$, where we compose with degree 2 cover $E\to ...
1
vote
1answer
233 views

Manin-Drinfeld and constructing a finite morphism with two given ramification points

Fix a smooth projective connected curve $X$ over $\overline{\mathbf{Q}}$ of genus $g\geq 1$ and distinct points $x,y \in X$ such that $x-y$ has infinite order in the Jacobian. Can we always find a ...
4
votes
1answer
471 views

References for bad reduction of Jacobians of modular curves?

Hi, Where can I learn about the reduction of the Jacobians of modular curves such as X_0(N) and X_1(N) at primes p dividing N? Thanks!
3
votes
1answer
277 views

parabolic-Eisenstein decomposition of cohomology of modular curve

Hi, Fix $N > 3$ and consider the modular curve $X(N)$ parametrizing elliptic curves with full level N structures. Let $\pi : E(N)\to X(N)$ be the universal elliptic curve. Then $V=R^1\pi_*\mathbf ...
7
votes
2answers
512 views

Hecke algebra generated by a single element

Let $\mathbb{T}_{\mathbb{Z}}$ be a $\mathbb{Z}$-module generated by Hecke operators $T_n$ acting on the space of cups forms $S_{k}(\Gamma,\mathbb{C})$ for the congruent subgroup satisfying ...
6
votes
1answer
775 views

Status of Ihara's lemma for Shimura curves over totally real fields?

What is the status of Ihara's lemma for Shimura curves over totally real fields? In particular, why is it not implicit in the exact sequence of Rajaei, "On the levels of mod $l$ Hilbert modular forms" ...
10
votes
1answer
802 views

Construction of abelian varieties from Hilbert modular forms?

Some experts tell me that the construction of abelian varieties from Hilbert modular forms is an (apparently difficult) open problem. However, in view of the construction of $l$-adic Galois ...
2
votes
0answers
235 views

Why is the Eisenstein quotient a quotient of the new part of the Jacobian?

Dear MO Community, Let $X = X_0(N)_{/\mathbb{Q}}$, and $J$ its jacobian. Mazur defines the Eisenstein quotient of $J$, denoted $\widetilde{J}$, as \[ 0 \rightarrow \gamma_IJ \rightarrow J ...
6
votes
0answers
608 views

Can you get Siegel's theorem “for free” from modularity and Mazur's Eisenstein Ideal paper?

There is a well-known theorem of Shafarevich that given a finite set $S$ of primes the number of isomorphism classes of elliptic curves over $\Bbb Q$ with everywhere good reduction outside $S$ is ...
6
votes
2answers
616 views

Rankin-Selberg convolutions of motivic L-series

Background: Let $M_{f_i}, i=1,2$ be two modular motives associated to cusp forms $f_i \in S_{w_i}(\Gamma_0(N_i))$ of weight $w_i$ and level $N_i$ respectively. The Rankin-Selberg convolution ...
13
votes
1answer
765 views

Eisenstein series as sections of line bundles on moduli spaces

It is well known that a modular form of weight k and level \Gamma is a global section of k-power of a Hodge line bundle over some modular curve. e.g. H^0(X,E^k). My question is How to characterize ...
17
votes
2answers
3k views

Intuition behind the Eichler-Shimura relation?

The modular curve $X_0(N)$ has good reduction at all primes $p$ not dividing $N$. At such a prime, the Eichler-Shimura relation expresses the Hecke operator $T_p$ (as an element of the ring of ...
12
votes
2answers
1k views

Galois representations attached to newforms

Suppose that $f$ is a weight $k$ newform for $\Gamma_1(N)$ with attached $p$-adic Galois representation $\rho_f$. Denote by $\rho_{f,p}$ the restriction of $\rho_f$ to a decomposition group at $p$. ...