# Tagged Questions

**8**

votes

**0**answers

189 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

**1**answer

616 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

**4**answers

774 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

**2**answers

370 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

**1**answer

236 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

**1**answer

483 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

**1**answer

283 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

**2**answers

522 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

**1**answer

792 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

**1**answer

836 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

**0**answers

241 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

**0**answers

624 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

**2**answers

638 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

**1**answer

783 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 ...

**19**

votes

**2**answers

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

**2**answers

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 $p$. ...