All Questions
Tagged with galois-representations modular-forms
13 questions
15
votes
1
answer
983
views
When is the image of a 2-dim l-adic representation associated to a modular form open
I know the following theorems by Serre:
1, The 2-dim l-adic representation associated to a non-CM elliptic curve is open.
2, The 2-dim l-adic representation associated the weight-12 cusp form $\...
- 871
13
votes
1
answer
1k
views
Reference for: CM Hilbert Modular forms arise from Hecke characters
For classical modular forms, the correspondence between the form having CM by an imaginary quadratic field $K$ and it being induced from a Hecke character on $K$ is well-known. (Ribet's paper is a ...
- 659
27
votes
6
answers
5k
views
Where can I find a comprehensive list of equations for small genus modular curves?
Does there exist anywhere a comprehensive list of small genus modular curves $X_G$, for G a subgroup of GL(2,Z/(n))$? (say genus <= 2), together with equations? I'm particularly interested in genus ...
- 10.5k
8
votes
2
answers
1k
views
When do the Galois reps of modular forms have open image?
Suppose f is a newform (with coefficients generating some number field E), and $\rho_{f,\lambda}: {\rm Gal}(\overline{\mathbb{Q}} / \mathbb{Q}) \to {\rm GL}_2(E_\lambda)$ the associated Galois rep (...
- 37k
23
votes
3
answers
4k
views
Subgroups of GL(2,q)
I have been wondering about something for a while now, and the simplest incarnation of it is the following question:
Find a finite group that is not a subgroup of any $GL_2(q)$.
Here, $GL_2(q)$ is ...
- 295
22
votes
1
answer
3k
views
The difficulties in proving modularity lifting theorems over non-totally real fields
First of all, let me apologize in advance for the terseness of this question.
It seems that by now there are well-developed techniques (the "Taylor-Wiles-Kisin" method) for proving modularity lifting ...
- 13.1k
17
votes
1
answer
2k
views
Representations attached to p-adic modular forms
A theorem of Gouvea and Hida (or rather a consequence of it) states that there exist a Galois representation attached to a $p$-adic eigenform $f$ provided the residual representation attached to a ...
- 995
11
votes
1
answer
646
views
Modularity of higher genus curves
The modularity conjecture for elliptic curves over number fields is well known, and indeed, is a theorem for all elliptic curves over $\mathbb{Q}$, and at least potentially, over any CM field.
What ...
- 1,267
8
votes
2
answers
1k
views
Field generated by the Fourier coefficients of a modular form
Let $f = \sum_n a_n q^n$ be a cuspidal newform of weight $k$ on $\Gamma_0(N)$ for some $N$. Let $K_f$ be the number field generated by the $a_q$ as $q$ runs over all primes.
My question: if we ...
- 115
7
votes
3
answers
2k
views
Free subquotient of Galois representations coming from Hida theory
Let $\mathbf{T}$ be the reduced nearly ordinary Hecke algebra of level $N$ of Hida theory for $\operatorname{GL}_{2}$ over $\mathbb{Q}$ (or more generally over a totally real field $F$). Then $\mathbf{...
- 10.9k
7
votes
1
answer
914
views
Are coefficients of Maass forms of eigenvalue 1/4 known to be algebraic?
I would really like to know whether the following famous conjecture has been solved. I've read in a few places that it has been solved, but I have been unable to find a reference. I do know that ...
- 18.7k
7
votes
1
answer
368
views
How large is Dcris of certain twists of modular forms?
I want to determine $\mathrm D_{\mathrm{cris}}$ of certain twists of the Galois representations attached to modular forms. For one particular twist it is not clear to me how $\mathrm D_{\mathrm{cris}}$...
- 734
2
votes
1
answer
294
views
About the restriction of a modular representation to a decomposition subgroup
Let $f$ be an eigenform of level $\Gamma_1(N)$ and let $p$ be a prime that does not divide $N$. It is well know that there is a $2$-dimensional representation
$$
\rho_f \colon G_{\mathbb Q} \to \...
- 21