All Questions
15 questions
3
votes
0
answers
91
views
Reference request: étale local system on $\Gamma\backslash\mathcal H$ for $\Gamma$ non-congruence
Suppose $\mathcal H$ is the upper half plane, and $\Gamma\subset \operatorname{PSL}_2(\mathbb Z)$ is a finite index subgroup. Then by Belyi's theorem we know that $\Gamma\backslash\mathcal H$ is an ...
- 775
2
votes
1
answer
243
views
$\pi$-adic Galois representations of attached to newforms at $p \nmid N$ are crystalline
Is [Scholl, Motives for modular forms, Theorem 1.2.4 (ii)] proven for any $p$ independent of the weight?
Concretely, let $f$ be a normalized eigenform of weight $w$. Let $p$ be a prime not dividing ...
user471019
2
votes
0
answers
229
views
Reference Request - New proof of Ribet's level lowering by Khare and Wintenberger
I'm currently following the note of Sug Woo Shin's course at Berkeley with notes taken by Rong Zhou. In Section 24.3 (Page 86), Ribet's level lowering theorem is stated:
[Theorem 24.7] $E = E_{a^{\...
- 639
3
votes
0
answers
152
views
Finiteness of points over the cyclotomic extension for modular forms
Let $\rho(f):G_\mathbb{Q} \rightarrow GL_2(K_f)$ be the Galois representation attached to some cuspidal modular form $f$ where $K_f$ is a finite extension of $\mathbb{Q}_p$.
Let $V_f$ be the vector ...
- 313
4
votes
1
answer
412
views
Motivations of families of modular forms, elliptic curves and Galois representations?
I want to know some reference, why do some number theorists study the families of the elliptic curves, modular forms or Galois representations? As far as I know, I always consider the Galois ...
user141691
6
votes
1
answer
380
views
Applications of Level Lowering
What are some applications/consequences of level lowering of Galois representations? I understand the application of Ribet's theorem in the proof of Fermat's last theorem but I am wondering what other ...
- 1,629
2
votes
1
answer
223
views
Level-Lowering in Weight 2
Let $N$ and $p$ be relatively prime integers with $p$ a prime. Suppose $f$ is a weight $k=2$ (normalized, cuspidal, etc) newform of level $\Gamma_1(N) \cap \Gamma_0(p)$. I seem to recall the existence ...
- 1,422
4
votes
2
answers
450
views
Is there a version of Serre's modularity conjecture for projective representations?
Serre's modularity conjecture asserts that a continuous odd irreducible representation $$\overline{\rho} : G_\mathbb{Q} \rightarrow \mathrm{GL}_2(\mathbb{F}_q)$$ must be modular, in the sense that ...
- 3,142
2
votes
1
answer
338
views
Finite Flat Group Schemes for Modular Forms of Higher Weight
Let $f$ be a newform (normalized, cuspidal) of weight $k \ge 2$. Then for a prime $\ell$ there is an $\ell$-adic Galois representation associated to $f$. If $k=2$, this comes from an abelian variety, ...
- 15.4k
1
vote
0
answers
162
views
Construction of RM abelian variety from eigenform
Let $f$ be a normalized eigenform of weight $2$ level $N$. If the Fourier coefficients of $f$ generate a totally real field $F$, then we associate to $f$ a system of $\ell$-adic Galois representations ...
- 15.4k
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
16
votes
0
answers
11k
views
Deligne's letter to Jean-Pierre Serre
I'm looking for another letter of Pierre Deligne, this time to Jean-Pierre Serre (from around 1974 I think), in which he proves that the Galois representation associated to a certain Hecke eigenform ...
8
votes
1
answer
464
views
Image of complex conjugation by modular representations in characteristic 2
The question I am going to ask looks well-known, and I even may have heard things about it (but since I used to be deaf to anything in characteristic 2, whatever I heard has never been recorded in my ...
- 26k
1
vote
1
answer
819
views
Reference for "Gal represenations attached to CM eigenforms"
I seem to recall that the construction of Gal representations associated to eigenforms with CM was done much before the general cases due to Eichler-Shimura, Deligne-Serre and Deligne. Was this done ...
- 659
4
votes
3
answers
1k
views
Companion forms
What is the best known result concerning the existence of companion forms for classical modular forms? Gross' tameness criterion paper is always mentioned with a "unchecked compatibility" caveat? ...
- 173