All Questions
Tagged with galois-representations modular-forms
26 questions with no upvoted or accepted answers
23
votes
0
answers
832
views
Eichler-Shimura over Totally Real Fields
By Eichler-Shimura over totally real fields I mean the following conjecture.
Conjecture. Let $K$ be a totally real field. Let $f$ be a Hilbert eigenform with rational eigenvalues, of parallel weight $...
- 3,142
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 ...
9
votes
0
answers
596
views
Tameness criterion in the reducible case
Dear MO,
This is a follow up to a previous question here in MO, but I will make this question self-contained for convenience. Those already familiar with the following paper [G] by Gross can safely ...
- 1,694
7
votes
0
answers
309
views
List of applications of the Deformation theory of Galois Representations in contexts other that $R=T$
The Deformation theory of Galois representations developed by Mazur has been extensively used in proving that Galois representations with special properties are modular/automorphic through the ...
user130124
6
votes
0
answers
142
views
Congruence between modular forms
This might be a very vague question since I am not very familiar with the theory of automorphic forms. Let $G$ be a connected reductive algebraic group defined over $F$ (a number field). Suppose we ...
- 453
5
votes
0
answers
195
views
Is there some computational evidence of the $pq$ analog of Serre's conjecture?
The $pq$ analog of Serre's conjecture (see "Mod pq Galois representations and Serre's Conjecture"- Khare, Kiming) states that if $\bar{\rho}_1:G_{\mathbb{Q}}\rightarrow \text{GL}_2(\mathbb{F}_p)$ is a ...
user130124
5
votes
0
answers
174
views
Is there a conjectural analog of Ribet's theorem (Converse to Herbrand's Theorem) for Imaginary Quadratic fields?
For $p$ a prime, let $Cl(\mathbb{Q}(\mu_p))$ denote the class group of the extension of $\mathbb{Q}$ obtained by adjoining a primitive $p$th root of unity. Associated to an eigenform of weight 2 and ...
user130124
5
votes
0
answers
512
views
Benedict Gross's paper on companion forms
In the page 458 of his paper(A tameness criterion for Galois representations associated to modular forms), Gross wrote the following
"A detailed analysis of $U_p(Af)+V_p(<p>f)$ shows that it
...
- 51
4
votes
0
answers
238
views
Diophantine consequences of the Buzzard–Diamond–Jarvis conjecture
Serre's modularity conjecture famously implies Fermat Last Theorem. More generally, Serre's conjecture implies that certain generalized Fermat equations have no non-trivial solutions (see Section 4.1 ...
- 595
4
votes
0
answers
303
views
Criterion for constancy in Mazur's Eisenstein ideal paper
In the paper of Mazur, Modular curves and the Eisenstein ideal (1977), he showed the following (at page 57):
Lemma (3.4) (Criterion for constancy) Let $G$ be an etale admissible $p$-group over $\text{...
- 335
4
votes
0
answers
279
views
On De Shalit's Lemma in Wiles' proof of R=T
In Wiles' proof of ``$R = {\Bbb T}$", if we associate the extra primes the set of which is denoted $D$, there is a famous result in the following$\colon$
Theorem(De Shalit's Lemma). Let ${\Bbb T}_N$ ...
- 781
4
votes
0
answers
330
views
Local splitting of modular Galois representations as $p$ varies
If $f$ is a classical eigenform of weight $\geq 2$ and ordinary at distinct, odd primes $p$ and $q$ which do not divide the level is it true that the restriction (as a $q$-adic representation) $\rho_{...
- 659
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
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
3
votes
0
answers
177
views
A question on a paper by Ribet
I'm reading the article On the equation $a^p + 2^\alpha b^p + c^p = 0$ by Ribet (http://math.berkeley.edu/~ribet/Articles/acta.pdf), but I'm having trouble understanding his proof of Theorem 3. For ...
- 31
3
votes
0
answers
285
views
modular forms with lots of companion forms
Let $f$ be a modular form (say of weight 2) and let $S$ be the set of primes $p$, the reductions modulo $p$ of $\rho_{f,p}$
restricted to $G_p$ splits. Equivalently, $S$ is the set of prime $p$ such ...
- 818
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
2
votes
0
answers
108
views
Deforming Modular Symbols
This is probably a silly question, as I don't really know a whole lot about modular symbols over arbitrary rings.
How do modular symbols over a finite field square with Katz modular forms? If they ...
user130124
2
votes
0
answers
212
views
Local components of quaternionic modular forms
Let $D$ be a totally definite quaternion algebra over a totally real number field $F$. Let $U$ be an open compact subgroup of $D(\mathbb{A}_F)^\times$, maximal compact almost everywhere. Consider ...
- 469
2
votes
0
answers
176
views
Residue fields of attached to coefficients of modular forms
Let $f = \sum_n a_n q^n$ be a cuspidal newform of some weight and level. Here I want to view the $a_n$ of $p$-adic numbers (by embedding $\overline{{\bf Q}}_p$ in ${\bf C}$ in some way).
Let $k_f$...
- 115
1
vote
0
answers
76
views
Global minimal discriminants of elliptic curves and Galois representations
Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $\Delta$ be the global minimal discriminant. By A. Wiles et al. $E$ is modular so that there is a corresponding modular form $f$. I am wondering if ...
- 1,428
1
vote
0
answers
254
views
Why does Deligne's construction of the Galois representation attached to the new cuspidal forms require that the Kuga-Sato manifold be regular?
The origin of this question is related to the construction of Galois representations of Deligne attached to $f$ a new cuspidal form (of weight $k\geq 2$). To do this, we consider the fiber product $k$-...
- 1,270
1
vote
0
answers
178
views
Compute dimension of space of modular forms by counting Galois representations
It is known that we can compute the dimension of the space $S_{k}^{\mathrm{new}}(N, \chi)$ of new forms of weight $k\geq 2$ and level $N$ and Nebentypus $\chi$ via Riemann-Roch theorem or using ...
- 2,215
1
vote
0
answers
2k
views
Necessary and sufficient condition for a prime to be represented by an arbitrary positive definite binary quadratic form $ax^2+bxy+cy^2$
Given an arbitrary (but fixed) positive definite primitive integral binary quadratic form $g(x, y)=ax^2+bxy+cy^2$, and let $m$ be an arbitrary integer. We will denote the discriminant of $g$ by $D=D_g=...
1
vote
0
answers
138
views
Hilbert modular form as a representation of Hecke algebra
I am reading some notes by Snowden and I don't understand a sentence.
Clearly, if we have an appropriate $R = T$
theorem then we get a modularity lifting theorem, as $\rho$ defines a homomorphism ...
- 11
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