All Questions
12 questions
3
votes
1
answer
131
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Depth of the filtration of higher ramification groups in the ramified case in Serre's modularity conjecture
I am studying Serre's paper "Sur les représentations modulaires de degré 2 de $\mathrm{Gal(\overline{\mathbb Q}/\mathbb Q)}$" and I have some questions about Serre's definition of "peu ...
8
votes
1
answer
567
views
Symmetric power lift of modular forms
Let $f_1$ and $f_2$ be two cuspforms of weights $k_1$ and $k_2$ and nebentypus $\epsilon_1$ and $\epsilon_2$ respectively such that $f_1 \neq f_2 \otimes \chi$ for some Dirichlet character $\chi$ of ...
- 424
5
votes
2
answers
237
views
Irreducibility of the $n$th symetric power of the reduction of the Galois representation of a non-CM newform
In "On $\ell$-adic representations attached to modular forms II", Ribet proved that the $\ell$-adic representation $\rho_{f,\ell}$ attached to a non-CM newform form $f$ satisfies
$${\rm SL}...
- 303
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
5
votes
1
answer
276
views
Properties of Mod $\ell^m$ Galois representation associated to modular form
(Sorry for my poor english..)
Let $F(z)\in S_{2k}(SL_2(\mathbb{Z})$) be a newform and $\ell$ be a prime larger than $3$. Let $K$ be a some number field and $v$ be a prime of $K$ over $\ell$. Let $K_v$...
- 661
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
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
3
votes
1
answer
529
views
Conductor of Galois representation attached to newform
(Sorry for poor my english skill..)
Let $k$ and $N$ be positive integers and $\chi$ be a Dirichlet character modulo $N$. Let $F$ be a newform with number field $K_{F}$. (All coefficients of $F$ in $...
- 661
5
votes
1
answer
306
views
Galois representation associated to CM-newforms
Let $f(z)=\sum_{n\ge 1}a(n)e(nz)$, be a newform of CM-type, and let $\psi_f$ be the associated Hecke character, so that,
$$
f(z)=\sum_{\mathfrak{a}}\psi_f(\mathfrak{a})e(N(\mathfrak{a})z),
$$
and let ...
- 400
3
votes
1
answer
510
views
$\ell$-conductor of a two-dimensional $\ell$-adic Galois representation
Let $\ell$ be a prime number, denote by $K_\ell$ the maximal algebraic extension of $\Bbb{Q}$ ramified only at $\ell$. Let $f = \sum a_n q^n$ be a Hecke eigenform of level $1$ with integer ...
- 31
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