All Questions
Tagged with galois-representations modular-forms
104 questions
15
votes
2
answers
994
views
Explicit Chebotarev and Langlands - irreducibility of X^5-X-1 mod primes
Is there an explicit infinite set of primes, modulo which $X^5 - X - 1$ is irreducible?
Since our polynomial's Galois group over $\mathbb{Q}$ is $S_5$, Chebotarev's density theorem implies that there ...
- 11.3k
6
votes
1
answer
412
views
Computing an eigencuspform in $S_2(\Gamma_0(1776))$
Consider
$$\bar{\rho}:G_{\mathbb Q}\longrightarrow\operatorname{GL}_2(\mathbb F_7)$$
the residual 7-adic Galois representation attached to the elliptic curve $y^2=x^3+x^2-4x-4$ of conductor 48. Then $...
- 10.9k
11
votes
1
answer
1k
views
Hodge–Tate structures of modular forms
The title refers to the paper of Faltings:
Hodge-Tate structures and modular forms.
Math. Ann. 278 (1987), no. 1-4, 133–149.
The main theorem in the paper says that the associated Galois rep to a ...
- 111
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
3
votes
1
answer
223
views
Some clarifications regarding Deligne's paper on $\ell$-adic representations arising from modular forms
I've posted this question few days ago on math.stackexchange because it seems quite superficial. However, since I've got no responses at all, I'm posting it here. If the question is not suitable, ...
- 2,227
11
votes
1
answer
762
views
Eichler-Shimura congruence
I'm trying to understand the Eichler-Shimura congruence which relates the Hecke operator $T_p$ to Frobenius at $p$ in characteristic $p$.
Two possible ways to compute $T_p$ mod $p$ seem to be:
A) ...
- 113
8
votes
1
answer
752
views
Adelic open image for modular forms?
There's a famous theorem of Serre that if $E$ is a non-CM elliptic curve over $\mathbf{Q}$, and $\rho_{E, \ell} : Gal(\overline{\mathbf{Q}}/{\mathbf{Q}}) \to GL_2(\mathbf{Z}_\ell)$ is its $\ell$-adic ...
- 37k
8
votes
2
answers
782
views
Field of definition of Galois representations of weight 1 modular forms
Let $f$ be a weight 1 modular form (let's say cuspidal, new, normalized, and a Hecke eigenform). Then there's an associated Artin representation $\rho_f: \operatorname{Gal}(\overline{\mathbf{Q}} / \...
- 37k
2
votes
1
answer
198
views
About the restriction of a modular representation to a decomposition subgroup II
This question is a variant of this one.
Let $f$ be as in the other question, but suppose that we look at the $\ell$-adic representiation attached to $f$:
$$
\rho_f : G_{\mathbb Q} \to \operatorname{...
- 77
2
votes
1
answer
294
views
About the restriction of a modular representation to a decomposition subgroup
Let $f$ be an eigenform of level $\Gamma_1(N)$ and let $p$ be a prime that does not divide $N$. It is well know that there is a $2$-dimensional representation
$$
\rho_f \colon G_{\mathbb Q} \to \...
- 21
12
votes
2
answers
2k
views
Modularity theorem for abelian varieties
There's alredy two posts on MO about the extension of modularity to elliptic curves over fields other than $\mathbb{Q}$ ([1], [2]), and another one about general algebraic varieties [3].
What is ...
- 17.6k
10
votes
1
answer
618
views
Universal deformations of modular Galois representations
Let $\bar\rho$ be an odd, absolutely irreducible, 2-dimensional mod $p$ representation of $\operatorname{Gal}(\overline{\mathbf{Q}} / \mathbf{Q})$ (with coefficients in some finite extension $k / \...
- 37k
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
7
votes
1
answer
1k
views
Properties of representations attached to p-adic modular forms
I found an old MOF post about representations attached to p-adic modular forms: Representations attached to p-adic modular forms and I have some follow up questions on the same topic.
If we have a ...
- 621
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
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
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
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
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
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
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
10
votes
2
answers
961
views
What is a(n algebro-geometric) family of modular forms?
We know that a family of elliptic curves is a morphism of schemes $f:X \to Y$ such that the fiber of every point of $Y$ is an elliptic curve (and we usually require the morphism to be smooth, proper, ...
- 15.4k
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
5
votes
1
answer
546
views
Level lowering for weight 1 forms
I'm interested in knowing to what extent is level lowering known to hold in weight 1. Specifically, let's say I have an eigenform $f$ in $S_1(N,\chi)$ and a prime $p$ which doesn't divide $N$. Let's ...
- 1,680
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
4
votes
1
answer
1k
views
Duals and Tate twists of Galois representations attached to modular forms.
Let $f$ be a normalized newform of weight $k\geq 2$, $p$ a prime, and $V$ the associated Galois representation (with coefficients in a finite extension $K$ of $\mathbf{Q}_p$ with ring of integers $\...
- 3,380
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
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
28
votes
1
answer
3k
views
Are there Maass forms where the expected Galois representation is $\ell$-adic?
Recall that by theorems of Deligne and Deligne--Serre, there is the following dichotomy:
Modular forms on the upper half plane of level $N$ and weight $k\geq 2$ correspond to representations $\rho:\...
- 18.7k
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
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
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
4
votes
1
answer
572
views
Explicitly computing D(V)^{psi=1} for (phi,Gamma)-modules
In the paper Construction of some families of 2-dimensional crystalline representations by Berger-Li-Zhu, a very explicit description is given for the $(\phi,\Gamma)$-module attached to some 2-...
- 173
0
votes
1
answer
329
views
On the image of the residual representation attached to a CM form
It is a classical result of Ribet that if an eigenform has CM the its residual projective image is "small" (cyclic or dihedral.) Is the converse true, i.e, if f is a form whose associated residual Gal ...
- 659
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
4
votes
2
answers
453
views
Number of modular lifts with prescribed parameters
Let $\bar{\rho} : Gal(\bar{\mathbb{Q}}/ \mathbb{Q} ) \rightarrow GL_2(\bar{\mathbb{F}}_p)$ be an odd, irreducible Galois representation mod $p$ which is unramified outside $S$, where $S$ is a finite ...
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
5
votes
2
answers
944
views
Elliptic curves over $\mathbf{Q}$ with isogenous mod $\ell$ reductions, for several $\ell$
Characteristic polynomials of Hecke operators $T_\ell$, with $\ell$ prime, acting on cusp forms $S_k$ of level one and weight $k$ "appear to be" squarefree (even irreducible!).
This can be ...
- 2,406
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
6
votes
2
answers
590
views
Intersection of field extensions of torsion points of non-isogenous elliptic curves
Let $E$ and $E'$ be non-isogenous elliptic curves over a field $k$ (characteristic 0) such that $Gal(k(E[p^{\infty}])/k)=Gal(k(E'[p^{\infty}])/k) = SL_2(\mathbb{Z}_p)$ with $p \geq 5$ (where $E[p^{\...
- 1,905
14
votes
2
answers
3k
views
Why is there a weight 2 modular form congruent to any modular form
I got my copy of Computational Aspects of Modular Forms and Galois Representations in the mail yesterday. The goal of the book is "How one can compute in polynomial time the value of Ramanujan's tau ...
- 4,593
1
vote
1
answer
615
views
Galois Cohomology maps
Let $\bar{\rho}$ be a residual ordinary and locally split galois representation associated to a weight $k$ level $1$ form (more specifically a mod $p$ companion form). In the sense of deformation ...
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
3
votes
2
answers
400
views
Why does $H^1(G_p/I_p,\mathbb{F}(\delta\epsilon^{-1}))$ vanish?
For a finite field $\mathbb{F}$ of char $p$ $\geq 2$, let $f$ be a normalised eigenform of weight $k\geq 2$ that is ordinary at $p$ and $\overline{\rho}_f$ be the mod $p$ Gal representation attached ...
- 659
6
votes
2
answers
846
views
Serre's conjecture for mod-p^n representations?
I think this may be a silly question, but here goes. Let $\rho:\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to \mathrm{GL}_2(\overline{\mathbf{F}_p})$ be a representation; say $\rho$ is of S-type ...
- 13.1k
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
8
votes
1
answer
470
views
Is there an R=T type result for modular forms with additive reduction?
Let E be an elliptic curve over the rationals with conductor $Mp^2$ with p>5 and M and p coprime, and let $\rho$ be the Galois representation attached to the p-torsion points of E. Is there a way to ...
- 818
18
votes
2
answers
2k
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 $p$. ...
- 1,609