All Questions
73 questions
28
votes
1
answer
2k
views
Is there a "Langlands philosophy" reason for the fact that the L-function of the Jacobi theta function is (almost) the Riemann zeta function?
The Jacobi theta function $\theta(z) = 1 + 2 \sum_{n = 1}^\infty q^{n^2}$, with $q = e^{\pi i z}$ is a (twisted) modular form with weight $1/2$. It has an associated $L$-function $L(\theta, s) = \sum_{...
- 3,058
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
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
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
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 ...
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
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
14
votes
1
answer
1k
views
A question on a paper of K. Murty
Let $f=\sum_{n\ge 1}a_nq^n$ be a normalized Hecke eigenform which is not of CM-type, of weight $k\ge 2$ for the congruence subgroup $\Gamma_0(N)$. Let $a\in\mathbb{Z}$ and define
$$
\pi_f(x,a):=\#\{p\...
- 303
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
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
11
votes
3
answers
944
views
"Extra Euler factors" in one definition of the L-function of a twist of a modular form
Let $(\rho_{f,\lambda})_\lambda$ be the system of Deligne's $\ell$-adic representations attached to a modular newform $f$ (where $\lambda$ runs over the finite places of the number field $K$ generated ...
- 734
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
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
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
10
votes
1
answer
676
views
Level raising by prime powers
Suppose $f$ is a weight $2$ level $N$ cusp form. When can we realize the mod-$\ell$ representation of $f$ in a form of weight $2$ and level $Np^3$, where $p$ is some prime not dividing $N$? I assume ...
- 13.1k
10
votes
1
answer
595
views
Type of a modular form
Let $f$ be an arbitrary weight 1 newform. We know by Serre-Deligne that there is an odd 2-dimensional irreducible Artin representation $\rho$ such that $L_f(s)=L(\rho,s)$.
I was wondering how much ...
- 17.6k
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
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
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
8
votes
2
answers
781
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
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
2
answers
403
views
Mazur's Question on Mod $N$ Galois representations
In Rational Isogenies of Prime Degree, Mazur poses:
"the problem of determining all elliptic curves $E'/\mathbb{Q}$ with symplectic $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ isomorphisms $E'[N]\...
- 901
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
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
1
answer
963
views
Deligne-Scholl's motives for modular forms: Hecke operators vs. transposed Hecke operators
EDIT: I moved my original question down to the bottom. The question here at the top is related, at least I suppose that the same phenomenon is behind both of them.
In the article "Valeurs de ...
- 734
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
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
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
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
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)$$
the residual 7-adic Galois representation attached to the elliptic curve $y^2=x^3+x^2-4x-4$ of conductor 48. Then $...
- 10.9k
6
votes
1
answer
380
views
Applications of Level Lowering
What are some applications/consequences of level lowering of Galois representations? I understand the application of Ribet's theorem in the proof of Fermat's last theorem but I am wondering what other ...
- 1,629
6
votes
1
answer
499
views
Galois representation and weight one Hilbert modular form
Let $f$ be a primitive weight one Hilbert modular form for the totally real field $F$ (the weight of $f$ is parallel). Assume that $f$ is $N$-new, $p$-stable and cuspidal. Let $\rho:G_F \rightarrow \...
- 1,066
5
votes
1
answer
306
views
Galois representation associated to CM-newforms
Let $f(z)=\sum_{n\ge 1}a(n)e(nz)$, be a newform of CM-type, and let $\psi_f$ be the associated Hecke character, so that,
$$
f(z)=\sum_{\mathfrak{a}}\psi_f(\mathfrak{a})e(N(\mathfrak{a})z),
$$
and let ...
- 400
5
votes
2
answers
944
views
Elliptic curves over $\mathbf{Q}$ with isogenous mod $\ell$ reductions, for several $\ell$
Characteristic polynomials of Hecke operators $T_\ell$, with $\ell$ prime, acting on cusp forms $S_k$ of level one and weight $k$ "appear to be" squarefree (even irreducible!).
This can be ...
- 2,406
5
votes
1
answer
801
views
Local Galois representation associated to twist of modular form
Let $f$ be a modular newform of weight $k \geq 2$, level $N$ (square free) and trivial nebentypus. Let $V_{f}$ be the $p$-adic (p odd) Galois representation associated $f$. We denote by $V_{f,l}:= V_{...
- 53
5
votes
1
answer
380
views
Smoothness of Hecke algebras
First I will introduce some notation and definitions.
Fix a level $N$ (take $N=1$ if it makes things easier) and a prime $p$. Let $k$ be a finite field of characteristic $p$ and let $\mathcal{C}$ be ...
- 597
5
votes
1
answer
546
views
Level lowering for weight 1 forms
I'm interested in knowing to what extent is level lowering known to hold in weight 1. Specifically, let's say I have an eigenform $f$ in $S_1(N,\chi)$ and a prime $p$ which doesn't divide $N$. Let's ...
- 1,680
5
votes
1
answer
281
views
Does the $p$-part of the level of a newform appear in its attached $p$-adic representation?
Let $f$ a newform of weight $2$ on $\Gamma_0(Np^r)$, $N$ coprime to $p$, and consider its $p$-adic Galois representation
$$
\rho:G_{\mathbb Q}\longrightarrow GL_2(\bar{\mathbb Q}_p)
$$
It's a theorem ...
- 51
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
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 ...
4
votes
1
answer
430
views
About the proof of Weil-Langlands theorem
The statement of the theorem is as follows:
Let $\rho$ be an irreducible two-dimensional representation of $G_\mathbb{Q}=Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ with Artin conductor $N$.
Suppose that $\...
- 135
4
votes
1
answer
807
views
Meaning of Atkin-Lehner eigenvalues
Suppose I have $f\in S_2(\Gamma_0(N))$ a classical modular newform of level $N$. I want to understand what information (if any) is carried by its Atkin-Lehner eigenvalues for primes $p\mid N$, as ...
- 51
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
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
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
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