20
votes
1answer
772 views

Is the Modularity Theorem (currently) effective?

The Modularity Theorem says every elliptic curve over $\mathbb{Q}$ can be gotten from the classic modular curve $X_0(N)$ by a rational map. Here $N$ is the conductor, easily calculable from a ...
1
vote
0answers
107 views

Cube-root of j-invariant [closed]

Cube-root of j-invariant is a modular function of level 3, does it have similar property as j-invariant? How about the its minimal polynomial? In particular, are it's coefficients much smaller like ...
4
votes
2answers
400 views

Gross's paper on Heegner points

I try to read Gross's paper on Heegner points and it seems ambiguous for me on some points: Gross (page 87) said that $Y=Y_{0}(N)$ is the open modular curve over $\mathbb{Q}$ which classifies ordered ...
1
vote
0answers
210 views

in the preface of “A first course in modular forms”

In the preface of "A first course in modular forms", the author considered the quadratic equation $Q: x^2=d, \ \ d \in \mathbb{Z}, d \not=0$, and for each prime $p$ define an integer $a_p(Q)=\#\tilde ...
1
vote
1answer
209 views

Example of non-modular elliptic surface?

In "On elliptic modular surfaces", Shioda proves some interesting theorems on smooth elliptic surfaces (admitting a section); he then focuses on "modular elliptic surfaces" and proves some more ...
6
votes
0answers
211 views

Modular interpretation of Ramanujan theta operator?

I'm a beginner to the theory of modular forms trying to understand a certain construction from the point of view of elliptic curves. Let $f(q) = \sum a_n q^n$ be a formal power series. Define $\theta ...
4
votes
0answers
182 views

Evaluation of $E_{\ell,2}$ on supersingular curves over $\mathbb{F}_{p^2}$

As mentioned in an answer to Modularity of $E_2$ on congruence subgroups, there exist modular forms $E_{\ell,2}$ of level $\Gamma_{0}(\ell)$ and weight 2, with $q$-expansion ...
4
votes
1answer
210 views

Rational points on $X_0(15)$

The modular curve $X_0(15)$ has a canonical model over $\mathbf{Q}$, and it has genus $1$. As the cusp $\infty$ is rational, it is an elliptic curve. Roughly, my question is whether we can find all ...
5
votes
1answer
285 views

Where do the product expansions of modular forms come from?

It is well-known that many modular forms can be expressed as infinite products. For instance, the most famous one is probably the expansion $$\Delta(q) = q \prod_{n=1}^\infty (1-q^n)^{24}$$ for the ...
4
votes
1answer
296 views

equivalence between katz and classical modular forms

$\newcommand{\CC}{\mathbb{C}}$ $\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\PP}{\mathbb{P}}$ $\newcommand{\QQ}{\mathbb{Q}}$ $\newcommand{\hH}{\mathcal{H}}$ $\newcommand{\eE}{\mathcal{E}}$ ...
3
votes
1answer
291 views

Is there an elliptic surface over $Y(1)$?

Actually I have a few related questions. Here, by $Y(1)$ I mean the affine $j$-line $\text{SL}_2(\mathbb{Z})\backslash\mathcal{H}$. I know $Y(1)$ is only a coarse moduli space, so there isn't a ...
8
votes
2answers
531 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, ...
0
votes
2answers
275 views

The origin of the root number $w(C/ℚ)=±1$ (the sign of the functional equation)

The motivation for this question is the same as in my previous question in MO: http://mathoverflow.net/questions/115179/real-root-1-of-the-hasse-weil-l-function-of-c-over I am just curious to know ...
3
votes
1answer
371 views

Imaginary quadratic field contained in Hecke orbit field?

Let $\tau$ in the upper half plane lie in an imaginary quadratic field $K$. Then is $K \subset \mathbb{Q}(\{j(g \tau) \ | \ g \in GL_2^+(\mathbb{Q}) \})$? (here $j$ is the modular $j$-function, and ...
0
votes
1answer
195 views

A question regarding the j-function

Consider $\tau$ and $\tau'$ in the upper half plane such that $j(g \tau) = j(g \tau')$ for all $g \in GL_2^{+}(\mathbb{Q})$, where $j$ is the modular $j$-function and $GL_2^{+}(\mathbb{Q})$ acts as ...
5
votes
1answer
413 views

Some help in digesting a paragraph in the introduction of Deligne/Rapoport's “Les Schemas de Modules de Courbes Elliptique”

http://www.springerlink.com/content/04x54gr171v556m4/fulltext.pdf On page 149 (DeRa-7), in the middle of the page, I can translate the middle paragraph that starts "3. La surface de Riemann ..." as ...
9
votes
3answers
2k views

A recommended roadmap to Fermat's Last Theorem

I was inspired to undertake math as a career after watching a documentary on the proof of Fermat's Last Theorem. As such it's been a small goal of mine to understand Wiles et al's proof. In a ...
8
votes
2answers
637 views

Extensions of the modularity theorem

In 1995 (if I'm not mistaken) Taylor and Wiles proved that all semistable elliptic curves over $\mathbb{Q}$ are modular. This result was extended to all elliptic curves in 2001 by Breuil, Conrad, ...
5
votes
1answer
293 views

Modular curve parametrizing two cyclic subgroups of an elliptic curve

The aim of this question is to better understand the following moduli space/modular curve, for which I propose (temporarily) the name $Y_0(M,N)$. We define $Y_0(M,N)$ as the moduli space parametrizing ...
7
votes
0answers
287 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 ...
2
votes
2answers
359 views

branch points of modular parametrization of an elliptic curve

Let $E$ be an elliptic curve over a number field K. Then there is a morphism $\phi:X_0(n) \to E$. Consider composition $f:X_0(n)\to \mathbf{P}^1_K$, where we compose with degree 2 cover $E\to ...
3
votes
1answer
495 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 ...
3
votes
0answers
332 views

Cyclotomic fields and singular moduli

Let $\mu$ be the roots of unity and $S$ be the image under the modular $j$-function of all imaginary quadratic $\tau$. Then what is $\mathbb{Q}(\mu)\cap\mathbb{Q}(S)$?
4
votes
2answers
473 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 ...
1
vote
1answer
183 views

On pseudo rational modular forms of weight 2 and level N

So consider the $\mathbb{Q}$-vector space $V$ of functions which satisfy the following conditions (1) $f:\mathbb{H}\rightarrow\mathbb{C}$ is holomorphic. Here $\mathbb{H}$ stands for the upper half ...
11
votes
5answers
2k views

Why does the definition of modularity demand weight 2?

Allow me to quote a definition from Gelbart in "Modular Forms and Fermat's Last Theorem": Definition. Let $E/\mathbb{Q}$ be an elliptic curve. We say that $E$ is modular if there is some normalised ...
9
votes
1answer
514 views

Historical question about modularity of CM curves

I'm looking for the answer of who first proved modularity of CM curves? That is if $E$ is an elliptic curve over $\mathbb{Q}$ which has complex multiplication then there's a non-constant morphism ...
3
votes
1answer
262 views

Tunnel like thereom: is there an interesting function with fourier coefficients related to $L'(E_n,1)$ instead of $L(E_n,1)$?

Tunnel's result on the congruent number problem hinges on the fact that there are modular forms with fourier coefficients related to the values $L(E_n,1)$. Is there an interesting function that has ...
32
votes
0answers
1k views

Constructing non-torsion rational points (over Q) on elliptic curves of rank > 1.

Consider an elliptic curve E defined over Q. Assume that the rank of E(Q) is >=2. (Assume the Birch-Swinnerton-Dyer conjecture if needed, so that analytic rank = algebraic rank.) How do you construct ...
18
votes
1answer
531 views

How to explicitly compute lifting of points from an elliptic curve to a modular curve?

Say $E$ is an elliptic curve over the rationals, of conductor $N$. There's a covering of $E$ by the modular curve $X_0(N)$, and if you rig it right then you can define this map over $\mathbf{Q}$: ...
5
votes
2answers
991 views

Growth of Coefficients of cusp forms

Hi I am curious about the growth of coefficients of cusp forms. I am aware of Ramanujan-Petersson conjecture/theorem in general terms but I was hoping for a more detailed description of precise ...
11
votes
1answer
619 views

Which elliptic curves over totally real fields are modular these days?

As the title says. In particular, every elliptic curve over $\mathbb{Q}$ is modular; but what is the current state of the art for general totally real number fields? I assume the answer is ...
6
votes
0answers
608 views

Can you get Siegel's theorem “for free” from modularity and Mazur's Eisenstein Ideal paper?

There is a well-known theorem of Shafarevich that given a finite set $S$ of primes the number of isomorphism classes of elliptic curves over $\Bbb Q$ with everywhere good reduction outside $S$ is ...
4
votes
1answer
465 views

A bound for the Manin constant

I recall that the Manin constant for a strong elliptic curve is a rational integer $c_E$ such that, for a modular parametrization $\phi: X_1(N) \to E$, one has $\phi^*(\omega_E)= 2\pi i c_E ...
9
votes
4answers
695 views

Geometric meaning of fiber of modular parameterization over a point of an elliptic curve?

Given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, parameterization $\psi : X_0(N) \rightarrow E$, and a point $P \in E$, take the fiber $\psi^{-1}(P)$. Its points, being on $X_0(N)$, correspond ...
8
votes
3answers
453 views

Fourier coefficients for elliptic curves on average

Fix a prime p, and look at elliptic curves in some family (e.g. all elliptic curves ordered by height). How often do the Fourier coefficients a_p occur? Are there any conjectures?
10
votes
1answer
648 views

Hecke operators acting as correspondences?

This question is inspired by Relation between Hecke Operator and Hecke Algebra I remember having heard of yet another way of looking at Hecke operators acting on the spaces of modular forms for ...
7
votes
3answers
1k views

Convergence of L-series

I remember to have read that the L-function of an elliptic curve, which a priori only converges for $\Re s > \frac{3}{2}$ also converges at $s=1$ provided that the $L$-function satisfies the ...
7
votes
1answer
367 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 ...
20
votes
4answers
2k views

Modular curves of genus zero and normal forms for elliptic curves

This is maybe the first question I actually need to know the answer to! Let $N$ be a positive integer such that $\mathbb{H}/\Gamma(N)$ has genus zero. Then the function field of ...
16
votes
3answers
2k views

Conceptual understanding of the Gross-Zagier theorem.

The Gross-Zagier paper "Heegner points and derivatives of $L$-series", is really computational and hard to plow through. It seems it is futile to read it as such and one must look for a more ...
5
votes
2answers
253 views

Alternate expresion of L-series coefficients

I was hoping that someone could help clarify a source of confusion for me, I must be doing and saying something wrong but I just don't know what: Let $E$ be an elliptic curve over $\mathbb{Q}$ and let ...
7
votes
5answers
1k views

Very strong multiplicity one for Hecke eigenforms

In Invent. math. 116, 645-649 (1994) Dinakar Ramakrishnan proves a theorem which I understand to imply that the following statement (in light of the fact that elliptic curves over Q are modular): ...
6
votes
1answer
410 views

Ways to characterize supersingular primes?

I've read the definition, and it basically says p is a supersingular prime iff the fundamental domain of a group generated by \Gamma(p) and a matrix ((0, 1), (-p, 0)) is rational. And there's a ...
10
votes
4answers
1k views

Mystery of the Monstrous Moonshine

There's a very famous group, the largest sporadic simple finite group, sometimes called a monster whose size is quoted below. What's the explanation that the primes appearing in it, ...
6
votes
2answers
564 views

Logarithmic structures on moduli of elliptic curves over Z

I've heard it stated that if you take the moduli of elliptic curves with some level structure imposed (as a moduli scheme over Spec(Z)), there is a logarithmic structure that you can impose at the ...
7
votes
2answers
765 views

Elliptic curve over spectra?

Filling the gaps in my knowledge to understand the tmf question. So, what is the analogue of elliptic curve over the category of spectra?