# Tagged Questions

**20**

votes

**1**answer

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**

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**0**answers

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

**2**answers

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**

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**0**answers

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

**1**answer

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**

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**0**answers

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**

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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**

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**1**answer

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

**1**answer

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

**1**answer

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**

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**1**answer

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**

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**2**answers

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**

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**2**answers

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

**1**answer

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

**1**answer

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**

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**1**answer

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**

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**3**answers

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 ...

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**2**answers

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**

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**1**answer

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 ...

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**0**answers

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**

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**2**answers

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**

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**1**answer

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**

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**0**answers

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**

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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**

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**1**answer

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**

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**5**answers

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**

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**1**answer

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

**1**answer

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**

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**0**answers

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**

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**1**answer

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

**2**answers

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**

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**1**answer

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**

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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**

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**1**answer

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**

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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**

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**3**answers

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**

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**1**answer

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**

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**3**answers

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

**1**answer

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**

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**4**answers

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

**3**answers

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

**2**answers

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

**5**answers

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

**1**answer

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

**4**answers

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

**2**answers

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

**2**answers

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?