Tagged Questions

1
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1answer
34 views

Modularity theorem and some results

Let $C$ be an elliptic curve over rationals. Then we can attach to $C$ an L-series $L(C,s)$. I read about the Modularity theorem http://en.wikipedia.org/wiki/Modularity_theorem In the section ...
3
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0answers
37 views

Good source of problems for Knapp's Elliptic Curves?

I'm studying elliptic curves (and eventually modular forms) out of Knapp's book because of the softer algebraic geometry prereqs. It's incredibly accessible but the problem is that I don't know I can ...
3
votes
1answer
72 views

Stark's formula for the j-invariant

In his paper On the "gap" in Heegner's proof (which you can find : here) Stark gives the following formula for the $j$-invariant (for some $\tau \in \mathcal{H}$ and $q=e^{2i\pi\tau}$) $$ j(\tau) = ...
5
votes
1answer
78 views

Direct proof of the non-zeroness of an Eisenstein series

Question: Can you show directly from its formula that $G_4(i)\neq0$? Recall that the holomorphic Eisenstein series of weight $2k$ is defined by: $$G_{2k}(\tau)= \sum_{(m,n)\in\mathbb{Z}^2\setminus ...
0
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0answers
90 views

finding torsion points of elliptic curves on MAGMA

I'm trying to learn how to use MAGMA for computing torsion points (i.e. n-torsion subgroups) of elliptic curves in various field. So far, I've looked in the documentation and other resources and I'm ...
0
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0answers
27 views

Transcendence of map induced from modular parametrization

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ having a modular parametrization $\Phi_N:X_0(N)\to E(\mathbb{C})$. In reading through an article by Henri Darmon, I came across a statement ...
2
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1answer
125 views

Congruence subgroups and modular curves of type (M,N)

I would like to study the "modular curve" $Y(M,N)$, parametrizing an elliptic curve $E$ together with $p \in E[M]$ and $q \in E[N]$ (here and in the following $M$ divides $N$). Let $\Gamma(M,N)$ be ...
5
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1answer
159 views

A question about modular curves and base change

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Suppose that the curve $X\times_{K,\sigma} \mathbf{C}$ is a modular curve for some $\sigma:K\to \mathbf{C}$. Can ...
2
votes
2answers
118 views

Epsilon conjecture analog

Recently this question caught my eye. Is there a relation to the modularity problem of elliptic curves over $\mathbb{Q}(\zeta_m)$ and this problem? Namely, if all elliptic curves over ...
6
votes
1answer
130 views

How can I determine in practice whether two elliptic curves over $\mathbb{Q}$ have isomorphic $p$-torsion?

Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{Q}$ with good, ordinary reduction at an odd prime $p$. I'm wondering how to determine whether $E_1[p]$ and $E_2[p]$ are isomorphic ...
6
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1answer
168 views

Euler Product of Modular L-series

This question is probably really easy to someone that knows the theory of modular forms well, so I apologize if this is obvious. Suppose $E_1$ and $E_2$ are elliptic curves over $\mathbb{Q}$ and the ...
1
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1answer
52 views

Does the following define a point of the modular curve $X_1(n)$

Let $X$ be a compact connected Riemann surface of genus $1$ and suppose that there is a finite morphism $X\to \mathbf{P}^1$ of degree $n$ which ramifies totally over $0$ and $\infty$. Let $f^{-1}(0)$ ...
0
votes
1answer
159 views

Nice formulas for the lambda invariant of an elliptic curve

Where can I find some nice formulas for the lambda invariant of an elliptic curve? I vaguely recall there's a nice product formula in terms of $q$, but a google search didn't give me much. Also, are ...
2
votes
1answer
96 views

Real elliptic curves in the fundamental domain of $\Gamma(2)$

An elliptic curve (over $\mathbf{C}$) is real if its j-invariant is real. The set of real elliptic curves in the standard fundamental domain of $\mathrm{SL}_2(\mathbf{Z})$ can be explicitly ...
3
votes
1answer
124 views

Finding a pencil of elliptic curves parametrized by a given modular surface

The following is an attempt to formulate a couple of questions which have been lurking in the back of my mind for a while. I'm sorry if this is long, or if my terminology is not correct, or if my ...
1
vote
1answer
191 views

Modular functions and elliptic functions

Does anybody know of an equation formally equating modular functions and elliptic functions similar to Euler's equation for exponential and trigonometric functions? Any advice much appreciated. ...
7
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2answers
485 views

A question on FLT and Taniyama Shimura

Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain ...