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Results tagged with elliptic-curves
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user 2024
An elliptic curve is an algebraic curve of genus one with some additional properties. Questions with this tag will often have the top-level tags nt.number-theory or ag.algebraic-geometry. Note also the tag arithmetic-geometry as well as some related tags such as rational-points, abelian-varieties, heights. Please do not use this tag for questions related to ellipses; instead use conic-sections.
6
votes
3
answers
719
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When is an affine part of an elliptic curve isomorphic to an affine part of a norm equation?
Given a cubic number field and a basis $\{\gamma_1,\gamma_2,\gamma_3\}$ for it over the rationals, we can write down the norm equation $N(x_1\gamma_1+x_2\gamma_2+x_3\gamma_3)=1$. For almost all substi …
- 4,593
19
votes
Accepted
Are there any rational solutions to this equation?
[Complete revamp of answer. It is based on the one before, but is better!]
In Tim's hyperelliptic equation, make the change of variables $y$ to $y/(-7)^2$, and $x$ to $x/(-7)$, to get:
$$y^2=x(x+7)(x …
- 4,593
7
votes
0
answers
503
views
Constructing large rank elliptic curves by multiplying quadratic imaginaries by cubes so tha...
I've been thinking of the relation between elliptic curves of large rank and quadratic imaginary fields with large 3-rank class groups. There are quite a few papers constructing infinitely many quadra …
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7
votes
0
answers
486
views
What is the right basis of solutions of the Picard-Fuchs equation of the Legendre family aro...
I have been trying to reconstruct some elliptic curves theory computationally and have gotten stuck on some period computations.
Specifically, let $$E_\lambda:\ y^2=x(x-1)(x-\lambda)$$ be the Legendr …
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4
votes
1
answer
385
views
Tunnel like theorem: is there an interesting function with fourier coefficients related to $...
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 c …
- 4,593
10
votes
2
answers
1k
views
Legitimacy of reducing mod p a complex multiplication action of an elliptic curve?
I scoured Silverman's two books on arithmetic of elliptic curves to find an answer to the following question, and did not find an answer:
Given an elliptic curve E defined over H, a number field, wit …
- 4,593
28
votes
Accepted
Why does the definition of modularity demand weight 2?
$\newcommand\Q{\mathbf{Q}}$
$\newcommand\Qbar{\overline{\Q}}$
$\newcommand\Gal{\mathrm{Gal}}$
$\newcommand\C{\mathbf{C}}$
$\newcommand\Sym{\mathrm{Sym}}$
$\newcommand\E{\mathcal{E}}$
$\newcommand\Bett …
26
votes
7
answers
6k
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When is a product of elliptic curves isogenous to the Jacobian of a hyperelliptic curve?
David's question Families of genus 2 curves with positive rank jacobians reminded me of a question that once very much interested me: when is a product of elliptic curves isogenous to the jacobian of …
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1
vote
1
answer
802
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Is the direct limit of Weil restriction of an elliptic curve a scheme?
In a discussion today on the Shafarevich-Tate group of an elliptic curve, the following structure and question came up. I will abuse many notations and be very vague about some things, but am very ope …
- 4,593
16
votes
4
answers
1k
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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 …
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