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Results tagged with elliptic-curves
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user 4213
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.
2
votes
Homomorphism of Legendre curve
Pete's is certainly the right way to look at this problem,
but in this example one can argue naively using explicit
calculations. One loses no generality by assuming $c=0$
(by replacing $x$ by $x+c$). …
- 20.8k
7
votes
Repeated digits of squares in different bases
Your equation $n^2=(b+1)(b^2+1)$ defines an elliptic curve. By
Siegel's theorem
http://en.wikipedia.org/wiki/Siegel%27s_theorem_on_integral_points
the set of integer solutions will be finite. Mordell' …
- 20.8k
23
votes
Why is an elliptic curve a group?
A proof I like is that the group of points on the curve is the classgroup
of the ring $R=k[x,\sqrt{x^3+Ax+B}]$ where $k$ is the field you're working over.
Set $y=\sqrt{x^3+Ax+B}\in R$ and let $K$
deno …
- 20.8k
12
votes
Accepted
isogeny of elliptic curves
Yes. In zero characteristic the image of an isogeny of elliptic curves
is determined up to isomorphism by its kernel. Your isogeny has the same kernel
as the doubling map from $E$ to itself.
- 20.8k
7
votes
Additive reduction of elliptic curves
If $p\ge5$ then $E$ has equation $y^2=x^3+Ax+B$
with $p\mid A$ and $p\mid B$. A quadratic twist alters
the discriminant, essentially $4A^3+27B^2$, by a sixth power,
so for it to have good reduction $v …
- 20.8k
5
votes
How to generate the n-torsion group in an elliptic curve
Under the assumption that $E[n]\subseteq E(\mathbb{F}_q)$, to compute
$E[n]$ I'd avoid using division polynomials, as they rapidly become cumbersome.
Rather I would generate random elements of $E[n]$ …
- 20.8k
10
votes
Algorithms for finding rational points on an elliptic curve?
There is a whole industry devoted to this. The basic method is by
descent, which is a formalized version of the infinite descent proofs
of Fermat and Euler. It helps if there are rational 2-torsion po …
- 20.8k
5
votes
Accepted
Analysis of a quadratic diophantine equation
One thing to do is to try to express these in terms of squares. Note that
$$12x(3x-1)=36x^2-12x=(6x-1)^2-1$$
so that your equations become
$$a_1^2+b_1^2=c_1^2+1$$
and
$$a_1^2-b_1^2=d_1^2-1$$
where $a_ …
- 20.8k
9
votes
Accepted
Generalizations of Belyi's theorem
The compactification is the usual one coming up in
the theory of modular forms, with the cusps being
orbits of $\Gamma$ on $\mathbb{Q}\cup\{\infty\}$.
As for the proof, I like
this paper by Bernhard …
- 20.8k