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Results tagged with elliptic-curves
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user 2757
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.
5
votes
Accepted
Supersingular curves over $\mathbb{F}_q$ and the splitting of $p$
Here is a solution that avoids explicit use of $\mathbf{Q}_p$ and in particular does not require knowing that $(\operatorname{End} E) \otimes \mathbf{Q}_p$ is a division ring. The key is to use insep …
- 23.8k
22
votes
Accepted
Images of action of Galois on the Tate module of Elliptic Curve,
Let $\Delta$ be the discriminant of $E$. Then the action of $G_{\mathbf{Q}}$ on $E[2]$ determines the action on $\sqrt{\Delta}$. On the other hand, the action of $G_{\mathbf{Q}}$ on $E[n]$ determine …
- 23.8k
6
votes
Homomorphism of Legendre curve
Here is a conceptual explanation that applies to any $E/k$ with $\operatorname{char} k \ne 2$ and $E[2] \subseteq E(k)$, and that explains why $x-a$ and $x-b$ are the relevant rational functions (the …
- 23.8k
16
votes
Accepted
What's the Hilbert class field of an elliptic curve?
EDIT: This is a completely new answer.
I will prove that your specific suggestion of defining a Hilbert class field of an elliptic curve $E$ over $K$ does not work. I am referring to your proposal t …
- 23.8k
9
votes
Accepted
Cyclic extensions coming from E[p] \equiv F[p],
The answer is that in fact this construction does not produce cyclic extensions! The problem is that $X_E(p^2) \to X_E(p)$ is not generically Galois; it is so only after extension of the ground field …
- 23.8k
24
votes
Are most cubic plane curves over the rationals elliptic?
Your question (as explained in the second paragraph) is not vague at all! In fact, it appears for instance after Conjecture 2.2 in http://www-math.mit.edu/~poonen/papers/random.pdf , which is Random …
- 23.8k