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Results tagged with elliptic-curves
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user 3927
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.
16
votes
Accepted
The order of the discriminant of a good-reduction elliptic curve
I will give an intrinsic characterization below for what this unit class modulo 12th powers means, which may be viewed as an answer of sorts: it expresses the obstruction to extracting the 12th root …
- 7,108
15
votes
Existence of fine moduli space for curves and elliptic curves
If you want to work over a base ring such as $\mathbf{Z}[1/n]$ rather than over $\mathbf{Q}$ or $\mathbf{C}$ then the relevant numerical condition is that the part of $N$ coprime to $n$ not be "too sm …
- 7,108
7
votes
Accepted
Is the direct limit of Weil restriction of an elliptic curve a scheme?
I agree with Adam Smith that the question seems a bit misguided, but let me show anyway that the answer is negative away from certain silly cases. Well, first to make a more well-posed question, one …
10
votes
Accepted
components of E[p], E universal in char p.
Speaking of "connected components" is a delicate thing since you really mean in a relative sense, and more specifically the etale quotient $H$ can have its open and closed non-identity part with very …
- 7,108
48
votes
Example of connected-etale sequence for group schemes over a Henselian field?
For concepts related to algebraic geometry when the base is not a field, it can be difficult for a beginner to reconcile the approach in Silverman with the approach via schemes. I wasted a lot of time …
- 7,108
64
votes
1
answer
5k
views
Is there a "classical" proof of this $j$-value congruence?
Let $j: \mathbf{C} - \mathbf{R} \rightarrow \mathbf{C}$ denote the classical $j$-function from the theory of elliptic functions. That is, $j(\tau)$ is the $j$-invariant of the elliptic curve $\mathbf{ …
13
votes
Modular curves of genus zero and normal forms for elliptic curves
As I mentioned in connection with an answer to another question, it is not generally true for elliptic curves $f:E \rightarrow S$ over a base $S$ that there is a global embedding of $E$ into $\mathbf{ …
- 7,108