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Results tagged with elliptic-curves
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user 35416
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.
26
votes
Accepted
Examples of elliptic curves over $\mathbb{Q}$
In general, for any integer $N$ and any fixed elliptic curve $E$, the elliptic curves $E'$ for which $E[N]\cong E'[N]$ as Galois modules (and such that the isomorphism respects the Weil pairing) are p …
- 13k
17
votes
Accepted
Order of Ш (Sha)
No, there are no such examples known. In fact, with the current technology, the two questions are more or less equally hard. That's because for any given prime $p$, you can, in principle, establish fi …
- 13k
16
votes
Accepted
Regulators of Number fields and Elliptic Curves
Let me first give you a heuristic "reason", why the regulator in the class number formula looks different from the regulator in the Birch and Swinnerton-Dyer conjecture. It is often more convenient (a …
- 13k
7
votes
Accepted
Does the equality of ranks imply equality of analytic ranks?
In general, it does not seem to be any easier to compare two elliptic curves than to say something about each individual curve. In other words, the only results in the direction that you are asking ab …
- 13k
7
votes
Accepted
Analogue of j-invariant for CM fields
The simplest generalisation to abelian surfaces is, I believe, the statement (which is a theorem, not a conjecture) that the Igusa invariants of an abelian surface with CM by $K$ generate an abelian u …
- 13k
6
votes
The parity conjecture
Let me add a few remarks to the very nice CW answer already given.
The parity conjecture (i.e. algebraic rank equals analytic rank modulo 2) is known for all elliptic curves over number fields (not …
- 13k
6
votes
Accepted
The cyclic twist of elliptic curve is a principally polarized abelian variety
Usually, $E^L$ is not principally polarised. See E. Howe, Isogeny Classes of Abelian Varieties with no Principal Polarizations, where it is shown that under some mild hypotheses every polarisation of …
- 13k
5
votes
What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil...
Dear Felipe,
as long as we restrict ourselves to elliptic curves over $\mathbb{Q}$, the only results known in this direction are that the 2, 3, 5, 7 and 13 primary torsion of Tate-Shafarevich groups c …
- 13k
4
votes
Mazur's Question on Mod $N$ Galois representations
In addition to Felipe's reference, you can also have a look at Tom Fisher's papers https://www.dpmms.cam.ac.uk/~taf1000/papers/congr7and11.html and https://www.dpmms.cam.ac.uk/~taf1000/papers/congr9.h …
- 13k
2
votes
Rank of jacobians of twists of hyperelliptic curves of genus one
If you are willing to assume finiteness of Sha, then the answer is "yes".
The first claim, that for infinitely many $d$ (square-free, presumably?) this has a rational point, is easy and does not need …
- 13k
2
votes
Accepted
Cassels Pairing for Fine Selmer groups
Flach has defined in [1] a Cassels-Tate pairing on very general Selmer groups, and the kernels of the pairing on both sides are the maximal divisible subgroups. But you need to be careful when applyin …
- 13k