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Results tagged with elliptic-curves
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user 17907
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
Accepted
Quadratic twist of an elliptic curve given by non-Weierstrass model
Let's assume the characteristic of the ground field $k$ is not $2$.
If $C$ is of the form $y^2=f(x)$ with $f$ a separable quartic, and $E$ is the Jacobian of $C$ (hence $E$ is an elliptic curve), the …
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6
votes
1
answer
314
views
E an elliptic curve over Z[1/N], how many p such that E(Z/p^2) = (Z/p)^2?
For a fixed elliptic curve $E$ over $\mathbb{Z}[1/N]$, is there a non-trivial upper bound in terms of $x$ for the number of primes $p \leq x$ with $p \nmid N$ for which $E(\mathbb{Z}/p^2\mathbb{Z}) \c …
- 4,746
8
votes
Accepted
distinguishing E(K)/E_0(K) groups of order 4
This is not a complete answer, since there are some subtleties in residue characteristic $2$, but in all other cases there is a simple answer.
In residue characteristic $p>2$, you can simply look at …
- 4,746
14
votes
2
answers
2k
views
Surjectivity of reduction maps of elliptic curves over Q
Let $E/\mathbf{Q}$ be an elliptic curve of rank $>0$. It is easy to see that there is a positive-density set of primes $p$ such that the reduction map $\mathrm{red}_p : E(\mathbf{Q}) \rightarrow \wide …
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9
votes
The boundedness of the rank of twists of a fixed curve
This is not a complete answer to your question (since I think it isn't known), but you might still find this interesting.
Mestre has shown in Rang de courbes elliptiques d'invariant donné (http://arx …
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15
votes
Elliptic Curves over Rings?
Elliptic curves can be defined over arbitrary base schemes $S$. In particular, for every (commutative!) ring $R$ one can talk about elliptic curves over (the spectrum of) $R$. Loosely speaking, what o …
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78
votes
0
answers
3k
views
The exponent of Ш of $y^2 = x^3 + px$, where $p$ is a Fermat prime
For $d$ a non-zero integer, let $E_d$ be the elliptic curve
$$
E_d : y^2 = x^3+dx.
$$
When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD,
$$
\# Ш(E_p …
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5
votes
Possible $p$-torsion subgroup of $E(\mathbb{Q}_p)$, and if there is a theorem to say which c...
I was interested in this question myself a while back, particularly for the additive reduction case. I wrote up a little note about my results here. The main result was a nice looking numerical criter …
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