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Results tagged with elliptic-curves
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user 5101
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
Mordell-Weil rank of some algebraic surface
I assume that $k$ is algebraically closed; if you are interested in non-closed fields then one needs to calculate the Picard group over the algebraic closure first then use Galois theory to try to dec …
- 21.4k
8
votes
2
answers
2k
views
Explicit $2$-descent on elliptic curves
Let $k$ be a field of characteristic $0$ and let
$$E: y^2 = f(x)$$
be an elliptic curve over $k$, with $\mathrm{deg}(f) = 3$. Kummer theory yields a map
$$\varphi:\mathrm{H}^1(k, E[2]) \to \mathrm{H} …
- 21.4k
14
votes
1
answer
727
views
$S$-Tate-Shafarevich groups of elliptic curves
Let $S$ be a finite set of places of a number field $k$ and let $E$ be an elliptic curve over $k$. Define the ''$S$-Tate-Shafarevich group" of $E$ to be
$$Ш(E,S) = \ker\left(H^1(k,E) \to \prod_{v \no …
- 21.4k
16
votes
0
answers
396
views
Quadratic non-residues in elliptic divisibility sequences
Let $E: y^2 = x^3 + ax + b$ be an elliptic curve over $\mathbb{Q}$ with $a,b \in \mathbb{Z}$. Recall that any rational point $P = (x,y)$ can be written uniquely as $P = (u/d^2, v/d^3)$ with $u,v,d \in …
- 21.4k
11
votes
Accepted
Embedding torsors of elliptic curves into projective space
Suppose that $C \subset X$ is a smooth projective curve of genus $1$ embedded in a Brauer-Severi surface over a field $k$. We have $C^2 = 9$ since this holds after passing to the algebraic closure, wh …
- 21.4k
6
votes
Accepted
Is an Isomorphism from an Abelian variety to a Shimura variety always defined over a solvabl...
In good cases, the ''isomorphism functor'' $\mathrm{Isom}(X,Y)$ is representable by a scheme. Hence, you are asking that if this scheme is non-empty (i.e. contains a geometric point), whether it conta …
- 21.4k
9
votes
Accepted
Elements of arbitrary large order in the first Galois cohomology of an elliptic curve
Here is the kind of method I had in mind.
We have the elliptic curve Kummer sequence
$$0 \to E[n] \to E \to E \to 0,$$
Here I denote by $E[n]$ the $n$-torsion group scheme of $E$. Applying Galois coh …
- 21.4k
9
votes
Accepted
Does every non-isotrivial 1-parameter family of elliptic curves have a positive rank special...
I suspect that this is unknown in general. I would guess that any method which produces at least one elliptic curve of positive rank should also produce infinitely many of positive rank, which as you …
- 21.4k
13
votes
Possible groups of K-rational points for elliptic curves over arbitrary fields
I assume in the question that $C = E$ is an elliptic curve.
First your claim that $E(\mathbb{R}) = U(1)$ is false; I mean $E(\mathbb{R})$ can be disconnected. The correct result is that $E(\mathbb{R} …
- 21.4k