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Here is a rather long comment in which I try to justify why I think that the majority will have a surjective map on the Mordell-Weil group. I would not want to guess what the % is. Let $E$ be an elli …

The comments above give already the answer, but for the sake of completeness let us be a bit more precise. Let $E/\mathbb{Q}$ be an elliptic curve. Let $\Omega^{+}$ be the smallest positive element i …

Question 3: Yes, for plenty of curves one can calculate the 3-Selmer group and show that it is equal to the contribution from the Mordell-Weil group and hence that $Ш(E/\mathbb{Q})[3^{\infty}]$ is tri …

In less fancy language, you are asking for the ramification of the extension of $K$ where you adjoin the coordinates of the $n$-torsion points of $A$. Under your assumption of everywhere good reductio …

Let us assume that $E$ has complex multiplication by a maximal order $\mathcal{O}$. Then $E[\ell]$ is a free rank $1$ module over $\mathcal{O}/\ell\mathcal{O}$. The image $G$ of $\rho_{E,\ell}$ is con …

There is the Stickelberg element $\Theta$ considered by Mazur and Tate which gives more information in this direction. It is conjectured to be in the Fitting ideal and hence in the annihilator of the …

In other words, given an elliptic curve $E/\mathbb{Q}_p$ with additive reduction, you wish to know whether there is a quadratic extension $F/\mathbb{Q}_p$ such that $E/F$ has good reduction. Of course …

Let $E/\mathbb{Q}$ be an elliptic curve with a rational $\ell$-torsion point $P$ for $\ell\in\{5,7\}$. Let $p$ be a prime different from $\ell$. Then the formal group $\hat{E}(p\mathbb{Z}_p)$ has no …

There is a formula but it involves both cubic twists. Let $E: y^2 = x^3+B$ be an elliptic curve over $\mathbb{Q}$ with $j=0$ as the one in the question. Let $D$ be a cubefree integer. Set $E_1: y^2=x^ …

As in the question $K$ is a number field and $E/K$ an elliptic curve. Let me start by saying that I think the best analogues are the following two short exact sequences (the first two "$/n$" means mod …

I fear you wish for too much here. If $Ш$ is finite, then we can represent each element by a torsor; each torsor has good reduction away from a finite set and the union of all bad places would then be …

The Kodaira symbol characterises the geometry of the special fibre, that is the structure over the algebraically closed field $\bar{\mathbb{F}}_p$. Good reduction means type I${}_0$, multiplicative re …

All your groups are torsion, so we may split it into primary parts. Let $\ell$ be a prime. First the map $a\colon E(\mathbb{Q})\otimes \mathbb{Q}_{\ell}/\mathbb{Z}_{\ell} \to \prod_p E(\mathbb{Q}_p)\o …

It is in Lang's book "Fundamentals of diophantine geometry", chapter 6: google book preview

Often it is, but not always. For instance if $K=\mathbb{Q}$ then the map is injective if and only if the rank of $E(\mathbb{Q})$ is at most $1$. This is because the $p$-adic elliptic logarithm of a no …