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Let me expand Felipe's answer a bit. Vélu's formulae given in the very readable short paper [1] are very easy to use. For instance if you are given a $n$-torsion point one can immediately write down …

This minimal ramification index is the order of the Serre-Tate group $\Phi$, defined in their article "Good reduction of abelian varieties". It is shown in the proof of theorem 2 there that $\Phi$ is …

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 …

I get that the Kodaira types of $E$ is II and for $E'$ it is IV${}^{*}$. This means that $\mathcal{E}$ is connected (or in simple terms no point over $\mathbb{Q}_3^{\text{unr}}$ reduces to the singula …

Fix the prime $\ell$. I hope: There exists a constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ the image of the Galois representation $\rho\colon \operatorname{Gal} ( \bar{\mathbb{Q}}/\math …

I believe this is the answer in the split case: Let $E$ be the Tate curve with parameter $q$. Let $n>1$. We look for isogenies with cyclic kernel of order $n$. We may suppose that $n$ is prime. Firs …

With the transformation $X = -n/x$ and $Y= ny/x$, the curve becomes isomorphic to the Weierstrass model $$ E_n\colon \ \ Y^2 - X\ Y - n\ Y = X^3.$$ The points in question are exactly the integral poi …

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 …

Well, yes, but no. So the function obtained by removing from the usual $L$-series of $E$ all the supersingular factors is probably not a very nice function. I doubt that it has an analytic continuatio …

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 …

Let me expand that sentence, which may hint at the confusion in this and the subsequent question. Let $K=\mathbb{C}(t)$ and see the ring $R=\mathbb{C}[t]$ as a Dedekind ring. Let $v$ be a place of $R$ …

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

If your elliptic curve $E$ has rank $1$ over $\mathbb{Q}$, then there is a point $P$, a Heegner point, in $E(\mathbb{Q})$ with a special point $x=(E_x,C_x)$ in the fibre of $\psi$, i.e. such that $E_x …