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Let $K$ be a finite extension of $\mathbb{Q}_p$ with $p\neq 2$. Suppose that $E/K$ is an elliptic curve with additive reduction and such that $E$ has full $4$-torsion over $K$. By the Kodaira classif …

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 …

The original proof is by Tate in WC groups over $\mathfrak{p}$-adic fields. Nowadays, it is often derived from local Tate duality $H^1\bigl(K_v,E[n]\bigr)\times H^1\bigl(K_v,E[n]\bigr)\to \mathbb{Z}/n …

Again an early source explaining how this is done in practice is Cremona's book. Specifically section 3.8. One current implementation for finding the isogeny class of an elliptic curves over a number …

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 …

(Not sure any of these questions are at the right level for this forum, but here the comments that may help.) question : Inflation-restriction sequence. question : The target can be identified with …

Let $\sigma$ be the non-trivial element of the Galois group of the quadratic extension $L/K$. Let $\phi \colon E \to E_D$ be the isomorphism defined over $L$. First, if $P \in E(\bar L)$ then $\sigma\ …

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 …

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^ …

Well spotted. This is a problem. After quite a bit of fiddling I found that the error is in Denis Simon's script used by Sage. In fact, when executed with higher values of the parameters so that the s …

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 …

The remark added to the question shows that the kernel of $Ш(E/F) \to Ш(E/L)^G$ is finite where $G$ is the finite Galois group of $L/F$. $\DeclareMathOperator{\coker}{coker}$ Here is an argument why t …

I start by explaining a tiny problem that appears even for quadratic twists, the same will appear for cubic twists even if they really should be the same. Let $E$ be an elliptic curve and $D$ a square …

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 …

Not having been mentioned before, I would recommend the two books "Fermat's Last Theorem, Basic Tools" and "Fermat's Last Theorem, The Proof" by Takeshi Saito. https://bookstore.ams.org/mmono-243 and …