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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 …

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 $R$ …

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 $\sigma$ 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 …

As far as I understand the question (I know elliptic curves, but I don't know what MU and BP are), the task is to express the coefficients of the Weierstrass equation given the series of multiplicatio …

Let $S$ be a finite set containing all places where an elliptic curve $E$ has bad reduction as well as $p$ and $\infty$. Write $T$ for $T_pE$ and $G_S$ for the Galois group of the maximal extension o …

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 …

(Edit: I revise most of my question as my first answer overlooked that $d_1$ and $d_2$ are odd.) $\DeclareMathOperator{\res}{res}$ Let $E$ be an elliptic curve over $\mathbb{Q}$ with $E[2]\subset E(\m …

Look at Section 18.4 in Ireland-Rosen "A classical introduction to modern number theory". Note $p\equiv 1 \pmod{4}$ and $p = \pi\cdot \bar{\pi}$ with $\pi = 1- iu \equiv 1 \pmod{2+2i}$. Let $\lambda: …

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 …

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 …

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 …

When there is a $2$-torsion point present then one should indeed use these isogenies to do a descent. One can do this over a number field as long as it is not too difficult to calculate the class grou …

On the second question: There is the inequality $12\sqrt{3}A\leq P^2$ that needs to be satisfied first of all to get a triangle. Using Heron's formula for a triangle with sides $x$ and $y$, you are …