# Questions tagged [tate-shafarevich-groups]

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### Large 2-part of Tate–Shafarevich group over $\Bbb{Q}$ with small number of prime factor of discriminants

$\newcommand{\Sha}{\operatorname{Sha}}$Let $E/\mathbb{Q}$ be an elliptic curve, and let $\Sha(E/\mathbb{Q})$ denote the Tate–Shafarevich group of $E/\mathbb{Q}$. It is known that the 2-primary ...
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### Norm map of Tate-Shafarevich group $\mathrm{Sha}(E/K)\to \mathrm{Sha}(E/\Bbb{Q})$

Let $K$ be a quadratic number field. Let $\sigma$ be a generator of Galois group of $K/\Bbb{Q}$. Let $E$ be an elliptic curve defined over $\mathbb{Q}$. Let $\mathrm{Sha}(E/K)$ denote the Tate-...
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### Tate-Shafarevich groups over finitely generated fields

Let $G$ be an algebraic group over a number field $k$. One defines the Tate-Shafarevich set of $G$ to be $$Ш(k,G) = \ker\left(H^1(k,G) \to \prod_{v} H^1(k_v,G)\right),$$ where the product is over all ...
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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 $$Ш(E,S) = \ker\left(H^1(k,E) \to \prod_{v \... 78 votes 0 answers 3k views ### The exponent of Ш of y^2 = x^3 + px, where p is a Fermat prime For d a non-zero integer, let E_d be the elliptic curve$$ E_d : y^2 = x^3+dx. $$When we let d be p = 2^{2^k}+1, for k \in \{1,2,3,4\}, sage tells us that, conditionally on BSD,$$ \# Ш(E_p)...
Hello everyone, I was deeply attracted by the Hasse and Siegel's theorems while studying $p$-adic analysis. While reading a paper B.J. Birch and H.P.F. Swinnerton-Dyer - Notes on elliptic curves. I, ...