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Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$. Let $D\in \Bbb{Z}$ be a square free integer and $E_D/\Bbb{Q}$ be its quadratic twist. It is widely known that for all $E/\Bbb{Q}$: elliptic …

Let $A$ be an algebraic variety over a field $K$, which is finite extension of $ \Bbb{Q}$. We say local global principle holds if $A(K_v) \neq \emptyset$ implies $A(K) \neq \emptyset$, where $K_v$ is …

Let $p$ be a prime number and $n$ be positive integer. Let $E_{p,n}:y^2=x^3+p^nx$ be an elliptic curve. LMFDB reads in the case $(p,n)=(73,3)$ , $\#Sha(E_{p,n})=64$. This is the biggest size of $Sha(E …

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-Shafare …

$\newcommand{\Sha}{\operatorname{Sha}}$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 compone …

$C/ \Bbb{Q}: 3X^3 + 4Y^3 + 5Z^3 = 0$ is known to be a nontrivial element of the Tate–Shafarevich group of the elliptic curve $E/\Bbb{Q}:X^3 + Y^3 + 60Z^3 = 0$. It is also an example of an abelian var …

$\DeclareMathOperator{\Sha}{Ш}$ Let $E/K$ be an elliptic curve over a number field $K$. Let $\Sha (E/K)$ be the Tate-Shafarevich group, and let $n\ge 2$ be an integer. According to Theorem 15 in the 2 …

Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$. Tate-Shafarevich group $\mathit{Sha}(E/\Bbb{Q})$ is defined as follows. $$\mathit{Sha}(E/\Bbb{Q})\stackrel{\text{def}}{=} \operatorname{Ker …

Let $E/\mathbb{Q}$ be an elliptic curve.Let $\operatorname{Sha}(E/\Bbb{Q})$ be a Tate-Shafarevich group. Milne's 'Arithmetic Duality Theorems' Remark 6.14(b) describe the following exact sequence. Su …

$\DeclareMathOperator\Sha{Sha}\DeclareMathOperator\Gal{Gal}$Let $L/K$ be a quadratic extension of number field $K$. Let $\sigma$ be a generator of $\Gal(L/K)$. Let $E/K$ be an elliptic curve defined o …

$\DeclareMathOperator\Sha{Sha}$I calculated the Tate–Shafarevich group $\Sha(E/K)[2]$ of the elliptic curve $E:y^2=x^3+17x$ over $K=\Bbb{Q}(\sqrt{-37})$. I calculated that by hand and I reached the co …

Let $E/K$ be a number field. For quadratic field extensions $L/K$, it is known that $\operatorname{Ш}(E/L)[2]$ can be arbitrarily large (cf. P. L. Clark and S. Sharif, "Period, index and potential Sha …