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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 …

$\DeclareMathOperator{\Sha}{Ш}$ 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 $E/K$ be an elliptic curve over a number field $K$. Let $M_K$ be the set of all places of $K$. Let $K_v$ be a completion of $K$ at $v$. I'm searching for a reference for the statement of the follo …

Let $E/K$ be an elliptic curve over number field $K$. Let $M_K$ be a set of all places of $K$. My question is, Does there exist a finite set $S\subset M_K$ such that $\forall C$: $E/K$-torsor, $\fo …

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 …

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 …

$\DeclareMathOperator\Sha{Sha}\DeclareMathOperator\Gal{Gal}$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 …

Let $E/\mathbb{Q}$ be an elliptic curve defined by $E: y^2 = x^3 + b$ (where $b \in \mathbb{Q}$). Let $E_D$ be an elliptic curve defined by $E_D: y^2 = x^3 + bD^2$. $E$ and $E_D$ are isomorphic over $ …

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

Let $E/\Bbb{Q}$ be an elliptic curve defined over $\Bbb{Q}$. Let $E_D$ be a quadratic twist of $E$ also defined over $\Bbb{Q}$. Is there a known elliptic curve, $E/\Bbb{Q}$, such that the rank of $E_D …

Let $L/K$ be a quadratic Galois extension of number fields. Let $E$ be an elliptic curve. Consider the natural map $$ F: H^1(\operatorname{Gal}(L/K), E(L)) \to \bigoplus_{v \in M_K} H^1(\operatorname{ …

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 …

$\DeclareMathOperator\Sha{Sha}$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 $E:y^2=x^3+17x$ be an elliptic curve. In this MO page(Infinitely many elliptic curve with twist rank more than $1$ in specific case), Nulhomologous's and other's comment reads from parity conjectu …

Let $E/\Bbb{Q}$ be an elliptic curve. Let $D$ be a square free negative integer. It is conjectured that 50％ of twist of elliptic curve $E_D$ has rank $0$ and $50％$ has rank $1$. But is some particular …