Behaviors of the Tate--Shafarevich group of elliptic curves under quadratic field extensions

Abstract: Let $E$ be an elliptic curve defined over $\Bbb{Q}$. We study the behavior of the Tate--Shafarevich group of $E$ under quadratic extensions $\Bbb{Q}(\sqrt{D})/\Bbb{Q}$. First, we determine the cokernel of the restriction map $H^1(\mathrm{Gal}(\overline{\Bbb{Q}}/\Bbb{Q}),E)[2] \to \bigoplus_{p}H^1(\mathrm{Gal}(\overline{\Bbb{Q}_p}/\Bbb{Q}_p),E)[2]$. Using this result, without assuming the finiteness of the Tate--Shafarevich group, we prove that the ratio $\frac{#\Sha(E/\Bbb{Q}(\sqrt{D}))[4]}{#\Sha(E_D/\Bbb{Q})[2]}$ can grow arbitrarily large, where $E_D$ denotes the quadratic twist of $E$ by $D$. For elliptic curves of the form $E : y^2 = x^3 + px$ with $p$ an odd prime, assuming the finiteness of the relevant Tate--Shafarevich groups, we prove two results: first, that $\Sha(E_D/\Bbb{Q})[2] = 0$ for infinitely many square-free integers $D$, and second, that $#\Sha(E/\Bbb{Q}(\sqrt{D}))[2] \leq 4$ for infinitely many imaginary quadratic fields $\Bbb{Q}(\sqrt{D})$, although $\Sha(E/\Bbb{Q}(\sqrt{D}))[2]$ cannot be made trivial in general.