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

Let $E\mathrm{/}\mathbb{Q}$ be an elliptic curve. We study the behavior of the Tate–Shafarevich group of $E$ under quadratic extensions $\mathbb{Q}(\sqrt{D})\mathrm{/}\mathbb{Q}$. By analyzing the cokernel of the restriction map, without assuming the finiteness of the Tate–Shafarevich group, we prove that the ratio $\frac{\mathrm{\#}\text{Sha}(E\mathrm{/}\mathbb{Q}(\sqrt{D}))[4]}{\mathrm{\#}\text{Sha}(E_D\mathrm{/}\mathbb{Q})[2]}$ and $\mathrm{\#}\text{Sha}(E_D\mathrm{/}\mathbb{Q})[2]$ can, under some conditions on $E\mathrm{/}\mathbb{Q}$, grow arbitrarily large simultaneously, where $E_D$ denotes the quadratic twist of $E$ by $D$. For elliptic curves of the form $E:y^2=x^3+px$ with $p\equiv 14$ being an odd prime, assuming the finiteness of the relevant Tate–Shafarevich groups, we prove that $\mathrm{\#}\text{Sha}(E\mathrm{/}\mathbb{Q}(\sqrt{D}))[2]\le 4$ and $\text{Sha}(E_D\mathrm{/}\mathbb{Q})[2]=0$ for infinitely many square-free integers $D$ with $-D$ being a prime number. Additionally, $\text{Sha}(E\mathrm{/}\mathbb{Q}(\sqrt{-D}))[2]\mathrm{\ne }0$ for all $D$ when $p=257$.