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

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve. We study the behavior of the Tate--Shafarevich group of $E$ under quadratic extensions $\mathbb{Q}(\sqrt{D})/\mathbb{Q}$. By analyzing the cokernel of the restriction map, without assuming the finiteness of the Tate--Shafarevich group, we prove that the ratio $\frac{\#\Sha(E/\mathbb{Q}(\sqrt{D}))[4]}{\#\Sha(E_D/\mathbb{Q})[2]}$ and $\#\Sha(E_D/\mathbb{Q})[2]$ can, under some conditions on $E/\mathbb{Q}$, 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 $p\equiv 1 \bmod 4$ being an odd prime, assuming the finiteness of the relevant Tate--Shafarevich groups, we prove that $\#\Sha(E/\mathbb{Q}(\sqrt{D}))[2] \leq 4$ and $\Sha(E_D/\mathbb{Q})[2] = 0$ for infinitely many square-free integers $D$ with $-D$ being a prime number. Additionally, $\Sha(E/\mathbb{Q}(\sqrt{-D}))[2]\neq 0$ for all $D$ when $p=257$.