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