2 problems

Let $E$ be an elliptic curve as above, let $N$ be a prime with $N\equiv 3\pmod{4}$, and suppose $\beta_{-N}\ne 0$. Let $E^{-N}$ denote the quadratic twist of $E$ associated with th…

Let $E$ be an elliptic curve over a number field $K$, and let $\Sha_{E/K}$ be its Tate–Shafarevich group. Weak Tate–Shafarevich conjecture. The dimension … is even. The source pres…