The 2-parity conjecture for elliptic curves with isomorphic 2-torsion

The Birch and Swinnerton--Dyer conjecture famously predicts that the rank of an elliptic curve can be computed from its $L$-function. In this article we consider a weaker version called parity prove following. Let $E_1$ $E_2$ two curves defined over number field $K$ whose 2-torsion groups are isomorphic as Galois modules. Assuming finiteness Shafarevich-Tate $E_2$, show Swinnerton-Dyer correctly $E_1\times E_2$. Using result, complete proof $p$-parity for totally real fields.