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 of this conjecture called the parity conjecture and prove the following. Let $E_1$ and $E_2$ be two elliptic curves defined over a number field $K$ whose 2-torsion groups are isomorphic as Galois modules. Assuming finiteness of the Shafarevich-Tate groups of $E_1$ and $E_2$, we show that the Birch and Swinnerton-Dyer conjecture correctly predicts the parity of the rank of $E_1\times E_2$. Using this result, we complete the proof of the $p$-parity conjecture for elliptic curves over totally real fields.