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 𝐸₁ and 𝐸₂ 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 𝐸₁ and 𝐸₂, we show that the Birch and Swinnerton-Dyer conjecture correctly predicts the parity of the rank of 𝐸₁×𝐸₂. Using this result, we complete the proof of the 𝑝-parity conjecture for elliptic curves over totally real fields.