Abstract
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.
Original language | English |
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Article number | 20220112 |
Number of pages | 16 |
Journal | Proceedings of the Royal Society A: Mathematical and Physical Sciences |
Volume | 478 |
Early online date | 7 Sept 2022 |
DOIs | |
Publication status | Published - 28 Sept 2022 |