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

Celine Maistret*, Holly M Green*

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

4 Citations (Scopus)

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 languageEnglish
Article number20220112
Number of pages16
JournalProceedings of the Royal Society A: Mathematical and Physical Sciences
Volume478
Early online date7 Sept 2022
DOIs
Publication statusPublished - 28 Sept 2022

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