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

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 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 × E 2 . Using this result, we complete the proof of the p -parity conjecture for elliptic curves over totally real fields.

Root numbers and parity phenomena

“…We will confine ourselves to the setting of elliptic curves, although the parity conjecture is formulated for abelian varieties. For some recent results on these conjectures, see [7, 8, 19, 22–24, 28, 32, 41, 49–53]; for results on root numbers of abelian varieties, see [2, 58].…”

2‐Selmer parity for hyperelliptic curves in quadratic extensions

“…For elliptic curves over $$\mathbb {Q}$$, Dokchitser–Dokchitser [16] have shown that the $$p$$‐parity conjecture holds for all primes $$p$$. Subsequently, Nekovář [44] extended this result to all totally real number fields, excluding some elliptic curves with potential complex multiplication; these exceptional cases have recently been treated by Green–Maistret [22]. For a general number field $$K$$, Česnavičius [8] has shown that the $$p$$‐parity conjecture holds for elliptic curves over $$K$$ possessing a $$p$$‐isogeny, whilst work of Kramer–Tunnell [27] and Dokchitser–Dokchitser [17] proves that the 2‐parity conjecture holds for an arbitrary elliptic curve $$E/K$$, not over $$K$$ itself, but over any quadratic extension of $$K$$.…”