On the p-adic Birch and Swinnerton-Dyer conjecture for elliptic curves over number fields

摘要

We formulate a multivariable p-adic Birch and Swinnerton-Dyer conjecture for p-ordinary elliptic curvesAover number fieldsK. It generalizes the one-variable conjecture ofMazur, Tate, and Teitelbaum, who studied the caseK =Qand the phenomenon of exceptional zeros.We discuss old and new theoretical evidence toward our conjecture and in particular we fully prove it, undermild conditions, in the following situation: K is imaginary quadratic, A = EK is the base change toK of an elliptic curve over the rationals, and the rank of A is either 0 or 1. The proof is naturally divided into a few cases. Some of them are deduced from the purely cyclotomic case of elliptic curves over Q, which we obtain from a refinement of recent work of Venerucci alongside the results of Greenberg, Stevens, Perrin-Riou, and the author. The only genuinely multivariable case (rank 1, two exceptional zeros, three partial derivatives) is newly established here. Its proof generalizes to show that the “almost-anticyclotomic” case of our conjecture is a consequence of conjectures of Bertolini and Darmon on families of Heegner points, and of (partly conjectural) p-adic Gross–Zagier andWaldspurger formulas in families.