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

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Abstract

We formulate a multivariable p-adic Birch and Swinnerton-Dyer conjecture for p-ordinary elliptic curves A over number fields K. It generalizes the one-variable conjecture of Mazur, Tate, and Teitelbaum, who studied the case K = Q and 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 and Waldspurger formulas in families.

Original languageEnglish
Pages (from-to)473-510
Number of pages38
JournalKyoto Journal of Mathematics
Volume60
Issue number2
DOIs
StatePublished - 1 Jun 2020

ASJC Scopus subject areas

  • Mathematics (all)

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