TY - JOUR

T1 - On the p-adic Birch and Swinnerton-Dyer conjecture for elliptic curves over number fields

AU - Disegni, Daniel

N1 - Funding Information:
The author was partially supported by a public grant from the Fondation Mathématique Jacques Hadamard. This article was written while the author held a fellowship at the Centre de Recherches Mathématiques, Montréal.
Publisher Copyright:
© 2020 by Kyoto University.

PY - 2020/6/1

Y1 - 2020/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85090884015&partnerID=8YFLogxK

U2 - 10.1215/21562261-2018-0012

DO - 10.1215/21562261-2018-0012

M3 - Article

AN - SCOPUS:85090884015

VL - 60

SP - 473

EP - 510

JO - Kyoto Journal of Mathematics

JF - Kyoto Journal of Mathematics

SN - 0023-608X

IS - 2

ER -