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June 2020 On the p-adic Birch and Swinnerton-Dyer conjecture for elliptic curves over number fields
Daniel Disegni
Kyoto J. Math. 60(2): 473-510 (June 2020). DOI: 10.1215/21562261-2018-0012


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, under mild conditions, in the following situation: K is imaginary quadratic, A=EK is the base change to K 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.


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Daniel Disegni. "On the p-adic Birch and Swinnerton-Dyer conjecture for elliptic curves over number fields." Kyoto J. Math. 60 (2) 473 - 510, June 2020.


Received: 28 February 2017; Revised: 7 November 2017; Accepted: 1 December 2017; Published: June 2020
First available in Project Euclid: 29 February 2020

zbMATH: 07223242
MathSciNet: MR4094741
Digital Object Identifier: 10.1215/21562261-2018-0012

Primary: 11G40
Secondary: 11F85

Rights: Copyright © 2020 Kyoto University


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Vol.60 • No. 2 • June 2020
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