2016/09/08 by Daniel Disegni · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Combinatorics #Conjecture #Elliptic curve #Geometry #Mathematical analysis #Mathematics #Pure mathematics #Quadratic equation #Rank (graph theory) #Rational number #Variable (mathematics) #math.NT #msc:11G40
paper · pdf · doi:10.1215/21562261-2018-0012
published as Kyoto J. Math. 60, no. 2 (2020), 473-510 · 28 pages, comments welcome
arxiv created 2016/09/08 · openalex publication_date 2020/02/29 · arxiv updated 2020/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.