We formulate a multivariable -adic Birch and Swinnerton-Dyer conjecture for -ordinary elliptic curves over number fields . It generalizes the one-variable conjecture of Mazur, Tate, and Teitelbaum, who studied the case 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: is imaginary quadratic, is the base change to of an elliptic curve over the rationals, and the rank of is either 0 or .
The proof is naturally divided into a few cases. Some of them are deduced from the purely cyclotomic case of elliptic curves over , 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 , 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) -adic Gross–Zagier and Waldspurger formulas in families.
"On the -adic Birch and Swinnerton-Dyer conjecture for elliptic curves over number fields." Kyoto J. Math. 60 (2) 473 - 510, June 2020. https://doi.org/10.1215/21562261-2018-0012