A p-adic Waldspurger Formula and the Conjecture of Birch and Swinnerton-Dyer

Abstract

About a decade ago Bertolini–Darmon–Prasanna proved a p-adic Waldspurger formula, which expresses values of an anticyclotomic p-adic L-function associated to an elliptic curve \(E_{/{\mathbb {Q}}}\) outside its defining range of interpolation in terms of the p-adic logarithm of Heegner points on E. In the ensuing decade an insight of Skinner based on the p-adic Waldspurger formula has initiated a progress towards the Birch and Swinnerton-Dyer conjecture for elliptic curves over \({\mathbb {Q}}\), especially rank one aspects. In this note we outline some of this recent progress.