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

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/Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{/{\mathbb {Q}}}$$\end{document} 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 Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Q}}$$\end{document}, especially rank one aspects. In this note we outline some of this recent progress.