Heegner points and $p$-adic $L$-functions for elliptic curves over certain totally real fields

Abstract

For an elliptic curve $E$ over $\mathbb{Q}$ satisfying suitable hypotheses, Bertolini and Darmon have derived a formula for the Heegner point on $E$ in terms of the central derivative of the two variable $p$-adic $L$-function associated to $E$. In this paper, we generalize their work to the setting of totally real fields in which $p$ is inert. We also use this generalization to improve the results obtained by Bertolini–Darmon in the case of an elliptic curve defined over the field of rational numbers.