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.