Derived p-adic heights and p-adic L-functions

TL;DR

This paper generalizes Rubin's relation between p-adic heights and derivatives of p-adic L-functions for elliptic curves, introducing derived heights and linking degeneracies to Iwasawa module properties.

Contribution

It provides an alternative definition of p-adic height pairing and extends Rubin's theorem to higher derivatives of p-adic L-functions, relating degeneracies to Iwasawa module semi-simplicity.

Findings

Generalized Rubin's relation to higher derivatives of p-adic L-functions.

Connected degeneracies in derived heights to Iwasawa module properties.

Proposed an alternative definition of p-adic height pairing.

Abstract

If E is an elliptic curve defined over a number field and p is a prime of good ordinary reduction for E, a theorem of Rubin relates the p-adic height pairing on the p-power Selmer group of E to the first derivative of a cohomologically defined p-adic L-function attached to E. Bertolini and Darmon have defined a sequence of "derived" p-adic heights. In this paper we give an alternative definition of the p-adic height pairing and prove a generalization of Rubin's result, relating the derived heights to higher derivatives of p-adic L-functions. We also relate degeneracies in the derived heights to the failure of the Selmer group of E over a Z_p-extension to be "semi-simple" as an Iwasawa module, generalizing results of Perrin-Riou.