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

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 Zp-extension to be “semi-simple ” as an Iwasawa module, generalizing results of Perrin-Riou. 0. Introduction and notation. Fix forever a rational prime p> 2. By a coefficient ring we mean a commutative ring which is complete, Noetherian, and local with residue field of characteristic p. Fix a finite set of places Σ of a number field F containing all archimedean places and all primes above p, and le