p-adic Rankin L-series and rational points on CM elliptic curves

Massimo Bertolini, Henri Darmon and Kartik Prasanna

Vol. 260 (2012), No. 2, 261–303

DOI: 10.2140/pjm.2012.260.261

Abstract

This article presents a new proof of a theorem of Karl Rubin relating values of the Katz p-adic L-function of an imaginary quadratic field at certain points outside its range of classical interpolation to the formal group logarithms of rational points on CM elliptic curves. The approach presented here is based on the p-adic Gross–Zagier type formula proved by the three authors in previous work. As opposed to the original proof which relied on a comparison between Heegner points and elliptic units, it only makes use of Heegner points, and leads to a mild strengthening of Rubin’s original result. A generalization to the case of modular abelian varieties with complex multiplication is also included.

Keywords

p-adic L-functions, elliptic curves, rational points

Mathematical Subject Classification 2010

Primary: 11G05, 11G40

Milestones

Received: 8 August 2012

Revised: 16 October 2012

Accepted: 16 October 2012

Published: 30 November 2012