Vol. 260, No. 2, 2012

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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
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
Authors
Massimo Bertolini
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50
I-20133 Milano
Italy
Henri Darmon
Department of Mathematics
McGill University
Montreal, QC  H3W 1Z4
Canada
Kartik Prasanna
Department of Mathematics
University of Michigan
530 Church Street, 2074 East Hall
Ann Arbor, MD 48109
United States