p-ADIC RANKIN L-SERIES AND RATIONAL POINTS ON CM ELLIPTIC CURVES

The aim of this article is to present a new proof of a theorem of Karl Rubin (see [Ru] and Thm. 1 below) 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. This theorem has been seminal in providing a motivation for Perrin-Riou’s formulation ([PR2], [PR3]) of the p-adic Beilinson conjectures. The new proof described in this work is based on the p-adic Gross-Zagier type formula of [BDP-gz], and only makes use of Heegner points (as opposed to the original proof which relied on on a comparison between Heegner points and elliptic units). Hence it should be adaptable to more general situations, for example to the setting of general CM fields.