 
Summary: ON RUBIN'S VARIANT OF THE pADIC BIRCH AND
SWINNERTONDYER CONJECTURE II
A. AGBOOLA
Abstract. Let E/Q be an elliptic curve with complex multiplication by the ring of integers
of an imaginary quadratic field K. In 1991, by studying a certain special value of the Katz
twovariable padic Lfunction lying outside the range of padic interpolation, K. Rubin
formulated a padic variant of the Birch and SwinnertonDyer conjecture when E(K) is
infinite, and he proved that his conjecture is true for E(K) of rank one.
When E(K) is finite, however, the statement of Rubin's original conjecture no longer
applies, and the relevant special value of the appropriate padic Lfunction is equal to zero.
In this paper we extend our earlier work and give an unconditional proof of an analogue of
Rubin's conjecture for the case in which E(K) is finite.
1. Introduction
The goal of this article is to extend the results of [1] to give an unconditional proof of a
certain variant of the padic Birch and SwinnertonDyer conjecture for elliptic curves with
complex multiplication.
Let E/Q be an elliptic curve with complex multiplication by OK , the ring of integers of
an imaginary quadratic field K (this implies that K is necessarily of class number one). Let
p > 3 be a prime of good, ordinary reduction for E; then we may write pOK = pp # , with
p = #OK and p # = # # OK .
