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 .

Source: Agboola, Adebisi - Department of Mathematics, University of California at Santa Barbara

Collections: Mathematics