On Rubin's variant of the p-adic Birch and Swinnerton-Dyer conjecture II

TL;DR

This paper extends Rubin's p-adic Birch and Swinnerton-Dyer conjecture to cases where the elliptic curve's rational points over an imaginary quadratic field are finite, providing an unconditional proof of the conjecture's analogue.

Contribution

It offers the first unconditional proof of Rubin's conjecture analogue for elliptic curves with finite rational points over an imaginary quadratic field.

Findings

Proves the conjecture analogue unconditionally for finite E(K)

Extends previous work on the p-adic BSD conjecture

Provides new insights into special values of p-adic L-functions

Abstract

Let E be an elliptic curve over Q 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 two-variable p-adic L-function lying outside the range of $p$-adic interpolation, K. Rubin formulated a p-adic variant of the Birch and Swinnerton-Dyer 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 $p$-adic L-function is equal to zero. In this paper we extend our earlier work and give an unconditional proof of an analogue of Rubin's conjecture when E(K) is finite.