In the early 1990's Rubin formulated a variant of the p-adic Birch and Swinnerton-Dyer conjecture for CM elliptic curves of rank at least one; this involved studying special values of the Katz two-variable p-adic L-function lying outside the range of interpolation. By relating his conjecture to the usual p-adic Birch and Swinnerton-Dyer conjecture, he was able to give a striking p-adic construction of rational points of infinite order on CM elliptic curves of rank one. I shall discuss the picture that emerges if one considers CM elliptic curves of rank zero instead.