2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/122298We study Rubin's variant of the $p$-adic Birch and Swinnerton-Dyer conjecture for CM elliptic curves concerning certain special values of the Katz two-variable $p$-adic $L$-function that lie outside the range of $p$-adic interpolation.Final version. To appear in Compositio MathematicaNumber TheoryAlgebraic Geometrytext