Rubin's conjecture on local units in the anticyclotomic tower at inert primes

We prove a fundamental conjecture of Rubin on the structure of local units in the anticyclotomic Z(p)-extension of the unramified quadratic extension of Q(p) for p >= 5 a prime. Rubin's conjecture underlies Iwasawa theory of the anticyclotomic deformation of a CM elliptic curve over the CM field at primes p of good supersingular reduction, notably the Iwasawa main conjecture in terms of the p-adic L -function. As a consequence, we prove an inequality in the p-adic Birch and Swinnerton-Dyer conjecture for Rubin's p-adic L -function. Rubin's conjecture is also an essential tool in our exploration of the arithmetic of Rubin's p-adic L -function, which includes a Bertolini-Darmon-Prasanna type formula.