We prove a fundamental conjecture of Rubin on the structure of local units in the anticyclotomic Zp-extension of the unramified quadratic extension of Qp 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 super-singular 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.
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