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

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