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

Abstract

We prove a fundamental conjecture of Rubin on the structure of local units in the anticyclotomic $\mathbb{Z}_p$-extension of the unramified quadratic extension of $\mathbb{Q}_p$ for $p\geq 5$ 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.