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 $\mathbb{Z}_p$-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.


Ashay A. Burungale

California Institute of Technology, Pasadena, CA and The University of Texas at Austin, Austin, TX

Shinichi Kobayashi

Kyushu University, Fukuoka, Japan

Kazuto Ota

Graduate School of Science, Osaka University, Toyonaka, Osaka, Japan