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 $p$-adic $L$-function. As a consequence, we prove an inequality in the $p$-adic Birch andSwinnerton-Dyer conjecture for Rubin's $p$-adic $L$-function. Rubin'sconjecture is also an essential tool in our exploration of the arithmeticof Rubin's $p$-adic $L$-function, which includes a Bertolini--Darmon--Prasanna type formula.
Citation
Ashay Burungale. Shinichi Kobayashi. Kazuto Ota. "Rubin's conjecture on local units in the anticyclotomic tower at inert primes." Ann. of Math. (2) 194 (3) 943 - 966, November 2021. https://doi.org/10.4007/annals.2021.194.3.8
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