Rubin's conjecture on local units in the anticyclotomic tower at inert primes
type: Item
instance of: scholarly article
title: Rubin's conjecture on local units in the anticyclotomic tower at inert primes
zbMATH Open document ID: 1487.11100
author: Ashay A. Burungale
author: Kazuto Ota
author: Shinichi Kobayashi
published in: Annals of Mathematics. Second Series
publication date: 2021-12-09T00:00:00Z
full work available at URL: https://authors.library.caltech.edu/112316/
review text: Let \(p \geq 5\) be a prime. The authors prove a conjecture of \textit{K. Rubin} [Invent. Math. 88, 405--422 (1987; Zbl 0623.14006)] on the structure of local units along the anticyclotomic \(\mathbb{Z}_p\)-extension of the unramified quadratic extension \(\Phi\) of \(\mathbb{Q}_p\). \par We briefly sketch Rubin's conjecture. Fix a Lubin-Tate formal group \(\mathcal{F}\) over the ring of integers \(\mathcal{O}\) in \(\Phi\) for the uniformizer \(\pi := -p\). Let \(\Phi_n\) be the field obtained from \(\Phi\) by adjoining all \(\pi^{n+1}\)-torsion points of \(\mathcal{F}\) and set \(\Phi_{\infty} := \bigcup_n \Phi_n\). Then there is a natural isomorphism \(\kappa:\mathrm{Gal}(\Phi_{\infty}/\Phi) \simeq \mathcal{O}^{\times}\). Moreover, one has a decomposition \[ \mathrm{Gal}(\Phi_{\infty}/\Phi_0) \simeq G^+ \times G^-, \] where both \(G^{\pm}\) are isomorphic to \(\mathbb{Z}_p\) and \(\mathrm{Gal}(\Phi/\mathbb{Q}_p)\) acts upon \(G^+\) and \(G^-\) by \(+1\) and \(-1\), respectively. We may identify \(G^+\) and \(G^-\) with the Galois groups of the cyclotomic and the anticyclotomic \(\mathbb{Z}_p\)-extension of \(\Phi\), respectively. \par Let \(U_n\) be the group of principal units in \(\Phi_n\) and consider the inverse limit \(\varprojlim_n U_n \otimes_{\mathbb{Z}_p} \mathcal{O}\) with respect to the norm maps. We denote the part of this limit upon which \(\mathrm{Gal}(\Phi_0/\Phi)\) acts via the Teichmüller character by \(U_{\infty}\). Then \(U_{\infty}\) is free of rank \(2\) over the Iwasawa algebra \(\mathcal{O}[[\mathrm{Gal}(\Phi_{\infty}/\Phi_0)]]\). The main object of study is the quotient \[ V^{\ast}_{\infty} := U_{\infty}^{\ast} / (\sigma-1), \] where \(U_{\infty}^{\ast}\) is \(U_{\infty}\) with the Galois action twisted by \(\kappa^{-1}\) and \(\sigma\) is a topological generator of \(G^+\). Then \(V^{\ast}_{\infty}\) is a free \(\Lambda := \mathcal{O}G^-\)-module of rank \(2\). Rubin defines two subspaces \(V^{\ast, \pm}_{\infty}\) of \(V^{\ast}_{\inf…
reviewed by: Andreas Nickel
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11G07
Mathematics Subject Classification ID: 11G15
zbMATH DE Number: 7441739
zbMATH Keywords: CM elliptic curves
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: local units
zbMATH Keywords: \(p\)-adic \(L\)-functions
Wikidata QID: Q113692557
MaRDI profile type: Publication
OpenAlex ID: W3211807101
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DOI: 10.4007/ANNALS.2021.194.3.8
description: scientific article; zbMATH DE number 7441739