Iwasawa theory and the Eisenstein ideal
type: Item
instance of: scholarly article
title: Iwasawa theory and the Eisenstein ideal
zbMATH Open document ID: 1131.11068
DOI: 10.1215/S0012-7094-07-13713-X
author: Romyar T. Sharifi
published in: Duke Mathematical Journal
publication date: 2007-04-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0501236
review text: Let \(p\) be an odd prime, \(F = {\mathbb Q}(\mu_p)\), \({\mathcal G}\) = the Galois group over \(F\) of the maximal pro-\(p\) unramified outside \(p\) extension of \(F.\) In their paper [Duke Math. J. 120, 269--310 (2003; Zbl 1047.11106)], \textit{W. G. McCallum} and \textit{R. T. Sharifi} studied the cup product \(H^1({\mathcal G}, \mu_p) \times H^1({\mathcal G}, \mu_p) \to A_F \otimes \mu_p\), where \(A_F\) is the \(p\)-class group of \(F\). In particular, they conjectured that the pairing induced by this cup product on the \(p\)-units \(\xi_F\) of \(F,\) \((\ldotp, \ldotp): \xi_F \times \xi_F \to A_F \otimes \mu_p,\) is surjective on the minus part. In this paper, the second named author shows that the conjecture holds for \(p < 10^3.\) The result itself is less interesting than the method of proof, which reveals an intriguing relationship between the structure of certain Iwasawa modules over Kummer extensions and the structure of ordinary Hecke algebras of modular forms localized at the Eisenstein ideal, in the same streamline as Ribet's proof of the converse of Herbrand's theorem (1976), Mazur-Wiles' proof of the Main Conjecture over \({\mathbb Q}\) (1984), and Ohta's revisitation of Mazur-Wiles using Hida theory (2000). More specifically, fix an even integer \(k < p\) such that \(p \mid B_k,\;p \nmid B_{p+1-k}\) and, for simplicity of this description, \(p^2 \nmid B_k.\) The author studies the action of \(G_{\mathbb Q}\) on an Eisenstein component \(\mathfrak X\) of a particular inverse limit of cohomology groups of modular curves previously considered by \textit{M. Ohta} [J. Reine Angew. Math. 585, 141--172 (2005; Zbl 1081.11035)]. This \(\mathfrak X\) is a free module of rank 2 over \({\mathfrak H},\) the localization of Hida's ordinary cuspidal Hecke algebra at the maximal ideal containing the Eisenstein ideal \({\mathcal J}\) with character \(\omega^k.\) Let \(f : G_{\mathbb Q} \to \Aut_{\mathfrak H}(\mathfrak X), \;\sigma \mapsto \left(\begin{smallmatrix…
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 5145499
zbMATH Keywords: cup product
zbMATH Keywords: Eisenstein ideal
reviewed by: Thong Nguyen Quang Do
MaRDI profile type: Publication
arXiv ID: math/0501236
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description: scientific article