Iwasawa theory and \(p\)-adic Hodge theory
type: Item
instance of: scholarly article
title: Iwasawa theory and \(p\)-adic Hodge theory
zbMATH Open document ID: 0798.11050
author: Kazuya Kato
published in: Kodai Mathematical Journal
publication date: 1993-09-07T00:00:00Z
review text: The Iwasawa main conjecture for varieties (or motives) over arbitrary number fields is formulated using \(p\)-adic Hodge theory. The classical Iwasawa main conjecture gives a relation between the special values of partial Riemann zeta-functions to the Galois module structures of the ideal class groups of cyclotomic fields over \(\mathbb{Q}\). In this paper, the Iwasawa main conjecture for motives is formulated, where \(\mathbb{Q}\) is replaced by an arbitrary number field \(K\), and a cyclotomic field over \(\mathbb{Q}\) by a finite abelian extension \(L\) of \(K\), and partial Riemann zeta-functions by partial \(L\)-functions \(L_ S (M, \sigma\)-part, \(s)\) of a motive \(M\) over \(K\) for \(\sigma \in \text{Gal} (L/K)\). (Here \(S\) is a finite set of finite places of \(K\) including ``bad primes'', and \(L_ S\) means the \(L\)-function without the \(S\)-part.) The main conjecture of Iwasawa in this setting describes the special values of \(L_ S (M, \sigma\)-part, \(s)\) in terms of the \(\text{Gal} (L/K)\)- module structures of the etale cohomology of \(\text{Spec} ({\mathcal O}_{L,S})\) with coefficients in an etale sheaf coming from \(M\). (Here \({\mathcal O}_{L,S}\) denotes the ring of elements of \(L\) which are integral outside \(S\).) In order to state the main conjecture, some notation ought to be fixed. Let \(G=\text{Gal} (L/K)\), and let \(M\) be a pure motive over \(K\) of weight \(\leq - 1\). Let \(p\) be a prime fixed once and for all. Let \(S\) be a finite set of finite places of \(K\) containing all finite places at which \(M\) has bad reduction and all finite places which ramify in \(L/K\). Let \(L_ S (M,s) = \sum_{\mathfrak A} a({\mathfrak A}) N({\mathfrak A})^{-s}\) be the Hasse-Weil \(L\)-function of \(M\) without Euler factors for places in \(S\). Here the sum runs over all nonzero ideals \({\mathfrak A}\) of \({\mathcal O}_ K\), \(a({\mathfrak A}) \in \mathbb{Q}\), and \(N( {\mathfrak A})\) denotes the norm of \({\mathfrak A}\). For \(\sigma…
reviewed by: Noriko Yui
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 409639
zbMATH Keywords: special values of \(L\)-series
zbMATH Keywords: \(p\)-adic Hodge theory
zbMATH Keywords: partial \(L\)- functions
zbMATH Keywords: Iwasawa main conjecture for motives
zbMATH Keywords: Galois module structures
zbMATH Keywords: analytic zeta element
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039701
OpenAlex ID: W2048205489
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DOI: 10.2996/KMJ/1138039701
description: scientific article; zbMATH DE number 409639