On some modules attached to the Lubin-Tate formal groups.
type: Item
instance of: scholarly article
title: On some modules attached to the Lubin-Tate formal groups.
zbMATH Open document ID: 1188.11063
author: Kazuhito Kozuka
author: Hirofumi Tsumura
published in: Tokyo Journal of Mathematics
publication date: 2002-07-24T00:00:00Z
review text: Let \(p\) be a prime number, \({\mathbb C}_p\) the completion of the algebraic closure of \({\mathbb Q}_p\) with the ring of integers \({\mathcal O}\). Let \(F(X,Y)\) be a Lubin-Tate formal group over \({\mathbb Z}_p\) and \(h(X)\) a meromorphic series in \({\mathcal O}((X))^\times\). Let \(\chi\) be a primitive Dirichlet character with conductor a power of \(p\). In an earlier work [Kyushu J. Math. 51, No. 2, 381--391 (1997; Zbl 0909.11053)], the first author constructed a meromorphic function \(L_p(s,\chi,F,h)\) in the case that \(F(X,Y)\) is a relative Lubin-Tate formal group defined over the ring of integers of an unramified extension of \({\mathbb Q}_p\). If \(F(X,Y)\) is the formal multiplicative group \(G_m(X,Y)=(X+1)(Y+1)-1\) and \(h(X)=X\), then \(L_p(s,\chi,G_m,X)\) coincides with the Kubota-Leopoldt \(p\)-adic \(L\)-function \(L_p(s,\chi)\). Iwasawa proved that \(L_p(s,\chi)\) is related to the Galois structure of the local units modulo the closure of the cyclotomic units. The main purpose of this paper is to generalize the result of Iwasawa for the function \(L_p(s,\chi,F,h)\), in the case that \(F\) is defined over \( {\mathbb Z}_p\) and that \(h(X)\) satisfies certain appropriate conditions.
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11S31
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 11S20
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 1744150
zbMATH Keywords: Lubin-Tate formal group
zbMATH Keywords: \(p\)-adic \(L\)-function
zbMATH Keywords: units
zbMATH Keywords: Galois module structure
zbMATH Keywords: Iwasawa theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1255958186
OpenAlex ID: W2014258737
DOI: 10.3836/TJM/1255958186
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description: scientific article; zbMATH DE number 1744150