On the \(\mu\)-invariant of \(p\)-adic \(L\)-functions attached to elliptic curves with complex multiplication
type: Item
instance of: scholarly article
title: On the \(\mu\)-invariant of \(p\)-adic \(L\)-functions attached to elliptic curves with complex multiplication
zbMATH Open document ID: 0615.12018
DOI: 10.1016/0022-314X(87)90013-8
author: Leila Schneps
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: The aim of this paper is to prove the vanishing of \(\mu\) (the Iwasawa invariant) in the ordinary elliptic case; let \(K\) be a principal imaginary quadratic field and \(p\) a prime \(\ne 2,3\) which splits as \(p=\mathfrak pp^-_{\infty}\) be obtained by adjoining to \(K\) all the \({\mathfrak p}^ n\)-division points on \(E\) \((n=1,\ldots)\) and \(M_{\infty}\) the maximal abelian \(p\)-extension of \(F_{\infty}\) unramified outside \({\mathfrak p}\); the result means that \(\mathrm{Gal}(M_{\infty}/F_{\infty})\) has no \({\mathbb{Z}}_ p\)-torsion. Its proof uses a result of algebraic independence for the elliptic curve. The method is similar (in this much more difficult case) to the one introduced by \textit{W. Sinnott} [Invent. Math. 75, 273--283 (1984; Zbl 0531.12004)]. The result has been generalized by the reviewer [J. Reine Angew. Math. 358, 76--91 (1985; Zbl 0551.12011)].
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 3995832
zbMATH Keywords: p-adic L-functions
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: vanishing of \(\mu \)-invariant
zbMATH Keywords: elliptic case
zbMATH Keywords: imaginary quadratic field
zbMATH Keywords: complex multiplication
zbMATH Keywords: maximal abelian p-extension
zbMATH Keywords: elliptic curve
reviewed by: Roland Gillard
MaRDI profile type: Publication
Wikidata QID: Q55982883
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description: scientific article; zbMATH DE number 3995832