The Iwasawa \(\mu \)-invariant of \(p\)-adic Hecke \(L\)-functions
type: Item
instance of: scholarly article
title: The Iwasawa \(\mu \)-invariant of \(p\)-adic Hecke \(L\)-functions
zbMATH Open document ID: 1223.11131
author: Haruzo Hida
published in: Annals of Mathematics. Second Series
publication date: 2010-09-06T00:00:00Z
review text: For an odd prime \(p,\) the author computes the \(\mu\)-invariant of the anticyclotomic Katz \(p\)-adic \(L\)-function of a \(p\)-ordinary \(CM\) field \(M\) if the conductor of the branch character is a product of primes split over the maximal real subfield \(F.\) Except for rare cases where the root number of the \(p\)-adic functional equation is congruent to \(-1 \bmod p\), the \(\mu\)-invariant vanishes. More precisely, assume the ``ordinary hypothesis'', i.e., every \(p\)-place of \(F\) splits in \(M.\) Fix a prime to \(p\) \({\mathcal O}_M\)-ideal \({\mathfrak G} = {\mathcal F} {\mathcal F}' {\mathcal J},\) where \({\mathcal F} {\mathcal F'}\) (resp. \({\mathcal J})\) consists of split (resp. inert or ramified) primes of \(F,\) \({\mathcal F} + {\mathcal F}' = {\mathcal O}_F\) and \({\mathcal F} \subset ({\mathcal F}')^c\) (where \(c\) denotes complex conjugation). Following Katz (who did the case \({\mathfrak G} = (1)),\) the author and \textit{J. Tilouine} [Ann. Sci. Éc. Norm. Supér. (4) 26, No. 2, 189--259 (1993; Zbl 0778.11061)] constructed a unique measure \(\varphi\) on the ray class group \(Z ({\mathfrak G})\) mod \({\mathfrak G} p^\infty\) of \( M\) which interpolates (in a precise way which we do not recall here) Hecke \(L\)-values. Fix a \(p\)-adic valuation ring \(W\) which is finite flat over the ring of Witt vectors \(W (\overline{\mathbb F}_p).\) Let \(\Delta\) be the maximal torsion subgroup of \(Z({\mathfrak G}).\) A character \(\psi : \Delta \to W^\times\) is called a branch character. The \(\psi\)-branch \(\varphi_\psi\) of the measure \(\varphi\) is defined on \(\Gamma = Z ({\mathfrak G})/\Delta\) by \(\int_\Gamma \;\Phi \;d \;\varphi_\psi = \int_{Z({\mathfrak G})} \;\psi \;\Phi\;d \;\varphi.\) The main result of this paper (theorem 5.1) computes \(\mu(\varphi^-_\psi)\) explicitly in terms of the branch character \(\psi\) when \({\mathcal J} = (1).\) This implies, when \(p\) is unramified in \(F,\) that \(\mu (\varphi^-_\psi) = 0\) unless th…
reviewed by: Thong Nguyen Quang Do
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 11S40
zbMATH DE Number: 5779355
zbMATH Keywords: mu invariant
zbMATH Keywords: anticyclotomic Katz \(p\)-adic \(L\)-function
MaRDI profile type: Publication
full work available at URL: annals.2010.172.41
OpenAlex ID: W2067651832
Wikidata QID: Q96632229
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DOI: 10.4007/ANNALS.2010.172.41
description: scientific article; zbMATH DE number 5779355