On Artin's conjecture and the class number of certain CM fields. I,II
type: Item
instance of: scholarly article
title: On Artin's conjecture and the class number of certain CM fields. I,II
zbMATH Open document ID: 0711.11042
DOI: 10.1215/S0012-7094-89-05925-5
published in: Duke Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Let K be a CM field, i.e., a totally complex quadratic extension of a totally real field k, with \([k:{\mathbb{Q}}]=n\). The problem of finding an effective lower bound for \(h_ K\), the class number of K, has a very similar nature to the corresponding problem for an imaginary quadratic field (the long-standing Gauss conjecture proved by Gross and Zagier), in view of the analogous factorization: \(\zeta_ K(s)=\zeta_ k(s)L(s,\chi),\) where in the imaginary quadratic case, L(s,\(\chi\)) denotes the Dirichlet L-series with the Kronecker symbol, while in the CM field case, L(s,\(\chi\)) denotes the real ray class L-series. The problem amounts to obtaining as good an effective lower estimate for the distance from 1 of a possible real (Siegel) zero \(\beta\) as possible. In the imaginary quadratic case, the above-mentioned recent important result of (Goldfeld-) Gross-Zagier gives effectively \(h_ K\gg (\log D_ k)^{1- \epsilon}\) for \(n=1\), where in general \(D_ M\) denotes the absolute value of the discriminant of the number field M. In the CM field case, under the restriction that k belongs to the class \({\mathcal N}\) of all totally real fields that are attainable by a sequence \({\mathbb{Q}}=k_ 0\subset k_ 1\subset...\subset k_{\ell}=k\) with each field normal over the preceding one (in particular, k normal over \({\mathbb{Q}}\) also belong to \({\mathcal N})\), \textit{H. M. Stark} [Invent. Math. 23, 135-152 (1974; Zbl 0278.12005)] has shown that \[ (1)\quad 1-\beta >\min (D_ k^{-1/n} f^{-1/2n},\quad (\log A_ nD_ k^ 2f)^{-1}), \] where \(A_ n=1\) and f denotes the norm of the conductor of \(\chi\). \textit{A. M. Odlyzko} [ibid. 29, 275-286 (1975; Zbl 0299.12010)] and further the first author [ibid. 55, 37-47 (1979; Zbl 0474.12009)] have settled the problem of proving \(h_ K\to \infty\) as \(n\to \infty\), not covered by Stark's result. For \(k\not\in N\), Stark's method yields (1) effectively with \(A_ n=n!\) whose presence, however, forces the restriction \[ (2)\q…
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 4168794
zbMATH Keywords: analytic estimates on class numbers of CM fields
zbMATH Keywords: zero-free region
author: Jeff Hoffstein
author: Naomi Jochnowitz
reviewed by: Shigeru Kanemitsu
MaRDI profile type: Publication
Wikidata QID: Q123235062
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description: scientific article; zbMATH DE number 4168794