On Artin's conjecture and the class number of certain CM fields. I,II
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 \[ (2)\quad D_ k>(Cn)^{2n},\quad C>0 \] in order that the methods of Odlyzko and Hoffstein might show \(h_ k\to \infty\) as \(n\to \infty\). (2) could be eliminated with these methods only if Artin's holomorphy conjecture is assumed. The purpose of these papers under review are to improve the zero-free region (1) in the other extreme case that k is not normal over \({\mathbb{Q}}\) and belongs to the class \({\mathcal S}\) of all totally real fields k with the property that the Galois group of the Galois closure e of k over k is \(S_ n\), the full symmetric group. In Part I, the authors obtain an improvement over (1) to the effect that n! is replaced by \(4^ n\), and as a consequence thereof, an effective lower bound for \(h_ K\) with a weaker restriction than (2), under the condition that \(k\in {\mathcal S}\) and K does not contain any imaginary quadratic field. The idea of proof hinges on the use of the well-known fact that if \(\beta\) is a multiple zero, then 1-\(\beta\) is bounded below by (log \(D_ K)^{-1}\) at some stage in pulling the zero up through a series of extensions of k, and for this, the interaction of poles and zeros at \(\beta\) of Artin L-series corresponding to the extension eK/k is closely studied by appealing to the representation theory combined with the correspondence between multiplicative relationship of Artin L-series and linear combination of characters. In Part II by more complicated methods (in the similar spirit) the restriction on K is removed and \(4^ n\) is improved to \(\alpha^ n\), \(\alpha\approx 5.8\), and the final result reads (Theorem 0.1, (b)): given \(\delta >0\), there is an effective constant \(C>0\) such that if \(D_ k>C^ n\), then \(h_ K>(1+\delta)^ n\).
analytic estimates on class numbers of CM fields
scientific article; zbMATH DE number 4168794