Lower bounds for the height and size of the ideal class group in CM-fields
The authors prove that under the assumption of the generalized Riemann hypothesis, the exponent of the ideal class group of a CM-field goes to infinity with its absolute discriminant. A weaker version of this result, namely restricting to CM-fields with a given degree, was proved by \textit{S. Louboutin} and \textit{R. Okazaki} [Math. Z. 243, 155--159 (2003; Zbl 1049.11122)]. The main result of this paper can be stated as follows, where \(d_k\), \(\Delta_k\), and \(E_k\) denote the degree, discriminant, and exponent of the ideal class group for the CM-field \(K\), respectively: for any \(\xi> 0\) and under the assumption of the generalized Riemann hypothesis for the Dedekind zeta-function of \(K\), \[ E_k\gg_\xi\max\{d^{-1}_k\log|\Delta_k|- \log d_k, d^{1-\varepsilon}_k\}/\log\log|\Delta_k|, \] where the implicit constant in \(\gg_\varepsilon\) depends only on \(\varepsilon\) and is effectively computable. A combination of algebraic, analytic, and Diophantine results related to the link between primes of small norm and algebraic numbers of small heights in CM-fields, lower bounds for the Weil absolute logarithmic height, and the effective version of the Chebotarev density theorem proved by \textit{J. C. Lagarias} and \textit{A. M. Odlyzko} in [Algebr. Number Fields, Proc. Symp. London math. Soc., Univ. Durham 1975, 409--464 (1977; Zbl 0362.12011)] are used to establish the main result. The primary result used to establish the main result is related to the size of the multiplicative relations in the class group of the CM-field, which is described as follows. Let \(G\) be a group and let \(L\) be a positive integer; define \(M_G(L)\) as the least positive integer \(A\) such that for all \(g_1,\dots, g_L\in G\) there exists \(\underline a\in Z^l\setminus\{0\}\) such that \(g^{a_1}_1,\dots,g^{a_L}_{g_L}= e\) and \(\sum_j| a_j|\leq A\). Under the assumption of the generalized Riemann hypothesis for the Dedekind zeta-function of the CM-field \(K\), the author proves that \[ M_G(L)\gg_\varepsilon \max\{d^{-1}_k\log|\Delta_k|- \log d_k, d^{1-\varepsilon}_k\}/\log L+\log\log| \Delta_k|. \] Under the same assumption the following corollary is then derived, where \(\lambda_1,\lambda_2,\dots, \lambda_n\) are the invariants of \(G\), i.e. \(K\) is isomorphic to the direct product of cyclic groups of order \(\lambda_1,\lambda_2,\dots, \lambda_n\) with \(\lambda_n\mid\lambda_{n-1} \mid\cdots \mid\lambda_1\), and \(\lambda_{n+1}= 1\), \(\varepsilon> 0\), \(j\in\{1,\dots,n+1\}\). The corollary states that \[ \lambda_j\log ((\lambda_1\cdots \lambda_{j-1}/ \lambda^{j-1}_j)\log |\Delta_k |)\gg_\varepsilon\max\{d^{-1}_k \log |\Delta_k|- \log d_k, d^{1-\varepsilon}_k\}. \] By choosing \(j=1\) the authors then obtain the main result of this paper. \{Remark: Note that reference [7] \textit{D. Simon} (2002) has been published in Indag. Math. 12, 505--517 (2001).\}
scientific article; zbMATH DE number 1903794