Ideal class groups, Hilbert's irreducibility theorem, and integral points of bounded degree on curves
type: Item
instance of: scholarly article
title: Ideal class groups, Hilbert's irreducibility theorem, and integral points of bounded degree on curves
zbMATH Open document ID: 1216.11101
author: Aaron Levin
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2008-07-21T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10808
review text: From the text: In 1922, \textit{T. Nagell} [Abh. Math. Semin. Univ. Hamb. 1, 140--150 (1922; JFM 48.0170.03)] proved that for any integer \(m\), there exist infinitely many imaginary quadratic number fields with class number divisible by \(m\). This result has since been reproved by a number of different authors. Nearly fifty years later, working independently, Yamamoto and Weinberger extended Nagell's class number divisibility result to real quadratic fields and Uchida proved the analogous result for cubic cyclic fields. The class number divisibility problem for number fields of arbitrary degree was resolved in 1984 by \textit{T. Azuhata} and \textit{H. Ichimura} [J. Fac. Sci., Univ. Tokyo, Sect. I A 30, 579--585 (1984; Zbl 0532.12006)]. In fact, they proved that for any integers \(m, n > 1\) and any nonnegative integers \(r_1, r_2\), with \(r_1 + 2r_2 = n\), there exist infinitely many number fields \(k\) of degree \(n = [k : \mathbb Q]\) with \(r_1\) real places and \(r_2\) complex places such that (1.1) \(\text{rk}_m \text{Cl}(k) \geq r_2\), where \(\text{Cl}(k)\) is the ideal class group of \(k\) and \(\text{rk}_m \text{Cl}(k)\) denotes the largest integer such that \((\mathbb Z/m\mathbb Z)\text{rk}_m \text{Cl}(k)\) is a subgroup of \(\text{Cl}(k)\). The right-hand of (1.1) was later improved to \(r_2 + 1\) by \textit{S. Nakano} [Tokyo J. Math. 6, 389--395 (1983; Zbl 0529.12005), J. Reine Angew. Math. 358, 61--75 (1985; Zbl 0559.12004)]. Choosing \(r_2\) as large as possible, we thus obtain, for any \(m\), infinitely many number fields \(k\) of degree \(n > 1\) with (1.2) \(\text{rk}_m \text{Cl}(k)\geq \lfloor\frac n2\rfloor +1\), where \(\lfloor\cdot\rfloor\) denotes the greatest integer function. Furthermore, improving on previous work of Ishida and Ichimura, Nakano proved the existence of infinitely many number fields \(k\) of degree \(n > 1\) with (1.3) \(\text{rk}_m \text{Cl}(k)\geq n\). Recently, progress has been made on obtaining quantitative results on…
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11G30
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 11R21
zbMATH DE Number: 5302786
MaRDI profile type: Publication
OpenAlex ID: W2170788603
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DOI: 10.5802/JTNB.598
description: scientific article; zbMATH DE number 5302786