Ideal class groups, Hilbert's irreducibility theorem, and integral points of bounded degree on curves
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 counting the number fields in the above results. Let \[ N_{m,n,s}(X) = \#\{k\subset\overline{\mathbb Q}\mid [k:\mathbb Q] = n,\quad \text{rk}_m \text{Cl}(k)\geq s,\quad |\text{Disc}_{k/\mathbb Q}| < X\}. \] Up to now general results have been proven only for \(s=1\) (of course, for \(m\) square-free, \(N_{m,n,1}(X)\) just counts number fields of degree \(n\) with class number divisible by \(m\)). The first such result, due to Murty, is that \(N_{m,2,1}(X)\gg X^{\frac 12+ \frac 1m}\). He also proved a result for real quadratic fields. Soundararajan improved Murty's result to \(N_{m,2,1}(X)\gg X^{\frac 12+ \frac 2m +\varepsilon}\) if \(m\equiv 0 \bmod 4\) and \(N_{m,2,1}(X)\gg X^{\frac 12+ \frac 3{m+2} +\varepsilon}\) if \(m\equiv 2 \bmod 4\), \(m\neq 2\). For cubic fields, Hernández and Luca proved \(N_{m,3,1}(X)\gg X^{\frac 1{6m}}\). \textit{Yu. F. Bilu} and \textit{F. Luca} [J. Reine Angew. Math. 578, 79--91 (2005; Zbl 1072.11084)] improved on this, as the special case \(n=3\), giving the first result for every \(n>1\), (1.4) \(N_{m,n,1}(X)\gg X^{\frac 1{2m(n-1)}}\). In this paper the authors prove the following. Theorem 1.1. Let \(m, n > 1\) be positive integers. Let \(s_1 =\lfloor\frac n2\rfloor\), \(s_2 =\lceil\lfloor\frac{n+1}2\rfloor+\frac n{m-1}-m\rceil\), where \(\lceil\cdot\rceil\) denotes the least integer functions, respectively. Then \[ N_{m,n,s_1}(X)\gg \frac{X^{\frac 1{m(n-1)}}}{\log X} \tag{1.5}, \] \[ N_{m,n,s_2}(X)\gg \frac{X^{\frac 1{m(n+1)n-1}}}{\log X}\tag{1.5}, \] if \(n>(m-1)^2\).
scientific article; zbMATH DE number 5302786
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