On imaginary quadratic fields with non-cyclic class groups
Yi Ouyang, Qimin Song, Chenhao Zhang
TL;DR
We study when the class group of an imaginary quadratic field $H$ contains a fixed abelian group $H$, focusing on two cases: $H_2=(Z/2Z)^l \times Z/g_1Z$ with $l\ge 2$ and $g_1$ odd, $g_1\ge 3$, and $H_3=(Z/gZ)^2$ with $g\ge 5$. The authors adapt Soundararajan's lower-bound method, using genus theory to force a $(Z/2Z)^l$ factor and a discriminant-driven order $g_1$ element, and for $H_3$ employ a multivariate square-free polynomial framework. They prove an unconditional bound $N_{H_2}(X) \gtrsim X^{\frac{1}{2}+\frac{3}{2g_1+2}-\varepsilon}$ and, under a square-free-density conjecture for integral multivariate polynomials, a conditional bound $N_{H_3}(X) \gtrsim X^{\frac{1}{g-1}-\varepsilon}$ for $g\ge 5$. These results extend prior lower bounds for $N_g(X)$ and $N^-(g^2;X)$ and provide a discriminant-construction framework to realize large $H$-subgroups in $CL(-d)$ across infinitely many $d$.
Abstract
For a fixed abelian group $H$, let $N_H(X)$ be the number of square-free positive integers $d\leq X$ such that H is a subgroup of $CL(\mathbb{Q}(\sqrt{-d}))$. We obtain asymptotic lower bounds for $N_H(X)$ as $X\to\infty$ in two cases: $H=\mathbb{Z}/g_1\mathbb{Z}\times (\mathbb{Z}/2\mathbb{Z})^l$ for $l\geq 2$ and $2\nmid g_1\geq 3$, $H=(\mathbb{Z}/g\mathbb{Z})^2$ for $2\nmid g\geq 5$. More precisely, for any $ε>0$, we showed $N_H(X)\gg X^{\frac{1}{2}+\frac{3}{2g_1+2}-ε}$ when $H=\mathbb{Z}/g_1\mathbb{Z}\times (\mathbb{Z}/2\mathbb{Z})^l$ for $l\geq 2$ and $2\nmid g_1\geq 3$. For the second case, under a well known conjecture for square-free density of integral multivariate polynomials, for any $ε>0$, we showed $N_H(X)\gg X^{\frac{1}{g-1}-ε}$ when $H=(\mathbb{Z}/g\mathbb{Z})^2$ for $ g\geq 5$. The first case is an adaptation of Soundararajan's results for $H=\mathbb{Z}/g\mathbb{Z}$, and the second conditionally improves the bound $X^{\frac{1}{g}-ε}$ due to Byeon and the bound $X^{\frac{1}{g}}/(\log X)^{2}$ due to Kulkarni and Levin.