On imaginary quadratic fields with non-cyclic class groups

For a fixed abelian group $H$, let $N_H(X)$ be the number of square-free positive integers $d\le X$ such that H is a subgroup of $CL(\mathbb{Q}(\sqrt{-d}))$. We obtain asymptotic lower bounds for $N_H(X)$ as $X\to \mathrm{\infty }$ in two cases: $H=\mathbb{Z}\mathrm{/}{g}_1\mathbb{Z}\times (\mathbb{Z}\mathrm{/}2\mathbb{Z}{)}^l$ for $l\ge 2$ and $2\nmid g_1\ge 3$, $H=(\mathbb{Z}\mathrm{/}g\mathbb{Z}{)}^2$ for $2\nmid g\ge 5$. More precisely, for any $\epsilon >0$, we showed $N_H(X)\gg X^{\frac{1}{2}+\frac{3}{2g_1+2}-\epsilon }$ when $H=\mathbb{Z}\mathrm{/}{g}_1\mathbb{Z}\times (\mathbb{Z}\mathrm{/}2\mathbb{Z}{)}^l$ for $l\ge 2$ and $2\nmid g_1\ge 3$. For the second case, under a well known conjecture for square-free density of integral multivariate polynomials, for any $\epsilon >0$, we showed $N_H(X)\gg X^{\frac{1}{g-1}-\epsilon }$ when $H=(\mathbb{Z}\mathrm{/}g\mathbb{Z}{)}^2$ for $g\ge 5$. The first case is an adaptation of Soundararajan’s results for $H=\mathbb{Z}\mathrm{/}g\mathbb{Z}$, and the second conditionally improves the bound $X^{\frac{1}{g}-\epsilon }$ due to Byeon and the bound $X^{\frac{1}{g}}\mathrm{/}(\log X{)}^2$ due to Kulkarni and Levin.