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 $\epsilon >0$, we showed $N_H(X)\gg X^{ \frac{ 1}{ 2}+\frac{ 3}{ 2g_1+2}-\epsilon}$ 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 $\epsilon >0$, we showed $N_H(X)\gg X^{ \frac{ 1}{ g-1}-\epsilon}$ 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 $X^{ \frac{ 1}{ g}-\epsilon}$ due to Byeon and the bound $X^{ \frac{ 1}{ g}}/(\log X)^{ 2}$ due to Kulkarni and Levin.
Categories: math.NT