For a prime number $p \geq 5$, we explicitly construct a family of imaginary quadratic fields $K$ with ideal class groups $Cl_{ K}$ having $p$-rank ${ { \rm{ rk}}_{ p}(Cl_{ K})}$ at least $2$. We also quantitatively prove, under the assumption of the $abc$-conjecture, that for sufficiently large positive real numbers $X$ and any real number $\varepsilon$ with $0 < \varepsilon < \frac{ 1}{ p - 1}$, the number of imaginary quadratic fields $K$ with the absolute value of the discriminant $d_{ K}$ $\leq X$ and ${ { \rm{ rk}}_{ p}(Cl_{ K})} \geq 2$ is $\gg X^{ \frac{ 1}{ p - 1} - \varepsilon}$. This improves the previously known lower bound of $X^{ \frac{ 1}{ p} - \varepsilon}$ due to Byeon and the recent bound $X^{ \frac{ 1}{ p}}/(\log X)^{ 2}$ due to Kulkarni and Levin.
Categories: math.NT