On the $p$-ranks of the ideal class groups of imaginary quadratic fields

Abstract

For a prime number $p\ge 5$, we explicitly construct a family of imaginary quadratic fields $K$ with ideal class groups $C{l}_K$ having $p$-rank ${\mathrm{rk}}_{\mathrm{p}}(\mathrm{C}{\mathrm{l}}_{\mathrm{K}})$ 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 $\epsilon$ with $0<\epsilon <\frac{1}{p-1}$, the number of imaginary quadratic fields $K$ with the absolute value of the discriminant $d_K$ $\le X$ and ${\mathrm{rk}}_{\mathrm{p}}(\mathrm{C}{\mathrm{l}}_{\mathrm{K}})\ge 2$ is $\gg X^{\frac{1}{p-1}-\epsilon }$. This improves the previously known lower bound of $X^{\frac{1}{p}-\epsilon }$ due to Byeon and the recent bound $X^{\frac{1}{p}}/(\log X{)}^2$ due to Kulkarni and Levin.

arXiv comment10 pages. Comments are welcome