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 $$Cl_{K}$$ having p-rank $${\mathrm{{rk}}_{p}(Cl_{K})}$$ at least 2. We also quantitatively prove, under 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 $$d_{K}\le X$$ and $${\mathrm{{rk}}_{p}(Cl_{K})} \ge 2$$ is $$\gg X^{\frac{1}{p - 1} - \varepsilon }$$ . This conditionally 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.