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

TL;DR

This paper constructs families of imaginary quadratic fields with ideal class groups having high p-rank and provides quantitative lower bounds on their density, assuming the abc conjecture, improving previous bounds.

Contribution

It explicitly constructs imaginary quadratic fields with class groups of p-rank at least 2 and establishes improved lower bounds on their density under the abc conjecture.

Findings

Constructed explicit families with p-rank ≥ 2

Proved lower bounds on the count of such fields under abc conjecture

Improved previous density bounds for imaginary quadratic fields

Abstract

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.