Counting Imaginary Quadratic Fields with an Ideal Class Group of 5-rank at least 2

TL;DR

This paper establishes a new lower bound on the number of imaginary quadratic fields with discriminant up to X having an ideal class group of 5-rank at least 2, improving previous results using genus 2 curves.

Contribution

It introduces a novel construction of genus 2 curves with specific torsion properties to improve lower bounds on class group ranks of quadratic fields.

Findings

New lower bound of (X^{1/3})/(( ext{log} X)^2) for fields with 5-rank 

Construction of genus 2 curves with rational Weierstrass points and 5-rank Jacobians

Improved quantitative results linking curves to class group properties

Abstract

We prove that there are $\gg\frac{X^{\frac{1}{3}}}{(\log X)^2}$ imaginary quadratic fields $k$ with discriminant $|d_k|\leq X$ and an ideal class group of $5$-rank at least $2$. This improves a result of Byeon, who proved the lower bound $\gg X^{\frac{1}{4}}$ in the same setting. We use a method of Howe, Lepr\'{e}vost, and Poonen to construct a genus $2$ curve $C$ over $\mathbb{Q}$ such that $C$ has a rational Weierstrass point and the Jacobian of $C$ has a rational torsion subgroup of $5$-rank $2$. We deduce the main result from the existence of the curve $C$ and a quantitative result of Kulkarni and the second author.