Counting Imaginary Quadratic Fields with an Ideal Class Group of 5-rank at least 2
Kollin Bartz, Aaron Levin, Aman Dhruva Thamminana
TL;DR
The paper addresses counting imaginary quadratic fields with a 5-torsion-rich ideal class group by linking class-group growth to the arithmetic of genus 2 curves. It uses the Howe–Leprévost–Poonen geometric construction to produce genus 2 curves $C$ with a $\mathbb{Q}$-rational Weierstrass point and a Jacobian whose rational torsion has $5$-rank at least $2$, leveraging a $(2,2)$-isogeny to a product of elliptic curves with rational $5$-torsion. Discriminant conditions on $E_t,E_u$ reduce to a solvable equation $\Delta_{10}(t)z^2=\Delta_{10}(u)$, yielding explicit sextic models for $C$ with the desired torsion properties. A computational search identifies many such curves, including explicit examples with torsion $\mathbb{Z}/5\mathbb{Z}\times\mathbb{Z}/10\mathbb{Z}$, and applying a quantitative KL bound gives the lower bound $\mathcal{N}^-(5^2;X) \gg \frac{X^{1/3}}{(\log X)^2}$, improving previous results and advancing understanding of how geometry informs class-group distributions in imaginary quadratic fields.
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, Leprévost, 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.