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

Abstract

We prove that there are $\gg \frac{X^{\frac{1}{3}}}{(\log X{)}^2}$ imaginary quadratic fields $k$ with discriminant $|d_k|\le X$ 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.