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

2025/02/02 by Bartz, Kollin, Levin, Aaron, Thamminana, Aman Dhruva
#11R11 #11R29 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.00845

Abstract

We prove that there are ≫\fracX^(1)/(3)(log X)² imaginary quadratic fields k with discriminant |d_k|≤ X and an ideal class group of 5-rank at least 2. This improves a result of Byeon, who proved the lower bound ≫ X^(1)/(4) in the same setting. We use a method of Howe, Leprévost, and Poonen to construct a genus 2 curve C over 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.

Related

• Quadratic fields with a class group of large 3-rank
• On the p-ranks of the ideal class groups of imaginary quadratic fields
• Class groups of imaginary quadratic fields of 3-rank at least 2
• Class number divisibility for imaginary quadratic fields
• Divisibility of the class numbers of imaginary quadratic fields