Quadratic fields with a class group of large 3-rank

TL;DR

This paper demonstrates the existence of infinitely many imaginary quadratic fields with large 3-rank class groups, improving previous results by constructing fields with at least 5 in 3-rank, using advanced elliptic curve and Jacobian torsion methods.

Contribution

It introduces new constructions of quadratic fields with high 3-rank class groups, extending prior work to achieve at least 5 in 3-rank and exploring related torsion phenomena.

Findings

Existence of infinitely many quadratic fields with 3-rank ≥ 5.

Construction methods based on Mestre's elliptic curves and Jacobian torsion.

Analysis of rational 3-torsion in hyperelliptic Jacobians.

Abstract

We prove that there are >>X^{1/30}/(log X) imaginary quadratic number fields with an ideal class group of 3-rank at least 5 and discriminant bounded in absolute value by X. This improves on an earlier result of Craig, who proved the infinitude of imaginary quadratic fields with an ideal class group of 3-rank at least 4. The proofs rely on constructions of Mestre for j-invariant 0 elliptic curves of large Mordell-Weil rank, and a method of the first author and Gillibert for constructing torsion in ideal class groups of number fields from rational torsion in Jacobians of curves. We also consider analogous questions concerning rational 3-torsion in hyperelliptic Jacobians.