Quadratic fields with a class group of large 3-rank

2019/10/27 by Levin, Aaron, Shengkuan, Yan, Wiljanen, Luke
#14H40 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Primary 11R29 #Secondary 11G30

paper · doi:10.48550/arxiv.1910.12276

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.

Related

• Counting Imaginary Quadratic Fields with an Ideal Class Group of 5-rank at least 2
• Class groups of imaginary quadratic fields of 3-rank at least 2
• Class number divisibility for imaginary quadratic fields
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• Ideal class groups and torsion in Picard groups of varieties