Class groups of imaginary quadratic points on X₁(16)

2025/07/07 by Derickx, Maarten
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.04604

Abstract

The main result is to show that if K \ncong \mathbb Q(√(-15)) is an imaginary quadratic field and E is an elliptic curve over K with a torsion point of order 16, then the class number of K is divisible by 10. This gives an affirmative answer to a 12 year old question by David Krumm. This is done by setting up a more general framework for studying divisibility of class groups of imaginary quadratic points on hyper-elliptic curves and applying it to X₁(16).

Citations

• Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville
• Pulling back torsion line bundles to ideal classes

Related

• Class number divisibility for imaginary quadratic fields
• Elliptic curves and lower bounds for class numbers
• Divisibility of the class numbers of imaginary quadratic fields
• Quadratic fields with a class group of large 3-rank
• Indivisibility of class numbers of imaginary quadratic fields