Class groups of imaginary quadratic points on $X_1(16)$

TL;DR

This paper proves that for certain imaginary quadratic fields with elliptic curves having a 16-torsion point, the class number is divisible by 10, answering a longstanding question and establishing a new framework for studying class group divisibility.

Contribution

It introduces a general framework for analyzing class group divisibility of imaginary quadratic points on hyper-elliptic curves and applies it to resolve a 12-year-old question about $X_1(16)$.

Findings

Class number divisible by 10 under specified conditions

Framework for studying class group divisibility on hyper-elliptic curves

Resolution of a longstanding open question

Abstract

The main result is to show that if $K \ncong \mathbb Q(\sqrt{-15})$ 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 $X_1(16)$.