A geometric approach to the Cohen-Lenstra heuristics

TL;DR

This paper introduces a geometric framework linking torsion elements in quadratic field class groups to polynomial resultants and Selmer groups of genus 1 curves, offering new insights into Cohen-Lenstra heuristics.

Contribution

It provides a novel geometric description of n-torsion in class groups using polynomial resultants and connects this to Selmer groups of singular genus 1 curves.

Findings

Characterization of n-torsion via polynomial resultants

Geometric parameterization linking class groups and Selmer groups

New perspective on Cohen-Lenstra heuristics

Abstract

We give a new geometric description of when an element of the class group of a quadratic field, thought of as a quadratic form $q$, is $n$-torsion. We show that $q$ corresponds to an $n$-torsion element if and only if there exists a degree $n$ polynomial whose resultant with $q$ is $\pm 1$. This is motivated by a more precise geometric parameterization which directly connects torsion in class groups of quadratic fields to Selmer groups of singular genus $1$ curves.