A geometric approach to the 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.