A new parametrization for ideal classes in rings defined by binary forms, and applications

TL;DR

This paper introduces a new parametrization of ideal classes in rings defined by binary forms, enabling progress on longstanding problems in number theory related to class groups, Selmer groups, and representation of squares.

Contribution

It develops a novel integral model of a higher composition law that facilitates analysis of ideal classes and their applications in number theory.

Findings

Bounded average size of 2-class groups in families of number fields

Proved most odd-degree binary forms do not primitively represent a square

Bound the second moment of 2-Selmer groups of elliptic curves

Abstract

We give a parametrization of square roots of the ideal class of the inverse different of rings defined by binary forms in terms of the orbits of a coregular representation. This parametrization, which can be construed as a new integral model of a ``higher composition law'' discovered by Bhargava and generalized by Wood, was the missing ingredient needed to solve a range of previously intractable open problems concerning distributions of class groups, Selmer groups, and related objects. For instance, in this paper, we apply the parametrization to bound the average size of the $2$-class group in families of number fields defined by binary $n$-ic forms, where $n \geq 3$ is an arbitrary integer, odd or even; in the paper [41], we applied it to prove that most integral odd-degree binary forms fail to primitively represent a square; and in the paper [11], joint with Bhargava and Shankar, we applied it to bound the second moment of the size of the $2$-Selmer group of elliptic curves.