On the number of cubic orders of bounded discriminant having automorphism group $C_3$, and related problems

TL;DR

This paper establishes a bijection between integer orbits of a specific representation and cubic orders with fixed lattice shape, enabling asymptotic counting of such orders and fields with bounded discriminant, and revealing distribution patterns among shapes.

Contribution

It introduces a novel orbit-parameterization linking cubic orders to a prehomogeneous vector space, facilitating new counting formulas and distribution results for cubic fields and orders.

Findings

Asymptotic count of cubic orders with bounded discriminant and nontrivial automorphism group.

Distribution of lattice shapes among cubic orders and fields with fixed discriminant.

Equidistribution of shapes in the class group of binary quadratic forms.

Abstract

For a binary quadratic form $Q$, we consider the action of $\mathrm{SO}_Q$ on a two-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and of a coregular space whose underlying group is not semisimple. We show that the nondegenerate integer orbits of this representation are in natural bijection with orders in cubic fields having a fixed "lattice shape". Moreover, this correspondence is discriminant-preserving: the value of the invariant polynomial of an element in this representation agrees with the discriminant of the corresponding cubic order. We use this interpretation of the integral orbits to solve three classical-style counting problems related to cubic orders and fields. First, we give an asymptotic formula for the number of cubic orders having bounded discriminant and nontrivial automorphism group. More generally, we give an asymptotic formula for the number of cubic orders that have bounded discriminant and any given lattice shape (i.e., reduced trace form, up to scaling). Via a sieve, we also count cubic fields of bounded discriminant whose rings of integers have a given lattice shape. We find, in particular, that among cubic orders (resp. fields) having lattice shape of given discriminant $D$, the shape is equidistributed in the class group $\mathrm{Cl}_D$ of binary quadratic forms of discriminant $D$. As a by-product, we also obtain an asymptotic formula for the number of cubic fields of bounded discriminant having any given quadratic resolvent field.