On binary cubic and quartic forms

TL;DR

This paper characterizes the automorphism groups of binary cubic and quartic forms with integer coefficients, derives asymptotic formulas for representable integers, and explores geometric properties of related surfaces.

Contribution

It provides a complete description of the rational automorphism groups and extends asymptotic counting results for integers represented by these forms.

Findings

Automorphism groups are described via quadratic covariants.

Asymptotic formulas for the count of representable integers are established.

The field of definition for lines on related cubic and quartic surfaces is determined.

Abstract

In this paper we determine the group of rational automorphisms of binary cubic and quartic forms with integer coefficients and non-zero discriminant in terms of certain quadratic covariants of cubic and quartic forms. This allows one to give precise asymptotic formulae for the number of integers in an interval representable by a binary cubic or quartic form and extends work of Hooley. Further, we give the field of definition of lines contained in certain cubic and quartic surfaces related to binary cubic and quartic forms.