Binary quartic forms with vanishing $J$-invariant

TL;DR

This paper derives an asymptotic count for irreducible binary quartic forms with integer coefficients that have a zero J-invariant and specific Hessian properties, advancing understanding of integral orbits under group actions.

Contribution

It provides the first asymptotic formula for counting such binary quartic forms with particular invariants and Hessian conditions, expanding orbit counting techniques.

Findings

Established an asymptotic formula for the count of forms.

Identified conditions on Hessians related to reducible or positive definite quadratic forms.

Extended orbit counting methods to a new class of algebraic forms.

Abstract

We obtain an asymptotic formula for the number of $\operatorname{GL}_2(\mathbb{Z})$-equivalence classes of irreducible binary quartic forms with integer coefficients with vanishing $J$-invariant and whose Hessians are proportional to the squares of reducible or positive definite binary quadratic form. These results give a case where one is able to count integral orbits inside a relatively open real orbit of a variety closed under a group action of degree at least three.