Binary quartic forms with vanishing $J$-invariant

Abstract

We obtain an asymptotic formula for the number of ${\mathrm{GL}}_2(\mathbb{Z})$-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.

Revised version corrects an error in the previous version

References in corpus (1)

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

Cited by in corpus (1)

• Quadrics in arithmetic statistics