A moduli interpretation of untwisted binary cubic forms

TL;DR

This paper provides a geometric interpretation of binary cubic forms by linking their orbits under GL_2 action to pairs of elliptic curves with specific invariants and Brauer classes, using Clifford algebras.

Contribution

It introduces a moduli interpretation of binary cubic forms, connecting algebraic forms to elliptic curves and Brauer classes through Clifford algebra constructions.

Findings

GL_2 orbits correspond to pairs of elliptic curves with j=0 and invariant Brauer classes

Binary cubic forms are classified via moduli of elliptic curves and algebraic invariants

Clifford algebra plays a central role in establishing the correspondence

Abstract

We give a moduli interpretation to the quotient of (nondegenerate) binary cubic forms with respect to the natural $\text{GL}_2$-action on the variables. In particular, we show that these $\text{GL}_2$ orbits are in bijection with pairs of $j$-invariant $0$ elliptic curves together with $3$-torsion Brauer classes that are invariant under complex multiplication. The binary cubic generic Clifford algebra plays a key role in the construction of this correspondence.