A moduli interpretation of untwisted binary cubic forms

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.