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 GL 2 -action on the variables. In particular, we show that these 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.