Twisted Bhargava cubes

Abstract

In his reinterpretation of Gauss’s composition law for binary quadratic forms, Bhargava determined the integral orbits of a prehomogeneous vector space which arises naturally in the structure theory of the split group ${Spin}_8$. We consider a twisted version of this prehomogeneous vector space which arises in quasisplit ${Spin}_8^E$, where $E$ is an étale cubic algebra over a field $F$. We classify the generic orbits over $F$ by twisted composition $F$-algebras of $E$-dimension $2$.