Vol. 8, No. 8, 2014

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Twisted Bhargava cubes

Wee Teck Gan and Gordan Savin

Vol. 8 (2014), No. 8, 1913–1957
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 $2$.

Keywords
Bhargava's cubes, twisted composition algebras
Mathematical Subject Classification 2010
Primary: 11S90
Secondary: 17A75, 17C40