On the arithmetic 4-orbifolds associated to integral quaternary quadratic forms of index 2

Abstract

Groups acting properly and discontinuously on the Cartesian product $\mathbb {H}^{2}\times \mathbb {H}^{2}$ of two hyperbolic planes are termed hyperabelian by Picard. The automorphism group $\mathrm {Aut}f$ of a quaternary integral quadratic form f of index 2 is an example of a hyperabelian group. Hence the quotient orbifold $Q_{f}$ of the action of $\mathrm {Aut}f$ on $\mathbb {H}^{2}\times \mathbb {H}^{2}$ is a 4-dimensional arithmetic orbifold, endowed with a natural $\mathbb {H}^{2}\times \mathbb {H}^{2}$-geometry. Plücker coordinates are used to understand $Q_{f}$. A real automorphism U of $\mathbb {R}^{4}$ induces a real automorphism $\mathbf {K(}U)$ of $(\mathbb {R}^{6},k)$ in such a way that if $U\in SL(4,\mathbb {Z})$ then $\mathbf {K(}U)\in SL(6,\mathbb {Z})$ is an automorph of the Klein quadratic form k. It is proved that the converse is true. That is, given an automorph $M\in SL(6,\mathbb {Z})$ of k there is $U\in SL(4,\mathbb {Z})$ such that $\mathbf {K(}U)=\pm M$, so that the proper automorphism group of the Klein quadric is isomorphic to $SL(4,\mathbb {Z})$ via $\mathbf {K}$. This is used to obtain the automorphism group of the quadratic line complex of line tangents to a quadric in projective space $P^{3}$. With this, a description is given of the automorphism group of a quaternary integral quadratic form of index 2.