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. Plucker 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.