Planes in quadratic 4-space and associated shapes of lattices

Abstract

Let $Q=-x_1^2-x_2^2-x_3^2+x_4^2$ be the standard signature $(1,3)$ quadratic form. To each non-degenerate rational plane $L$ in the four-dimensional quadratic space $({\mathbb{Q}}^4,Q)$ we can naturally attach a periodic geodesic on the Bianchi orbifold ${\mathrm{SL}}_2(\mathbb{Z}[i])\backslash {\mathbb{H}}^3$ which records the position of $L$ in the Grassmannian up to integer rotations. Moreover, each such plane $L$ defines a CM point and a periodic geodesic on the modular curve through restriction of $Q$ to $L$ and its orthogonal complement. Lastly, the local isomorphism between ${\mathrm{SO}}_{1,3}(\mathbb{R})$ and ${\mathrm{SL}}_2(\mathbb{C})$ gives rise to a further periodic geodesic on the Bianchi orbifold. In this article, we exhibit a natural coupling of all the above objects and prove simultaneous equidistribution under a Linnik-type splitting condition. The main ingredient is the classification of joinings of higher-rank diagonalizable actions on homogeneous spaces due to Einsiedler and Lindenstrauss.

38 pages