# Planes in four space and four associated CM points

## TL;DR

This paper studies the distribution of certain geometric and lattice objects associated with rational planes in four-dimensional space, proving their equidistribution under specific conditions using advanced ergodic theory results.

## Contribution

It introduces a new framework connecting rational planes, Grassmannians, and lattices, and applies recent ergodic theory results to prove their simultaneous equidistribution.

## Key findings

- Proves simultaneous equidistribution of associated objects under splitting conditions
- Establishes connections between rational planes and lattice structures in four-space
- Utilizes recent results on algebraicity of joinings for the proof

## Abstract

To any two-dimensional rational plane in four-dimensional space one can naturally attach a point in the Grassmannian Gr(2,4) and four lattices of rank two. Here, the first two lattices originate from the plane and its orthogonal complement and the second two essentially arise from the accidental local isomorphism between SO(4) and SO(3)xSO(3). As an application of a recent result of Einsiedler and Lindenstrauss on algebraicity of joinings we prove simultaneous equidistribution of all of these objects under two splitting conditions.