Abstract

In the 1950’s, Cassels and Swinnerton-Dyer conjectured that the values of products of n linear forms in n variables at integer points behave rather differently in n>=3 variables than in the case of 2 variables. The reason these two questions behave differently is that the symmetry group of such forms is an (n-1)-dimensional diagonal group, and (as conjectured independently from a different point of view by Furstenberg in 1967) higher rank diagonalizable actions, unlike one-parameter diagonalizable actions, have subtle rigidity properties which are still quite mysterious. One aspect of this question where the current state of knowledge is quite satisfactory is the study of joinings of such actions, where Einsiedler and I have a rather general classification of ergodic joinings.

This classification has several striking number theoretic applications, and I will describe two. Both relate to work of Linnik from around 70 years ago regarding the distribution of integer points of the sphere. In this direction, Aka, Einsiedler, and Shapira studied joint distribution of integer points on a two dimensional sphere together with the shape of its orthogonal lattice, and work of Khayutin regarding orbits of the class group of a number field on pairs of CM points.