# Genus Orbits of Torsion Points on Tori with Complex Multiplication

# Genus Orbits of Torsion Points on Tori with Complex Multiplication

The homogeneous space Y=SL2(Z)⋉Z^2 \ SL2(R)⋉R^2 parametrizes pairs (L,X) where L is a unimodular lattice in the complex plane C and X is a point in the torus C/L. A classical construction associates to an integer D>0 the rank 2 sub-lattices of Z^3 orthogonal to integral points of norm sqrt(D). A refined construction associates to D a finite collection of SO2(R) orbits of special points on Y. These special points are lattices L such that C/L is a Heegner point - its ring of complex endomorphisms is the quadratic order of discriminant -D, and X is a torsion point of order D.

Aka, Einsiedler and Shapira conjectured that these collections of SO2(R) orbits equidistribute in Y when D goes to infinity. I will discuss a proof of a stronger theorem implying their conjecture for sequences of D’s satisfying a splitting condition at two primes. Time permitting, I will present an arithmetic interpretation involving Gal(Qbar/Qab)-orbits of special points on connected Kuga-Sato varieties.