Equidistribution in the Heisenberg group
Joint with J. Parkkonen.
I view the Heisenberg group as a real quadric
In these coordinates, Haar measure is . The map , , is a group homomorphism.
1. Equidistribution of rational points
The field will play the role of the rationals, and the role of integers. requiring that and belong to defines the rational Heisenberg group . Elements of can be uniquely written as fractions with mutually prime Gaussian integers , , .
Theorem 1 At rational points of denominator of modulus (the number of them in a ball grows like ), put a unit Dirac mass. Then the normalized sum weakly converges to Haar measure up to the multiplicative constant
Corollary 2 Modulo the integer points, , The numbder of rational points of denominator is .
2. Counting arithmetic chains
Let be the signature Hermitian form in . The Poincaré hypersphere
is a compactification of . Indeed, embeds in it via as the complement of single point .
On we shall use the Cygan distance (aka Koranyi distance). It is the unique left-invariant on such that
And also a modified Cygan distance
It is equivalent to or the a Carnot-Caratheodory metric. We need this exact expression in order to state sharp estimates.
Definition 3 (von Staudt) A chain is the intersection of Poincaré’s hypersphere with a projective line.
In , those which contain are lines . The others are ellipses which projet to circles in .
2.2. Arithmetic groups and chains
on and preserves lines. The subgroup preserves the Hermitian form and therefore acts on Poincarés hypersphere. Let
This is a discrete subgroup of , which preserves chains.
Definition 4 A chain is arithmetic if its stabilizer in has a dense orbit in the chain.
Theorem 5 Let be an arithmetic chain. There exist constants and (depending on ) such that
3. Complex hyperbolic geometry
This enters the proof. Complex hyperbolic space has curvature between and . We use the Siegel domain model
and the projective model
related by the map . Taking closures, we recover the previously encountered and . The identity component of their isometry group is . The subset
is a horoball centered at . is transitive on horoballs.
Definition 6 A complex geodesic is the intersection of with a projective line.
Complex geodesics are totally geodesic with curvature . Chains coincide with ideal boundaries of complex geodesics . A chain is arithmetic if and only if is, i.e. the stabilizer of in has finite covolume on .
4. Equidistribution of common perpendiculars
Fix an arithmetic complex geodesic with ideal boundary . We project radially from onto via
This allows to identify the boundary of the model horoball with .
Consider common perpendiculars
- of and , horoballs.
- of and a complex geodesic.
I both cases, these are vertical segments.
Theorem 7 Put a unit Dirac mass at each footpoint of a common perpendicular of with whose height is . Renormalize by . Then the resulting sum weakly converges to Haar measure on up to a mltiplicative constant
The values of functions in previous theorem arive from these volumes.
Use a formula for the Poisson measure and the measure of maximal entropy in terms of the Cygan distance. Then establish exponential decay of correlations. Here is the crucial Lemma.
if the point at infinity of is .
whence the change of variable .
Next seminar on may 14th.