## Notes of Frederic Paulin’s lecture

Equidistribution in the Heisenberg group

Joint with J. Parkkonen.

I view the Heisenberg group as a real quadric

$\displaystyle \begin{array}{rcl} Heis_3=\{(w_0,w)\in{\mathbb C}\times{\mathbb C}\,;\,2\Re e(w_0)=|w|^2\}. \end{array}$

In these coordinates, Haar measure is ${d(\Im m(w_0))\frac{dwd\bar{w}}{i}}$. The map ${(w_0,w)\mapsto w}$, ${Heis_3\rightarrow{\mathbb C}}$, is a group homomorphism.

1. Equidistribution of rational points

The field ${{\mathbb Q}(i)}$ will play the role of the rationals, and ${\mathcal{O}={\mathbb Z}[i]}$ the role of integers. requiring that ${w_0}$ and ${w}$ belong to ${K}$ defines the rational Heisenberg group ${Heis_3({\mathbb Q})}$. Elements of ${Heis_3({\mathbb Q})}$ can be uniquely written as fractions ${(\frac{a}{c},\frac{b}{c})}$ with mutually prime Gaussian integers ${a}$, ${b}$, ${c\in{\mathbb Z}[i]}$.

Theorem 1 At rational points of denominator of modulus ${ (the number of them in a ball grows like ${s^{-4}}$), put a unit Dirac mass. Then the normalized sum weakly converges to Haar measure up to the multiplicative constant

$\displaystyle \begin{array}{rcl} c_1=2\pi\frac{|D_k|^{3/2}\zeta_K(3)}{\zeta(3)}. \end{array}$

Corollary 2 Modulo the integer points, ${Heis_3({\mathbb Z})}$, The numbder of rational points of denominator ${ is ${\sim \frac{1}{c_1}s^4}$.

2. Counting arithmetic chains

2.1. Chains

Let ${h=-z_0\bar{z}_2-z_2\bar{z}_0+|z_1|^2}$ be the signature ${(1,2)}$ Hermitian form in ${{\mathbb C}^3}$. The Poincaré hypersphere

$\displaystyle \begin{array}{rcl} \mathcal{HS}\{[z_0:z_1:z_2]\in P^2({\mathbb C})\,;\,h(z_0,z_1,z_2)=0\}. \end{array}$

is a compactification of ${Heis_3}$. Indeed, ${Heis_3}$ embeds in it via ${(w_0,w)\mapsto[1:w:w_0]}$ as the complement of single point ${\infty=[0:0:1]}$.

On ${Heis_3}$ we shall use the Cygan distance (aka Koranyi distance). It is the unique left-invariant on ${Heis_3}$ such that

$\displaystyle \begin{array}{rcl} d_{Cyg}((w_0,w),(0,0))=\sqrt{2|w_0|}. \end{array}$

And also a modified Cygan distance

$\displaystyle \begin{array}{rcl} d''_{Cyg}((w_0,w),(0,0))=\frac{2|w_0|}{\sqrt{|w|^2+2|w_0|}}. \end{array}$

It is equivalent to ${d_{Cyg}}$ 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 ${Heis_3}$, those which contain ${\infty}$ are lines ${\{w=const.\}}$. The others are ellipses which projet to circles in ${{\mathbb C}}$.

2.2. Arithmetic groups and chains

${SL(3,{\mathbb C})}$ on ${P^2({\mathbb C})}$ and preserves lines. The subgroup ${SU_h}$ preserves the Hermitian form ${h}$ and therefore acts on Poincarés hypersphere. Let

$\displaystyle \begin{array}{rcl} \Gamma=SU_h \cap SL_3({\mathbb Z}[i]). \end{array}$

This is a discrete subgroup of ${SU_h}$, which preserves chains.

Definition 4 A chain is arithmetic if its stabilizer in ${\Gamma}$ has a dense orbit in the chain.

Theorem 5 Let ${C_0}$ be an arithmetic chain. There exist constants ${\kappa>0}$ and ${c_2>0}$ (depending on ${C_0}$) such that

$\displaystyle \begin{array}{rcl} |Heis_3({\mathbb Z})\setminus\{c\in\Gamma C_0\,;\, \mathrm{diameter}_{d''_{Cyg}}(c)\geq\epsilon\}|\sim c_2 \epsilon^{-4}(1+O(\epsilon^\kappa)). \end{array}$

3. Complex hyperbolic geometry

This enters the proof. Complex hyperbolic space has curvature between ${-4}$ and ${-1}$. We use the Siegel domain model

$\displaystyle \begin{array}{rcl} \{(w_0,w)\in{\mathbb C}\,;\, 2\Re e(w_0)-|w|^2>0\} \end{array}$

and the projective model

$\displaystyle \begin{array}{rcl} \{[z_0:z_1:z_2]\in P^2({\mathbb C})\,;\, h(z_0,z_1,z_2)<0\} \end{array}$

related by the map ${(w_0,w)\mapsto [1:w:w_0]}$. Taking closures, we recover the previously encountered ${Heis_3}$ and ${\mathcal{HS}}$. The identity component of their isometry group is ${PSU_h}$. The subset

$\displaystyle \begin{array}{rcl} \{(w_0,w)\in{\mathbb C}\,;\, 2\Re e(w_0)-|w|^2>1\} \end{array}$

is a horoball centered at ${\infty}$. ${PSU_h}$ is transitive on horoballs.

Definition 6 A complex geodesic is the intersection of ${H^2({\mathbb C})}$ with a projective line.

Complex geodesics are totally geodesic with curvature ${-4}$. Chains ${c}$ coincide with ideal boundaries of complex geodesics ${L}$. A chain ${c}$ is arithmetic if and only if ${L}$ is, i.e. the stabilizer of ${L}$ in ${\Gamma}$ has finite covolume on ${L}$.

4. Equidistribution of common perpendiculars

Fix an arithmetic complex geodesic ${L_0}$ with ideal boundary ${C_0}$. We project ${H^2({\mathbb C})}$ radially from ${\infty}$ onto ${Heis_3}$ via

$\displaystyle \begin{array}{rcl} (w_0,w)\mapsto(\frac{1}{2}|w|^2+i\Im m(w_0),w). \end{array}$

This allows to identify the boundary of the model horoball ${H}$ with ${Heis_3}$.

Consider common perpendiculars

• of ${H}$ and ${H'}$, horoballs.
• of ${H}$ and a complex geodesic.

I both cases, these are vertical segments.

Theorem 7 Put a unit Dirac mass at each footpoint ${x_\gamma}$ of a common perpendicular of ${H}$ with ${\gamma H}$ whose height is ${>t}$. Renormalize by ${e^{-4t}}$. Then the resulting sum weakly converges to Haar measure on ${Heis_3}$ up to a mltiplicative constant

$\displaystyle \begin{array}{rcl} c_3=2\pi\frac{vol()vol()}{vol()}. \end{array}$

The values of ${\zeta}$ functions in previous theorem arive from these volumes.

4.1. Proof

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.

Lemma 8

$\displaystyle \begin{array}{rcl} d(H,\gamma H)=2\log|c| \end{array}$

if the point at infinity of ${\gamma H}$ is ${(\frac{a}{c},\frac{b}{c})}$.

$\displaystyle \begin{array}{rcl} d(H,\gamma L_0)=-\log(\frac{\mathrm{diameter}(\gamma C_0)}{\sqrt{2}}). \end{array}$

whence the change of variable ${s=e^t}$.

Next seminar on may 14th.