Counting and equidistribution in Heisenberg groups

We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$. We prove a Mertens' formula for the integer points over a quadratic imaginary number fields $K$ in the light cone of Hermitian forms, as well as an equidistribution theorem of the set of rational points over $K$ in Heisenberg groups. We give a counting formula for the cubic points over $K$ in the complex projective plane whose Galois conjugates are orthogonal and isotropic for a given Hermitian form over $K$, and a counting and equidistribution result for arithmetic chains in the Heisenberg group when their Cygan diameter tends to $0$.


Introduction
The aim of this paper is to give original asymptotic counting and equidistribution results with error terms of arithmetically defined points or circles in nilmanifolds covered by the Heisenberg groups.We refer for instance to [Bre, BF, GT, Kim, BeQ] or [EiW,Chap. 10] for other types of results.
The classical result, known as Mertens' formula, describing the asymptotic behaviour of the average order of Euler's function, or equivalently the asymptotic counting of Farey fractions, and its related equidistribution result of Farey fractions in the group R, can be stated in the following way.The additive group Z acts on Z × Z by horizontal shears (transvections): k • (u, v) = (u + kv, v).Then for some κ > 0 (see for example [HaW,Thm. 330], a better error term is due to Walfisz [Wal]).Furthermore, with ∆ x the unit Dirac mass at x, as s → +∞, Our first result is an analog of Mertens' formula for Heisenberg groups.Let K be an imaginary quadratic number field, O K its ring of integers, and tr, n its (absolute) trace and 1 arXiv:1402.7225v1[math.DG] 28 Feb 2014 norm.The nilpotent group Heis 3 (O K ) = {(w 0 , w) ∈ O K × O K : tr(w 0 ) = n(w)} with law (w 0 , w)(w 0 , w ) = (w 0 + w 0 + w w, w + w ) acts on O K × O K × O K by the shears (w 0 , w)(a, α, c) = (a + w α + w 0 c, α + w c, c) .
where D K is the discriminant and ζ K Dedekind's zeta function of K.
The 3-dimensional Heisenberg group with law (w 0 , w)(w 0 , w ) = (w 0 + w 0 + w w, w + w ), is the Lie group of R-points of a Q-group whose group of Q-points is Heis 3 ∩(K × K).We endow it with its Haar measure d Haar Heis 3 (w 0 , w) = d(Im w 0 ) d(Re w) d(Im w) . (1) The above counting result is a counting theorem of the rational points ( a c , α c ) over K (analogous to Farey fractions) in Heis 3 , and the following result is a related equidistribution theorem of the set of rational points over K in Heis 3 .Theorem 2 As s → +∞, we have These measure theoretic computations, with the above-described main tool, also allow us in Section 7 to prove an asymptotic counting result of arithmetic chains in hyperspherical geometry.Let q be the Hermitian form −z 0 z 2 −z 2 z 0 +|z 1 | 2 on C 3 .Its isotropic locus in the complex projective plane P 2 (C) is called by Poincaré the hypersphere [Poi].With [1 : 0 : 0] removed, the hypersphere identifies with the Heisenberg group Heis 3 , and carries a natural (slightly modified) Cygan's (sometimes called Koranyi's) distance d Cyg (see Section 3 for precise definitions).Recall that a chain, as introduced by von Staudt and developped in particular by E. Cartan, see for instance [Car] and [Gol,§4.3], is a nontrivial intersection with the hypersphere of a complex projective line.A chain is either a fiber of the canonical morphism Heis 3 → C (defined by (w 0 , w) → w) or an ellipse whose projection by this map is a circle.We say that a chain C 0 is arithmetic (over K) if the orbit of some point in C 0 under the stabiliser of C 0 in the arithmetic lattice PSU q (O K ) is dense in C 0 .The stabiliser PSU q (O K ) ∞ of [1 : 0 : 0] in PSU q (O K ) preserves the diameters of the chains for d Cyg .The picture below shows an orbit of arithmetic chains under the arithmetic lattice PSU q (Z[i]).
Theorem 3 Let C 0 be an arithmetic chain in the hypersphere.There exists a constant κ > 0 and an explicit constant C > 0 such that, as s → +∞, the number of chains modulo PSU q (O K ) ∞ in the PSU q (O K )-orbit of C 0 , with diameter at least , is equal to C −4 (1 + O( κ )).
We will also prove that the centers of the finite arithmetic chains equidistribute in the hypersphere.
An analogous method allows us in Section 6 to prove the following counting result of some arithmetic points in the complex projective plane P 2 (C).Let z 0 ∈ P 2 (C) be a cubic point over K, whose Galois conjugates z 0 , z 0 over K are isotropic and orthogonal to z 0 for the Hermitian form q. The arithmetic lattice PSU q (O K ) acts with infinitely many orbits on the set of such points.The inverse of the modified Cygan distance d Cyg between the two isotropic conjugates over K is a natural positive complexity c on the orbit of z 0 under PSU q (O K ), which is invariant under PSU q (O K ) ∞ .
Theorem 4 There exists a constant κ > 0 and an explicit constant C > 0 such that, as s → +∞, We refer to Section 6 for a more precise and more general statement, valid for instance for the congruence subgroups of PSU q (O K ).
Acknowledgement: The first author thanks the Université de Paris-Sud (Orsay) for a visit of a month and a half which allowed an important part of the writing of this paper, under the financial support of the ERC grant GADA 208091.We thank Y. Benoist and L. Clozel for their help with Proposition 18.

Geometric counting and equidistribution
In this Section, we briefly review a simplified version of the geometric counting and equidistribution results proved in [PaP5], whose arithmetic applications will be considered in the main part of this paper (see also [PaP4] for a review of related references).
Let M be a negatively curved rank one symmetric space (see [PaP5,§2] for a more general setting).In other words, M is a hyperbolic space H n F where F is the set R of real numbers, C of complex numbers, H of Hamilton's quaternions, or O of octonions, and n ≥ 2, with n = 2 if K = O see for instance [Mos,Par3]).We will normalise them so that their maximal sectional curvature is −1.
Let Γ be a discrete nonelementary group of isometries of M and let M = Γ\ M be the quotient orbifold.Let D − and D + be nonempty proper closed convex subsets of M , such that the families (γD − ) γ∈Γ and (γD + ) γ∈Γ are locally finite in M (see [PaP5,§3.3]for more general families).Let Γ D − and Γ D + be the stabilisers in Γ of the subsets D − and D + , respectively.
We denote by ∂ ∞ M the boundary at infinity of M , by ΛΓ the limit set of Γ and by (ξ, x, y) → β ξ (x, y) the Busemann function on ∂ ∞ M × M × M (see for instance [BrH]).For every v ∈ T 1 M , let π(v) ∈ M be its origin, and let v − , v + be the points at infinity of the geodesic line defined by v.We denote by ∂ 1 ± D ∓ the outer/inner unit normal bundle of ∂D ∓ , that is, the set of v ∈ T 1 M such that π(v) ∈ ∂D ∓ and the closest point projection on . For all γ, γ in Γ, the convex sets γD − and γ D + have a common perpendicular if and only if their closures γD − and γ D + in M ∪ ∂ ∞ M do not intersect.We denote by α γ, γ this common perpendicular (starting from γD − at time t = 0) and by (α γ, γ ) its length.The multiplicity of α γ,γ is which equals 1 when Γ acts freely on T 1 M (for instance when Γ is torsion-free).Let where Γ acts diagonally on Γ × Γ.When Γ has no torsion, N D − , D + (t) is the number (with multiplicities coming from the fact that Γ D ± \D ± is not assumed to be embedded in M ) of the common perpendiculars of length at most t between the images of D − and D + in M .We refer to [PaP5,§4] for the use of Hölder potentials on T 1 M to modify this counting function by adding weights, which could be useful for some further arithmetic applications.
Recall the following notions (see for instance [Rob]).The critical exponent of Γ is Note that in the right hand side of this equation, π(v) may be replaced by any point x on the geodesic line defined by v, since We will use this elementary observation in the proof of Lemma 10(ii).The measure m BM is nonzero, independent of x 0 ∈ M , is invariant under the geodesic flow, the antipodal map v → −v and the action of Γ.Thus, it defines a nonzero measure m BM on T 1 M which is invariant under the geodesic flow of M and the antipodal map, called the Bowen-Margulis measure on M = Γ\ M .When m BM is finite (for instance when M has finite volume or when Γ is geometrically finite), denoting the total mass of a measure m by m , the probability measure m BM m BM is then uniquely defined, and is the unique probability measure of maximal entropy for the geodesic flow (see [OP]).

Using the endpoint homeomorphism
we defined in [PaP3] (generalising the definition of Oh and Shah [OhS2,§1.2]when D − is a horoball or a totally geodesic subspace in and satisfies σ γD − = γ * σ D − for every γ ∈ Γ.Since the family (γD − ) γ∈Γ is locally finite in M , the measure γ∈Γ/Γ D − γ * σ D − is a well defined Γ-invariant locally finite (nonnegative Borel) measure on T 1 M .Hence, it induces a locally finite measure σ D − on T 1 M = Γ\T 1 M , called the skinning measure of D − in T 1 M .We refer to [OhS1,§5] and [PaP3,Theo. 9] for finiteness criteria of the skinning measure σ D − , in particular satisfied when M has finite volume and if either D − is a horoball centred at a parabolic fixed point of Γ or if D − is a totally geodesic subspace.
Remark 5 Let H be a horoball in M centred at ξ, and let ρ be the geodesic ray starting from any point in ∂H and converging to ξ.Then (see for instance [HeP1,§2.3])the weak-star limit exists, defines a measure on ∂ ∞ M − {ξ} which is invariant under the elements of Γ preserving H .The limit measure satisfies for all x ∈ M , η ∈ ∂ ∞ M − {ξ}, where x H , η is the intersection with ∂H of the geodesic line from η to ξ.Take x 0 = ρ(t) and let t go to +∞ in the definition of the Bowen-Margulis measure and the skinning measures.Then, for every v ∈ T 1 M such that v ± = ξ, we have Furthermore, for every v ∈ ∂ 1 + H , we have since The following result is a special case of [PaP5,Coro. 20,21,Theo.28] (we prove that the pairs of the initial/terminal tangent vectors of the common perpendiculars equidistribute in the product of the outer/inner tangent bundles of D − /D + ).We refer to [PaP4] for an review of the particular cases known before [PaP5] (due to Huber, Margulis, Herrmann, Cosentino, Roblin, Oh-Shah, Martin-McKee-Wambach, Pollicott, and the authors for instance).
For all t ≥ 0 and x ∈ ∂D − , let be the multiplicity of x as the origin of common perpendiculars with length at most t from D − to the elements of the Γ-orbit of D + .We denote by ∆ x the unit Dirac mass at a point x.
Theorem 6 Let M , Γ, D − , D + be as above.Assume that the measures m BM , σ D − , σ D + are nonzero and finite.Then ) for some κ > 0. Furthermore, the origins of the common perpendiculars equidistribute in the boundary of D − : for the weak-star convergence of measures on the locally compact space T 1 M .
When ∂D − is smooth, for smooth functions ψ with compact support on ∂D − , there is an error term in the equidistribution claim (5) when the measures on both sides are evaluated on ψ, of the form O(e −κt ψ ) where κ > 0 and ψ is the Sobolev norm of ψ for some ∈ N, as proved in [PaP5,Theo. 28].
When M has finite volume, the Bowen-Margulis measure m BM coincides up to a multiplicative constant with the Liouville measure on T 1 M , and the skinning measures of points, horoballs and totally geodesic subspaces D ± coincide with the (homogeneous) Riemannian measures on ∂ 1 ± D ∓ induced by the (Sasaki's) Riemannian metric of T 1 M .In Section 4, we explicit some of these proportionality constants when M is a complex hyperbolic space (see [PaP4,§7] for the real hyperbolic case).

Complex hyperbolic geometry
In this Section, we recall some background on the complex hyperbolic spaces, as mostly contained in [Gol], and, unless otherwise stated, we will follow the conventions therein.For all w, w in C n−1 , we denote by w • w = n−1 i=1 w i w i their standard Hermitian product, and we denote |w| 2 = w • w.Recall that for every n ≥ 1, the Siegel domain model of the complex hyperbolic n-space endowed with the Riemannian metric In accordance with Section 2, this metric is normalised so that its sectional curvatures are in [−4, −1], instead of in [−1, − 1 4 ] as in [Gol] and [Par4].Its boundary at infinity is A complex geodesic line in H n C is the image by an isometry of H n C of the intersection of H n C with the complex line C × {0}; with our normalisation of the metric, a complex line has constant sectional curvature −4.The boundary at infinity of a complex geodesic line is a topological circle, called a chain (see Section 7 for more informations).
Let q be the nondegenerate Hermitian form −z 0 z n − z n z 0 + |z| 2 of signature (1, n) on C × C n−1 × C with coordinates (z 0 , z, z n ).This is not the form considered in [Gol,p. 67], hence we need to do some computations with it, but it is better suited for our purposes.It is the one considered for instance in [PaP1], to which we will refer frequently.The Siegel domain H n C embeds in the complex projective n-space P n (C) by the map (using homogeneous coordinates) We identify H n C with its image by this map.This image, called the projective model of H n C when endowed with the isometric Riemannian metric, is the negative cone of q, that is {[z 0 : z : z n ] ∈ P n (C) : q(z 0 , z, z n ) < 0}.This embedding extends continuously to the boundary at infinity, by mapping The linear action of the special unitary group of q on C n+1 induces a projective action on P n (C).The quotient group PSU q = SU q /(U n+1 Id) of SU q by the kernel of the projective action, where U n+1 is the group of (n + 1)-th roots of unity, preserves H n C , and its restriction to H n C is the orientation-preserving isometry group of H n C .For instance by the paragraph above [PaP1,Lem. 6.3], an element γ ∈ SU q fixes ∞ if and only if γ is upper triangular (this is the reason, besides rationality problems, that we chose the Hermitian form q rather than the one in [Gol]), see for instance [Gol,p. 119], [FaP,§2.1] up to signs.
By for instance [PaP1,Eq. (42)], the intersection of SU q with the upper triangular subgroup of SL 3 (C) is and its image B q in PSU q is equal to the stabiliser in PSU q of ∞.
, that we will use from now on unless otherwise stated, of (w 0 , w) so that the Riemannian metric is given by In horospherical coordinates, the geodesic lines from (ζ, u, 0) ∈ ∂ ∞ H n C − {∞} to ∞ are, up to translations at the source, the map s → (ζ, u, e 2s ), by the normalisation of the metric.The closed horoballs centred at ∞ ∈ ∂ ∞ H n C are the subsets and the horospheres centred at ∞ are their boundaries for any s > 0. Note that, for every s ≥ 1, we have As introduced by [Par1, p. 297], the Cygan distance on The Heisenberg group Heis 2n−1 of dimension 2n − 1 is the real Lie group structure on that are isometries for both the Riemannian metric and the Cygan distance, and that preserve the horospheres centred at ∞.For every u ∈ R, the Heisenberg translation by (0, u) is called a vertical translation.
It is easy to see that the Cygan distance on Heis 2n−1 (see [Gol,page 160], it is called the Korányi distance by many people working in sub-Riemannian geometry, though Korányi [Kor] does attribute it to Cygan [Cyg]) is the unique left-invariant distance on Heis 2n−1 with d Cyg ((ζ, u), (0, 0)) = (|ζ| 4 + u 2 ) 1 4 .We introduced in [PaP1, Lem.6.1] the modified Cygan distance d Cyg as the unique left-invariant distance on Heis 2n−1 with , which is almost a distance on For every nonempty subset A of Heis 2n−1 , we define the diameter of A for this almost distance as We conclude this section by two geometric lemmas that will be useful in Section 4. See also [Kim,§3], with slightly different conventions, for a computation similar to Lemma 7 based on [Gol,p. 113].The Cygan distance, the Poisson kernel e β (ξ, r) , the Patterson measures µ x computed in the next section, and related quantities are useful in potential theory on the Heisenberg group and for the study of the hypoelliptic Laplacian in sub-Riemannian geometry, see for instance [FS, Kra].
x are replaced by other points on the horospheres centred at ∞ through them, and since the map s → (0, 0, e 2s ) is a geodesic line in It is easy to check that the map ι : (w 0 , w) Hence, with x = (w 0 , w) and x = (w 0 , w ), using Equation ( 7) and the fact that d Cyg (x, (0, 0)) 4 = 4|w 0 | 2 and d Cyg (x , (0, 0)) 4 = 4|w 0 | 2 , we have The Heisenberg translation τ by (ξ, r) preserves the last horospherical coordinates and the Cygan distances.Thus, In particular, the preimages by this orthogonal projection are the spheres of center (0, 0) for the Cygan distance on Heis 2n−1 .They are spinal spheres with complex spine For every parameter a ranging in ]0, +∞[ , consider the horosphere ∂H a centred at ∞.Its image by the isometric involution ι : (w, w 0 ) → ( 1 w 0 , w w 0 ) is, using Equation ( 7), the horosphere {(ξ, r, t) ∈ H n C : t = a 4 ((|ξ| 2 + t) 2 + r 2 )} centred at (0, 0).The image of this horosphere by the Heisenberg translation by (ζ, u) is the horosphere The orthogonal projection of (ζ, u) on the geodesic line from (0, 0) to ∞ is attained when the parameter a gives a double point of intersection (0, 0, t) between this horosphere and .The quadratic equation with unknown t has a double solution if and only if its re- 4 Measure computations in complex hyperbolic spaces In this Section, we give proportionality constants relating, on the one hand, Patterson, Bowen-Margulis and skinning measures associated to some convex subsets and, on the other hand, the corresponding Riemannian measures, in the complex hyperbolic case.
We will denote the standard Lebesgue measure on C n−1 by dζ, and the usual left Haar measure on Heis 2n−1 by In horospherical coordinates, the volume form of We begin by recalling a lemma that relates the volume of a Margulis cusp neighbourhood with the volume of its boundary.
Proof.Since the group of isometries of H n C acts transitively on the set of horospheres of H n C , we may assume that D = H 1 .The horosphere centred at ∞ through a point (ζ, u, t) ∈ H 1 is ∂H t and its orthogonal geodesic line at this point is s → (ζ, u, e 2s ), hence By Equation ( 10), we hence have ).The homeomorphism from ∂H t to ∂H 1 defined by (ζ, u, t) → (ζ, u, 1) commutes with the action of Γ.Thus, Vol(Γ\∂H 1 ) .
Let Γ be a lattice in Isom(H n C ).Its critical exponent is δ Γ = 2n (see for instance [CI,§6]).The Patterson density (µ x ) x∈H n C of Γ is uniquely defined up to a multiplicative constant, and is independent of Γ.We will choose the normalisation such that the measure µ ∞ defined in Remark 5 by the horoball This is possible since µ ∞ is invariant under the isometry group Lemma 10 Let Γ be a lattice in Isom(H n C ), and let µ ∞ be normalised as above.For all and for every horoball and, with m the order of the pointwise stabiliser of D − in Γ, (vi) for every complex geodesic line D − in H n C , we have and, with m the order of the pointwise stabiliser of D − in Γ, Proof.In the computations below, it is useful to note that Lemma 7 implies that (i) The geodesic line from (ξ, r) to ∞ goes through ∂H 1 at the point (ξ, r, 1).By the normalisation of dµ ∞ and by Equation ( 2), we hence have The result then follows from Equation ( 12).
(ii) Note that if x is on the geodesic line defined by v, then Hence, by Equation ( 12) and Assertion (i), by letting x converge to v − on the geodesic line defined by v, we have (iii) Recall that the Liouville measure vol T 1 H n C (which is the Riemannian measure for Sasaki's metric on T 1 H n C ) disintegrates under the fibration π : , with conditional measures the spherical measures on the unit tangent spheres: By homogeneity and by Assertion (i), we have, Using Mathematica for the first equation and the well known expression of Γ(n + 1/2) for the second one, we have .
The first claim of Assertion (iii) follows by computing the Jacobian at (0, 1, 0) of the map F , which is equal to 1 2 n , since at the point (0, 1, 0), we have The second claim follows from the facts that Vol(T 1 M ) = Vol(S 2n−1 ) Vol(M ) and that Vol(S 2n−1 ) = 2 π n (n−1)! .
The next step is to obtain a similar expression for the Riemannian measure of the submanifold ∂ 1 + D − of T 1 H n C (endowed with Sasaki's metric).For every x ∈ D − , let us denote by ν 1 x D − the fiber over x of the normal bundle map v → π(v) from ∂ 1 + D − to D − .We endow ν 1 x D − with the spherical metric induced by the scalar product of the tangent space T x H n C at x.The Riemannian measure of ∂ 1 + D − disintegrates under this fibration over the Riemannian measure of D − as .
The second one follows, since pushforwards of measures preserves their total mass, and since (vii) By the transitivity of the isometry group of H n C on the set of its complex geodesic lines, we may assume that D − is the complex geodesic line sending a normal unit vector v to its point at infinity v + = (ζ, u), we have, by Equation ( 12) and by Assertion (i), 7, In particular, For every x ∈ C, let us denote by ν 1 x C the fiber over x of the normal bundle map v → π(v) from ∂ 1 + C to C, endowed with the spherical metric induced by the scalar product of the tangent space T x H n C at x.The Riemannian measure of ∂ 1 + C disintegrates under this fibration over the Riemannian measure of C as Using the homeomorphism (u, t = |ζ| 2 ) → x = (0, u, t) from R×[0, +∞[ to C, and Equation (8), we have Hence The result follows as in the end of the proof of the previous Assertion.
By Theorem 6, we then have the following counting and equidistribution result of common perpendiculars.We first define some constants.
If D − and D + are horoballs in H n C centred at parabolic fixed points of a lattice Γ in Corollary 11 Let Γ be a discrete group of isometries of H n C such that the orbifold M = Γ\H n C has finite volume.In each of the above three cases, if m + is the cardinality of the pointwise stabiliser of D + , then If Γ is arithmetic, then there exists κ > 0 such that, as t → +∞, Furthermore, if D − is a horoball centred at a parabolic fixed point, then the origins of the common perpendiculars from D − to the images of D + under the elements of Γ equidistribute in ∂D − to the induced Riemannian measure: For smooth functions ψ with compact support on ∂D − , there is an error term in the equidistribution claim (19) when the measures on both sides are evaluated on ψ, of the form O(e −κt ψ ) where κ > 0 and ψ is the Sobolev norm of ψ for some ∈ N, by the remark following Theorem 6.

A Mertens' formula for Heisenberg groups
Let K be an imaginary quadratic number field.We will denote, in Sections 5 to 7, by O K its ring of integers, by D K its discriminant, by ζ K its zeta function, and by |O × K | the order of the unit group of O K .Let tr, n : K → Q be the (absolute) trace and norm of K, that is tr(z) = z + z = 2 Re z and n(z) = z z = |z| 2 .We denote by a, α, c the ideal of O K generated by a, α, c ∈ O K .
Let m be a nonzero ideal in O K .We endow the ring O K /m with the involution induced by the complex conjugation.Let SU q (O K /m) be the finite group of 3×3 matrices in O K /m, having determinant 1 and preserving the Hermitian form −z 0 z 2 − z 2 z 0 + z 1 z 1 on (O K /m) 3 .Let B q (O K /m) be its upper triangular subgroup.The action by shears on O K × O K × O K of the nilpotent group Heis 3 (O K ) defined in the Introduction preserves O K × m × m.In this section, we will study the asymptotic of the counting function Ψ m , where, for every s ≥ 0, Ψ m (s) is the cardinality of When m = O K , this map Ψ m is the counting function, in terms of their standard heights, of the rational points over K in the complex projective plane P 2 (C), that lie in Segre's hyperconic with equation 2 Re u − |v| 2 = 0 in the standard affine chart with coordinates (u, v).Theorem 12 As s → +∞, we have The particular case m = O K gives Theorem 1 in the Introduction.We will prove this result simultaneously with the next one.We endow the 3-dimensional Heisenberg group Heis 3 with its Haar measure Haar Heis 3 as in the introduction.The following result is an equidistribution result of the set of Q-points (satisfying some congruence properties) in Heis 3 .The particular case m = O K gives Theorem 2 in the introduction.
Theorem 13 As s → +∞, we have As said after Corollary 11, for smooth functions ψ with compact support on Heis 3 , there is an error term in this equidistribution result when the measures on both sides are evaluated on ψ, of the form O(s −κ ψ ) where κ > 0 and ψ is the Sobolev norm of ψ for some ∈ N.
Proof of Theorem 12 and Theorem 13.As a preliminary remark, the 3-dimensional Heisenberg group Heis 3 defined above contains Heis 3 (O K ) (by the definition of the norm and trace of K), as a (uniform) lattice, and identifies with the Heisenberg group Heis 3 defined in Section 3 by the change of variable so that the Haar measures Haar Heis 3 and λ 3 satisfy Let q be the Hermitian form −z 0 z 2 − z 2 z 0 + z 1 z 1 of signature (2, 1) on C × C × C with coordinates (z 0 , z 1 , z 2 ) (which is, up to isomorphism, the unique Hermitian form over K with signature (2, 1) and determinant −1, see [Sch,Ch. 10] for this cultural remark).As previously, we denote by SU q the special unitary group of q.Let Γ = SU q ∩ M 3 (O K ) be the Picard modular group of K, which is a nonuniform arithmetic lattice in SU q by a result of Borel and Harish-Chandra (see for instance [PaP1,§6.3]).As another cultural remark, every nonuniform arithmetic lattice in SU q is commensurable to a Picard modular group (see for instance [Sto,§ 3.1]).
Consider the map from Heis 3 to SU q defined, in the two sets of coordinates of Heis 3 defined in the Introduction and in Section 3, by This map is a Lie group isomorphism onto its image, by which we identify from now on Heis 3 and its image.Note that Heis 3 ∩ Γ is then exactly Heis 3 (O K ).
We denote by Γ m the Hecke congruence subgroup of Γ modulo m, that is the preimage, by the group morphism Γ → SL 3 (O K /m) of reduction modulo m, of the upper triangular subgroup of SL 3 (O K /m).
As previously, we denote by PSU q the quotient Lie group SU q /{id, j id, j 2 id} where j = e 2iπ 3 .For every subgroup G of SU q , we denote by G its image in PSU q , and again by g the image in PSU q of any element g of SU Q .
We denote by Γ H 1 the stabiliser in Γ m of the horoball Note that this agrees with our notation for horoballs introduced in Section 3. The group Γ H 1 is independent of m, since Γ m is Hecke's congruence subgroup of Γ.Note that Heis 3 (O K ) is contained in Γ H 1 , and that the projection map from Heis 3 (O K ) to Heis 3 (O K ) is injective.
Let B q be as defined in Section 3. Note that Γ H 1 = B q ∩Γ m = B q ∩Γ, since an isometry of H 2 C fixes ∞ if and only if it preserves H 1 .We claim that the index of Heis A separate treatment of the cases D K = −3, −4 and of the general case gives the result.A similar argument shows that the projection map from For all a, α, c ∈ O K and λ ∈ C, we have λa, λα, λc = a, α, c = O K if and only if λ ∈ O × K .Therefore the cardinality of the fibers of the projection map from {(a, α, c) Let g ∈ SU q be such that gH 1 and H 1 are disjoint (there are only finitely many double [PaP1,Lem. 6.3] and since the sectional curvature is normalised to have maximum −1, the length of the common perpendicular δ g between g H 1 and H 1 is then We use, in the last one of the following equalities, Equation ( 21) and Corollary 11 with n = 2, Γ = Γ m and D − = D + = H 1 (whose pointwise stabilisers in Γ m are trivial).We hence have, for some κ > 0, The next two lemmas are devoted to the computation of the two volumes that appear in the previous line.
Proof.We follow arguments similar to the ones in the reference [KK,§4], which uses the same convention as [Par2] for the Riemannian measure on Heis 2n−1 , see also [FaP,§3.1] when D K = −3.Let n = 2.As in [Par2], we endow Heis 3 with the left-invariant Riemannian metric whose Riemannian volume is vol Heis 3 = 4λ 3 .In particular, the Heis 3 -equivariant map from ∂H 1 to Heis 3 , which in horospherical coordinates maps (ζ, u, t) to (ζ, u), sends vol ∂H 1 to 1 2 λ 3 = 1 8 vol Heis 3 , by Equation ( 8).Let t K be the minimal vertical translation in Heis 3 (O K ), that is, the minimal s > 0 such that (w 0 = is 2 , w = 0) ∈ Heis 3 (O K ).In particular, Consider the following set The Cayley transform from the Siegel domain to the ball model of the complex hyperbolic space conjugates Γ to the Picard modular group Γ used by Holzapfel in loc.cit.. Hence . By the Main Theorem 4.9 of [Hol3, page 83], we have Hence, using the well known relation L K (s) = ζ K (s)/ζ(s) between Dirichlet's L-series and Dedekind's zeta function of K, the result follows since Theorem 12 follows from Lemma 15, Lemma 14 and Equation ( 23).
Let us prove now Theorem 13.The orthogonal projection map f : C − {∞} and the horospherical coordinates on ∂H 1 (see Equation ( 7)).Let x ∈ ∂H 1 be the origin of a common perpendicular of length at most t from H 1 to an element γH 1 for some γ ∈ Γ m not fixing ∞.By the previous arguments, x is the orthogonal projection on H 1 of the point at infinity of this horoball γH 1 .This point at infinity may be written [ a c : α c : 1] for some triple (a, α, c) ∈ O K × m × m with a, α, c = O K , tr(a c) = n(α) and 0 < n(c) ≤ 4 e 2t (using Equation ( 22)).There are exactly |O × K | such triples.Hence by Equation ( 19), considering the value of C(D − , D + ) for D − = D + = H 1 , using the horospherical coordinates on ∂H 1 , we have, as t → +∞, The image of the Haar measure Haar Heis 3 (defined in Equation ( 1)) by f is, by Equation ( 8), f * Haar Heis 3 = vol ∂H 1 .
Using the change of variables s = 4 e 2t , the identification of ∂ ∞ H 2 C − {∞} with Heis 3 and the continuity of the pushforward by f −1 of the measures on ∂H 1 applied to Equation (25), we hence have, as s → +∞, Finally, Theorem 13 follows from this and from Lemma 15 and Lemma 14.
Remark.A result of Feustel [Feu] (see also [Hol2,page 280] and [Zin]) says that the map, which associates to a parabolic fixed point of Γ the fractional ideal generated by its homogeneous coordinates in O K , induces a bijection from the set of cusps (that is, of orbits under Γ of its parabolic fixed points) to the set of ideal classes of K.In Theorem 1 and Theorem 2, replacing in its proof D + = H 1 by a horoball centred at a parabolic fixed point p of Γ not in the orbit of ∞ (which hence exists if and only if D K = −3, −4, −7, −8, −11, −19, −43, −67, −163), we can obtain a counting and equidistribution result with error term in Heis 3 of the points in Γ • p.But the volume of the quotient of this new D + by its stabiliser in Γ is not explicit for the moment, hence we would not have results as precise as in the case p = ∞.
Remark 16 Theorem 12 and Theorem 13 have generalisations in higher dimension.Let n ≥ 2, and let (w, w ) → w • w be the standard Hermitian scalar product on C n−1 .Let with law (w 0 , w)(w 0 , w ) = (w 0 + w 0 + w • w, w + w ), be the Heisenberg group of dimension 2n − 1, which identifies with the boundary at infinity of the Siegel domain H n C with ∞ removed.Let q be the Hermitian form defined in Section 3. Let SU q be its special unitary group and Γ = SU q ∩ M n (O K ), which is an arithmetic lattice in SU q .Then Corollary 11 (which is valid in any dimension), applied with the image Γ of Γ in PSU q and with D − = D + the horoball of points in H n C with last horospherical coordinates at least 1, gives a counting and equidistribution result with error term in Heis 2n−1 of the points in Γ • ∞ − {∞}.The volume of Γ\H n C could be computed using [EmS], up to computing the index of Γ in a principal arithmetic subgroup containing it.But the volume of the cusp corresponding to ∞ in Γ\H n C is not explicit for the moment, hence we would not have results as precise as in the case n = 2.
Other counting and equidistribution results of arithmetically defined points in the Heisenberg group Heis 2n−1 may be obtained by varying the integral Hermitian form q of signature (1, n) and the arithmetic lattice Γ in SU q .
6 Counting cubic points over quadratic imaginary fields in the projective plane Let K be an imaginary quadratic number field.Let q be the Hermitian form −z 0 z 2 − z 2 z 0 +|z 1 | 2 on C 3 (the following result could be adapted to any Hermitian form on K 3 with complex signature (1, 2), see for instance [Sch,Ex. 1.6(iv), p. 351] for their classification).
We will say that a point in the complex projective plane P 2 (C) is isotropic (respectively that two projective points are orthogonal) if the corresponding complex lines in C 3 are isotropic (respectively orthogonal) for q.
The Galois group Gal(C|K) acts naturally on P 2 (C) by σ[z 0 : z 1 : z 2 ] = [σz 0 : σz 1 : σz 2 ] using homogeneous coordinates.A point z ∈ P 2 (C) will be called Hermitian cubic over K if it is cubic over K (that is, if its orbit under Gal(C|K) has exactly three points), and if its other conjugates z , z over K are isotropic and orthogonal to z.
We will denote by (γ, z) → γ • z the projective action of SL 3 (C) or PSL 3 (C) on P 2 (C).Recall that SU q is the real Lie group of linear automorphisms of C 3 having determinant 1 and preserving q.Let Γ = SU q (O K ) = SU q ∩ SL 3 (O K ), which is an arithmetic lattice in SU q .The action of Γ on P 2 (C) preserves the sets of rational, cubic and Hermitian cubic points of P 2 (C) over K.The number of orbits of Hermitian cubic points is infinite, which explains why we will restrict ourselves to a given orbit below.Let Γ ∞ be the stabiliser Let z ∈ P 2 (C) be Hermitian cubic over K, and let z , z be its other conjugates over K. Since z , z are distinct, irrational over K, and isotropic for q, they lie in Heis 3 , and we may define the complexity c(z) of z as the inverse of (a modification of) the Cygan distance between its conjugates over K: The complexity of z is in particular invariant under Γ ∞ .It is part of the proof of the following result that the number of projective points, that are Hermitian cubic over K, belong to a given orbit of Γ = SU q (O K ), and have complexity at most s, is finite for every s ≥ 0. It turns out that the use of d Cyg instead of the actual Cygan distance allows, in this Section as well as in Section 7, to give precise asymptotic results with error terms, instead of upper/lower estimates that differ by a multiplicative constant.
Let us now state and prove an asymptotic estimate as s → +∞ of the counting function of the set of points in the projective plane that are Hermitian cubic over K, in a given orbit of for instance a congruence subgroup of SU q (O K ), with complexity at most s.
Theorem 17 Let K be an imaginary quadratic number field.Let z 0 ∈ P 2 (C) be Hermitian cubic over K. Let G be a finite index subgroup of PSU q (O K ), and let G ∞ be the stabiliser of ∞ = [1 : 0 : 0] in G. Then there exists κ > 0 such that, as s → +∞, where |λ 0 | is the smallest modulus > 1 of an eigenvalue of an element in G fixing the other two conjugates z 0 , z 0 of z 0 over K, where ι 0 = 2 if G contains an element exchanging z 0 and z 0 and ι 0 = 1 otherwise, and where n 0 is the cardinality of the pointwise stabiliser in G of the projective line through z 0 and z 0 .
Proof.The group G, which has finite index in the arithmetic lattice Γ = PSU q (O K ), is also an arithmetic discrete group of isometries with finite covolume in the complex hyperbolic plane H 2 C .By Lemma 15, we have Proposition 18 A point z 0 ∈ P 2 (C) is Hermitian cubic over K if and only if there exists γ 0 ∈ PSU q (O K ) of infinite order and K-irreducible such that z 0 is the only fixed point of γ 0 that belongs to the positive cone of q in P 2 (C).
Recall that an element of PSU q (K) is K-irreducible if it does not preserve a point or a line defined over K in P 2 (C).
Proof.(Y.Benoist) Assume first that γ 0 ∈ PSU q (O K ) has infinite order and is Kirreducible.Then γ 0 is not an elliptic element, since elliptic elements of PSU q (O K ) have finite order.It is not parabolic, since the fixed points in ∂ ∞ H 2 C of the parabolic elements of PSU q (O K ) are rational over K (see [Hol1] or [Hol2,page 290]).Hence, it is loxodromic, and fixes exactly two distinct points z 0 and z 0 in ∂ ∞ H 2 C .In particular, z 0 and z 0 are isotropic for q.Since γ 0 belongs to PSL 3 (C) and preserves ∂ ∞ H 2 C , it preserves the unique complex projective lines L and L tangent to ∂ ∞ H 2 C at the points z 0 and z 0 , respectively.Note that L and L are exactly the sets of points in P 2 (C) which are orthogonal to z 0 and z 0 , respectively.The projective lines L and L meet at exactly one point z 0 , which belongs to the complement in This complement is exactly the positive cone of q in P 2 (C).The fixed points z 0 , z 0 , z 0 of γ 0 are at most cubic over K, since γ 0 has coefficients in K.They are exactly cubic and conjugates, since γ 0 is K-irreducible.Hence z 0 is Hermitian cubic, and is the only fixed point of γ 0 in the positive cone of q.
Conversely, let z 0 ∈ P 2 (C) be Hermitian cubic over K, and let z 0 , z 0 be its other two Galois conjugates.Let G be the linear algebraic group defined over Q, such that G(Z) = PSU q (O K ) and G(R) = PSU q .It has R-rank one.The pointwise stabiliser of {z 0 , z 0 } in G is the centraliser Z(T ) of a maximal algebraic torus T in G, since z 0 , z 0 are distinct and isotropic.Note that Z(T ) also fixes z 0 , since z 0 , being orthogonal to z 0 and z 0 , belongs to the complex projective lines tangent to the null cone of q in P 2 (C) at z 0 and z 0 , and as seen above, these two projective lines meet at exactly one point.The algebraic group Z(T ), being the pointwise stabiliser of {z 0 , z 0 , z 0 } which is invariant under Gal(C/K), is defined over K.The torus T has rank one and has finite index in Z(T ).Hence T is also defined over K, and is isomorphic to C × over C. It has no nontrivial Q-character (as it is one-dimensional, it is not defined over Q, overwise its fixed points z 0 , z 0 , z 0 would individually be defined over Q).Hence by the Borel and Harish-Chandra theorem (see [BoH,Th. 9.4], though the particular case we use here is due to Ono), T (Z) is a lattice in T (R).Such a lattice contains an element of infinite order γ 0 .The set of fixed points of γ 0 in P 2 (C) is {z 0 , z 0 , z 0 }.Any proper nonzero linear subspace of C 3 invariant under γ 0 is the sum of one or two geodesic lines in {z 0 , z 0 , z 0 }, hence is not defined over K since z 0 is cubic.Therefore γ 0 is K-irreducible.As seen above, z 0 is the only fixed point of γ 0 in the positive cone of q.
Let γ 0 be as in Proposition 18 for z 0 given by the statement of Theorem 17.As seen in the above proof, γ 0 is loxodromic.Let D + be the geodesic line in the projective model of H 2 C with endpoints the other two Galois conjugates z 0 , z 0 of z 0 over K, and let G D + be its stabiliser in G. Up to replacing γ 0 by a power, we may assume that γ 0 ∈ G.We may also assume that γ 0 is primitive in G, so that if λ 0 is its eigenvalue with modulus > 1, then its translation length in H 2 C is ln |λ 0 | (see Equation ( 6)), and with ι 0 defined in the statement of Theorem 17.
Let G z 0 be the stabiliser of z 0 in G, which is also the stabiliser of z ⊥ 0 = D + , hence coincides with G D + .Let g ∈ G be such that the geodesic line gD + is disjoint from H 1 (which is the case except for finitely many double classes in G H 1 \G/G D + ).Let z , z be the endpoints of gD + .Let δ g be the common perpendicular from H 1 to gD + .The length of δ g is, by [PaP2,Lem. 3.4], by Equation ( 9), and by the definition of the complexity, Therefore, by Corollary 11 (with n = 2, in the case D − = H 1 is a horoball, whose pointwise stabiliser is trivial, and D + is a geodesic line, with pointwise stabiliser of order n 0 as defined in the statement of Theorem 17), and by Equations ( 26), ( 27) and ( 28), This proves Theorem 17.

Counting arithmetic chains in hyperspherical geometry
Let us consider again the Hermitian form q = −z 0 z 2 − z 2 z 0 + |z 1 | 2 of signature (1, 2) on C 3 with coordinates (z 0 , z 1 , z 2 ).Following Poincaré [Poi] (who was rather using the diagonal form ), see also [Car], we will call hypersphere the projective isotropic locus of q, that is the subspace of the complex projective plane P 2 (C) with homogeneous coordinates [z 0 : z 1 : z 2 ].It is a real analytic submanifold, diffeomorphic to the 3-sphere S 3 .The subgroup PSU q of PSL 3 (C), acting projectively on P 2 (C), preserves the hypersphere.
In Section 3, we introduced a natural modification d Cyg of Cygan's distance on H S − {∞}, with ∞ = [1 : 0 : 0], as follows.We identified H S − {∞} with the real quadric (called a hyperconic by Segre) Heis 3 = {(w 0 , w) ∈ C 2 : 2 Re w 0 − |w| 2 = 0} by the map (w 0 , w) → [w 0 : w : 1].Then d Cyg is the unique map from (H S − {∞}) 2 to [0, +∞[, invariant under the diagonal action of the unipotent radical of the stabiliser of ∞ in PSU q , such that A complex projective line L in P 2 (C) intersects the hypersphere either in the empty set, or a one point set (in which case L is the unique complex projective line tangent to H S at this point, giving at this point the canonical contact structure of H S ), or a real analytic circle, called a chain (a notion attributed to von Staudt by [Car, footnote 3)]).A chain C separates the complex projective line L(C) containing it into two real discs D ± (C), which we endow with their unique Poincaré metric (of constant curvature −1) invariant under the stabiliser of C in PSU q .The diameter of a chain C is diam(C) = sup x, y ∈C : x =y d Cyg (x, y) .
It is finite if and only if C is a finite chain, that is, if it does not contain ∞ = [1 : 0 : 0].We refer to [Gol,§4.3] for more informations on the chains, including the following fact: the infinite chains are precisely the fibers of the vertical projection (w 0 , w) → w, and the finite chains are ellipses in the affine coordinates (2 Im w 0 , w) of Heis 3 whose vertical projections are circles.
Let K be an imaginary quadratic number field.For every finite index subgroup G of the arithmetic lattice PSU q (O K ), we will denote by G C the stabiliser of C in G, by G ∞ the stabiliser of ∞ in G, and by Covol G (C) the (common) volume of G C \D ± (C).A chain C will be called arithmetic (over K) if PSU q (O K ) C has a dense orbit in C.
Theorem 19 Let C 0 be an arithmetic chain in the hypersphere H S over an imaginary quadratic number field K. Let G be a finite index subgroup of PSU q (O K ).Then there exists a constant κ > 0 such that, as > 0 tends to 0, the number ψ C 0 , G ( ) of chains modulo G ∞ in the G-orbit of C 0 with diameter at least is equal to where n 0, G is the order of the pointwise stabiliser of C 0 in G.
Given a complex projective line L in P 2 (C), there is a unique order 2 complex projective map with fixed point set L, called the reflexion on L. Given a finite chain C, contained in the projective line L(C), the center of C (see for instance [Gol,4.3.3]),denoted by cen(C) ∈ H S − {∞} = Heis 3 , is the image of ∞ = [1 : 0 : 0] under the reflexion on L(C).
The following result is an equidistribution result in the Heisenberg group of the centers of the arithmetic chains in a given orbit under (a finite index subgroup of) PSU q (O K ).
Theorem 20 Let C 0 and G be as in Theorem 19.As > 0 tends to 0, we have * Haar Heis 3 .
Proof of Theorem 20 and Theorem 19.
As seen in Section 3, the hypersphere H S is the boundary at infinity of the projective model of the complex hyperbolic space H 2 C .The chains are precisely the boundary at infinity of the complex geodesic lines in H 2 C .The diameter of a chain is invariant under the stabiliser in PSU q of the horosphere ∂H 1 , hence is invariant under G ∞ .The counting function ψ C 0 , G is thus well defined.
Recall (see for instance [Bow], in particular for the terminology) that a discrete group of isometries Γ of a complete simply connected Riemannian manifold M with sectional curvature at most −1 is geometrically finite if and only if every limit point of Γ is either a bounded parabolic point or a conical limit point.Furthermore, the discrete groups Γ of isometries of M with finite covolume are the geometrically finite discrete groups of isometries Γ whose limit set is the whole sphere at infinity ∂ ∞ M of M ; then the orbit under Γ of every point in Let C be a chain in H S , and let D be the complex geodesic line (which is totally geodesic in H 2 C ) with ∂ ∞ D = C. Hence, C is arithmetic over K if and only if the stabiliser in PSL q (O K ) (or equivalently in G) of D has finite covolume on D.
We denote by D + the complex geodesic line in Let g ∈ G be such that the complex geodesic line gD + is disjoint from H 1 (which is the case except for g in finitely many double classes in G H 1 \G/G D + ).Let δ g be the common perpendicular from H 1 to gD + .Its length (δ g ) is the minimum of the distances from H 1 to a geodesic line between two points of ∂ ∞ (gD + ) = gC 0 .Hence, by [PaP2,Lem. 3.4] and Equation ( 9), we have (δ g ) = min x,y∈gC 0 , x =y By the definition of the diameter of a chain, we thus have Now, we apply Corollary 11 with n = 2, in the case D − = H 1 is a horoball, whose pointwise stabiliser is trivial, and D + is a complex geodesic line, whose pointwise stabiliser has order n 0, G as defined in the statement of Theorem 19.Respectively by the definition of the counting function ψ C 0 , G in the statement of Theorem 19, since the stabiliser of C 0 in G is equal to G D + , by Equation ( 30), by Corollary 11, and by Equations ( 26), ( 27) and ( 29), we have, as > 0 tends to 0, This proves Theorem 19.Let us now prove Theorem 20.
We apply the equidistribution result in Equation ( 19) (in the case D + is a complex geodesic line) of the origins or(δ g ) of the common perpendiculars δ g from D − = H 1 to the images gD + for g ∈ G.As t → +∞, we hence have Note that, for every chain C, if r C is the reflexion on the complex projective line containing C, then the geodesic line from ∞ to cen(C) = r C (∞), being invariant under r C , is orthogonal to the complex geodesic line with boundary at infinity C. Hence for every g ∈ G, we have f −1 (or(δ g )) = cen(gC 0 ) .
Let us use in Equation ( 31) the change of variables t = − ln √ 2 and the continuity of the pushforward of measures by f −1 .By Equations ( 26) and ( 29), as > 0 tends to 0, the measures and no two distinct elements of such a fiber are sent one to the other by an element of Heis 3 (O K ).By [PaP1, Prop.6.5 (2)], the orbit Γ m • ∞ of ∞ = [1 : 0 : 0] under Γ m is equal to the set of elements of P 2 (C) which can be written in homogeneous coordinates [a : α : c] where (a, α, c) ∈ O K × m × m, a, α, c = O K and tr(a c) = n(α).
H 2 C with ∂ ∞ D + = C 0 .Let G D + be the stabiliser of D + in G.By definition, we have Vol(G D + \D + ) = 4 Covol G (C 0 ) , (29)since the sectional curvature of D + is constant −4 and D + has real dimension 2.
last horospherical coordinate t.Since the group I x of isometries of H n Cfixing x acts transitively on T 1x H n C , and both µ x and the Riemannian measure vol T 1x