A classification of <span class="mathjax-formula">$\protect \mathbb{R}$</span>-Fuchsian subgroups of Picard modular groups

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CONFLUENTES MATHEMATICI

Jouni PARKKONEN and Frédéric PAULIN

A classification ofR-Fuchsian subgroups of Picard modular groups Tome 10, n^o2 (2018), p. 75-92.

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10, 2 (2018) 75-92

A CLASSIFICATION OF R-FUCHSIAN SUBGROUPS OF PICARD MODULAR GROUPS

JOUNI PARKKONEN AND FRÉDÉRIC PAULIN

Abstract. Given an imaginary quadratic extensionK of Q, we classify the maximal nonelementary subgroups of the Picard modular group PUp1,2;OKqpreserving a totally real totally geodesic plane in the complex hyperbolic planeH²_C. We prove that these maximal R-Fuchsian subgroups are arithmetic, and describe the quaternion algebras from which they arise. For instance, if the radius ∆ of the correspondingR-circle lies inN´ t0u, then the stabiliser arises from the quaternion algebra

´∆,|D_K| Q

¯

. We thus prove the existence of infinitely many orbits ofK-arithmeticR-circles in the hypersphere ofP2pCq.

1. Introduction

Lethbe a Hermitian form with signature p1,2qonC³. The projective unitary Lie group PUp1,2qofhcontains exactly two conjugacy classes of connected Lie subgroups locally isomorphic to PSL2pRq. The subgroups in one class are conjugate to PpSUp1,1q ˆ t1uq and they preserve a complex projective line for the projective action of PUp1,2qon the projective plane P2pCq, and those of the other class are conjugate to POp1,2qand preserve a maximal totally real subspace ofP2pCq. The groups PSL2pRqand PUp1,2qact as the groups of holomorphic isometries, respec- tively, on the upper halfplane model H²_R of the real hyperbolic space and on the projective model H²_Cof the complex hyperbolic plane defined using the formh.

If Γ is a discrete subgroup of PUp1,2q, the intersections of Γ with the connected Lie subgroups locally isomorphic to PSL2pRq are its Fuchsian subgroups. The Fuchsian subgroups preserving a complex projective line are calledC-Fuchsian, and the ones preserving a maximal totally real subspace are calledR-Fuchsian. In [16], we gave a classification of the maximalC-Fuchsian subgroups of the Picard modular groups, and we explicited their arithmetic structures, completing work of Chinburg- Stover (see Theorem 2.2 in version 3 of [3] and [4, Theo. 4.1]) and Möller-Toledo in [11], in analogy with the result of Maclachlan-Reid [10, Thm. 9.6.3] for the Bianchi subgroups in PSL2pCq. In this paper, we prove analogous results for R-Fuchsian subgroups, thus completing an arithmetic description of all Fuchsian subgroups of the Picard modular groups. The classification here is more involved, as in some sense, there are moreR-Fuchsian subgroups thanC-Fuchsian ones. Our approach is elementary, some of the results can surely be obtained by more sophisticated tools from the theory of algebraic groups.

LetK be an imaginary quadratic number field, with discriminantDK and ring of integers OK. We consider the Hermitian formhdefined by

pz₀, z₁, z₂q ÞÑ ´1

2z₀z₂´1

2z₂z₀`z₁z₁.

Math. classification:11F06, 11R52, 20H10, 20G20, 53C17, 53C55.

Keywords:Picard modular group, ball quotient, arithmetic Fuchsian groups, Heisenberg group, quaternion algebra, complex hyperbolic geometry,R-circle, hypersphere.

75

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ThePicard modular groupΓK“PUp1,2q XPGL3pOKqis a nonuniform arithmetic lattice of PUp1,2q.¹ In this paper, we classify the maximalR-Fuchsian subgroups of ΓK, and we explicit their arithmetic structures. The results stated in this introduction do not depend on the choice of the Hermitian formhof signaturep2,1q defined over K, since the algebraic groups over Q whose groups of Q-points are PUp1,2q XPGL3pKq depend up to Q-isomorphism only on K and not on h, see for instance [20, § 3.1], so that the Picard modular group ΓK is well defined up to commensurability.

LetI₃ be the identity matrix and letI_1,2be the matrix ofh. Let AHIpQq “ tY PM3pKq : Y^˚I_1,2Y “I_1,2 and Y Y “I₃u

be the set ofQ-points of an algebraic subset defined overQ, whose real points consist of the matrices of the Hermitian anti-holomorphic linear involutionszÞÑY zofC³. For instance,

Y∆“

¨

˝

0 0 _∆¹ 0 1 0

∆ 0 0

˛

‚

belongs to AHIpQq for every ∆ P K^ˆ. The group Up1,2q acts transitively on AHIpRqby

pX, Yq ÞÑX Y X^´1

for allX PUp1,2qandY PAHIpRq. In Section 4, we prove the following result that describes the collection of maximal R-Fuchsian subgroups of the Picard modular groups ΓK.

Theorem 1.1. — The stabilisers in ΓK of the projectivized rational points in AHIpQqare arithmetic maximal nonelementaryR-Fuchsian subgroups of ΓK. Ev- ery maximal nonelementary R-Fuchsian subgroup of ΓK is commensurable up to conjugacy inPUp1,2q XPGL3pKqwith the stabiliserΓK,∆inΓK of the projective class ofY∆, for some∆POK´ t0u.

A nonelementary R-Fuchsian subgroup Γ of PUp1,2q arises from a quaternion algebraQ overQifQ splits overRand if there exists a Lie group epimorphismϕ fromQpRq¹to the conjugate of POp1,2qcontaining Γ such that Γ andϕpQpZq¹qare commensurable. In Section 5, we use the connection between quaternion algebras and ternary quadratic forms to describe the quaternion algebras from which the maximal nonelementary R-Fuchsian subgroups of the Picard modular groups ΓK

arise.

Theorem 1.2. — For every ∆ P OK ´ t0u, the maximal nonelementary R- Fuchsian subgroup ΓK,∆ of ΓK arises from the quaternion algebra with Hilbert symbol`2 Tr_K{Q∆, N_K{Qp∆q |DK|

Q

˘ifTrK{Q∆‰0and from`_1,₁

Q

˘»M2pQqotherwise.

This arithmetic description has the following geometric consequence. Recall that an R-circle is a topological circle which is the intersection of the Poincaré hypersphere

HS “ trzs PP2pCq : hpzq “0u

1See for instance [6, Chap. 5] and subsequent works of Falbel, Parker, Francsics, Lax, Xie, Wang, Jiang, Zhao and many others, for information on these groups, using different Hermitian forms of signature (2,1) defined overK.

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with a maximal totally real subspace of P2pCq. It isK-arithmetic if its stabiliser in ΓK has a dense orbit in it.

Corollary 1.3. — There are infinitely many ΓK-orbits of K-arithmetic R- circles in the hypersphereHS.

The figure below shows the image under vertical projection fromB8H²_C toCof part of the ΓQpiq-orbit of the standard infinite R-circle, which is Qpiq-arithmetic.

The image of each finite R-circle is a lemniscate. We refer to Section 3 and [5,

§4.4] for an explanation of the terminology. See the main body of the text for other pictures ofK-arithmeticR-circles.

2. The complex hyperbolic plane Lethbe the nondegenerate Hermitian form onC³ defined by

hpzq “z^˚I_1,2z“ ´Repz₀z₂q ` |z₁|², whereI1,2 is the antidiagonal matrix

I_1,2“

¨

˝

0 0 ´¹₂

0 1 0

´¹₂ 0 0

˛

‚.

A point z “ pz0, z1, z2q PC³ and the corresponding element rzs “ rz0 : z1 :z2s P P2pCq (using homogeneous coordinates) is negative, null or positiveaccording to whether hpzq ă 0, hpzq “ 0 orhpzq ą 0. Thenegative/null/positive cone of his the subset of negative/null/positive elements of P2pCq.

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The negative cone ofhendowed with the distanceddefined by cosh²dprzs,rwsq “ |xz, wy|²

hpzqhpwq,

where x¨,¨y is the sesquilinear form associated with h, is the complex hyperbolic plane H²_C. The distance d is the distance of a Riemannian metric with pinched negative sectional curvature ´4 ď K ď ´1. The null cone of his the Poincaré hypersphere H S, which is naturally identified with the boundary at infinity of H²_C.

The Hermitian form h in this paper differs slightly from the one we used in [15, 17, 16] and from the main Hermitian form used by Goldman and Parker (see [5, 13, 14]). Hence we will need to give some elementary computations that cannot be found in the literature. This form is a bit more appropriate for arithmetic purposes concerning R-Fuchsian subgroups, as it allows us to consider Z-points of our linear algebraic groups and not their 2Z-points.

Let Up1,2qbe the linear group of 3ˆ3 invertible matrices with complex coefficients preserving the Hermitian formh. Let PUp1,2q “Up1,2q{Up1qbe its associated projective group, where Up1q “ tζPC : |ζ| “1uacts by scalar multiplication.

We denote byrXs “ raijs1ďi,jďn PPUp1,2qthe image ofX “ paijq1ďi,jďn PUp1,2q. The linear action of Up1,2qonC³induces a projective action of PUp1,2qonP2pCq that preserves the negative, null and positive cones ofhin P2pCq, and is transitive on each of them.

If

X “

¨

˝

a γ b α A β c δ d

˛

‚PM3pCq, then I_1,2^´1X^˚I1,2“

¨

˚

˝

d ´2β b

´^δ₂ A ´^γ₂ c ´2α a

˛

‹

‚. The matrixXbelongs to Up1,2qif and only ifXis invertible with inverseI_1,2^´1X^˚I1,2, that is, if and only if

$

’’

&

’’

%

ad`bc´¹₂δγ“1 dα`cβ´¹₂Aδ“0 cd`dc´¹₂|δ|²“0 AA´2αβ´2βα“1 ab`ba´¹₂|γ|²“0 bα`aβ´¹₂Aγ“0.

(2.1)

Remark 2.1. — A matrix X P Up1,2q in the above form is upper triangular if and only if c “0. Indeed, then the third equality in Equation (2.1) implies that δ“0. The first two equations then becomead“1 anddα“0, so that α“0.

TheHeisenberg group

Heis3“ rw0:w: 1s PP2pCq: Rew0“ |w|²(

with lawrw0 :w: 1srw¹₀, w¹ : 1s “ rw0`w¹₀`2w¹w, w`w¹ : 1s is identified with CˆR by the coordinate mapping rw₀ : w : 1s ÞÑ pw,Imw₀q “ pζ, vq. It acts isometrically on H²_C and simply transitively on HS ´ tr1 : 0 : 0su byHeisenberg

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translations

tζ,v“

» –

1 2ζ |ζ|²`iv

0 1 ζ

0 0 1

fi

flPPUp2,1q

with ζ P Cand v P R. Note that t^´1_ζ,v “t´ζ,´v and tζ,v “t_ζ,´v. The Heisenberg dilationwith factorλPC^ˆ is the element

hλ“

» –

λ 0 0 0 1 0 0 0 ¹_λ

fi

flPPUp1,2q,

which normalizes the group of Heisenberg translations. The subgroup of PUp1,2q generated by Heisenberg translations and Heisenberg dilations is called the group ofHeisenberg similarities.

We end this subsection by defining the discrete subgroup of PUp1,2qwhoseR- Fuchsian subgroups we study in this paper.

Let K be an imaginary quadratic number field, withD_K its discriminant, OK

its ring of integers, Tr : z ÞÑ z`z its trace and N : z ÞÑ |z|² “ z z its norm.

Recall² that there exists a squarefree positive integer d such that K “ Qpi? dq, that DK “ ´dand OK “Zr^1`i

?d

2 s ifd ” ´1 mod 4, and that DK “ ´4d and OK “Zri?

ds otherwise. Note that OK is stable under conjugation, and that Tr and N take integral values on OK. A unitin OK is an invertible element inOK. Since N :K^ˆ ÑR^ˆ is a group morphism, we have Npxq “1 for every unit xin OK.

ThePicard modular group

ΓK“PUp1,2;OKq “PUp1,2q XPGL3pOKq is a nonuniform lattice in PUp1,2q.

3. The space of R-circles

A (maximal) totally real subspace V of the Hermitian vector space pC³, hq is the fixed point set of a Hermitian antiholomorphic linear involution of C³, or, equivalently, a 3-dimensional real linear subspace of C³ such that V and JV are orthogonal, where J : C³ Ñ C³ is the componentwise multiplication by i. The intersection with H²_C of the image under projectivization inP2pCqof a totally real subspace is called anR-plane in H²_C. The group PUp1,2qacts transitively on the set ofR-planes, the stabiliser of eachR-plane being a conjugate of POp1,2q. Note that POp1,2qis equal to its normaliser in PUp1,2q.

AnR-circleCis the boundary at infinity of anR-plane. See [12], [5, §4.4] and [7,

§9] for references onR-circles (introduced by E. Cartan). AnR-circle isinfiniteif it contains8 “ r1 : 0 : 0s andfiniteotherwise. The group of Heisenberg similarities acts transitively on the set of finite R-circles and on the set of infinite R-circles.

Thestandard infiniteR-circleis

C₈“ rx0:x1:x2s : x0, x1, x2PR, x²₁´x0x2“0( ,

2See for instance [18].

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which is the boundary at infinity of the intersection withH²_Cof the image inP2pCq ofR³ĂC³. For everyDPC^ˆ, the set

CD“ rz0:x1:D z0s : z0PC, x1PR, x²₁´RepDz₀²q “0(

is a finiteR-circle, which is the boundary at infinity of the intersection withH²_Cof the fixed point set of the projective Hermitian anti-holomorphic involution

rz0:z1:z2s ÞÑ rz2

D :z1:D z0s. We callC1 thestandard finite R-circle.

LetCbe a finiteR-circle. ThecentercenpCqofCis the image of8 “ r1 : 0 : 0s by the unique projective Hermitian anti-holomorphic involution fixing C. The radius radpCq of C is λ² where λ P C^ˆ is such that there exists a Heisenberg translation t mapping 0 “ r0 : 0 : 1s to the center of C with C “ t˝h_λpC₁q.

For instance, cenpCDq “ 0 and radpCDq “ ¹

D, since the Heisenberg dilations preserve 0 and C_D “ h?1

D

pC₁q. For every Heisenberg translation t, we have cenptCq “ tcenpCq and radptCq “ radpCq. For every Heisenberg dilation hλ, we have cenphλCq “hλcenpCqand radphλCq “λ²radpCq.

The image of a finite R-circle under the vertical projection pζ, vq ÞÑ ζ from Heis3“ B₈H²C´ t8utoCis a lemniscate, see [5, §4.4.5]. The figure below shows on the left six images of the standard infinite R-circle under transformations in ΓQpωq where ω “ ^´1`i

?3

2 is the usual third root of unity, and on the right their images inCunder the vertical projection.

Let us introduce more notation in order to describe the space of R-circles, see [5, §2.2.4] for more background. A 3ˆ3 matrix Y with complex coefficients is called unitary-symmetricif it is Hermitian with respect to the Hermitian form h and invertible with inverse equal to its complex conjugate, that is, ifY^˚I_1,2Y “I_1,2

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andY Y “I3, whereI3is the 3ˆ3 identity matrix. Note that for instanceI3and, for every DPC^ˆ, the matrix

YD“

¨

˝

0 0 _D¹ 0 1 0

D 0 0

˛

‚

is unitary-symmetric.

Let

AHI“ tY PM3pCq : Y^˚I1,2Y “I1,2 and Y Y “I3u

be the set of unitary-symmetric matrices, which is a closed subset of Up1,2q, identified with the set of Hermitian anti-holomorphic linear involutionszÞÑY z ofC³. Note that|detY| “1 for anyY PAHI. Let

PAHI“ trYs PPUp1,2q : Y Y “I3u

be the image of AHI in PUp1,2q, that is, the quotient Up1qzAHI of AHI modulo scalar multiplications by elements of Up1q. The group Up1,2qacts transitively on AHI by

pX, Yq ÞÑX Y X^´1

for allX PUp1,2qandY PAHI, and the stabiliser ofI3 is equal to Op1,2q. For every Y PAHI, we denote by PY the intersection with H²_C of the image in P2pCq of the set of fixed points of z ÞÑ Y z. Note that PY is an R-plane, which depends only on the class rYs of Y in PUp1,2q. We denote by C_Y “ B₈P_Y the R-circle at infinity ofP_Y, which depends only onrYs. For instance,C₈“C_I₃ and C_D“C_Y_D.

LetC_Rbe the set ofR-circles, endowed with the topology induced by the Haus- dorff distance between compact subsets of B8H²_C,³ and let P_R be the set of R- planes⁴ endowed with the topology of the Hausdorff convergence on compact subsets ofH²_C.

The projective action of PUp1,2qon the set of subsets ofP2pCqinduces contin- uous transitive actions onCR andPR, with stabilisers ofC₈ “CI₃ andPI₃ equal to POp1,2q. We hence have a sequence of PUp1,2q-equivariant homeomorphisms

PUp1,2q{POp1,2q ÝÑ PAHI ÝÑ PR ÝÑ CR

rXsPOp1,2q ÞÝÑ “ XX^´1‰

P ÞÝÑ B₈P . rYs ÞÝÑ PY

(3.1)

Lemma 3.1. — LetY “

¨

˝

a γ b α A β c δ d

˛

‚PAHI.

(1) For everyrXs PPUp1,2q, we haverXsCY “C_{XY X}´1. (2) TheR-circleCY is infinite if and only ifc“0.

(3) If theR-circleCY is finite, then its center is cenpCYq “ rYs 8 “ ra:α:cs,

3for any Riemannian distance on the smooth manifoldB8H²_C

4which are closed subsets ofH²_C

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and its radius is

radpC_Yq “A c´α δ c² “ ´ c

c² detY . In particular,ˇ

ˇradpCYqˇ

ˇ“ |c|^´1.

Proof. — (1) This follows from the equivariance of the homeomorphisms in Equa- tion (3.1).

(2) Recall thatC_Y is the intersection withB₈H²_C of the image in the projective plane of the set of fixed points of the Hermitian anti-holomorphic linear involution z ÞÑY z. Hence8 “ r1 : 0 : 0s belongs to C_Y if and only if the image of p1,0,0q byY is a multiple ofp1,0,0q, that is, if and only ifα“c“0. Using Remark 2.1, this proves the result.

(3) The first claim follows from the fact that the center of theR-circleCY is the image of 8 “ r1 : 0 : 0s under the projective map associated with z ÞÑ Y z. In order to prove the second claim, we start by the following lemma.

Lemma 3.2. — For everyrYs PPAHI, the center ofC_Y is equal to0“ r0 : 0 : 1s if and only if there existsDPC^ˆ such thatrYs “ rY_Ds.

Proof. — We have already seen that cenpCYDq “ cenpCDq “ 0. By the first claim of Lemma 3.1 (3), if cenpC_Yq “0, we have a“α“0. By the penultimate equality in Equation (2.1), we haveγ“0. SinceY Y “I3, we haveb c“1,b δ“0, b d“0 andβ c“0, so thatY “

¨

˝

0 0 ¹_c

0 A 0

c 0 0

˛

‚with|A| “1. SincerYs “ r_A¹Ys, the

result follows withD“_A^c.

Now, let ζ “ ^α_c, v “ Im^a_c and X “

¨

˝

1 2ζ |ζ|²`iv

0 1 ζ

0 0 1

˛

‚. Note that since Y PUp1,2q, we have

|α|²´Repa cq “hpa, α, cq “hpYp1,0,0qq “hp1,0,0q “0. Hence

Re`a c

˘“ 1

|c|²Repa cq “ˇ ˇ ˇ α c

ˇ ˇ ˇ

2

“ |ζ|².

The Heisenberg translationt_ζ,v“ rXsmaps 0“ r0 : 0 : 1stor^a_c : ^α_c : 1s “cenpC_Yq. Since

cenpC_X´1Y Xq “cenpt^´1_ζ,vCYq “t^´1_ζ,vcenpCYq “0,

and by Lemma 3.2, the elementX^´1Y X PAHI is anti-diagonal. A simple computation gives

X^´1Y X “

¨

˝

0 0 ¹_c

0 A´ζδ 0

c 0 0

˛

‚.

IfD“ ^c

A´ζδ, we hence haverX^´1Y Xs “ rYDs. Therefore

radpCYq “radpt^´1ζ,vCYq “radpC_X´1Y Xq “radpCY_Dq “ 1 D .

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Since detX “1, we have detY “ ´^c_cpA´ζδq, so that D“ ´_c _det^c²_Y. The result

follows.

We end this section by describing the algebraic properties of the objects in Equa- tion (3.1). We refer for instance to [23, §3.1] for an elementary introduction to algebraic groups and their Zariski topology.

LetGbe the linear algebraic group defined overQ, with set ofR-points PUp1,2q and set of Q-points

PUp1,2;Kq “PUp1,2q XPGL3pKq.

We identify G with its image under the adjoint representation for integral point purposes, so thatGpZq “ΓK.

SinceI1,2 has rational coefficients, the set PAHI of unitary-symmetric matrices modulo scalars is the set of real points PAHI “ PAHIpRq of an affine algebraic subsetPAHI defined overQofG, whose set of rational points is

PAHIpQq “PAHIXGpQq “PAHIXPGL3pKq.

The action of GonPAHI defined byprXs,rYsq ÞÑ rX Y X^´1s is algebraic defined overQ. This notion of rational point inPAHI will be a key tool in the next section in order to describe the maximal nonelementaryR-Fuchsian subgroups of ΓK.

4. A description of the R-Fuchsian subgroups of ΓK

Our first result relates the nonelementary R-Fuchsian subgroups of the Picard modular group ΓK to the rational points inPAHI. The proof of this statement is similar to the one of its analog forC-Fuchsian subgroups in [16].

Proposition 4.1. — The stabilisers inΓK of the rational points in PAHIare maximal nonelementary R-Fuchsian subgroups of ΓK. Conversely, any maximal nonelementaryR-Fuchsian subgroupΓofΓK fixes a unique rational point inPAHI andΓ is an arithmetic lattice in the conjugate ofPOp1,2qcontaining it.

Proof. — LetrYs PPAHIpQqbe a rational point in PAHI. Since the action of G on PAHI is algebraic defined over Q, the stabiliser H of rYs in G is algebraic defined over Q. Note that H is semi-simple with set of real points a conjugate of (the normaliser of POp1,2q in PUp1,2q, hence of) POp1,2q. Therefore by the Borel-Harish-Chandra theorem [2, Thm. 7.8], the group StabΓKrYs “ HpZq is an arithmetic lattice in HpRq, and in particular is a maximal nonelementary R- Fuchsian subgroup of ΓK.

Conversely, let Γ be a maximal nonelementaryR-Fuchsian subgroup of ΓK. Since it is nonelementary, its limit set ΛΓ contains at least three points. Two R-circles having three points in common are equal. Hence Γ preserves a uniqueR-planeP. Let Y PAHI be such thatP “PY. By the equivariance of the homeomorphisms in Equation (3.1),rYsis the unique point inPAHI fixed by Γ.

Let H be the stabiliser in G of rYs, which is a connected algebraic subgroup of G defined over R, whose set of real points is conjugated to POp1,2q. Since a nonelementary subgroup of a connected algebraic group whose set of real points is isomorphic to PSL2pRqis Zariski-dense in it, and since the Zariski-closure of a subgroup of GpZqis defined over Q(see for instance [23, Prop. 3.1.8]), we hence have that H is defined over Q. The action of the Q-group G on the Q-variety

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PAHI is defined over Q, and the Galois group GalpC|Qqacts on PAHI and on G commuting with this action. For everyσPGalpC|Qq, we haveH^σ“H. Hence by the uniqueness of the point inPAHI fixed by a conjugate of POp1,2q, we have that rYs^σ “ rYsfor everyσPGalpC|Qq. ThusrYsis a rational point.

An R-circle C is K-arithmetic if its stabiliser in ΓK has a dense orbit in C. Proposition 4.1 explains this terminology: The stabiliser in ΓKof aK-arithmeticR- circle is arithmetic (in the conjugate of POp1,2qcontaining it). Withω“ ^´1`i

?3 2 , the figure below shows part of the ΓQpωq-orbit of the standard infiniteR-circleC₈, which isK-arithmetic.

The next result reduces, up to commensurability and conjugacy in PUp1,2;Kq, the class of nonelementary R-Fuchsian subgroups that we will study. Note that PUp1,2;Kqis the commensurator of ΓK in PUp1,2q, see [1, Theo. 2].

Proposition 4.2. — Any maximal nonelementary R-Fuchsian subgroup Γ of ΓK is commensurable up to conjugacy inPUp1,2;Kqwith the stabiliser in ΓK of the rational point rY∆s PPAHI⁵ for some∆POK. If∆PN´ t0uand if

γ₀“

»

— –

1`i 2?

∆ 0 ₂^1´i^?_∆

0 1 0

p1´iq?

∆

2 0 ^p1`iq₂^?^∆ fi ffi fl,

thenγ0PPUp1,2qand we haveStabΓ_KrY∆s “γ0POp1,2qγ0^´1XΓK.

Proof. — Let Γ be as in the statement. By Proposition 4.1, there exists a rational point rYs P PAHIpQq in PAHI such that Γ “ StabΓKrYs “ StabΓKC_Y. Up to

5or equivalently to the stabiliser in ΓK of theR-circleC_∆

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conjugating Γ by an element in ΓK, we may assume that theR-circle CY is finite.

The center of the finite R-circle CY belongs toP2pKq X pB₈H²_C´ t8uqby Lemma 3.1 (3). The group of Heisenberg translations with coefficients in K acts (simply transitively) onP2pKq X pB₈H²_C´ t8uq. Hence up to conjugating Γ by an element in PUp1,2;Kq, we may assume that the center of theR-circleCY is 0“ r0 : 0 : 1s. By Lemma 3.2 (and its proof), there exists ∆ P K´ t0usuch that rYs “ rY_∆s. Since for everyλPC^ˆ we havehλrY_∆shλ

´1“ rY

∆λ^´2s, up to conjugating Γ by a Heisenberg dilation with coefficients inK, we may assume that ∆POK.

Fixing square roots of ∆ and ∆ such thata

∆“

?∆ , let

γ₀¹ “

»

— –

1`i 2

?

∆ 0 ^1´i

2

?

0 1 0∆

p1´iq?

∆

2 0 ^p1`iq₂^?^∆ fi ffi fl.

One easily checks using Equation (2.1) that γ₀¹ PPUp1,2q. An easy computation proves thatγ₀¹rI3sγ¹₀^´1“γ₀¹ γ₀¹^´1“ rY∆s. Since the stabiliser ofrI3sfor the action of PUp1,2qonPAHI is equal to POp1,2q, the fact that

StabΓ_KrY∆s “γ₀¹POp1,2qγ0¹

´1XΓK

follows from the equivariance properties of the homeomorphisms in Equation (3.1).

Furthermore,γ¹₀is the only element of PUp1,2qsatisfying this formula, up to right multiplication by an element of POp1,2q. The last claim of Proposition 4.2 follows

sinceγ₀“γ₀¹ when ∆PN´ t0u.

Here is a geometric interpretation of the invariant ∆ introduced in Proposition 4.2: Since radpCY_∆q “radpC∆q “ ¹

∆ for every ∆PC^ˆ, the above proof shows that if theR-circleC_Γpreserved by Γ is finite, then we may take ∆POK´t0usquarefree (uniquely defined modulo a square unit, hence uniquely defined if D_K ‰ ´4,´3) such that

∆P

´radpCΓq

¯´1

pK^ˆq².

5. Quaternion algebras, ternary quadratic forms and R-Fuchsian subgroups

In this section, we describe the arithmetic structure of the maximal nonelemen- taryR-Fuchsian subgroups of ΓK. By Proposition 4.2, it suffices to say from which quaternion algebra theR-Fuchsian subgroup

ΓK,∆“StabΓKrY_∆s arises for any ∆POK´ t0u.

Let D, D¹ P Q^ˆ. The quaternion algebra Q “

´D,D¹ Q

¯ is the 4-dimensional central simple algebra overQwith standard generatorsi, j, ksatisfying the relations i²“D,j²“D¹ andij“ ´ji“k. Ifx“x0`x1i`x2j`x3kis an element ofQ, we denote itsconjugateby

x“x₀´x₁i´x₂j´x₃k , its (reduced)trace by

trx“x`x“2x₀,

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and its(reduced) normby

npx₀`x₁i`x₂j`x₃kq “x x“x²₀´Dx²₁´D¹x²₂`DD¹x²₃.

The group of elements in QpZq “ Z`iZ`jZ`kZ with norm 1 is denoted by QpZq¹. We refer to [22] and [10] for generalities on quaternion algebras.

The quaternion algebra Q splits over R if the R-algebra QpRq “ Qb_Q R is isomorphic to the R-algebra M2pRq of 2-by-2 matrices with real entries. We say that a nonelementaryR-Fuchsian subgroup Γ of PUp1,2qarises fromthe quaternion algebraQ“`D,D¹

Q

˘ifQsplits overRand if there exists a Lie group epimorphism ϕ from QpRq¹ to the conjugate of POp1,2q containing Γ, with kernel the center ZpQpRq¹qofQpRq¹, such that Γ andϕpQpZq¹qare commensurable.

Let A_Q be the set of isomorphism classes of quaternion algebras over Q. For every APA_Q, we denote by

A₀“ txPA : trx“0u

the linear subspace ofAofpure quaternions, generated byi, j, k. LetTQbe the set of isometry classes of nondegenerate ternary quadratic forms overQwithdiscrimi- nant⁶a square. It is well known (see for instance [10, §2.3–2.4] and [22, §I.3]) that the map Φ from AQ to TQ, which associates toAPAQ therestricted norm form n|A₀, is a bijection. The map Φ has the following properties, for everyAPAQ.

(1) Ifa, bP Q^ˆ and A is (the isomorphism class of) `_a,b

Q

˘, then ΦpAq is (the equivalence class of)´a x²₁´b x²₂`ab x²₃, whose discriminant is pabq². (2) Ifa, b, cPQ^ˆwithabca square inQand ifqPTQ is (the equivalence class

of) ´a x²₁ ´b x²₂`c x²₃, then Φ^´1pqq is (the isomorphism class of) `_a,b

Q

˘, since if abc“ λ² with λP Q, then the change of variables px¹₁, x¹₂, x¹₃q “ px₁, x₂,_ab^λx₃qoverQturnsqto the equivalent form´a x²₁´b x²₂`ab x²₃. (3) The quaternion algebraAsplits overRif and only if ΦpAqis isotropic over

R(that is, if the real quadratic form ΦpAqis indefinite), see [22, Coro I.3.2].

(4) The map ΘAfromApRq^ˆto the special orthogonal group SOΦpAqof ΦpAq, sending the class of an elementainApRq^ˆto the linear mapa0ÞÑaa0a^´1 from A0 to itself, is a Lie group epimorphism with kernel the center of ApRq^ˆ (see [10, Th. 2.4.1]). IfApZq “Z`Zi`Zj`Zkis the usual order inA, then ΘA sendsApZq¹to a subgroup commensurable with SOΦpAqpZq.

Proof of Theorem 1.2. — The setP∆of fixed points of the linear Hermitian anti- holomorphic involutionzÞÑY∆zfromC³ toC³ is a real vector space of dimension 3, equal to

P_∆“ tzPC³ : z“Y_∆zu “ pz0, z1, z2q PC³ : z1“z1, z2“∆z0

(. LetV be the vector space overQsuch thatVpRq “C³andVpQq “K³. Since the coefficients of the equations defining P_∆ are in Q, there exists a vector subspace W “W_∆ofV overQsuch thatWpRq “P_∆. The restriction toW of the Hermitian form h, which is defined over Q, is a ternary quadratic form q“q_∆ defined over Q, that we now compute.

SinceK“Q`ia

|DK|Q, we write

∆“u`ia

|DK|v

6the determinant of the associated matrix

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withu, vPQ, and the variableszj “xj`ia

|DK|yjwithxj, yjPRforjP t0,1,2u. Ifpz0, z1, z2q PP∆, we have

hpz0, z1, z2q “ ´Repz2z0q ` |z1|²“ ´Rep∆z02q ` |z1|²

“ ´u x²₀`u|DK|y₀²´2|DK|v x0y0`x²₁.

The right hand side of this formula is a ternary quadratic formq“q∆onP∆, whose coefficients are indeed inQ. It is nondegenerate and has nonzero discriminant´w, where

w“v²D²_K`u²|DK| “Np∆q|DK| PQ´ t0u.

By equivariance of the homeomorphisms in Equation (3.1) and as StabPUp1,2qrY∆s is equal to its normaliser, the map from StabPUp1,2qrY∆sto the projective orthogonal group POq of the quadratic spacepP∆, qq, induced by the restriction map from StabUp1,2qP∆to Opqq, sendingg tog_|P_∆, is a Lie group isomorphism. It sends the lattice ΓK,∆ to a subgroup commensurable with the lattice POqpZqin POq. If we find a nondegenerate quadratic formq¹“q_∆¹ equivalent toqoverQup to a rational scalar multiple, whose discriminant is a rational square, and which is isotropic over R, then ΓK,∆ arises from the quaternion algebra Φ^´1pq¹q, by Properties (3) and (4) of the bijection Φ.

First assume that u“0. By an easy computation, we have q“ ´

´

´x²₁´|D_K|v

2 px₀´y₀q²`|D_K|v

2 px₀`y₀q²

¯ .

The quadratic formq¹“ ´X₁²´^|D^K₂^|vX₂²`^|D^K₂^|vX₃²overQis equivalent toqover Qup to sign. Its discriminant is the rational square p^|D^K₂^|vq², andq¹ represents 0 overR. By Property (2) of the bijection Φ, we have Φ^´1pq¹q “`1,^|DK|v₂

Q

˘“`_1,1

Q

˘. Therefore if u“0, then ΓK,∆arises from the trivial quaternion algebraM2pQq.

Now assume thatu‰0. By an easy computation, we have q“ ´1

u

`´u x²₁´ pv²D²_K`u²|DK|qy₀²` pu x0` |DK|v y0q²˘

“ ´ 1 u²w

`´u²w x²₁´uw²y₀²`uwpu x0` |DK|v y0q²˘ .

The quadratic formq¹ “ ´uw²X₁²´wu²X₂²`uw X₃²is equivalent toqoverQup to a scalar multiple inQ. Its discriminant is the rational squarepuwq⁴and it represents 0 overR. By Property (2) of the bijection Φ, we have Φ^´1pq¹q “`_uw²_{, wu}²

Q

˘“`_{u, w}

Q

˘. Therefore ifu‰0, sinceu“¹₂Tr ∆ andw“Np∆q|D_K|, then ΓK,∆arises from the quaternion algebra`2 Tr ∆, Np∆q|D_K|

Q

˘. This concludes the proof of Theorem 1.2.

Corollary 5.1. — Let∆,∆¹ POK´ t0uwith nonzero traces. The maximal nonelementary R-Fuchsian subgroups ΓK,∆ and ΓK,∆¹ are commensurable up to conjugacy in PUp1,2qif and only if the quaternion algebras`2 Tr ∆, Np∆q |D_K|

Q

˘and

`_{2 Tr ∆}¹_{, Np∆}¹_{q |D}_K_|

Q

˘over Qare isomorphic.

Proof. — Since the action of PUp1,2qon the set of R-planesPR is transitive, this follows from the fact that two arithmetic Fuchsian groups are commensurable up to conjugacy in PSL2pRqif and only if their associated quaternion algebras are

isomorphic (see [21]).

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To complement Theorem 1.2, we give a more explicit version of its proof in the special case when ∆PN´ t0u.

Proposition 5.2. — Let∆PN´t0u. The maximal nonelementaryR-Fuchsian subgroupΓK,∆ arises from the quaternion algebra`∆,|DK|

Q

˘.

Proof. — Let ∆ P N´ t0u. Let D “ ^|D₄^K^| if D_K ” 0 mod 4 and D “ |D_K| otherwise, so that OKXR“Z andOK XiR“i?

DZ. Let D¹ “D∆. We have D, D¹ PN´ t0u. LetQ“`_D,´D¹

Q

˘. The matrices

e0“

ˆ0 ´1

0 0

˙ , e1“

ˆ1 0 0 ´1

˙ , e2“

ˆ0 0 1 0

˙

form a basis of the Lie algebra sl₂pRq “

" ˆ

x₁ ´x₀ x₂ ´x₁

˙

: x₀, x₁, x₂ P R

* of PSL2pRq. Note that

´detpx0e₀`x₁e₁`x₂e₂q “ ´x₀x₂`x²₁

is the quadratic form restriction of h to R³ Ă C³. We thus have a well known exceptional isomorphism between PSL2pRq and the identity component SO0p1,2q of Op1,2q, which associates to g P PSL2pRq the matrix in the basispe0, e1, e2q of the linear automorphism Adpgq: X ÞÑ gXg^´1, which belongs to GLpsl2pRqq. We denote by Θ : PSL2pRq ÑPUp1,2qthe group isomorphism onto its image POp1,2q obtained by composing this exceptional isomorphism first with the inclusion of SO0p1,2qin Up1,2q, then with the canonical projection in PUp1,2q. Explicitly, we have by an easy computation

Θ :

„a b c d

 ÞÑ

» –

a² 2ab b² ac ad`bc bd c² 2cd d² fi fl . We have a mapσ_D,´D1 :QÑM2pRqdefined by

px0`x1i`x2j`x3kq ÞÑ

ˆ x0`x1

?D px2`x3

?Dq? D¹

´px2´x3

?Dq?

D¹ x0´x1

?D

˙ . As is well-known⁷, the induced map σ : QpRq¹ Ñ PSL2pRq is a Lie group epimorphism with kernel ZpQpRq¹q, such that σpQpZq¹q is a discrete subgroup of PSL2pRq. With γ0 as in Proposition 4.2, for all x0, x1, x2, x3 P Z, a computation gives that the elementγ0Θ`

σpx0`x1i`x2j`x3kq˘

γ₀^´1 of PUp1,2qis equal to

»

—

— –

apxq bpxq cpxq{∆

dpxq?

∆ npxq dpxq {?

∆ cpxq∆ bpxq apxq

fi ffi ffi fl ,

7See for instance [8].

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where

apxq “x²₀`Dx²₁` p2D¹x2x3qi? D , bpxq “2px1x2`x0x3` px1x3`x0x2

D qi? Dq

?DD¹

?∆ , cpxq “DD¹x²₃`D¹x²₂`2x0x1i?

D , dpxq “ px0x3´x1x2` px₀x₂

D ´x1x3qi? Dq?

DD¹. Let us consider the orderO ofQ defined by

O “ tx₀`x₁i`x₂j`x₃kPQpZq : x₁, x₂, x₃”0 modDu. Since ^?^?^DD_∆¹ “D PZ and ?

DD¹∆ “D¹ PZ, the above computation shows that the subgroupγ₀ΘpσpO¹qqγ^´1₀ of PUp1,2qis contained in ΓK. Since

´D,´D¹ Q

¯

“

´|D_K|,´|D_K|∆ Q

¯

“

´|D_K|,∆ Q

¯ ,

the result follows.

Remark. — Note that by Hilbert’s Theorem 90, if ∆¹ P K satisfies |∆¹| “ 1, then there exists ∆²POK´ t0usuch that ∆¹“ ^∆²

∆², so that the Heisenberg dilation h_∆2 ´1 commensurates ΓK,∆¹ to ΓK, Np∆²q and Np∆²qbelongs to N´ t0u. Hence Proposition 5.2 implies that ΓK,∆¹arises from the quaternion algebra`Np∆²q,|D_K|

Q

˘. We conclude this paper by a series of arithmetic and geometric consequences of the above determination of the quaternion algebras associated with the maximal nonelementaryR-Fuchsian subgroups of the Picard modular groups. Their proofs follow closely the arguments in [9] pages 309 and 310, and a reader not interested in the arithmetic details may simply admit that they follow by formally replacing

´dbydin the statements of loc. cit.

Recall that given aP Z´ t0uand p an odd positive prime not dividinga, the Legendre symbol `_a

p

˘ is equal to 1 if a is a square mod p and to ´1 otherwise.

Recall⁸ that ifdPZ´ t0uis squarefree, a positive primepis either

‚ ramifiedin Qp

?

dqwhenpdifpis odd, and whend”2,3r4sifp“2,

‚ split in Qp

?

dq whenpffl dand`_d

p

˘“1 ifpis odd, and whend”1r8sif p“2,

‚ inert inQp?

dqwhenpffld, and`_d

p

˘“ ´1 ifpis odd, and whend”5r8s ifp“2.

Recall that a quaternion algebra A over Q is determined up to isomorphism by the finite (with even cardinality) set RAMpAqof the positive primespat whichA ramifies, that is, such thatAb_QQp is a division algebra.

Proposition 5.3. — LetAbe an indefinite quaternion algebra overQ. If the positive primes at whichAis ramified are either ramified or inert inQpa

|DK|q, then there exists a maximal nonelementaryR-Fuchsian subgroup ofΓKwhose associated quaternion algebra isA.

8See for instance [18, page 91].

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Proof. — Recall⁹ that for all a, b P Z´ t0u and for all positive primes p, the (p-)Hilbert symbolpa, bqp, equal to´1 if`a, b

Qp

˘is a division algebra and 1 otherwise, is symmetric ina, b, and satisfies pa, bcqp“ pa, bqppa, cqpand

pa, bq_p“

#p´1q^u´1² ^v´1² ^`α^v²⁸^´1^`β^u²⁸^´1 ifp“2, a“2^αu, b“2^βv, withu, vodd

`_a

p

˘ ifp‰2, pffla, pb, p²fflb .

(5.1) Letd“^|D₄^K^| ifDK ”0r4sandd“ |DK|otherwise, so thatdPN´t0uis squarefree. GivenAas in the statement, we may write RAMpAq “ tp1, . . . , pr, r1, . . . , rsu withpiinert inQp?

dqandriramified inQp?

dq, so that the prime divisors ofdare r1,¨ ¨ ¨ , rs, s1,¨ ¨ ¨ , sk, unless someri, sayr1, is equal to 2 andd”3 r4s, in which case the prime divisors ofdarer2,¨ ¨ ¨, rs, s1,¨ ¨ ¨, sk. As in [9, page 310], letq be an odd prime different from allpi, ri, si such that

‚ q ” p₁¨ ¨ ¨p_r r8s if no r_i is equal to 2, q ” 5p₁¨ ¨ ¨p_r r8s if r_i “ 2 and d”2r4sandq”3p₁¨ ¨ ¨p_rr8sifr_i“2 andd”3r4s,

‚ for every i“1, . . . , s, ifr_i is odd, then`_q

r_i

˘“ ´`_p₁_¨¨¨_p_r

r_i

˘,

‚ for every i“1, . . . , k, ifsi is odd, then`q s_i

˘“`p₁¨¨¨p_r s_i

˘.

With ∆“p1¨ ¨ ¨prq, which is a positive squarefree integer, let us prove thatA is isomorphic to `_d,_∆

Q

˘. This proves the result by Proposition 5.2. By the charac- terisation of the quaternion algebras over Q, we only have to prove that for every positive primetnot in RAMpAq, we havepd,∆qt“1 and for every positive primet in RAMpAq, we havepd,∆qt“ ´1. We distinguish in the first case betweent“q, t“siandt‰q, s1,¨ ¨ ¨, sk, and in the second case betweent“piandt“ri. Using several times Equation (5.1), the result follows by elementary computations.

Recall that thewide commensurabilityclass of a subgroupH of a given groupG is the set of subgroups ofGwhich are commensurable up to conjugacy toH. Two groups areabstractly commensurableif they have isomorphic finite index subgroups.

Corollary 5.4. — Every Picard modular groupΓK contains infinitely many wide commensurability classes in PUp1,2q of (uniform) maximal nonelementary R-Fuchsian subgroups.

Corollary 1.3 of the introduction follows from Corollary 5.4. Note that there is only one wide commensurability class of nonuniform maximal nonelementary R-Fuchsian subgroups of ΓK, by [10, Thm. 8.2.7].

Proof. — As seen in Corollary 5.1, two maximal nonelementaryR-Fuchsian subgroups are commensurable up to conjugacy in PUp1,2qif and only if their associated quaternion algebras are isomorphic. Two such quaternion algebras are isomorphic if and only if they ramify over the same set of primes. By Proposition 5.3, for every finite setIwith even cardinality of positive primes which are inert overQpa

|DK|q, the quaternion algebra with ramification set equal toIis associated with a maximal nonelementary R-Fuchsian subgroup of ΓK. Since there are infinitely many inert primes overQpa

|DK|q, the result follows.

9See for instance [22], in particular pages 32 and 37, as well as [19, chap. III].