A classification of $\mathbb C$-Fuchsian subgroups of Picard modular groups

Given an imaginary quadratic extension $K$ of $\mathbb Q$, we give a classification of the maximal nonelementary subgroups of the Picard modular group $\operatorname{PSU}_{1,2}(\mathcal O_K)$ preserving a complex geodesic in the complex hyperbolic plane $\mathbb H^2_\mathbb C$. Complementing work of Holzapfel, Chinburg-Stover and M\"oller-Toledo, we show that these maximal $\mathbb C$-Fuchsian subgroups are arithmetic, arising from a quaternion algebra $\Big(\!\begin{array}{c} D\,,D_K\\\hline\mathbb QQ\end{array} \!\Big)$ for some explicit $D\in\mathbb N-\{0\}$ and $D_K$ the discriminant of $K$. We thus prove the existence of infinitely many orbits of $K$-arithmetic chains in the hypersphere of $\mathbb P_2(\mathbb C)$.


Introduction
Let h be a Hermitian form with signature (1, 2) on C 3 .The projective special unitary subgroup PSU h of h contains two conjugacy classes of Lie subgroups isomorphic to PSL 2 (R).The subgroups in one class preserve a complex projective line for the projective action of PSU h on the projective plane P 2 (C), and those of the other class preserve a totally real subspace.The groups PSL 2 (R) and PSU h act as the groups of orientation preserving isometries, respectively, on the upper halfplane model of the real hyperbolic space and on the projective model of the complex hyperbolic plane defined using the form h. If Γ is a discrete subgroup of PSU h , the intersections of Γ with the Lie subgroups isomorphic to PSL 2 (R) are its Fuchsian subgroups and the Fuchsian subgroups preserving a complex projective line are called C-Fuchsian subgroups.We refer to Section 2 for more precise definitions and comments on the terminology.
Let K be an imaginary quadratic number field, with discriminant D K and ring of integers O K .We consider the Hermitian form h defined by The Picard modular group Γ K = PSU h (O K ) is a nonuniform arithmetic lattice of PSU h , see for instance [Hol2,Chap. 5] and subsequent works of Falbel, Parker, Francsics, Lax, Xie-Wang-Jiang, and many others, for information on these groups.In this paper, we classify the maximal C-Fuchsian subgroups of Γ K , and we explicit their arithmetic structures.
When G = PSL 2 (C), there is exactly one conjugacy class of Lie subgroups of G isomorphic to PSL 2 (R).When Γ is the Bianchi group PSL 2 (O K ), the analogous classification is due to Maclachlan and Reid (see [Mac,MR1] and [MR2,Chap. 9]).They proved that the maximal nonelementary Fuchsian subgroups of PSL 2 (O K ) are commensurable up to conjugacy with the stabilisers of the circles |z|2 = D for D ∈ N − {0}, when PSL 2 (C) acts projectively (by homographies) on the projective line P 1 (C) = C ∪ {∞}, and that all these subgroups arise from explicit quaternion algebras over Q.For information on Bianchi groups, see for instance [Fin] and the references of [MR1].
More generally, given a semisimple connected real Lie group G with finite center and without compact factor, there is a nonempty finite set of infinite conjugacy classes of Lie subgroups of G locally isomorphic to SL 2 (R), unless G itself is locally isomorphic to SL 2 (R).The structure of the set of these subgroups plays an important role for the classification of the linear representations of G, and for the classification of the groups G themselves, see for instance [Kna, Ser] among others.Given a discrete subgroup Γ of G, it is again interesting to study the Fuchsian subgroups of Γ, that is, the intersections of Γ with these Lie subgroups, to classify the maximal ones and to see, when Γ is arithmetic, if its maximal Fuchsian subgroups are also arithmetic (see Proposition 3.1 for a positive answer) with an explicit arithmetic structure.From now on, G = PSU h .
We first prove (see Proposition 3.2 and just after) that a nonelementary C-Fuchsian In Section 3, we prove the following classification result (see [MR1,Thm. 1] and [MR2,Thm. 9.6.2] in the Bianchi group case).
Theorem 1.1 Let D ∈ N − {0}.The set of Γ K -conjugacy classes of maximal nonelementary C-Fuchsian subgroups of Γ K with discriminant D is finite and nonzero.Every maximal nonelementary C-Fuchsian subgroup of Γ K with discriminant D is commensurable up to conjugacy in PSU h with Γ K,2D .
In the course of the proof of this result, we prove a criterion for when two groups Γ K,D for D ∈ N − {0} are commensurable up to conjugacy in PSU h .A further application of this condition shows that every maximal nonelementary C-Fuchsian subgroup of Γ K is commensurable up to conjugacy in PSU h with Γ K,D for a squarefree natural number D.
Recall (see for instance [Gol]) that a chain 2 is the intersection of the Poincaré hypersphere with a complex projective line (if nonempty and not a singleton).It is K-arithmetic if its stabiliser in Γ K has a dense orbit in it.
Corollary 1.2There are infinitely many Γ K -orbits of K-arithmetic chains in the hypersphere H S .
The figure below shows part of the image under vertical projection in the Heisenberg group of the orbit under Γ K of a K-arithmetic chain whose stabiliser has discriminant 10, We say that a subgroup of PSU h arises from a quaternion algebra A defined over Q if it is commensurable with σ(A(Z) 1 ) for some Q-algebra morphism σ : A → M 3 (C).In Section 4, we prove the following result (see [MR2,Thm. 9.6.3] in the Bianchi group case).
Theorem 1.3 Every nonelementary C-Fuchsian subgroup of Γ K of discriminant D is conjugate in PSU h to a subgroup of PSU h arising from the quaternion algebra D, D K Q .
The classification of the quaternion algebras over Q then allow to classify up to commensurability and conjugacy the maximal nonelementary C-Fuchsian subgroups of Γ K : two such groups, with discriminant D and D are commensurable up to conjugacy if and only if the quaternion algebras D, D K Q and D , D K Q are isomorphic.This holds for instance if and only if the quadratic forms D K x 2 +Dy 2 −DD K z 2 and D K x 2 +D y 2 −D D K z 2 are equivalent over Q.
As was mentioned to us by M. Stover after we posted a first version of this paper on ArXiv, the existence of a bijection between wide commensurability classes of C-Fuchsian subgroups of Γ K and isomorphism classes of quaternion algebras over Q unramified at infinity and ramified at all finite places which do not split in K/Q is a particular case of the 2011 unpublished preprint [CS] (see Theorem 2.2 in its version 3), which proves such a result for all arithmetic lattices of simple type in SU 2,1 .In particular, the existence of this bijection (and our Corollary 4.3) should be attributed to Chinburg-Stover (although they say it was known by experts).Möller-Toledo in [MT] (a reference we were also not aware of for the first draft of this paper) also give a description of the quotients by the maximal C-Fuchsian subgroups of the real hyperbolic planes they preserve, and more generally of all Shimura curves in Shimura surfaces of the first type.We believe that our precise correspondence brings interesting effective and geometric informations.
The negative cone of h endowed with the distance d defined by is the complex hyperbolic plane H 2 C .The distance d is the distance of a Riemannian metric with pinched negative sectional curvature −4 ≤ K ≤ −1.The linear action of the special unitary group of h on C 3 induces a projective action on P 2 (C) with kernel U 3 Id, where U 3 is the group of third roots of unity.This action preserves the negative, null and positive cones of h, and is transitive on each of them.The restriction to C .The null cone of h is the Poincaré hypersphere H S , which is naturally identified with the boundary at infinity of H 2 C .The Heisenberg group acts isometrically on H 2 C and simply transitively on H S − {[1 : 0 : 0]} by the action induced by the matrix representation The projective transformations induced by these matrices are called Heisenberg translations.
If a complex projective line meets H 2 C , its intersection with H 2 C is a totally geodesic submanifold of H 2 C , called a complex geodesic.The intersection of a complex projective line in P 2 (C) with the Poincaré hypersphere is called a chain, if nonempty and not reduced to a point.Each complex projective line L in P 2 (C) meeting H 2 C (or its associated complex geodesic L ∩ H 2 C , or its associated chain L ∩ H S ) is polar to a unique positive point ).This element P L is the polar point of the projective line L, of the complex geodesic L ∩ H 2 C and of the chain L ∩ H S .Conversely, for each positive point P , there is a unique complex projective line P ⊥ polar to P , the polar line of P .The intersection of P ⊥ with H 2 C is a complex geodesic.
An easy computation (using for instance Equation ( 42) in [PP1]) shows that In particular, , then by Equation ( 1), its stabiliser in PSU h is the direct product of a Lie group embedding of PSL 2 (R) in PSU h preserving the complex geodesic polar to P , with the group of complex reflections with fixed point set the projective line polar to P .The polar chain of P is that is C P ∩ Heis 3 is the set of [w 0 : w : 1] ∈ Heis 3 satisfying the equation When z 2 = 0, in the coordinates (w, 2 Im w 0 ) ∈ C × R of [w 0 : w : 1] ∈ Heis 3 , this is the equation of an ellipse, whose image under the vertical projection [w 0 : w : 1] → w is the circle with center z 1 z 2 and radius in C given by the equation If z 2 = 0, then C P ∩ Heis 3 is the vertical affine line over z 1 z 2 .We refer to Goldman [Gol,p. 67] and Parker [Par] for the basic properties of H 2 C .These references use different Hermitian forms of signature (1, 2) to define the complex hyperbolic plane, and the curvature is often normalised differently from our definitions.Our choices are consistent with [PP1] and [PP2].
3 Classification of C-Fuchsian subgroups of Γ K Before starting to study Fuchsian subgroups of discrete subgroups of PSU h , let us mention that it is a very general fact that the maximal nonelementary (that is, not virtually cyclic) Fuchsian subgroups of arithmetic subgroups of PSU h are automatically (arithmetic) lattices of the copy of PSL 2 (R) containing them.
Proposition 3.1 Let G be a semisimple connected real Lie group with finite center and without compact factor, and let Γ be a maximal nonelementary Fuchsian subgroup of an arithmetic subgroup Γ of G. Then Γ is an arithmetic lattice in the copy of the group locally isomorphic to SL 2 (R) containing it.
One of the main points of what follows will be to determine explicitly the arithmetic structure of Γ, that is the Q-structure thus constructed on the group locally isomorphic to SL 2 (R) containing it, relating it to the arithmetic structure of Γ, that is the given Q-structure on G.
Proof.Let G be a semisimple connected algebraic group defined over Q, let H be an algebraic subgroup of G defined over R locally isomorphic to SL 2 , and assume that Γ = H(R) ∩ G(Z) is nonelementary in H(R).As a nonelementary subgroup of a group locally isomorphic to SL 2 is Zariski-dense in it, and as the Zariski closure of a subgroup of G(Z) is defined over Q, we hence have that H is defined over Q. Therefore by the Borel-Harish-Chandra theorem [BHC,Thm. 7.8], Γ = H(Z) is an arithmetic lattice in H(R).Since the copies of subgroups of G locally isomorphic to SL 2 (R) are algebraic, the result follows.
Let K be an imaginary quadratic number field, with D K its discriminant, O K its ring of integers, tr : z → z + z its trace and N : z → |z| 2 its norm.The Picard modular group of K, that we denote by Γ K = PSU h (O K ), consists of the images in PSU h of matrices of SU h with coefficients in O K .It is a nonuniform arithmetic lattice by the result of Borel and Harish-Chandra cited above.
A discrete subgroup Γ of PSU h is an extended C-Fuchsian subgroup if it satisfies one of the following equivalent conditions (1) Γ preserves a complex projective line of P 2 (C) meeting H 2 C , (2) Γ fixes a positive point in P 2 (C), (3) Γ preserves a chain.Many references, see for example [FaP1], do not use the word "extended".But as defined in the introduction, in this paper, a C-Fuchsian subgroup is a discrete subgroup of PSU h preserving a complex geodesic in H 2 C and inducing the parallel transport on its unit normal bundle.It is the image of a Fuchsian group (that is, a discrete subgroup of PSL 2 (R)) by a Lie group embedding of PSL 2 (R) in PSU h .The extended C-Fuchsian subgroups are then finite extensions of C-Fuchsian subgroups by finite groups of complex reflections fixing the projective line or positive point or chain in the definition above.In particular, up to commensurability, the notions of extended C-Fuchsian subgroups and of C-Fuchsian subgroups coincide.The C-Fuchsian lattices have been studied under a different viewpoint than our differential geometric one, as fundamental groups of arithmetic curves on ball quotient surfaces or Shimura curves in Shimura surfaces, by many authors, see for instance [Kud,Hol1,Hol2,MT] and their references.
An element of Γ K is K-irreducible if it does not preserve a point or a line defined over K in P 2 (C).An element of P 2 (C) is rational if it lies in P 2 (K).Note that the polar line of a positive rational point of P 2 (C) is defined over K.The group PSU h (K), image of SU h (K) = SU h ∩ SL 3 (K) in PSU h , preserves P 2 (K), but in general acts transitively on neither the positive, the null nor the negative points of P 2 (K).
The Galois group Gal(C|K) acts on P 2 (C) by σ[z 0 : z 1 : z 2 ] = [σz 0 : σz 1 : σz 2 ], and fixes P 2 (K) pointwise.A positive point z ∈ P 2 (C) is 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 Galois conjugates z , z over K are null elements in the polar line of z.
Proposition 3.2 A nonelementary extended C-Fuchsian subgroup Γ of Γ K fixes a unique rational point in P 2 (C).This point is positive and it is the polar point of the unique complex geodesic preserved by Γ.
C be its repelling and attracting fixed points, and let α 0 be its positive fixed point.Since the two projective lines tangent to the hypersphere H S at α − and α + are invariant under α, their unique intersection point is fixed by Γ, therefore is equal to α 0 .In particular, α 0 is polar to the complex projective line through α − , α + (see also [Par,Lemma 6.6] for a more analytic proof).
Let L be the complex projective line preserved by Γ, which meets H 2 C .As Γ is not elementary, there are loxodromic elements α, β ∈ Γ such that their sets of fixed points in ∂ ∞ H 2 C ∩ L are disjoint.Since L passes through α − , α + as well as through β − , β + , and by the uniqueness of the polar point to L, we hence have α 0 = β 0 .
As α and β have infinite order, one of them cannot be K-irreducible.Otherwise, if both were K-irreducible, then by [PP2,Prop. 18], the point α 0 = β 0 would be Hermitian cubic and its orbit under Gal(C|K) would be {α − , α + , α 0 } = {β − , β + , β 0 }, a contradiction.Assume then for instance that α preserves a line or a point defined over K.As any projective subspace preserved by α is a combination of α − , α + and α 0 , and as α − and α + are not defined over K, it follows that α 0 is rational.
Let Γ be a nonelementary extended C-Fuchsian subgroup of Γ K .By the previous proposition, Γ fixes a unique rational point P Γ in P 2 (C), which may be written P Γ = [z 0 : z 1 : z 2 ] with z 0 , z 1 , z 2 ∈ O K relatively prime.Such a writing is unique up to the simultaneous multiplication of z 0 , z 1 , z 2 by a unit in O K .Since the units in O K have norm 1, and since the trace and norm of K take integral values on the integers of K, the number is well defined, we call it the discriminant of Γ.As P is positive, we have ∆ Γ ∈ N−{0}.The radius of the vertical projection of the polar chain of P Γ is hence . The discriminant of Γ depends only on the conjugacy class of Γ in Γ K : for every γ ∈ Γ K , since by uniqueness we have P γΓγ −1 = γP Γ , we have A chain C is (K-)arithmetic if its stabiliser in Γ K has a dense orbit in C. The following result along with Proposition 3.2 justifies this terminology.This result is well known, and it is the other direction of [MT,Lem. 1.2], see also [Hol1,Prop. 1.5,§III.1]and [Kud,§3].We give a proof, which is a bit different, for the sake of completeness.
Proposition 3.3 The stabiliser Stab Γ K P of any positive rational point P ∈ P 2 (K) is a maximal nonelementary extended C-Fuchsian subgroup of Γ K , whose invariant chain is arithmetic.
Proof.Let G be the linear algebraic group defined over Q, such that G(Z) = PSU h (O K ) and G(R) = PSU h .We endow P 2 (C) with the Q-structure X whose Q-points are P 2 (K) so that the action of G on X is defined over Q.
As seen in Section 2, the set of real points of Stab G P is isomorphic to S 1 × PSL 2 (R) as a real Lie group.The group Stab G P is reductive and it has a (semisimple) Levi subgroup H defined over Q, such that H(R) is isomorphic to PSL 2 (R).By a theorem of Borel-Harish-Chandra [BHC], the group H(Z) is an arithmetic lattice in H(R), which (preserves the projective line polar to P and) is contained in Stab Γ K P .As H(Z) is a lattice in H(R), the group Stab Γ K P is nonelementary and has a dense orbit in the chain P ⊥ ∩ H S .
Recall that in the coordinates (w, −2 Im w 0 ) of Heis 3 , the chains are ellipses whose images under the vertical projection are Euclidean circles (see also [Gol,§4.3]).The figure in the introduction is the vertical projection of part of the orbit under Γ K of the chain [−5 : 0 : 1] ⊥ ∩ H S when K = Q[i], so that Γ K is the Gauss-Picard modular group, whose generators have been given by [FFP].The figure shows the square | Re z|, | Im z| ≤ 1.5 in C with projections of chains whose diameter is at least 1.In the figures below, where ω is a primitive third root of unity, so that Γ K is the Eisenstein-Picard modular group, whose generators have been given by [FaP2] The second part of Theorem 1.1 concerns the classification up to commensurability and conjugacy.Given a group G and a subgroup H of G, recall that two subgroups Γ, Γ of H are commensurable if Γ ∩ Γ has finite index in Γ and in Γ , and are commensurable up to conjugacy in G (or commensurable in the wide sense) if there exists g ∈ G such that Γ and gΓg −1 are commensurable.For any positive natural number D, let The group Γ K,D is, by Proposition 3.3, a maximal nonelementary extended C-Fuchsian subgroup, which preserves the projective line [−D : 0 : 1] ⊥ .Its discriminant is 2D.We will prove that every element of F C with discriminant D is commensurable up to conjugacy in PSU h with Γ K,2D .
Proof of Theorem 1.1.(1) Let D ∈ N − {0} and let By Proposition 3.3, the stabiliser in Γ K of the positive rational point P D is a maximal nonelementary extended C-Fuchsian subgroup of Γ K , with discriminant D. Hence F C (D) is nonempty.
Let G be the semisimple connected linear algebraic group defined over Q such that is the linear action of SU h on C 3 .Let X D be the closed algebraic submanifold of V with equation h = D.In particular, X D is defined over Q, and X D (R) is homogeneous under G(R) = SU h , by Witt's theorem.The map , is well defined by Proposition 3.3 and G(Z)-equivariant, and its image contains F C (D). Hence the finiteness of Γ K \ F C (D) follows from the finiteness of the number of orbits of G(Z) on X D (Z), see [BHC,Thm. 6.9].
(2) Let Γ ∈ F C , and let D ∈ N − {0} be its discriminant.By Propositions 3.2 and 3.3, and by maximality, there is a unique positive rational point P = [z 0 : Assuming this claim for the moment, we conclude the proof of the second part of Theorem 1.1:The groups γΓγ −1 and Γ K,2D are commensurable, since by a standard argument of reduction to a common denominator.
The following result, useful for the proof of the above claim, also gives a natural condition for when two such groups Γ K,D for D ∈ N − {0} are commensurable up to conjugacy in PSU h .A necessary and sufficient condition will be given as a consequence of Section 4. Assume first that D K ≡ 0 mod 4, so that It is easy to check using Equation (1) and since Z, the same argument works when γ in the above proof is replaced by the matrix and the equation N = x 2 + xy + 1−D K 4 y 2 with x, y ∈ Z.
Proof of the claim.As the lattice Γ K does not preserve the complex geodesic with equation z 2 = 0, we may assume that z 2 is nonzero, up to replacing P by an element in its orbit under Γ K , which does not change the discriminant D of Γ.Let γ 1 be the Heisenberg translation by the element which belongs to PSU h (K).An easy computation shows that Let γ 2 be the image in PSU h (K) of the diagonal element Q is the 4-dimensional central simple algebra over Q with standard generators i, j, k satisfying the relations i 2 = a, j 2 = b and ij = −ji = k.The (reduced) norm of an element of A is The group of elements in A(Z) = Z + iZ + jZ + kZ with norm 1 is denoted by A(Z) 1 .We refer to [Vig] and [MR2] for generalities on quaternion algebras.
Proof of Theorem 1.3.By Theorem 1.1, we only have to prove that the maximal C-Fuchsian subgroup F D of Γ K stabilising [−2D : 0 : 1] (which has finite index in Γ K,2D ) arises from the quaternion algebra D, D K Q .It is easy to check that the element A straightforward computation gives This matrix has coefficients in O K if and only if (2) and c, b ∈ iR is equivalent to s , t ∈ iR.Hence u, v ∈ Z and there exists s, t ∈ Z such that s = s D K , t = t D K .Therefore .
The following corollaries follow from the arguments in [Mac], pages 309 and 310.Corollary 1.2 of the introduction follows from Corollary 4.3 below.Proposition 4.2 Let A be an indefinite quaternion algebra over Q.There exists an arithmetic C-Fuchsian subgroup of Γ K whose associated quaternion algebra is A if and only if the primes at which A is ramified are either ramified or inert in K. Corollary 4.4 Any arithmetic Fuchsian group whose associated quaternion algebra is defined over Q is contained (up to commensurability) as a C-Fuchsian subgroup of some Picard modular group Γ K .
Corollary 4.5 For all quadratic irrational number fields K and K , there are infinitely many commensurability classes of arithmetic Fuchsian subgroups with representatives in both Picard modular groups Γ K and Γ K .
. The first figure shows part of the orbit of [−1 : 0 : 1] ⊥ ∩ H S and the second figure shows part of the orbit of [−2 : 0 : 1] ⊥ ∩ H S .They both show the square | Re z|, | Im z| ≤ 1 in C with projections of chains whose diameter is at least 0.5 in the first figure and at least 0.75 in the second.The first part of Theorem 1.1 in the introduction concerns the classification up to conjugacy of the maximal nonelementary C-Fuchsian subgroups of Γ K .Consider the set F C of maximal nonelementary C-Fuchsian subgroups of Γ K , on which the group Γ K acts by conjugation.We will prove that the discriminant map Γ → ∆ Γ on F C induces a finite-to-one map from Γ K \ F C onto N − {0}.

Lemma 4. 1
The map σ = σ a,b : A → M 3 (C) defined by to SU h and maps [0 : 1 : 0] to [−2D : 0 : 1].Hence, using Equation (1), a matrix M ∈ SU h (O K ) has its image (by the canonical projection SU h → PSU h ) in F D if and only if there exists a, d ∈ R and b, c ∈ iR with ad − bc = 1 such that M = γ 0 and D K differ by a square, the result follows.