On arithmetic quotients of the group SL 2 over a quaternion division k -algebra

: Given a totally real algebraic number field k of degree s , we consider locally symmetric spaces X G / Γ associated with arithmetic subgroups Γ of the special linear algebraic k -group G = SL M 2 ( D ) , attached to a quaternion division k -algebra D . The group G is k -simple, of k -rank one, and non-split over k . Using reduction theory, one can construct an open subset Y Γ ⊂ X G / Γ such that its closure Y Γ is a compact manifold with boundary ∂Y Γ , and the inclusion Y Γ → X G / Γ is a homotopy equivalence. The connected components Y [ P ] of the boundary ∂Y Γ are in one-to-one correspondence with the finite set of Γ-conjugacy classes of minimal parabolic k -subgroups of G . We show that each boundary component carries the natural structure of a torus bundle. Firstly, if the quaternion division k -algebra D is totally definite, that is, D ramifies at all archimedean places of k , we prove that the basis of this bundle is homeomorphic to the torus T s − 1 of dimension s − 1, has the compact fibre T 4 s , and its structure group is SL 4 s (ℤ) . We determine the cohomology of Y [ P ] . Secondly, if the quaternion division k -algebra D is indefinite, thus, there exists at least one archimedean place v ∈ V k , ∞ at which D v splits over ℝ , that is, D v ≅ M 2 (ℝ) , the fibre is homeomorphic to T 4 s , but the base space of the bundle is more complicated.


The arithmetic groups to be considered
Let A be a central simple algebra defined over an algebraic number field k with ring of integers O k .We may associate with a given a maximal O k -order Λ in A an affine O k -group scheme SL Λ of finite type.In this construction (see Section 3 for details), the reduced norm map nrd A/k plays a decisive role.This gives rise to an integral structure on the special linear algebraic k-group SL A = SL Λ × O k k obtained by extension of scalars.Our object of concern is the case A = M 2 (D), where D is a quaternion division k-algebra.Then the k-group G := SL M 2 (D) is a k-simple simply connected algebraic k-group of k-rank one which is non-split over k.Indeed, it is a k-form of the special linear k-group SL 4 .
We denote by s s (resp.s r ) the number of real places of k at which D splits (resp.ramifies), and s = s s + s r (resp.t) denotes the number of real (resp.complex) places of k.Then the real Lie group G ∞ of real points of the ℚ-group Res k/ℚ (G) obtained from G by restriction of scalars takes the form of the finite direct product where the product ranges over the set V k,∞ of all archimedean places of k, and G v denotes the real Lie group G σ v (k v ) obtained from G by extension of scalars from k to the completion k v of k at the place v ∈ V k,∞ via the corresponding embedding σ v : k → k v .The special linear group SL 2 (ℍ) over the non-commutative ℝ-algebra ℍ of Hamilton quaternions is usually denoted by SU * (4).This group is the real form of SL 4 (ℂ) associated with the complex conjugation σ : SL 4 (ℂ) → SL 4 (ℂ), defined by g  → η t 2 gη 2 , where η 2 = ( 0 E 2 −E 2 0 ), with E 2 the identity matrix of size two, and where g stands for conjugating each entry of the matrix g.
For each place v ∈ V k,∞ , let X v be the symmetric space associated with G v , described as , where d(G v ) = dim G v − dim K v , the space X v is contractible.We define as the product of the symmetric spaces X v , and we let d(G) = ∑ v∈V k,∞ d(G v ).Since the real Lie group G ∞ acts properly from the right on X G , a given arithmetic subgroup Γ of G(k), being viewed as a discrete, thus closed subgroup of G ∞ , acts properly on X G as well.If Γ is torsion-free, the action of Γ on X G is free, and the quotient X G /Γ is a smooth manifold of dimension d(G).In fact, there is a G ∞ -invariant Riemannian metric on X G , and the homogenous space X G /Γ carries the structure of a Riemannian manifold of finite volume.
Using reduction theory, one can construct an open subset Y Γ ⊂ X G /Γ such that its closure Y Γ is a compact manifold with boundary ∂Y Γ , and the inclusion Y Γ → X G /Γ is a homotopy equivalence.The connected components of the boundary ∂Y Γ are in one-to-one correspondence with the finite set, to be denoted P/Γ, of Γ-conjugacy classes of minimal parabolic k-subgroups of G.If P is a representative for a class in P/Γ, we denote the corresponding connected component in ∂Y Γ by Y [P]  .Then we have as a disjoint union ∂Y Γ = ∐ [P]∈P/Γ Y [P]  .
Under the assumption that k is a totally real field of degree s = s s + s r , we are concerned with the geometric structure of the boundary components and their cohomology.We have to distinguish two cases: (1) The quaternion division k-algebra D is totally definite, that is, by the very definition, k is a totally real field, and D ramifies at all archimedean places v ∈ V k,∞ , thus, D v ≅ ℍ, and s = s r .Consequently, G ∞ = SU * (4) s , and the corresponding symmetric space is the product of s copies of hyperbolic 5-space.This is due to the fact that the symmetric space of type AII attached to the pair (SU * (4), Sp(2)) coincides with the symmetric space of type BDI attached to the pair (SO(5, 1) 0 , SO(5) × SO(1)) (cf.[7,Chapter X]).It is of dimension five and rank one.Therefore, for any v ∈ V k,∞ , the symmetric space X v can be identified with the hyperbolic 5-space.
(2) The quaternion division k-algebra D is indefinite, thus, there exists at least one archimedean place v ∈ V k,∞ at which D v splits over ℝ, that is, D v ≅ M 2 (ℝ).Therefore, G v ≅ SL 4 (ℝ), and X v is the corresponding symmetric space SO(4)\SL 4 (ℝ) of dimension nine.

Results
In general, given a boundary component Y [P]  , starting off from a Levi decomposition P = LN of P into a semidirect product of its unipotent radical N and a Levi subgroup L, the component Y [P]  admits the structure of a fibre bundle whose fibre is N ∞ /(N ∞ ∩ Γ), where N ∞ = Res k/ℚ (N)(ℝ), and whose base is the homogenous space Z L /Γ L , where Z L denotes the symmetric space attached to the subgroup L  [P] admits the structure of a fibre bundle equivalent to the fibre bundle

Proposition. Given a representative P for a Γ-conjugacy class of minimal parabolic k-subgroups of G, the boundary component Y
which is associated by the natural action of Γ L on the compact fibre N ∞ /(N ∞ ∩ Γ), induced by inner automorphisms, to the universal covering Z L → Z L /Γ L .
The action of the fundamental group Γ L on N ∞ /(N ∞ ∩ Γ) via inner automorphisms extends to an action on the cohomology H * (N ∞ /(N ∞ ∩ Γ), ℚ) of the fibre.By an algebraic version of van Est's theorem (see [22,Section 9.3]), we may replace H * (N ∞ /(N ∞ ∩ Γ), ℚ) by the cohomology H * (n, ℚ) of the Lie algebra of N ∞ .Therefore the cohomology of the fibre is independent of the arithmetic group.
This gives rise to a local coefficient system, to be denoted H * (n, ℂ), on the pathwise connected base space Z L /Γ L .By the general result [20,Theorem 2.7], the spectral sequence in cohomology attached to the structure of Y [P]  as a fibre bundle degenerates, and we have H * (Y [P]  , ℂ) ≅ H * (Z L /Γ L , H * (n, ℂ)).In the specific case of a totally definite quaternion division k-algebra, we show that the bundle structure of Y [P]  is the one of a torus bundle over a torus.This enables us to derive a more precise result regarding the cohomology of Y [P]  .Its proof uses a general constructive approach regarding torus bundles over tori and their cohomology.
Theorem.Let D be a totally definite quaternion division k-algebra over a (necessarily totally real) field k of degree s.Given a representative P for a Γ-conjugacy class of minimal parabolic k-subgroups of G, the base space of the fibration Y [P] → Z L /Γ L is the torus T s−1 = (ℝ/ℤ) s−1 , and the fibre is the torus T 4s .The cohomology of the Y [P] is given as where H * (N ∞ /(N ∞ ∩ Γ), ℂ) Γ L denotes the space of elements invariant under the action of Γ L .
In this case, the arithmetic group Γ L may be viewed as a subgroup of the group of units ) occurs only in degrees 1, s, 2s, 3s, and 4s.
The theorem is a generalisation of the analogous result in the classical case of the special linear group over an algebraic number field, stated in [6, Proposition 1.1].A proof is given in [23].
If D is an indefinite quaternion k-algebra, the base space of the fibration Y [P] → Z L /Γ L is no longer a torus.Depending on the number of places at which D splits one has to add a second component given as a compact arithmetic quotient covered by a product of copies of H × H, where H denotes the upper half plane.
This paper has to be viewed as the third in a sequence of treatments of the cohomology of the general linear group GL 2 (or SL 2 ) attached to a division algebra D over k (see [4], [21]).Our discussion of the geometric structure of the boundary components Y [P]  and their cohomology plays a role in the construction of cohomology classes at infinity in H * (X G /Γ, ℂ) by means of the theory of Eisenstein series (see [24]).These classes supplement the construction of square-integrable classes given in [21].

Notation and conventions
Let k be an algebraic number field, and let O k denote its ring of integers.The set of places of k will be denoted by V k , and V k,∞ (resp.V k,f ) refers to the subsets of archimedean (resp.non-archimedean) places of k.Given a place v ∈ V k , the completion of k with respect to v is denoted by Suppose the extension k/ℚ has degree m = [k : ℚ].Let Σ be the set of distinct embeddings σ i : k → ℂ, 1 ≤ i ≤ m.Among these embeddings some factor through k → ℝ.Let σ 1 , . . ., σ s denote these real embeddings k → ℝ.Given one of the remaining embeddings σ : k → ℂ, σ(k) ̸ ⊂ ℝ, to be called imaginary, there is the conjugate one σ : k → ℂ, defined by x  → σ(x), where z denotes the usual complex conjugation of the complex number z.Then the number of imaginary embeddings is an even number, which we denote by 2t.We number the m = s + 2t embeddings σ i : k → ℂ, i = 1, . . ., m, in such a way that, as above, σ i is real for 1 ≤ i ≤ s, and The set V ∞ of archimedean places of k is naturally identified with the set of embeddings {σ i } 1≤i≤s+t ⊂ Σ.We denote by σ v the embedding which corresponds to v ∈ V k,∞ .
Let  k (resp. k ) be the ring of adèles (resp.the group of idèles) of k.We denote by  k,∞ = ∏ v∈V k,∞ k v the archimedean component of the ring  k , and by  k,f the finite adèles of k.There is the usual decomposition of  k into the archimedean and the non-archimedean part In this section we briefly review some basic facts in the theory of central simple algebras defined over an algebraic number field k and their O k -orders.For details we refer to [18] and [27].

Quaternion algebras
Given a field F of characteristic zero, a quaternion algebra Q over F is a central simple F-algebra of degree two.Viewed as a vector space over F, Q has a basis e 0 , e 1 , e 2 , e 3 , where e 0 is the multiplicative identity element, subject to the relations e 2 1 = a, e 2 2 = b, and e 1 e 2 = −e 2 e 1 for some elements a, b ∈ F × . Although the quaternion algebra does not uniquely determine the elements a, b ∈ F × , we may also use the notation Q = Q(a, b|F).Note that a quaternion algebra is either isomorphic to the F-algebra M 2 (F) of (2 × 2)-matrices with entries in F or it is a division algebra.
Let Q be a quaternion algebra over an algebraic number field k.Given a place v ∈ V k , the local analogue A given quaternion k-algebra splits at all but a finite number of places, and the set Ram(Q) = {v ∈ V k | Q ramifies at v ∈ V k } has even cardinality.The isomorphism class of the algebra Q over k is determined by the ramification set Ram(Q).Furthermore, given a set of places S ⊂ V k \ {v ∈ V k | v complex place}, where S has even cardinality, there exists a unique up to isomorphism quaternion k-algebra with ramification set equal to S.

Central simple k-algebras
Let A be a central simple algebra of degree d over an algebraic number field k.Given a place v ∈ V k , there exist a positive number r v and a central division algebra Δ v over k v of degree d v ≥ 1 (uniquely determined up to isomorphism) so that and, by r we get that d is even in this case.
Let A be a finite-dimensional central simple k-algebra, and let ℓ be a splitting field for A. Thus, we may fix an ℓ-algebra isomorphism α : A ℓ := A ⊗ k ℓ → M n (ℓ) for some n.Given a ∈ A, we define its reduced characteristic polynomial by redχ a,A/k = χ α(a⊗1),M n (ℓ) , that is, via the characteristic polynomial of α(a ⊗ 1) ∈ M n (ℓ).Note that the right-hand side does not depend on the choice of the isomorphism α.Moreover, the definition of the reduced characteristic polynomial is independent of the splitting field ℓ, and the polynomial redχ a,A/k has coefficients in k.With a given choice of a splitting α : We call nrd A/k (a) := det(α(a ⊗ 1)) the reduced norm of a, and redtr A/k (a) := tr(α(a ⊗ 1) the reduced trace of a ∈ A. This gives rise to the reduced norm map nrd A/k : A → k, a  → nrd A/k (a), resp.the reduced trace map redtr A/k : A → k, a  → redtr(a).The reduced norm map is multiplicative, whereas the reduced trace map is k-linear.For any a ∈ A, we have the relation In an analogous way, given a place v ∈ V k , one constructs for the central simple This map is multiplicative, and it shares the properties of nrd A/k .With regard to its image we have nrd In order to understand under which local conditions a given x ∈ k is in the image of the reduced norm map, we define This forms a subgroup of k × .The Hasse-Schilling Theorem (see e.g.[18,Theorem 33.15]) states that an element

Totally definite quaternion algebras
We single out a specific class of central simple k-algebras which play the role of exceptions to the theory of simple algebras over number fields, in particular, with regards to orders (see [18,Section 34]).

Definition 2.1. A central simple k-algebra is called a totally definite quaternion algebra if A ramifies at every archimedean place, and if furthermore
Consequently, the k-algebra A has degree d = 2, and every archimedean place is a real place.
To have a family of examples at hand, we determine all quaternion algebras Q over the field ℚ of rational numbers which ramify exactly at a given prime p and the unique archimedean place, to be denoted ∞.A nonarchimedean place v ∈ V ℚ corresponds to a unique prime in ℤ, and ℚ v is the field ℚ p of p-adic numbers.Using the device of the Hilbert symbol and the related reciprocity law one derives the following (see, e.g., [27, 14.2]):

Proposition 2.2. Given a prime p the quaternion algebras
, where q is a prime such that q ≡ 3 mod 4 and q is not a quadratic residue mod p ramify exactly at the places {∞, p}.Each quaternion ℚ-algebra Q whose ramification set is {∞, p} is isomorphic to one of the quaternion algebras as listed.

Orders in a central simple k-algebra
Let A be a central simple k-algebra, and let Λ be an O k -order in A. By definition, Λ is a subring of A with 1 Λ = 1 A and such that Λ is a complete O k -lattice in A. Then Λ is a finitely generated projective O k -module.For each x ∈ Λ the reduced characteristic polynomial redχ x,A/k has coefficients in O k .In particular, given the reduced Given two maximal O k -orders Λ and Λ  in A, then we have locally Λ v = Λ  v for all but finitely many places v ∈ V k .Therefore we may attach to A the restricted direct product A  = ∏ v∈V k (A v : Λ v ), endowed with pointwise addition and multiplication.This ring, to be called the ring of adeles of A, is locally compact, inheriting the topology from the local components in the usual way.We denote by A ×  the locally compact group of invertible elements in A  .It is endowed with the topology induced by viewing A ×  as a subset of the topological product A  × A  via the injection (x)  → (x, x −1 ).The group A ×  can also be viewed as the restricted topological product for some m (see [18,Theorem 10.6]).Given two left Λ-ideals M, N in A, we say that M and N are isomorphic if M ≅ N as left Λ-modules.If Λ is a maximal O k -order in A, the set LF 1 (Λ) of isomorphism classes of left Λ-ideals in A is a finite set, and its cardinality is independent of the choice of Λ (see [18,Theorem 26.4]).Therefore, we may define h A := |LF 1 (Λ)| for any maximal O k -order Λ in A; it is called the class number of the central simple k-algebra A. By [3, Theorem 1], the set LF 1 (Λ) can be parametrised by the space of double cosets 3 The algebraic k-group schemes to be considered Given an algebraic number field k with ring of integers O k , we associate with a given maximal O k -order Λ in a central simple k-algebra A an affine O k -group scheme SL Λ of finite type.In this construction, the reduced norm map nrd A/k plays a decisive role.One obtains an integral structure on the special linear k-group where D is a central division k-algebra.In this case, the group We refer to [22,Section 8.3] for the construction in a more general context.

General constructions
Let k be a commutative ring with identity.Let Alg k be the category of commutative associative k-algebras with identity.An algebraic k-group is an affine k-group scheme G that is of finite type as an affine scheme over k.
For any R ∈ Alg k , we denote its R-rational points by G(R).If k is a field, we additionally assume the defining condition that G is smooth.
The additive k-group scheme is denoted by  a .The group of R-valued points is (R, +) for all R ∈ Alg k .It is an algebraic k-group.The multiplicative affine k-group scheme is defined by the functor  m with  m (R) = R × , the group of units of R, for all R ∈ Alg k .
Any finitely generated projective module M over k defines a k-group functor M a over k with M a (R) = (M ⊗ k R, +) for all k-algebras R. In fact, M a is an affine k-group scheme of finite type with coordinate ring k[M a ] = Sym(M ∨ ), the symmetric algebra over the dual k-module M ∨ . If the k-module M is endowed with an additional ring structure such that M is a k-algebra, then M a : Alg k → (Rings) is a scheme of rings.Now we assume that k is a field.Let A be a finite-dimensional associative k-algebra.By applying the construction above, we have the ring functor A a : Alg k → (Rings).In addition, we define the k-group functor GL , and, given u ∈ Hom(R, S), GL A (u) = Id ⊗ k u.The norm map n A/k : A → k gives rise to the k-scheme map, denoted by the same letter, is the endomorphism algebra over a finite-dimensional k-vector space, we obtain GL V := GL End k (V) the general linear group of V.In particular, if A = M n (k), we have GL n = GL A .
We have a similar construction in the following case: Let A be a central simple algebra of degree d over an algebraic number field k.Given its ring of integers O k , let Λ be an O k -order in A. Then Λ is a finitely generated projective O k -module.There is an O k -group functor GL Λ :

The case A = M 2 (D)
Next we suppose that A = M 2 (D), where D is a finite-dimensional central division k-algebra.The group of k-rational points of the connected reductive k-algebraic group GL A coincides with the group GL(2, D) of (2 × 2)-matrices with entries in D. The group Z(k) of k-rational points of the centre Z of GL A is given by the group of scalar diagonal matrices.We fix a maximal k-split torus S in GL A subject to We denote its group of rational characters by X * (S) k .Let Φ k = Φ(GL A , S) ⊂ X * (S) k be the set of roots of GL A with respect to S. A basis of Φ k is given by the non-trivial character α : S/k →  m /k, defined by the assignment ( λ 0 0 μ )  → λμ −1 .We denote by Q 0 the minimal parabolic k-subgroup of GL A which is determined by {α}.We have a Levi decomposition of Q 0 into the semi-direct product Q 0 = Z GL A (S)N 0 of its unipotent radical N 0 by the centraliser L Q 0 = Z GL A (S) of S. The group of k-rational points of the centraliser Z GL A (S) of S is given by We may and will identify Z GL A (S) with the algebraic k-group D × × D × .Recall that the k-rational points of the group The group SL A is a k-simple simply connected algebraic group of k-rank one.We fix the maximal k-split torus T of SL A , whose k-rational points are given by T(k) = SL A (k) ∩ S(k), hence, A basis for the set of roots for SL A with respect to T is given by the restriction of α on T, denoted by the same letter.The minimal parabolic k-subgroup which corresponds to α is denoted by P 0 with Levi decomposition then M 0 is the largest connected anisotropic subgroup of L 0 , the intersection M 0 (k) ∩ T(k) is finite, and L 0 = M 0 T. Therefore, we have P 0 = M 0 TN 0 .The k-rational points of L 0 = Z SL A (T) resp.M 0 are given by We call P 0 the standard minimal parabolic subgroup of G.
Moreover, if the order Λ D is maximal, then Λ is maximal as well (see [18,Theorem 21.6]).In the latter case, the associated affine O k -group scheme SL Λ of finite type is smooth The group viewed as the topological product of the groups For each place v ∈ V k,∞ , let X v be the symmetric space associated with G v , described as the space of maximal compact subgroups of G v .In fact, all of these are conjugate to one another, thus, we may write , where as the product of the symmetric spaces X v , and we let d(G) = ∑ v∈V k,∞ d(G v ).Since the real Lie group G ∞ acts properly from the right on X G , a given arithmetic subgroup Γ of G(k), being viewed as a discrete, thus closed subgroup of G ∞ , acts properly on X G as well.If Γ is torsion-free, the action of Γ on X G is free, and the quotient X G /Γ is a smooth manifold of dimension d(G).

The case of a quaternion division algebra
We now focus on a quaternion division k-algebra D and the algebraic k-group G = SL Λ × O k k attached to a maximal order Λ in the central simple k-algebra A = M 2 (D).The group G is a k-simple simply connected algebraic k-group of k-rank one.We denote by s s (resp.s r ) the number of real places of k at which D splits (resp.ramifies), and t denotes the number of complex places of k.Then the real Lie group G ∞ takes the form of the finite direct product

.2)
Remark 4.1.The special linear group SL n over the non-commutative algebra ℍ of Hamilton quaternions is usually denoted by SU * (2n).This group is the real form of SL 2n (ℂ) associated with the complex conjugation σ : SL 2n (ℂ) → SL 2n (ℂ), defined by g  → η t n gη n , where η n = ( 0 E n −E n 0 ), with E n the identity matrix of size n, and where g stands for conjugating each entry of the matrix g.Let ψ : ℂ 2n → ℂ 2n be the real linear transformation defined by the assignment (x 1 , . . ., x n , x n+1 , . . ., x 2n )  → (x n+1 , . . ., x 2n , −x 1 , . . ., x n ).The real Lie group SU * (2n) is also realised as {g ∈ SL 2n (ℂ) | gψ = ψg}.Its intersection with the maximal compact subgroup U(2n) of GL 2n (ℂ) is the group Sp(n) = {g ∈ SU * (2n) | gg t = g t g = 1} (see [7, X, Lemma 2.1]).The symmetric space There is a G ∞ -invariant Riemannian metric on X G .Given an arithmetic subgroup Γ of G(k), we are interested in the homogenous space X G /Γ.If Γ is torsion-free, the space X G /Γ carries the structure of a Riemannian manifold of finite volume.Since G ∞ is not compact and the k-group G is k-simple simply connected, the group G has the strong approximation property (see [9]).Therefore, G(k) is dense in the locally compact group G( k,f ), or, equivalently, Let p be a prime ideal in O k , and let v 0 ∈ V k,f be the corresponding non-archimedean place of k.Given a proper ideal a ⊂ O k let ν p (a) be the maximal exponent e such that p e divides the ideal a.Thus, we have Using the strong approximation property of the algebraic k-group G, we conclude that the continuous map which is equivariant under the action of G ∞ .
We single out the following case: If D is a totally definite quaternion divison k-algebra, then, by the very definition, k is a totally real field, and D ramifies at all archimedean places v ∈ V k,∞ , thus, s = s r and t = 0. Consequently, , and the corresponding symmetric space is the product of s copies of hyperbolic 5-space.This is due to the fact that the symmetric space X 2 of type AII attached to the pair (SU * (4), Sp(2)) coincides with the symmetric space of type BDI attached to the pair (SO(5, 1) 0 , SO(5) × SO( 1)).It is of dimension 5 and rank one.This originates from the exceptional isomorphism SU * (4) ∼  → Spin(5, 1) 0 of real Lie groups where the latter one is the connected component of the spin group attached to a real quadratic form of signature (5, 1) (cf.[7, Chapter X]).Therefore, X 2 can be identified with the hyperbolic 5-space, to be denoted H 5 .

Reduction theory -The boundary components
As Let P be a minimal parabolic k-subgroup of the algebraic k-group G. Let T be a maximal k-split torus of P, and let N be the unipotent radical of P. In fact, P is the semi-direct product Z G (T)N.Since G has k-rank one, the set Δ of simple k-roots of G with respect to the pair (P, T) and the order associated with N consists of a single element.
We consider the standard minimal parabolic k-subgroup P 0 = L 0 N 0 of G as defined in Section 3.2.Any k-character χ : L 0 →  m induces a homomorphism Given an archimedean place v ∈ V k,∞ , we denote by | ⋅ | v the absolute value on k v = ℝ if v is real resp.the square of the absolute value on k v = ℂ if v is complex.The norm homomorphism is defined by The compositum | ⋅ | ∘ χ can be canonically extended to a homomorphism We apply this construction to the character ρ : L 0 →  m , given by Moreover, since the image of the arithmetic group Γ under ρ is an arithmetic subgroup of  m (k), thus, contained in O × k , we have |ρ|(γ) = 1 for every γ ∈ P 0,∞ ∩ Γ.It follows that P 0,∞ ∩ Γ = P (1) 0,∞ ∩ Γ.Given any other minimal parabolic subgroup P of G, there is a g ∈ G(k) such that gP(k)g −1 = P 0 (k).Therefore, we can define P Y [P]  , and the component Y [P] is diffeomorphic to the double coset space (K ∩ P (1) ∞ /(P ∞ ∩ Γ), where K denotes a maximal compact subgroup of G ∞ .
Proof.This is a specific case, namely of k-rank one, of the general results in [5,Section 1.2].A different approach in the k-rank one case to this result is carried through in [1,Theorem 17.10].In fact, X G /Γ is identified with the interior of a compact manifold with boundary, that is, the boundary components as constructed are added "at infinity".
We are interested in the geometric structure of such a boundary component Y [P]  .The morphism P = LN → P/N gives rise to a surjective morphism p : ∞ under this projection is a maximal compact subgroup in L (1) ∞ .We write ∞ for the associated manifold of right cosets.The preimage of a point in ∞ .The group Γ L acts properly and freely on Z L , and the double coset space Z L /Γ L is a manifold with universal cover Z L .The projection p : it is a locally trivial fibration with fibre the compact manifold N ∞ /(N ∞ ∩ Γ).
Proposition 4.3.Given a representative P for a Γ-conjugacy class of minimal parabolic k-subgroups of G, the corresponding boundary component Y [P] , diffeomorphic to the double coset space ∞ /(P ∞ ∩ Γ), admits the structure of a fibre bundle which is equivalent to the fibre bundle This bundle is associated by the natural action of Γ L on the compact fibre N ∞ /(N ∞ ∩ Γ), induced by inner automorphisms, to the universal covering Z L → Z L /Γ L .
Proof.The action of the group ∞ is proper and free.Since P is the normaliser of N in G, the group N ∞ ∩ Γ is a normal subgroup in P ∞ N ∞ as a semi-direct product, induced by the semi-direct product P = LN, this space can be viewed as the product space We have that P is the normaliser of N, thus, the group P ( ∞ ∩ Γ acts via inner automorphisms on N ∞ .It follows, since N is commutative, that there is an induced action of the quotient group Γ P/N via diffeomorphisms on N ∞ /(N ∞ ∩ Γ).The group Γ P/N is isomorphic to Γ L .In view of (4.5), the fibration in question is equivalent to the fibre bundle which is associated by the natural action of Γ L on N ∞ /(N ∞ ∩ Γ), induced by inner automorphisms, to the universal covering Z L → Z L /Γ L .
Remark 4.4.In various contexts, for example in cohomological questions regarding X G /Γ, it is of interest to determine the number, say cs Λ , of SL Λ (O k )-conjugacy classes of minimal parabolic k-subgroups of G.If this quantity would be known, then one would be in the position to indicate the number of connected components in the boundary of the compactification of the space X G /Γ associated with a torsion-free arithmetic subgroup In the classical case of the special linear group SL 2 /k over an algebraic number field, given the unique maximal order O k , this number is equal to the class number h k of k (see [25,Proposition 20]).
In our case at hand, the situation is more complicated.The results obtained so far in determining cs Λ are strongly interwoven with the arithmetic of the division algebra D and its field of definition k.We refer for some background on arithmetic orders to [3].There one finds a detailed analysis of the relation between the set LF 1 (Λ) of left Λ-ideals in D, the class group Cl(Λ) of Λ, to be defined as the set of stable isomorphism classes of left Λ-ideals, and the ray class group Cl + D (k) attached to D. First, in the case of an indefinite quaternion division algebra, as shown in [13,Chapter 5] or, with a different proof, in [10, Theorem 4], we have cs Λ = h D .In particular, this quantity is independent of the choice of the maximal order Λ.
Second, in the case of a totally definite quaternion algebra over k, a simple closed formula for cs Λ does not yet exist.However, as proved in [10, Theorem 5], we have: If D is a totally definite quaternion algebra over ℚ, then cs Λ = h 2 D (see [11,Satz 2.1]).This type of result is more generally correct (see [10,Lemma 5]) in the case of a number field for which h + k = h k = 1, where h + k denotes the narrow class number of k (see Remark 5.3 below for definition).
Next, as an intriguing example, we consider the quaternion algebra Q(−1, −1|ℚ( √ 6)).The real quadratic field ℚ( √ 6) has class number h k = 1 whereas h + k = 2.With the help of Magma it is shown in [10,Section 5.4] that h D = 3, and cs Λ = 5, that is, cs Λ is not an integral multiple of h D .open subset Y Γ ⊂ X G /Γ such that its closure Y Γ is a compact manifold with boundary ∂Y Γ , and the connected components of the boundary ∂Y Γ carry the structure of a fibre bundle.We determine the geometric structure of the fibre and the base space of such a component Y [P]  , where P is a representative for a Γ-conjugacy class of minimal parabolic k-subgroups of G.

The fibre
The first result concerns the compact fibre N ∞ /(N ∞ ∩ Γ) of the fibration (4.3) of a boundary component in ∂Y Γ .We restrict our attention to the case of a totally real number field k of degree m.Lemma 5.1.Given a boundary component Y [P] which corresponds to the Γ-conjugacy class [P] of the minimal parabolic k-subgroup P = LN of G, the fibre N ∞ /(N ∞ ∩ Γ) of the locally trivial fibration (4.3) with total space Y [P]  is diffeomorphic to the torus T 4m .
Proof.We may assume that P is the standard minimal parabolic k-subgroup P 0 = L 0 N 0 whose group of k-points The group of k-points of its unipotent radical is commutative, and, since m = s s + s r we obtain as additive groups Note that the group N 0,∞ ∩ Γ as a discrete subgroup of N 0,∞ forms a complete lattice in ℝ 4m , therefore the claim follows.

Unit groups of O k -orders in D
In order to study the base space Z L /Γ L of the fibration (4.3), we need some insight in the structure of the unit group of an O k -order Λ D in a quaternion algebra D defined over an arbitrary algebraic number field k.Let μ k be the group of roots of unity in k.Recall that the group of units O × k of k is a finitely generated ℤ-module and Given an O k -order Λ D , the group of units Λ × D contains O × k as its centre, and the restriction of the reduced norm map nrd on Λ D = {±1}.Given the quaternion algebra D, we denote by θ ⊂ V k,∞ the set of all real places of k at which D ramifies.Then the set k + θ := {x ∈ k × | x v > 0 for all v ∈ θ} forms a subgroup of k × .We define Proof.Suppose D is a totally definite quaternion algebra.Then k is a totally real field, θ ⊂ Ram(D), and D v ≅ ℍ for each v ∈ θ.For any v ∈ θ, the group D (1) v ≅ ℍ (1)  is compact and contains Λ D as a discrete subgroup, therefore, Λ

The fibre bundle -The case of a totally definite quaternion algebra
Let Γ ⊂ SL Λ (O k ) be a torsion-free subgroup of finite index in the group of integral points of the O k -group scheme SL Λ attached to a maximal O k -order Λ in the central simple k-algebra A = M 2 (D), where D is a totally definite quaternion algebra defined over a totally real algebraic number field k of degree s = [K : ℚ].Given a representative P for a Γ-conjugacy class of minimal parabolic k-subgroups of G = SL Λ × O k k, our object of concern is the base space of the fibre bundle structure of the boundary component Y [P]  of ∂Y Γ .We will seek to understand its geometric structure.Taking Lemma 5.1 into account, we have: Theorem 5.4.We retain the previous notation and assumptions introduced above.The boundary component Y is the total space of a fibre bundle with fibre N ∞ /(N ∞ ∩ Γ)) ≅ T 4s , base space Z L /Γ L ≅ T s−1 , and structure group Γ L .Therefore it is a torus bundle over a torus.The structure group Γ L of the fibre bundle is a totally disconnected commutative group.
Proof.First, we deal with the universal cover Z ∞ of the base space Z L /Γ L of the fibration (4.3).We may assume that P = P 0 is the standard minimal parabolic k-subgroup of G. Since D is a totally definite quaternion k-algebra, we have the identification Passing over to the group L (1) 0,∞ , we obtain a diffeomorphism The mapping L 0,∞ = (ℍ × ) s → (ℝ × >0 ) s , defined by (h 1 , . . ., h s )  → (nrd(h 1 ), . . ., nrd(h s )), is a surjective homomorphism.It gives rise to a surjective homomorphism ψ (1)  : L (1) . For the following we may assume that P = P 0 is the standard parabolic k-subgroup.The k-rational points of its unipotent radical are given by N 0 (k) = {g = ( 0 x 0 0 )     x ∈ D}.
Moreover, upon identifying N 0 (k) with D, we see that N 0 (k) ∩ Γ = Δ is a complete O k -lattice in D. Passing over to the real points of the group Res k/ℚ (N 0 ), we obtain Therefore, the underlying structure as a vector space over ℝ, endowed with the Euclidean topology, is ℍ ≅ ℝ 4s .
The group N + 0,∞ ∩ Γ is a discrete subgroup of maximal rank in N + 0,∞ ≅ ℝ 4s .It follows that N + 0,∞ ∩ Γ is freely generated over ℤ by 4s vectors u 1 , . . ., u 4s which are linearly independent over ℝ.We fix such a basis u = {u 1 , . . ., u 4s } of ℝ 4s . With regard to the basis u, the action of N + 0,∞ ∩ Γ on N + 0,∞ ≅ ℝ 4s is the standard action of ℤ 4s on ℝ 4s .This allows us to describe the action of Γ L on N 0,∞ /(N 0,∞ ∩ Γ) in terms of matrices with integral entries.It is induced by the operation of Γ L on N 0,∞ via inner automorphisms.The group Γ L is commensurable with the group O × k , hence, viewed as a finitely generated ℤ-module of rank s − 1.Given a set {α 1 , . . ., α s−1 } of generators, each of them acts on N 0,∞ ≅ ℝ 4s with respect to the basis u by an integral matrix A i ∈ SL n (ℤ), i = 1, . . ., s − 1, since α i leaves N 0,∞ ∩ Γ invariant.Since Γ L is commutative, the matrices A i , i = 1, . . ., s − 1, commute with one another.Thus, following the construction (and notation) introduced in Section 6, we have to deal with a torus bundle T(A 1 , . . ., A s−1 ) with fibre T 4s and basis T s−1 , determined by the integral matrices A i , i = 1, . . ., s − 1.As a result we have .The proof of this result is by induction over the number of integral matrices A i , i = 1, . . ., s − 1.Indeed, in the case of a bundle (E, S 1  , F, π) over S 1 with characteristic homomorphism A, the Wang sequence (see [15,Lemma 8.4.]) gives rise to a short exact sequence 0 → coker(H q−1 (A) − Id) → H q (E, ℂ) → ker(H q (A) − Id) → 0.
This sequence splits and one gets a direct sum decomposition H q (E, ℂ) = ker(H q (A) − Id) ⊕ coker(H q−1 (A) − Id).This isomorphism is not canonical but depends on the choice of a basis.However, in the case at hand, the endomorphism H * (A) is semi-simple, thus there is a canonical identification coker(H q−1 (A) − Id) = ker(H q−1 (A) − Id).Taking into account that ker(H q−1 (A) − Id) ≅ ker(H q−1 (A) − Id) ⊗ H 1 (S 1 , ℂ) resp.ker(H q (A) − Id) ≅ ker(H q (A) − Id) ⊗ H 0 (S 1 , ℂ) together with the identity ker(H q (A) − Id) = H * (F) A brings the result in this case.The induction step deserves a careful analysis.It finally relies on the fact that the matrices A 1 , A 2 , . . ., A s−1 commute with one another and that each of the homomorphisms H * (A i ), i = 1, . . ., s − 1, are semi-simple endomorphisms over ℂ.

Remark 7.2.
There is a more classical analogue to this result: Given a torsion-free arithmetic subgroup Γ of the special linear group SL 2 /k over an algebraic number field of degree m, the corresponding locally symmetric space X SL 2 /Γ is homotopy equivalent to a compact manifold W Γ with boundary ∂W Γ .As discussed in [6, Section 1], its connected components are torus bundles whose base space B is T r and whose fibre F is T m , where r The connected components of the boundary ∂Y Γ = ∐ Y [P]  are in one-to-one correspondence to the finitely many Γ-conjugacy classes of minimal parabolic k-subgroups of G. Therefore we can replace H * (∂Y Γ , ℂ) by ⨁ [P]∈P/Γ H * (Y [P]  , ℂ). and apply Theorem 7.1.It is possible, using the theory of automorphic forms for congruence subgroups of deep enough level, to construct non-vanishing classes in the cohomology H * (Y Γ , ℂ) which restrict under r * trivially to H q (∂Y Γ , ℂ).This is achieved in [21], in combination with the transfer of automorphic representations under central isogenies in [12].In which way the theory of Eisenstein series is essential in constructing non-vanishing classes in H * (Y Γ , ℂ) at infinity will be dealt with in [24].This permits us to draw conclusions concerning the image of the restriction r * : H * (Y Γ , ℂ) → H q (∂Y Γ , ℂ).

∞
of L ∞ modulo the action of the image Γ L of P ∞ ∩ Γ under the natural projection (see Section 4.3 for details).
where n Λ/O k denotes the scheme-theoretic norm map.This functor defines an affine O k -group scheme of finite type which is smooth (see e.g.[22, Proposition 8.3.1])Thereduced norm map nrd A/k : A → k is a polynomial function on A. Given an O k -order Λ in A, its restriction on Λ takes values in O k .Therefore, the reduced norm map gives rise to a morphism nrd Λ/O k : GL Λ →  m /O k of O k -group schemes.The scheme-theoretic kernel, to be denoted SL Λ , of the morphism nrd Λ/O k of smooth algebraic O k -groups is an O k -group scheme of finite type.If Λ is a maximal O k -order in A, then the O k -group scheme SL Λ is smooth.The group SL Λ (O k ) of integral points of SL Λ is an arithmetic subgroup of the k-points of the algebraic k-group SL A .
. The algebraic k-group SL Λ × O k k obtained by base change is the group G := SL A .Occasionally we write Γ Λ := SL Λ (O k ) for the group of integral points of SL Λ .Given any proper ideal a ⊂ O k the corresponding principal congruence subgroup of level a is defined by Γ Λ (a) := ker(SL Λ (O k ) → SL Λ (O k /a)).(3.1)It gives rise to an arithmetic subgroup Γ Λ (a) of G(k).For almost all choices of the ideal a the group Γ Λ (a) is torsion-free.Given an algebraic number field k and a central division k-algebra D of degree d, the algebraic k-group G = SL Λ × O k k attached to a maximal order Λ in the central simple k-algebra A = M 2 (D) is a connected semisimple algebraic k-group.For every archimedean place v ∈ V k,∞ , together with the corresponding embedding σ v : k → k, there are given a field k v = ℝ or ℂ and a real Lie group X n := Sp(n)\SU * (2n), n ̸ = 1, attached to the Riemannian symmetric pair (SU * (2n), Sp(n)) of non-compact type is of type AII.It is a simply connected space of dimension (n − 1)(2n + 1) and of rank n − 1.
before we consider a quaternion division k-algebra D and the algebraic k-group G = SL Λ × O k k attached to a maximal order Λ in the central simple k-algebra A = M 2 (D).As an application of the main results in reduction theory (see [5, Theorems 1.2.2 and 1.2.3]), in the case of a torsion-free arithmetic subgroup Γ ⊂ G(k), one can construct an open subset Y Γ ⊂ X G /Γ such that its closure Y Γ is a compact manifold with boundary ∂Y Γ , and the inclusion Y Γ → X G /Γ is a homotopy equivalence.The connected components of the boundary ∂Y Γ are in one-to-one correspondence to the Γ-conjugacy classes of minimal parabolic k-subgroups of G.We are concerned with the geometric structure of the boundary components.Since the k-rank of G is one, all proper parabolic k-subgroups of G are minimal, all of these are conjugate under G(k).This conjugacy class falls into finitely many Γ-conjugacy classes (see[1, Proposition 15.6]).

Theorem 4 . 2 .
Given a torsion-free arithmetic subgroup Γ ⊂ G(k), there exists an open subset Y Γ ⊂ X G /Γ such that its closure Y Γ is a compact manifold with boundary ∂Y Γ , and the inclusion Y Γ → X G /Γ is a homotopy equivalence.The connected components of the boundary ∂Y Γ are in one-to-one correspondence with the finite set, to be denoted P/Γ, of Γ-conjugacy classes of minimal parabolic k-subgroups of G.If P is a representative for a Γ-conjugacy class of minimal parabolic k-subgroups of G, we denote the corresponding connected component in ∂Y Γ by Y[P] .Then we have as a disjoint union ∂Y Γ = ∐[P]∈P/Γ

Proposition 4 . 5 .
Let D be a definite quaternion algebra over a totally real algebraic number field k.Given two maximal orders Λ = M 2 (Λ D ) and Λ  = M 2 (Λ  D ), where Λ D and Λ  D are maximal orders in D, we have cs Λ = cs Λ  .
Given a quaternion division k-algebra D, any maximal O k -order in the central simple k-algebra A = M 2 (D) defines an algebraic O k -group scheme SL Λ .The associated algebraic k-group obtained by base change isG = SL Λ × O k k.Let Γ ⊂ SL Λ (O k) be a torsion-free subgroup of finite index.By Theorem 4.2, there exists an H n (T(A 1 , A 2 , . . ., A s−1 ), ℂ) = ⨁ q+r=n H q (T s−1 , ℂ) ⊗ H r (T 4s , ℂ) A * 1 ,A * 2 ,...,A * s−1 ,where H r (T 4s , ℂ) A * 1 ,A * 2 ,...,A * s−1 denotes the subspace of elements in H r (T 4s , ℂ) which are invariant under the endomorphisms A * 1 , A * 2 , . . ., A * s−1 of H * (T 4s , ℂ) induced by the action of the integral matrices A 1 , A 2 , . . ., A s−1 on T 4m D is finite.Since in the exact sequence (5.2) the term U + θ /O × k is finite.Conversely, suppose there exists a place w ∈ V k,∞ at which D splits, that is, D w ≅ M 2 (ℝ).Then we can find a finite field extension ℓ/k, where ℓ ⊂ ℝ such thatD ℓ ≅ M 2 (ℓ).If M ℓ denotes the algebraic closure of O k in ℓ, then the finitely generated ℤ-module M × ℓ /O × k containsan element, say μ, of infinite order.By comparing the O k -lattices M ℓ and M ℓ ∩ Λ D we see that M × ℓ /(M ℓ ∩ Λ D ) × has finite order.Therefore, a suitable power μ  ∈ M × ℓ of μ lies in (M ℓ ∩ Λ D ) × .This implies μ  ∈ Λ × Endowed with the multiplication of ideals, the set I O k of non-zero fractional ideals in the ring Its image is the group of fractional principal ideals, to be denoted P O k .The ideal class group of k is defined to be Cl k = O k /P O k .The ideal class group of k is finite.We call its order h k the class number of k.An element x ∈ k is said to be totally positive if x v ∈ ℝ is positive for all real archimedean places in V k,∞ .The totally positive elements form a subgroup k + of k × , and we write U + k := O × k ∩ k + for the group of totally positive units.If a principal ideal of O k can be written in the form xO k where x ∈ k + , it is said to be totally positive.The totally positive principal ideals form a subgroup P + O k of P O k , and we define the narrow ideal class group of k to be Cl + k := I O k /P + O k .There is a natural surjective homomorphism π k : Cl + k → Cl k .The cokernel of the injective morphism p k : P + O k → P O k is isomorphic to ker π k .arrowsareinjective.The structure of the quotient groupO × k /U + k is related to the kernel of the homomorphism π k : Cl + k → Cl k .To each x ∈ k ×we assign the vector (sign(σ v 1 (x)), ..., sign(σ v s (x)), where the σ v i range over the real embeddings k → ℝ.This gives rise to a homomorphism ω : k × → ⟨±1⟩ s.By the weak approximation theorem the homomorphism ω is surjective, and ker ω = k + .From the diagram (5.3) and the isomorphism coker(p k ) ≅ ker π k we get an exact sequence1 → O × k /U + k → ⟨±1⟩ s → ker π k → 1.Since the quotient group O × k /U + k contains at least the non-trivial element −1, the group ker π k is a 2-group whose order divides 2 s−1.It follows that the narrow ideal class group Cl + k is a finite group.We call h + of integers O k of an algebraic number field forms a free abelian group, generated by the non-zero prime ideals.Each element x ∈ k × defines a fractional principal ideal (x), and the map k × → I O k , defined by x  → (x), is a group homomorphism.k := |Cl + k | the narrow class number of k.