MAXIMAL ORDERS IN CENTRAL SIMPLE ALGEBRAS AND BRUHAT–TITS BUILDINGS

ANNE SCHWARTZ AND THOMAS R. SHEMANSKE

Abstract. We study the affine building for SL_{n} over a local field and give a characterization of distance involving Hecke operators. For n = 3 we give an explicitly computable distance formula. We use this local information to show that the class number of a maximal order in a central simple algebra of dimension n^{2} over a number field K is equal to the number of orbits of a group of isometries (related to the unit group of the maximal order) acting on a Bruhat-Tits building for SL_{n}(K). This generalizes results of Serre and Vignéras who considered the quaternion case in which the Bruhat-Tits building is a tree.

1. Introduction

We begin with a summary of the quaternion case. Let F be a non-archimedean local field with valuation ring R, A the quaternion algebra M_{2}(F), and V a 2-dimensional vector space over F. If we identify A with End_{F} (V), then it is well known that every maximal order in A is of the form Λ = End_{R}(L) where L is a free R-module (lattice) of rank 2. Moreover, any two maximal orders are conjugate by an element of A^{×}, and if M is another lattice of rank 2, then End_{R}(L) = End_{R}(M) if and only if L = λM for some nonzero element λ of F^{×}. Thus the maximal orders in A are in 1–1 correspondence with classes of lattices [L], where [L] = [M] if and only if L = λM as above.

Given two lattices L and M on V, Serre [7, p 70] defines the distance between them using elementary divisors: If π is a uniformizing parameter for R, then by the elementary divisor theorem there is a basis {e_{1}, e_{2}} of L such that {π^{a}e_{1}, π^{b}e_{2}} is a basis for M. The distance between L and M is defined to be the absolute value |a−b|. It is trivial to observe that the distance is unchanged by replacing a lattice with another in the same class; hence the distance function can be viewed as a function on classes of lattices, or equivalently on the set X of maximal orders in A. We can view the elements of X as the vertices of a graph in which two maximal orders are connected by an edge if they are distance one apart. Serre shows that X is a tree and that it is (q + 1)-regular, where q is the cardinality of the residue class field R/πR.

Date: April 2, 1995.

1991 Mathematics Subject Classification. Primary 11R52; Secondary 11R54, 11S45, 16H05, 20E42.

Key words and phrases. Maximal Order, Central Simple Algebra, Bruhat–Tits Building.

Vignéras [9] continues the examination of the quaternion case by shifting to a consideration of quaternion algebras over the ground field ℚ. Let A be a positive definite rational quaternion algebra, Λ a maximal order in A, and let p be a prime which splits in A, i.e., for which A_{p} = A ⊗_{ℚ} ℚ_{p} ≅ M_{2}(ℚ_{p}). Serre’s tree is defined in the local quaternion algebra A_{p}. Vignéras shifts the focus to the global algebra by means of the local-global correspondence for orders: We let X be the set of maximal orders Γ in A for which Γ_{q} = Λ_{q} for all primes q ≠ p. The elements of X are in 1–1 correspondence with the maximal orders in A_{p}, and we may transport the distance function for maximal orders in A_{p} to A by designating the distance between two global orders Γ and Γ′ to be the distance between their localizations at p: Γ_{p} and Γ′_{p}. As in the case of Serre, we can make the set X into a graph by designating the maximal orders as vertices and by placing edges between vertices which are distance one apart. Setting Λ^{(p)} = Λ ⊗_{ℤ}ℤ[\frac{1}{p}], we see that there is a natural action of the unit group Λ^{(p)×} on the graph X. Under this action, Vignéras shows that the class number of Λ (and hence of A) is equal to the number of orbits of Λ^{(p)×} in X.

Finally, we note that the classical Hecke operators (for SL_{2}(ℤ)) make an appearance at this stage. Here we remark only that it is not a coincidence that the (p + 1)- regularity of the graph X is equal to the degree of the Hecke operator T_{p}. We shall explore this interaction carefully in our generalization of these results.

In this paper, we extend the above results to arbitrary central simple algebras (of dimension n^{2}) over a number field K. In general, our interest is in n ≥ 3, both because the quaternion case generalizes formally to the number field setting, and also because there is a fundamental difference between the structure which results on the set X of maximal orders when n = 2 or when n ≥ 3. The structure one obtains is always isomorphic to the Bruhat-Tits building for SL_{n}(F), F a non-archimedean local field. However, in the case n = 2 (the quaternion case), the building is a tree, while in general it is isomorphic to an (n − 1)-dimensional simplicial complex on the set X. The fundamental difference between the quaternion and the higher rank setting is most easily visible in the difference between the Coxeter diagrams (see [1, p 148]), which we shall discuss in more detail later.

This paper consists of several parts. The first part establishes a correspondence between the set of maximal orders in the central simple algebra and the vertices of the Bruhat-Tits building for SL_{n}(K). It will once again turn out that we may think interchangeably of the vertices in the building as maximal orders or classes of lattices. An important consequence is that we are able to characterize the distance between classes of lattices (in terms of elementary divisors) in a way which is compatible with the natural distance function on the building. For n = 3 we give an explicit distance formula. Next we establish the regularity of the underlying graph by computing the valence of each vertex in terms of the degree of rank n Hecke operators. For ranks n = 3, 4, 5, we compute the valence explicitly. Finally, we show that the class number of maximal orders is equal to the number of orbits of a certain unit group acting on the set X of maximal orders. The proof is an adelic argument requiring an application of Eichler’s norm theorem which characterizes principal ideals in the algebra in terms of their norms.

Preliminaries

Let K be a number field with ring of integers O. For a prime 𝔓 (finite or infinite) of K, let K_{𝔓} denote its completion with respect to the usual 𝔓-adic valuation, and when 𝔓 is finite, let O_{𝔓} denote the valuation ring of K_{𝔓}. For any Dedekind domain R, O ⊂ R ⊂ K, let R_{𝔓} denote the closure of R in K_{𝔓}. Throughout, let π denote a uniformizing parameter for O_{𝔓}.

For later convenience in applying the Eichler norm theorem (Theorem 5.2), we want to choose a Dedekind domain R, O ⊂ R ⊂ K with the property that R is a PID and yet with the property that for all but a finite number of primes 𝔓 of K, R_{𝔓} = O_{𝔓}. This choice also obviates the need to restrict to number fields of class number one. In fact, it is convenient to construct a Dedekind domain containing O which has strict class number one. Let S be a finite set of primes of K including all the infinite ones. Put O^{(S)} = {x ∈ K | x ∈ O_{𝔓} for all 𝔓 ∉ S}. Note that O^{(S)} is not a localization of O; rather it is the intersection of all localizations O_{(𝔓)} for 𝔓 ∉ S. In particular, when K = ℚ and S consists of the infinite prime, O^{(S)} = ℤ.

Lemma 1.1. There exists a finite set of primes S, which contains all the infinite primes, and for which O^{(S)} is a PID. Moreover, the set S can be chosen in such a way that O^{(S)} has strict class number one.

Proof. For any set S of the required form, O^{(S)} is a Dedekind domain (see Chapter II of [3]), so we need only show that O^{(S)} has strict class number one for some set S. Let S consist of the infinite primes together with one prime from each strict ideal class of O. Let I be a fractional O^{(S)}-ideal. Then I is a fractional O-ideal, and so can be written as I = 𝔓_{1}^{e_{1}}𝔓_{2}^{e_{2}} ⋯ 𝔓_{r}^{e_{r}} · 𝔔 where the 𝔓_{i} ∈ S and where 𝔔 is a fractional O-ideal relatively prime to the primes in S. It is easy to see that IO^{(S)} = 𝔔O^{(S)} since 𝔓O^{(S)} = O^{(S)} for any 𝔓 ∈ S. Finally by the assumptions on the set S, 𝔔 is in the same strict ideal class as some prime 𝔓 in S, so 𝔔 = 𝔓α for some totally positive α ∈ K^{×}. Thus I = IO^{(S)} = O^{(S)}α, whence O^{(S)} has strict class number one.

Remark 1.2. Observe that O_{𝔓}^{(S)} = O_{𝔓} for all 𝔓 ∉ S, and that for any finite set T ⊃ S, O^{(T)} ⊃ O^{(S)} and O^{(T)} also has strict class number one. Finally, we note that the reader will suffer little loss of understanding in assuming throughout that K = ℚ and O = O^{(S)} = ℤ.

Let A be a central simple algebra of dimension n^{2} over K, n ≥ 3. For any Dedekind domain R with O ⊂ R ⊂ K, we call a subset Λ ⊂ A an R-order if Λ is a subring of A having the same multiplicative identity as A, and Λ is a finitely generated R-module such that K ⊗_{R} Λ ≅ A. When R is a PID, an order is a free R-module of rank n^{2}.

Throughout we shall keep R = O^{(S)} with S fixed and chosen as above so that all Rorders are free R-modules. Fix a maximal R-order Λ in A, and choose a prime 𝔓 ∉ S which is unramified in A (see §32 of [4]). Then A_{𝔓} ≅ M_{n}(K_{𝔓}) and by Theorem 17.3 of [4], Λ_{𝔓} is conjugate to M_{n}(O_{𝔓}). We may assume without loss of generality that A_{𝔓} is identified with M_{n}(K_{𝔓}) in such a way that Λ_{𝔓} = M_{n}(O_{𝔓}). In particular, let V be an n-dimensional vector space over K_{𝔓} and L a free O_{𝔓}-module of rank n. Then with respect to some basis, we assume that A_{𝔓} = End_{K_{𝔓}}(V) and Λ_{𝔓} = End_{O_{𝔓}}(L).

2. The Bruhat-Tits Building for SL_{n}(K_{𝔓})

In this section we characterize the Bruhat-Tits Building for SL_{n}(K_{𝔓}) as an (n − 1)-dimensional simplicial complex on the set of maximal orders in A_{𝔓}, and give a characterization of the distance between maximal orders (vertices in the building) in terms of the elementary divisors of the lattices whose classes determine the orders.

To begin, recall that the affine building Δ_{n} for SL_{n}(K_{𝔓}) has type Ã_{n−1}(K_{𝔓}) and can be realized as an (n − 1)-dimensional simplicial complex on the set of classes of lattices as follows (for complete details see [1, Chapter V.8] or [6, Chapter 9.2]): The type of the building specifies a certain Coxeter diagram, which implicitly specifies a collection of positive integers m_{ij} used to construct the (n − 1)-simplices. Associated to the Coxeter diagram is a Coxeter group, W, which is given by generators and relations as

W = ⟨r_{i} | r_{i}^{2} = (r_{i}r_{j})^{m_{ij}} = 1 for 1 ≤ i, j ≤ n⟩.

The building is comprised of apartments, each of which is a Coxeter complex whose geometric realization is a tesselation of ℝ^{n−1} by (n−1)-simplices called chambers. The (n − 1)-simplices which are used in the tesselation have dihedral angles π/m_{ij} where the integers m_{ij} are specified by the corresponding Coxeter diagram. For example, in the building for SL_{3}(K_{𝔓}), each apartment can be visualized as the Euclidean plane tesselated by equilateral triangles, and for SL_{4}(K_{𝔓}), the apartments can be visualized as Euclidean 3-space tesellated by tetrahedra with two dihedral angles π/3 and the other two π/2.

For most of the important results, attention may be restricted to apartments, so we now give a more careful description of them. The Coxeter group W acts simplytransitively on the set of chambers in an apartment. For convenience, we therefore specify a fundamental chamber on which W acts to generate the apartment. To characterize this notion in terms of classes of lattices we let, as in the previous section, V be an n-dimensional vector space over K_{𝔓}. For L, M lattices on V (free O_{𝔓}-modules of rank n), we say that L and M are in the same class if L = λM for some λ ∈ K_{𝔓}^{×}, and we denote the class of L by [L]. As a matter of notation, if the lattice L has O_{𝔓}-basis {v_{1}, …, v_{n}}, we will often write [v_{1}, …, v_{n}] for [L]. Fix a basis {v_{1}, …, v_{n}} for V. From [1, p 137], the fundamental chamber C of Δ_{n} can be taken to be the (n − 1)-simplex with vertices [v_{1}, …, v_{i−1}, πv_{i}, …, πv_{n}], i = 1, 2, …, n. By [1, p 148], the fundamental apartment Σ of Δ_{n} (the apartment generated by the action of W on C) has vertices [π^{a_{1}}v_{1}, …, π^{a_{n}}v_{n}] with the a_{i} ∈ ℤ.

Our principle interest is the study of maximal orders in central simple algebras, so we first show that the vertices of our building Δ_{n} may be taken to be maximal orders in the local algebra A_{𝔓}. From the previous section, we know that all maximal orders in A_{𝔓} are of the form End_{O_{𝔓}}(L) for some lattice L on V. First we determine all possible redundancy in this characterization.

Proposition 2.1. Let Γ_{1} = End_{O_{𝔓}}(L_{1}) and Γ_{2} = End_{O_{𝔓}}(L_{2}) be two maximal orders in A_{𝔓}. Then Γ_{1} = Γ_{2} if and only if L_{1} = αL_{2} for some α ∈ K_{𝔓}^{×}.

Proof. That End_{O_{𝔓}}(L) = End_{O_{𝔓}}(αL) is obvious since every element of End_{O_{𝔓}}(L) extends to a unique element of End_{K_{𝔓}}(V) by linearity. We establish the converse by showing the contrapositive. By the elementary divisor theorem, there is an O_{𝔓}-basis {e_{1}, e_{2}, …, e_{n}} of L_{1} and there are rational integers a_{1}, …, a_{n} such that {π^{a_{1}}e_{1}, π^{a_{2}}e_{2}, …, π^{a_{n}}e_{n}} is an O_{𝔓}-basis of L_{2}. Now L_{2} = αL_{1} iff L_{2} = π^{a}L_{1} for some a, so to begin the proof, we assume that L_{2} ≠ π^{a}L_{1}, and hence that there is an index j for which a_{j} ≠ a_{1}. Consider the element φ ∈ End_{O_{𝔓}}(L_{1}) induced by φ(e_{j}) = e_{1}, φ(e_{1}) = e_{j} and φ(e_{i}) = e_{i} for all i ≠ 1, j. Then φ(L_{2}) ⊄ L_{2} which completes the proof.

We see from Proposition 2.1 that there is a 1–1 correspondence between classes of lattices on V and maximal orders in A_{𝔓}, so we will speak of the vertices of our building interchangeably as maximal orders or as classes of lattices. Also from the proposition, we see that whenever convenient, given two maximal orders End_{O_{𝔓}}(L_{1}) and End_{O_{𝔓}}(L_{2}), we may assume that L_{1} ⊂ L_{2}. Hence by the elementary divisor theorem, we may assume that there is an O_{𝔓}-basis {e_{1}, e_{2}, …, e_{n}} of L_{2} and there are rational integers 0 ≤ a_{1} ≤ … ≤ a_{n} such that {π^{a_{1}}e_{1}, π^{a_{2}}e_{2}, …, π^{a_{n}}e_{n}} is an O_{𝔓}-basis of L_{1}.

We want to discuss the notion of distance in the building Δ_{n} and to show that the natural distance can be interpreted in terms of the elementary divisors of the lattices whose endomorphism rings are the maximal orders comprising the vertices in our building. As an end result, we shall show that the number of vertices at distance one from a given vertex corresponds to the degree of a certain Hecke operator (Theorem 3.3), and that the class number of a maximal order in the global algebra A is the number of orbits of a certain group acting as isometries on the building (Theorem 5.3).

First we discuss distance in the general framework of a building. Recall that two chambers in a building are adjacent if they share a codimension-one face (a panel, which determines a unique wall). Within an apartment, we can be more explicit. Since the apartments of a building are Coxeter complexes, the notion of adjacency within an apartment can be described via the Coxeter group as follows (see [6, p 10]): The chambers in an apartment are labeled by the elements of the Coxeter group W = ⟨r_{i} | r_{i}^{2} = (r_{i}r_{j})^{m_{ij}} = 1 for 1 ≤ i, j ≤ n⟩ in such a way that i-adjacency is defined by w ∼_{i} wr_{i}. Recall that W acts simply-transitively by left translation on the chambers in an apartment, and it is clear that this action preserves i-adjacency. For SL_{3}(K_{𝔓}), the apartment would be labelled (starting from an arbitrary chamber g ∈ W as in Figure 2.1. Notice that the lines of different shades and thicknesses represent the three types of adjacency.

Figure 2.1

A gallery in a building is a sequence of chambers (equilateral triangles for SL_{3}(K_{𝔓})), C_{0}, C_{1}, …, C_{d} in which consecutive chambers C_{i−1} and C_{i} (i = 1, … d) are adjacent. The length of the gallery is d. The distance between two chambers is defined to be the length of any minimal gallery between the two chambers. Finally, since two distinct adjacent chambers are separated by a unique wall, given any minimal gallery C_{0}, …, C_{d}, the walls crossed by the gallery are distinct and are precisely the walls separating C_{0} from C_{d}. In particular, the distance d between the two chambers C_{0} and C_{d} is equal to the number of walls separating them (see [1, p 73]). Since any two chambers are contained in an apartment Σ [6, Corollary 3.7], and any minimal gallery between those chambers lies within Σ, [6, Theorem 3.8], the distance between chambers can be computed within any apartment which contains both chambers. Thus we can rephrase the notion of distance again in terms of the labeling of the apartment by elements of the Coxeter group W. It is clear that the distance from chamber g to the chamber gw (where w is a reduced word in W) is the length of w.

Now we wish to give a formulation of distance which takes advantage of the building’s characterization as a simplicial complex on classes of lattices and use the fact that we have restricted our attention to an apartment. Recall that we have fixed a basis {v_{1}, …, v_{n}} for V. The fundamental chamber C of Δ_{n} is the (n − 1)-simplex with vertices [v_{1}, …, v_{i−1}, πv_{i}, …, πv_{n}], i = 1, 2, …, n, and the fundamental apartment Σ of Δ_{n} has vertices [π^{a_{1}}v_{1}, …, π^{a_{n}}v_{n}] with the a_{i} ∈ ℤ. Since the basis is fixed, we simplify the typography further by writing [a_{1}, …, a_{n}] for [π^{a_{1}}v_{1}, …, π^{a_{n}}v_{n}]. Then since we are concerned only with classes of lattices we observe that each class [a_{1}, …, a_{n}] has a unique representative of the form [0, b_{2}, …, b_{n}] (with b_{i} = a_{i} − a_{1}).

With this notation we again get a labelling of the apartment Σ, this time by a labelling of its vertices. For SL_{3}(K_{𝔓}), this is shown in Figure 2.2.

Figure 2.2

Remark 2.2. In practice we shall choose the basis {v_{1}, …, v_{n}} of V to facilitate the calculation of distance. For example, to find the distance between vertices v and w, first represent v = [L] and w = [M] where L and M are two lattices on V. By the elementary divisor theorem, there exists a basis {v_{1}, …, v_{n}} of L (and hence of V) and integers a_{1} ≤ ⋯ ≤ a_{n} so that L = ⊕O_{𝔓}v_{i} and M = ⊕O_{𝔓}π^{a_{i}}v_{i}. The collection {π^{a_{1}}, …, π^{a_{n}}} is called the set of elementary divisors of M in L, denoted {L : M}.

The basis {v_{1}, …, v_{n}} generates an apartment Σ containing v and w as above. For any other apartment Σ′ containing v and w, there is an isomorphism (preserving adjacency) which fixes v and w [1, IV.1]. In particular, the distance between v and w is independent of the apartment in which we consider v and w, so we may do our calculations within Σ. Since the vertices v and w are determined in Σ by the elementary divisors {L : M} (i.e., v = [0, …, 0] and w = [a_{1}, …, a_{n}]), the distance between v and w is completely determined by the elementary divisors {L : M}.

In the building Δ_{n}, the chambers correspond to maximal flags of lattices on V (see [6, Chapter 9.2]):

πL_{0} ⊂ L_{n−1} ⊂ L_{n−2} ⊂ ⋯ ⊂ L_{0}

where the vertices of the chamber are [L_{0}], [L_{1}], …, [L_{n−1}], and π is a uniformizing parameter for O_{𝔓}. Our goal is to define the distance between vertices in the building. To do so, we need to connect the notion of distance described in terms of galleries with that of the elementary divisors of the lattices. We proceed as follows. As in Chapter 9.2 of [6], we can realize the Coxeter group W as a group of reflections generated by reflections in the codimension-1 faces of the fundamental chamber. In particular, we can define reflections S_{i}, 1 ≤ i ≤ n in terms of their actions on elementary divisors (actually on classes of lattices):

S_{i}([α_{1}, …, α_{n}]) = [α_{1}, …, α_{i−1}, α_{i+1}, α_{i}, α_{i+2}, … α_{n}] for 1 ≤ i ≤ n − 1, and
S_{n}([α_{1}, …, α_{n}]) = [α_{n} − 1, α_{2}, …, α_{n−1}, α_{1} + 1].

The first n − 1 reflections merely interchange the i^{th} and i + 1^{st} coordinates. In terms of our standard representation of vertices, we have:

S_{1}([0, β_{2}, …, β_{n}]) = [0, −β_{2}, β_{3} − β_{2}, … β_{n} − β_{2}],
S_{i}([0, β_{2}, …, β_{n}]) = [0, β_{2}, …, β_{i−1}, β_{i+1}, β_{i}, β_{i+2}, …, β_{n}] for 2 ≤ i ≤ n − 1, and
S_{n}([0, β_{2}, …, β_{n}]) = [0, β_{2} − β_{n} + 1, …, β_{n−1} − β_{n} + 1, 2 − β_{n}].

To deduce the action on chambers, let C denote the fundamental chamber with vertices described by the flag

πL_{0} ⊂ L_{n−1} = [0, 1, …, 1] ⊂ L_{n−2} = [0, 0, 1, …, 1] ⊂
⋯ ⊂ L_{1} = [0, …, 0, 1] ⊂ L_{0} = [0, …, 0].

It is then straightforward to verify the action of each S_{i} on the chamber C. In particular, we see that S_{i} fixes L_{k} for k ≠ n − i. That is, each S_{i} fixes n − 1 of the n vertices of the chamber C, and so C and its image are reflections across a codimension-1 face of C. Now the S_{i} satisfy all the same relations as the generators of the Coxeter group W [6], so we associate the S_{i} with the corresponding r_{i} so that i-adjacency denoted by g ∼_{i} gr_{i} is compatible with the i-adjacency of C and S_{i}(C).

For later convenience, we now denote S_{i} by S_{r_{i}}, where r_{i} is the corresponding generator of W. Moreover, if w = r_{i_{1}} ⋯ r_{i_{k}} is a word in W, denote by S_{w} the composition S_{r_{i_{1}}} ∘ ⋯ ∘ S_{r_{i_{k}}}. We also continue the abuse of notation S_{w}(C) where we mean the chamber whose vertices are determined by the action of S_{w} on the vertices of C. Finally we can connect the notion of adjacency given in terms of the Coxeter group with that of the reflections described in terms of elementary divisors.

Proposition 2.3. Let C be the fundamental chamber in the apartment Σ with vertices [0, …, 0], [0, …, 0, 1], … [0, 1, …, 1], and associate to C a group element g ∈ W (g ↔ C). Then

1. For any element v ∈ W, the chamber gv ↔ S_{v}(C).

2. For any chamber labelled by h ∈ W, let C_{h} denote the corresponding maximal flag of lattices (i.e., h ↔ C_{h}). Then for any w ∈ W, the chamber hw ↔ S_{vwv^{−1}}(C_{h}), where h = gv.

Proof. We show gv ↔ S_{v}(C) by induction on the length of v. When v has length one, v = r_{i} for some i, and by design gr_{i} ↔ S_{r_{i}}(C). To complete the induction we need only show that g(vr) ↔ S_{vr}(C) where r is any generator r_{i} ∈ W. We have seen that C and S_{r}(C) are adjacent chambers, and by the base step of the induction, gr ↔ S_{r}(C). Therefore since S_{v} is a composition of reflections and hence a rigid motion, S_{v}(C) and S_{v}(S_{r}(C)) are adjacent chambers. Since g ↔ C and gr ↔ S_{r}(C) are “r-adjacent”, so are S_{v}(C) and S_{v}(S_{r}(C)). This means that if h ↔ S_{v}(S_{r}(C)) and ℓ ↔ S_{v}(C), then h = ℓr. By induction ℓ ↔ S_{v}(C) ↔ gv, hence S_{v}(S_{r}(C)) ↔ h = ℓr = (gv)r = g(vr) as required.

For the second part of the proposition, suppose that h ↔ C_{h}. Write h = gv for a unique v ∈ W. Then by the first part of the proposition, C_{h} = S_{v}(C) and hw = gvw ↔ S_{vw}(C). Observing S_{vw}(C) = S_{vwv^{−1}}(S_{v}(C)) = S_{vwv^{−1}}(C_{h}) completes the proof.

Now that we have a good grasp on different characterizations of distance between chambers, we would like to define the distance between two vertices in the building. An obvious choice is to define the distance to be the minimal length among all galleries whose initial chamber contains one vertex and whose ending chamber contains the other. First we show that this is well-defined.

Proposition 2.4. Let q be the cardinality of the residue class field k of K_{𝔓}. Then for n ≥ 2, every vertex in Δ_{n} is contained in precisely r_{n} chambers where

r_{n} = \frac{q^{n} − 1}{q − 1} · \frac{q^{n} − q}{q(q − 1)} · \frac{q^{n} − q^{2}}{q^{2}(q − 1)} ⋯ \frac{q^{n} − q^{n−2}}{q^{n−2}(q − 1)}.

Remark 2.5. For computational convenience, we note that r_{2} = q + 1 and r_{n+1} = (\frac{q^{n+1}−1}{q−1}) r_{n}.

Proof. A vertex is contained in a chamber (i.e., is a 0-simplex in the chamber) if and only if the chamber is part of the residue of the vertex. By [6, Chapter 9.2], such a residue is isomorphic to a building of type A_{n−1}(k). By [6, Chapter 1.2], the chambers of this spherical building are maximal flags in an n-dimensional vector space U over k. That is, the chambers are nested sequences of subspaces

0 ⊂ U_{1} ⊂ U_{2} ⊂ ⋯ ⊂ U_{n−1} ⊂ U (with dim(U_{i}) = i).

To count the number of sequences is elementary: Any 1-dimensional space is spanned by a single nonzero vector of which there are q^{n} − 1 in U. For each of these vectors, any nonzero multiple of it generates the same subspace, and this is the only possible redundancy. Thus there are (q^{n} − 1)/(q − 1) possible U_{1}. Suppose that U_{1} ⊂ ⋯ ⊂ U_{j} have been chosen. To find all possible U_{j+1} ⊃ U_{j}, we need only ask how many distinct subspaces of dimension j + 1 can be generated by adding a single vector to a basis of U_{j}. Similar to the first case, there are q^{n} − q^{j} choices of vectors in U not in U_{j}. To determine the redundancy, we observe that for u, w ∈ U \ U_{j}, U_{j} ⊕ ⟨u⟩ = U_{j} ⊕ ⟨w⟩ if and only if w = λu + v for v ∈ U_{j} and λ ≠ 0. This yields (q^{n} − q^{j})/q^{j}(q − 1) possible U_{j+1}. From here the proposition is clear.

This generalizes the well-known SL_{2} case in which the chambers are 1-simplices and the building Δ_{2} is a tree. The proposition states that the tree is (q + 1)-regular.

3. Hecke Operators and the Bruhat-Tits Building

In this section we demonstrate that the number of vertices in the building Δ_{n} at distance one from a given vertex is the degree of a certain Hecke operator. In the case of SL_{2}, this provides another (cf. Proposition 2.4) proof of the (q + 1)-regularity of the tree (building) Δ_{2}. As suggested by the special cases 3 ≤ n ≤ 5 in Proposition 3.5 below, for n ≥ 3, it is unlikely that the degree of the Hecke operator is equal to the number given in Proposition 2.4, however there does appear to be an intriguing relationship between the two numbers.

We shall need generalizations of a few theorems of Shimura [8]. Let V be our fixed n-dimensional vector space over K_{𝔓}. For two lattices L and M on V, let {L : M} denote the set of elementary divisors of M in L. Fix a lattice L on V and a basis of L which identifies GL(L) with GL_{n}(O_{𝔓}). Then the group G = GL_{n}(O_{𝔓}) acts (on the left) on the set of lattices on V.

The proof of the following proposition is essentially identical to Shimura’s Lemma 3.12 [8] with SL_{2}(ℤ) replaced by G, since he uses only that the ring over which he is working is a PID.

Proposition 3.1. Let L, M, N be lattices on V. Then {L : M} = {L : N} if and only if there exists a g ∈ G such that gM = N.

For elements α_{1}, …, α_{n} ∈ O_{𝔓}, let ⟨α_{1}, …, α_{n}⟩ denote the diagonal matrix in M_{n}(O_{𝔓}) with diagonal entries α_{1}, …, α_{n}. Let ξ = ⟨π^{a_{1}}, …, π^{a_{n}}⟩ ∈ M_{n}(O_{𝔓}) with the a_{i} ∈ ℤ, a_{1} ≤ … ≤ a_{n}. It is easily seen that ξ ∈ G̃, the commensurator of G, so by Proposition 3.1 of [8] we have

GξG = ∪_{i=1}^{d}Gα_{i} = ∪_{j=1}^{e}β_{j}G

where d = [G : G ∩ ξ^{−1}Gξ] and e = [G : G ∩ ξGξ^{−1}] are finite. The integer d is called the degree of the double coset GξG.

Let Δ(ℓ) = {M ∈ M_{n}(O_{𝔓}) | ord_{𝔓}(det(M)) = ℓ}. The existence of a Smith normal form tells us that for any α ∈ Δ(ℓ), GαG = G⟨π^{a_{1}}, …, π^{a_{n}}⟩G where 0 ≤ a_{1} ≤ … ≤ a_{n} are rational integers, and ∑_{1}^{n} a_{i} = ℓ. By the class number formula [2, pp 19-20], for any element α ∈ Δ(ℓ), GαG = ∪_{ν}Gα_{ν} with finitely many cosets. In particular, GαG has finite degree. With a proof identical to Lemma 3.13 of [8], we have

Proposition 3.2. Let ξ = ⟨π^{a_{1}}, …, π^{a_{n}}⟩ ∈ Δ(ℓ) as above, and let L be a lattice on V. Then there is a 1–1 correspondence between cosets Gξ_{ν} in GξG and lattices M on V such that {L : M} = {π^{a_{1}}, …, π^{a_{n}}}.

We now turn to our task of determining how many vertices of our building are at distance 1 from a fixed vertex. It follows from Proposition 2.4 that the number of chambers containing a given vertex is finite, so it makes sense to define the distance between two vertices as the minimal length among all galleries whose initial chamber contains one vertex and whose ending chamber contains the other.

Theorem 3.3. Let v be any vertex in the building Δ_{n}. Then the number of vertices in Δ_{n} which are distance one from v is equal to

1. the degree of G⟨1, π⟩G if n = 2, and

2. the degree of G⟨1, π, …, π, π^{2}⟩G if n ≥ 3.

Remark 3.4. In a global setting, with G = SL_{2}(ℤ) and π = p a prime, the double

coset G (\begin{matrix}1 & 0 \\ 0 & p \end{matrix}) G is the standard Hecke operator T_{p} which acts on modular forms of

integral weight. This is our motivation for referring to these double cosets as Hecke operators. Serre defined related operators as formal sums on the vertices of Δ_{2} [7]. These are also called Hecke operators as they satisfy recursion relations paralleling those of the standard Hecke operators.

Proof. The differences between the two cases stem principally from the very different Coxeter diagrams which describe the buildings for SL_{2} and for SL_{n}, n ≥ 3. Since the result stated in this proposition is known for n = 2, we start the proof with the case of n ≥ 3, and at the end of the proof comment about the differences which govern the SL_{2} case.

Let v and w be vertices in the building Δ_{n}. It is clear from the definition of distance given in terms of galleries that vertices v and w are distance one apart if and only if they are contained in chambers C_{v} and C_{w} which share a codimension-1 face, and where C_{w} is not in the residue of v (i.e. v ∉ C_{w}).

As we saw above, C_{v} and C_{w} lie in a common apartment Σ. Let L and M be lattices on V so that v = [L] and w = [M]. By Remark 2.2, we may assume that relative to some basis v = [0, …, 0] and w = [0, a_{2}, …, a_{n}]. The vertex v is contained in the fundamental chamber C of Σ with vertices [0, …, 0], [0, …, 0, 1], …,[0, 1, …, 1].

From the discussion preceding Proposition 2.3, we know the reflections S_{r_{i}} each fix [0, …, 0] if and only if 1 ≤ i ≤ n − 1, hence the residue of v in Σ consists of the chambers S_{u}(C), where u ∈ ⟨r_{i} | 1 ≤ i ≤ n − 1⟩. Thus C_{v} = S_{ξ}(C) for some ξ ∈ ⟨r_{i} | 1 ≤ i ≤ n − 1⟩. Let g ∈ W correspond to the fundamental chamber C. Then by Proposition 2.3, C_{v} ↔ gξ. Moreover, since C_{v} and C_{w} are adjacent, but C_{w} is not in the residue of C_{v}, C_{w} ↔ gξr_{n}, that is C_{w} = S_{ξr_{n}}(C) by Proposition 2.3. This means that w = S_{ξr_{n}}(v) = S_{ξr_{n}}([0, …, 0]) = S_{ξ}([0, 1, …, 1, 2]). On the other hand, each S_{r_{i}} for i < n simply permutes the coordinates. Thus w = [a_{1}, …, a_{n}], where {a_{1}, …, a_{n}} is simply a permutation of {0, 1, …, 1, 2}. By permuting the original basis for L, we may assume that v = [L] = [0, …, 0] and w = [M] = [0, 1, …, 1, 2].

Thus to find all vertices distance one from v, we need only seek all classes of lattices [M] for which [M] has elementary divisors [0, 1, …, 1, 2] relative to [L]. Since we are concerned only with classes of lattices, this is the same as finding all lattices M so that with respect to some O_{𝔓}-basis of L, the elementary divisors of M in L, {L : M}, are {1, π, …, π, π^{2}}. By Proposition 3.2, this number is the same as the degree of the double coset G⟨1, π, …, π, π^{2}⟩G. Note that if M_{1} and M_{2} are distinct lattices which satisfy {L : M_{1}} = {L : M_{2}}, then [M_{1}] ≠ [M_{2}] and so correspond to distinct vertices, for if M_{1} = λM_{2}, {L : M_{1}} = {λ, λπ, … λπ, λπ^{2}} = {L : M_{2}} if and only if λ ∈ O_{𝔓}^{×}, and hence M_{1} = λM_{2} = M_{2}.

For n = 2, there is a minor difference in the appearance of the final result arising from the difference of the Coxeter diagrams for Ã_{1} and Ã_{m} for m ≥ 2. Using the distance function described by Serre [7] and discussed in the introduction, two classes,

[L] and [M], of (rank 2) lattices are distance one apart if and only if {L : M} = {π^{a}, π^{a+1}}. Again since we are concerned only with classes of lattices, it suffices to find all lattices M with {L : M} = {1, π}. Hence the number of vertices at distance one from a given vertex in the building for SL_{2} is the degree of the double coset

G (\begin{matrix}1 & 0 \\ 0 & π \end{matrix}) G, which is q + 1. As mentioned above, with G = SL_{2}(ℤ) and π = p a

prime in ℤ, this is the standard Hecke operator T_{p} which acts on modular forms of integral weight.

We compute the degree of the Hecke operator in several cases.

Proposition 3.5. Let r_{n} be (as in Proposition 2.4) the number of chambers containing a given vertex in Δ_{n}, and ω_{n} be the number of vertices having distance one from a given vertex in Δ_{n}. Let q denote the cardinality of the residue class field O_{𝔓}/πO_{𝔓}. Then

1. r_{2} = ω_{2} = q + 1,

2. qr_{3} = q(1 + 2q^{1} + 2q^{2} + q^{3}) = ω_{3},

3. qr_{4} = (q + 1)ω_{4} = r_{2}ω_{4}, and

4. qr_{5} = (1 + 2q^{1} + 2q^{2} + q^{3})ω_{5} = r_{3}ω_{5}.

Remark 3.6. Based solely upon the relatively complicated relations occuring above (and not on any insight into why this relationship exists), it is natural to conjecture that for n ≥ 3, qr_{n} = r_{n−2}ω_{n}.

Proof. The result is obtained by a straightforward, but relatively tedious computation. The result for n = 2 is well-known. We indicate the method for n = 3, and leave the remaining details to the reader. By consideration of the Smith normal form, it is immediate that

Δ(3) = G⟨1, 1, π^{3}⟩G ∪ G⟨1, π, π^{2}⟩G ∪ G⟨π, π, π⟩G.

Moreover, by Theorem II.4 of [2], the class number (i.e. the total number of right cosets in the union) is equal to 1+q+2q^{2}+2q^{3}+2q^{4}+q^{5}+q^{6} where q is the cardinality of the residue class field O_{𝔓}/πO_{𝔓}. Obviously the 1 corresponds to the double coset G⟨π, π, π⟩G = G⟨π, π, π⟩, leaving the rest to distribute between the other two double cosets.

By Theorems II.2 and II.3 of [2], we may assume that every representative of the right cosets is in Hermite normal form. This done, we use Theorem II.10 of [2] to associate the right cosets to the appropriate double coset. In particular, we use that GαG = GβG if and only if α and β have the same determinantal divisors. The result now follows from a case by case analysis.

4. The case of SL_{3}

When n = 3, we know that the fundamental apartment is the Euclidean plane tiled by equilateral triangles. Recall that we may restrict our attention to galleries contained within the fundamental apartment Σ. Within the apartment, each vertex is contained in six equilateral triangles which together form a regular hexagon with the vertex at center. Moreover, each vertex is the center of a sequence of concentric hexagons (see Figure 4.1), expanding out from the vertex. This in turn gives a well defined notion of the radial distance from our vertex to any other vertex in the plane, the vertex at center having radial distance zero, the six vertices in the hexagon containing the vertex having radial distance 1, the next 12 vertices radially outward from the center having distance 2, and so on.

Because the fundamental apartment is a regular tiling of the plane, it is clear that a rigid motion will not alter distances, so there is no loss in considering the distance from an arbitrary vertex [0, a, b] to the “origin” [0, 0, 0]. Consider three fundamental classes of lattices: e_{1} = [0, 1, 0], e_{2} = [0, 0, 1], and e_{3} = [0, 1, 1]. We abuse the notation further by thinking of these classes of lattices as ordered triples in ℤ^{3}. Then for any class of lattices [0, a, b], we have:

[0, a, b] = ae_{1} + be_{2} = (a − b)e_{1} + be_{3} = (b − a)e_{2} + ae_{3}.

Proposition 4.1. The radial distance of a point [0, a, b] from [0, 0, 0] is given by

r = min{ |a| + |b|, |a − b| + |b|, |a − b| + |a| } = max{ |a|, |b|, |a − b| }.

Proof. Since any two of the three vectors e_{i} are linearly independent, the vertex [0, a, b] can be expressed as a unique linear combination of any two of them. Consider the

Figure 4.1

three sums of the absolute values of the coefficients corresponding to these linear combinations. The first equality of the proposition asserts that the radial distance is the smallest of these three sums.

Our three fundamental vectors e_{1}, e_{2}, and e_{3} determine three lines through the origin in our fundamental apartment Σ. Clearly if [0, a, b] = re_{i} for some i then r is the radial distance, and it is trivial to check that the coordinates satisfy the conditions of the proposition.

Otherwise [0, a, b] lies in the interior of one of the six sectors determined by the lines containing the e_{i}. The point is then in the interior of the region bounded by two of the six rays ε_{i}e_{i}, for some i = 1, 2, 3, and ε_{i} = ±1 (see Figure 4.2); call them f_{1} and f_{2}.

Since f_{1} and f_{2} are linearly independent, there are unique positive integers s and t for which [0, a, b] = sf_{1} + tf_{2}. Moreover, it is clear that the radial distance r = s + t, since each move of unit length parallel to either vector f_{i} crosses from one “radial band” to the next. That is, each such move of unit length proceeds from a vertex having radial distance r to one having radial distance r + 1.

Let f_{3} be whichever of the three vectors e_{i} is not a multiple of f_{1} or f_{2}. Note that vertices in the sector determined by the rays f_{1} and f_{2} which have a fixed

Figure 4.2

radial distance from [0, 0, 0] lie on line segments which are parallel to f_{3}. Express [0, a, b] = u_{i}f_{i} + v_{i}f_{3}, i = 1, 2. We will show that |u_{i}| + |v_{i}| ≥ s + t which will establish the first part of the proposition by the comments in the first paragraph of the proof.

To establish |u_{i}| + |v_{i}| ≥ s + t, we show that |u_{i}| = s + t. This is trivial since u_{i}f_{i} brings us to a vertex having radial distance |u_{i}| from [0, 0, 0]. Movement parallel to f_{3} is through a set of vertices having a fixed radial distance from [0, 0, 0], so the radial distance of [0, a, b] = u_{i}f_{i} + v_{i}f_{3} is |u_{i}| = s + t.

Now a simple algebraic argument establishes that

min{ |a| + |b|, |a − b| + |b|, |a − b| + |a| } = max{ |a|, |b|, |a − b| }.

Theorem 4.2. The distance from [0, 0, 0] to v = [0, a, b] is given by

d = \begin{cases}2(r − 1) & if v = λe_{i}, λ ∈ ℤ \\ 2(r − 2) + 1 & otherwise \end{cases}

where r is the radial distance from [0, 0, 0] to v.

Proof. First we observe that v has distance zero from [0, 0, 0] if and only if v = ±e_{i} and r = 1 since such vertices are vertices of the chambers which form the residue of [0, 0, 0] in the apartment Σ. This much agrees with the formula, so we assume r ≥ 2.

We consider two cases, radial and nonradial, corresponding to whether or not v = λe_{i}. Consider the radial case first. Then r = |λ| ≥ 2, and the situation is pictured in Figure 4.3. It is clear that a minimal gallery must lie within the “strip” containing the residues (hexagons) of [0, 0, 0] and v, for any gallery which leaves the strip must still cross at least as many walls (once outside the strip) and must also cross a wall to exit and one to reenter the strip. It is then clear from the figure that a minimal gallery has length 2(r − 1) which completes the proof in the radial case.

Figure 4.3

In the nonradial case, to proceed from [0, 0, 0] to a vertex at radial distance 2 requires crossing at least one wall. To proceed from this vertex to any vertex of radial distance 3 requires crossing at least two more walls. It is then clear that it requires crossing at least 1 + 2(r − 2) walls to proceed from [0, 0, 0] to any vertex of radial distance r (r ≥ 2). This provides a lower bound for the distance. To establish an upper bound, we simply provide a gallery of length 1 + 2(r − 2). Since v is not a radial point, it lies in a unique sector. As in Proposition 4.1, we can write v = [0, a, b] = sf_{1} + tf_{2} with s and t positive integers. Then the radial distance r = s + t. It is clear from Figure 4.4 that the highlighted gallery is simply the combination of a radial gallery followed by a “radial-like” gallery whose lengths are 2(s − 1) + 2(t − 1) + 1, where the “1” accounts for the crossing from the “s”-gallery to the “t”-gallery. We therefore have a gallery of length 2(s + t − 2) + 1 = 2(r − 2) + 1 which provides our upper bound and completes the proof.

Next we return to the question of the number of vertices at distance one from a given vertex. A general answer to the problem was given in Theorem 3.3 in terms of the degree of a double coset. Given an explicit distance function for the n = 3 case, we now give a brief alternate proof.

Fix a lattice L on V and let M be any other lattice on V. Recall that the vertices of our building are in 1–1 correspondence with the classes of lattices on V, so we may feel free to scale lattices. If {L : M} = {π^{a_{1}}, π^{a_{2}}, π^{a_{3}}} with a_{1} ≤ a_{2} ≤ a_{3},

Figure 4.4

then {L : π^{−a_{1}}M} = {1, π^{a_{2}−a_{1}}, π^{a_{3}−a_{1}}}. In the notation of §2, [L] = [0, 0, 0], and [M] = [0, a, b], where 0 ≤ a = a_{2} − a_{1} ≤ b = a_{3} − a_{1}.

Applying Proposition 4.1 and Theorem 4.2, we see that the radial distance from [M] to [L] is b, and that the distance d = 1 if and only if a = 1 and b = 2. Thus we need to find all lattices with elementary divisors {L : M} = {π^{a}, π^{a+1}, π^{a+2}}. Since we are concerned only with the classes of the lattices, it suffices to consider all lattices with elementary divisors {L : M} = {1, π, π^{2}}. By Proposition 3.2, there is a 1–1 correspondence between such lattices and the right cosets Gξ_{ν} in G⟨1, π, π^{2}⟩G. Also note for any two such lattices M_{1} and M_{2}, that [M_{1}] ≠ [M_{2}] since M_{2} = λM_{1} implies {L : M_{2}} = {λ, πλ, π^{2}λ} = {1, π, π^{2}} if and only if λ is a unit in O_{𝔓}, in which case M_{1} = M_{2}. We summarize this result as:

Proposition 4.3. In the Bruhat-Tits building for SL_{3}(K_{𝔓}) the number of vertices at distance one from a given vertex is equal to the number of of right cosets Gξ_{ν} in the double coset G⟨1, π, π^{2}⟩G.

5. Class Numbers

Now we return to the global setting and consider the global central simple algebra A over the number field K. Recall that we have chosen a finite set S of primes of K, containing all the infinite ones, for which R = O^{(S)} has strict class number one (when K = ℚ, we may assume that R = ℤ). Actually, we require this assumption only to invoke Eichler’s theorem in Theorem 5.3. We have fixed a maximal R-order Λ. Note that because R is a PID, Λ is a free R-module. As in §1, let 𝔓 be a finite prime of K not contained in the set S. We also assume that 𝔓 is unramified in A. Let R^{(𝔓)} = O^{(S∪{𝔓})}, and let Λ^{(𝔓)} = Λ ⊗_{R} R^{(𝔓)}. The goal of this section is to characterize the class number of Λ as the number of orbits of the unit group Λ^{(𝔓)}^{×} acting on a global analog of our Bruhat-Tits building for SL_{n}(K_{𝔓}).

Let X consist of the set of maximal R-orders Γ in A such that Γ_{𝔮} = Λ_{𝔮} for all primes 𝔮 of R, 𝔮 ≠ 𝔓. This is the same as the set of primes 𝔮 of O not in S ∪ {𝔓}. By the local-global correspondence for orders, the elements in X are in 1–1 correspondence with the vertices of the Bruhat-Tits building Δ_{n} for SL_{n}(K_{𝔓}). We make the set X into a graph by defining the distance between two orders to be the distance between their completions at the prime 𝔓, and by placing an edge between two orders in X if and only if they are distance one apart. Then from section 3 we know that X is a regular graph. Note that X is not necessarily a connected graph. For n = 3, the graph induced on an apartment is the union of three isomorphic connected components, each (combinatorially) isomorphic to the original apartment. We can actually say more about the structure of the graph X utilizing the correspondence between X and the building Δ_{n}; however, we do not require the knowledge of any additional structure for our current purposes.

Let J_{A} denote the ideles of the algebra A. As a set

J_{A} = {α̃ = (α_{𝔭}) ∈ ∏_{𝔭}A_{𝔭}^{×} | α_{𝔭} ∈ Λ_{𝔭}^{×} for almost all 𝔭}

where the product is over all primes 𝔭 of K including the infinite ones.

Since all maximal orders in the local algebras are conjugate (in fact, unique when A_{𝔭} is a division algebra), the local-global correspondence between orders tells us that the ideles J_{A} act transitively on the elements of X:

Λ ↔ {Λ_{𝔭}} ↔ {α_{𝔭}^{−1}Λ_{𝔭}α_{𝔭}} = {Γ_{𝔭}} ↔ Γ = α̃^{−1}Λα̃.

By a left Λ-ideal in A we mean an R-lattice I on A such that I_{𝔭} = Λ_{𝔭}α_{𝔭} for all finite 𝔭 ∈ R and α_{𝔭} ∈ A_{𝔭}^{×}. Using the Invariant Factor Theorem (Theorem 81.11 of [3]), it is easy to see that α_{𝔭} ∈ Λ_{𝔭}^{×} for almost all primes 𝔭, so as shorthand, we will write I = Λα̃ where α̃ = (α_{𝔭}) ∈ J_{A} is any idele giving the proper localizations at the finite places. Two left Λ-ideals I and J are in the same ideal class if I = J α for some α ∈ A^{×}. Let

U(Λ) = {α̃ = (α_{𝔭}) ∈ J_{A} | α_{𝔭} ∈ Λ_{𝔭}^{×} for all finite primes 𝔭 of R}.

Then the left Λ-ideals are in 1–1 correspondence with the cosets U(Λ)\J_{A}, and the left Λ-ideal classes are in 1–1 correspondence with the double cosets U(Λ)\J_{A}/A^{×}.

The number of such ideal classes is finite (e.g., see Theorem 26.4 of [4]), and is called the class number of Λ. We have now defined both objects we need to compare.

Consider the R^{(𝔓)}-order

Λ^{(𝔓)} = Λ ⊗_{R} R^{(𝔓)} = ∩_{𝔮≠𝔓}(Λ_{𝔮} ∩ A)

where the intersection is over all finite R-primes 𝔮. Note that every maximal order Γ in X generates the same R^{(𝔓)}-order since Λ_{𝔮} = Γ_{𝔮} for all primes 𝔮 ≠ 𝔓 of R. Let G = Λ^{(𝔓)}^{×}, the unit group of Λ^{(𝔓)}.

Proposition 5.1. There is a natural group action of the group G on the graph X in which G acts as a group of isometries.

Proof. As usual, embed A^{×} into J_{A} as the diagonal. Note that for all finite primes 𝔮 ≠ 𝔓 of K, Λ_{q}^{(𝔓)} = Λ_{𝔮} (and Λ_{𝔮} = A_{𝔮} for 𝔮 a finite prime in S). Consequently,

α ∈ Λ^{(𝔓)}^{×} implies that α ∈ Λ_{𝔮}^{×} for all 𝔮 ≠ 𝔓. Then G = Λ^{(𝔓)}^{×} acts by conjugation on the maximal orders in X: For Γ ∈ X, and α ∈ G, we have

α^{−1}Γ α ↔ \{α^{−1}Γ_{𝔮} α\} ↔ \begin{cases}Γ_{𝔮} = Λ_{𝔮} & if 𝔮 ≠ 𝔓 \\ α^{−1}Γ_{𝔓} α & if 𝔮 = 𝔓 \end{cases}\} ↔ \begin{cases}Λ_{𝔮} & if 𝔮 ≠ 𝔓 \\ α^{−1}Γ_{𝔓} α & if 𝔮 = 𝔓 \end{cases}\} ∈ X,

so G acts on the vertices of X. To show that the action is an isometry and in particular to show the action takes edges to edges, choose two vertices Γ, Γ′ ∈ X. By definition, the distance between two orders in X is defined to be the distance between their completions at 𝔓 in Δ_{n}. Thus if α ∈ G, we need only show that the distance between Γ_{𝔓} and Γ′_{𝔓} is the same as the distance between α^{−1}Γ_{𝔓} α and α^{−1}Γ′_{𝔓} α. Recall that all maximal orders in A_{𝔓} are conjugate to Λ_{𝔓}, so we set Γ_{𝔓} = γΛ_{𝔓}γ^{−1} and Γ′_{𝔓} = γ′Λ_{𝔓}γ′^{−1}, γ, γ′ ∈ A_{𝔓}^{×}. Let L be a lattice on the vector space V (over K_{𝔓}) so that Λ_{𝔓} = End_{O_{𝔓}}(L). Then Γ_{𝔓} = End_{O_{𝔓}}(γL) and Γ′_{𝔓} = End_{O_{𝔓}}(γ′L). From Remark 2.2, we see that the distance between orders is completely determined by the elementary divisors of the lattices which determine the orders. Thus the distance between Γ_{𝔓} and Γ′_{𝔓} is completely determined by the elementary divisors {γL : γ′L} while the distance between α^{−1}Γ_{𝔓} α and α^{−1}Γ′_{𝔓} α is completely determined by the elementary divisors {α^{−1}γL : α^{−1}γ′L}. Using the linearity of α, it is trivial to see that these sets of elementary divisors are equal; hence our group action determines an isometry.

Before stating our main result, it is convenient to remind the reader of Eichler’s norm theorem (e.g. see Theorem 34.9 of [4]).

Theorem 5.2 (Eichler). Let R be a Dedekind domain with quotient field K, and let A be a central simple algebra over K which satisfies the Eichler condition over R. Let L be a normal ideal in A. Then L is a principal ideal if and only if its reduced norm is a principal ideal Rα for α ∈ 𝔲(A).

Here, the Eichler condition is always satisfied for central simple algebras of dimension n^{2}, n ≥ 3 (see [4, Remark 34.4]). Also, a normal ideal is one whose left- (or equivalently right-) order is maximal, and

𝔲(A) = {α ∈ K^{×} | α_{𝔭} > 0 for all infinite 𝔭 for which A_{𝔭} is ramified}.

Note that because we constructed R to have strict class number one, the norm of any normal ideal in A is a principal ideal with totally positive generator, and hence automatically satisfies the requirements of Eichler’s theorem.

Theorem 5.3. The number of orbits in X under the action of G = Λ^{(𝔓)}^{×} is finite and equal to the class number of Λ.

Proof. We prove our result by establishing a bijection between the following sets:

U(Λ)\J_{A}/A^{×} ↔ Λ_{𝔓}^{×}K_{𝔓}^{×}\A_{𝔓}^{×}/Λ^{(𝔓)}^{×} ↔ X/G.

Since Λ is a maximal R-ideal, Λ^{(𝔓)} is a maximal R^{(𝔓)}-ideal (see e.g., Chapter IV of [5]). By Eichler’s theorem and our condition on R, we see that Λ^{(𝔓)} has class number one, that is,

|U(Λ^{(𝔓)})\J_{A}/A^{×}| = 1  or  J_{A} = U(Λ^{(𝔓)})A^{×}.

Also, since R has class number one, we have that the ideles of K, K_{A}^{×}, can be expressed K_{A}^{×} = U(R)K^{×}. For notational convenience, we “decompose” the groups U(Λ^{(𝔓)}) and U(R) and write

J_{A} = A_{∞}^{×} · A_{𝔓}^{×} · ∏_{𝔮≠𝔓} Λ_{q}^{(𝔓)}^{×} · A^{×}  and  K_{A}^{×} = K_{∞}^{×} · R_{𝔓}^{×} · ∏_{𝔮≠𝔓} R_{𝔮}^{×} · K^{×}.

where A_{∞}^{×} and K_{∞}^{×} denote (respectively) the product of A_{𝔭}^{×} (respectively K_{𝔭}^{×}) over the infinite primes of K. We recall that for finite primes 𝔮 ≠ 𝔓 we have Λ_{q}^{(𝔓)} = Λ_{𝔮}. For 𝔮 ∈ S (finite), Λ_{q}^{(𝔓)} = Λ_{𝔮} = A_{𝔮} and R_{𝔮} = K_{𝔮}, while for 𝔮 ∉ S (𝔮 ≠ 𝔓), Λ_{q}^{(𝔓)} = Λ_{𝔮} ≠ A_{𝔮} and R_{𝔮} = O_{𝔮}. In particular, for all finite primes 𝔮 ≠ 𝔓, R_{𝔮}^{×} ⊂ Λ_{𝔮}^{×}. Finally, R_{𝔓} = O_{𝔓}. Thus,

J_{A} = A_{∞}^{×} · A_{𝔓}^{×} · ∏_{𝔮≠𝔓} Λ_{𝔮}^{×} · A^{×}  and  K_{A}^{×} = K_{∞}^{×} · O_{𝔓}^{×} · ∏_{𝔮≠𝔓} R_{𝔮}^{×} · K^{×}.

Consistent with the above decompositions, write an element α̃ ∈ J_{A} as α̃ = (α_{∞}; α_{𝔓}, α_{0}) · a; similarly, for k̃ ∈ K_{A}^{×}, write k̃ = (k_{∞}; k_{𝔓}, k_{0}) · k. Consider the map

Λ_{𝔓}^{×}K_{𝔓}^{×}\A_{𝔓}^{×}/Λ^{(𝔓)}^{×} → U(Λ)\J_{A}/A^{×} defined by Λ_{𝔓}^{×}K_{𝔓}^{×}a_{𝔓}Λ^{(𝔓)}^{×} ↦ U(Λ)(1; a_{𝔓}, 1)A^{×}.

First we show that this map is well-defined. We only need to show that

U(Λ)(1; λ_{𝔓}k_{𝔓}a_{𝔓}λ, 1)A^{×} = U(Λ)(1; a_{𝔓}, 1)A^{×}

where λ_{𝔓} ∈ Λ_{𝔓}^{×}, k_{𝔓} ∈ K_{𝔓}^{×} and λ ∈ Λ^{(𝔓)}^{×}. Using K_{A}^{×} = U(R)K^{×}, we see that U(Λ)K_{A}^{×}K^{×} = U(Λ)K^{×}. This, together with the fact that K^{×} is in the center of A^{×} and A_{𝔮}^{×} for all primes 𝔮, yields U(Λ)(1; λ_{𝔓}k_{𝔓}a_{𝔓}λ, 1)A^{×} = U(Λ)(1; a_{𝔓}λ, 1)A^{×}. Now

U(Λ)(1; a_{𝔓}λ, 1)A^{×} = U(Λ)(λ^{−1}; a_{𝔓}, λ^{−1})A^{×} since Λ^{(𝔓)}^{×} ⊂ A^{×} is embedded in J_{A} as the diagonal. Since Λ^{(𝔓)}^{×} is contained in A_{∞}^{×} and Λ_{𝔮}^{×} for 𝔮 ≠ 𝔓, (λ^{−1}, 1, λ^{−1}) ∈ U(Λ) which yields the result.

Next we show that the map is onto: By Eichler’s theorem, we have that J_{A} = U(Λ^{(𝔓)})A^{×}, and since U(Λ) and U(Λ^{(𝔓)}) differ only in the 𝔓-th component, it is trivial to see that for any α̃ = (α_{𝔭}) ∈ J_{A}, U(Λ)α̃A^{×} = U(Λ)(1; α_{𝔓}, 1)A^{×}.

Finally we show that the map is one-to-one: Suppose that

U(Λ)(1; α_{𝔓}, 1)A^{×} = U(Λ)(1; β_{𝔓}, 1)A^{×}.

Then (1; α_{𝔓}, 1) = (λ_{∞}a, λ_{𝔓}β_{𝔓}a, λ_{𝔮}a) where (λ_{∞}; λ_{𝔓}, λ_{𝔮}) ∈ U(Λ) and a ∈ A^{×}. It follows that α_{𝔓} = λ_{𝔓}β_{𝔓}a where λ_{𝔓} ∈ Λ_{𝔓}^{×}, and a ∈ A^{×}, and λ_{𝔮}a = 1 for all 𝔮 ≠ 𝔓.

Thus a ∈ ∩_{𝔮≠𝔓}(Λ_{𝔮}^{×} ∩ A^{×}) = Λ^{(𝔓)}^{×}, and so α_{𝔓} ∈ Λ_{𝔓}^{×}K_{𝔓}^{×}β_{𝔓}Λ^{(𝔓)}^{×} which completes the proof of injectivity.

Now we show that there is a bijection between Λ_{𝔓}^{×}K_{𝔓}^{×}\A_{𝔓}^{×}/Λ^{(𝔓)}^{×} and the orbits X/G of G in X. Recall that the points in X are maximal R-orders Γ in A satisfying Γ_{𝔮} = Λ_{𝔮} for all 𝔮 ≠ 𝔓; thus the points in X are in 1–1 correspondence with maximal O_{𝔓}-orders in A_{𝔓}, all of which are of the form α_{𝔓}^{−1}Λ_{𝔓}α_{𝔓} for some α_{𝔓} ∈ A_{𝔓}^{×}. Now 𝔓 is unramified in A by choice, and we have chosen to identify A_{𝔓} with M_{n}(K_{𝔓}) in such a way that Λ_{𝔓} = M_{n}(O_{𝔓}) = M_{n}(R_{𝔓}). By Corollary 37.26 of [4], the normalizer of Λ_{𝔓} is Λ_{𝔓}^{×}·K_{𝔓}^{×}, so the points of X are in 1–1 correspondence with the cosets Λ_{𝔓}^{×}K_{𝔓}^{×}\A_{𝔓}^{×}. Let Γ ∈ X, with Γ_{𝔓} = α_{𝔓}^{−1}Λ_{𝔓}α_{𝔓} for some α_{𝔓} ∈ K_{𝔓}^{×}. An orbit of Γ under the action of G =

Λ^{(𝔓)}^{×} thus corresponds to the double coset Λ_{𝔓}^{×}K_{𝔓}^{×}α_{𝔓}Λ^{(𝔓)}^{×}. This correspondence is

obviously onto. Moreover, if Λ_{𝔓}^{×}K_{𝔓}^{×}α_{𝔓}Λ^{(𝔓)}^{×} = Λ_{𝔓}^{×}K_{𝔓}^{×}β_{𝔓}Λ^{(𝔓)}^{×}, then β_{𝔓} = λ_{𝔓}k_{𝔓}α_{𝔓}λ

for some λ_{𝔓} ∈ Λ_{𝔓}^{×}, k_{𝔓} ∈ K_{𝔓}^{×} and λ ∈ Λ^{(𝔓)}^{×}. Thus β_{𝔓}^{−1}Λ_{𝔓}β_{𝔓} = λ^{−1}α_{𝔓}^{−1}Λ_{𝔓}α_{𝔓}λ (since λ_{𝔓}k_{𝔓} is in the normalizer of Λ_{𝔓}) is in the same orbit as α_{𝔓}^{−1}Λ_{𝔓}α_{𝔓}. Hence the correspondence is one-to-one.

References

[1] K. Brown, Buildings, Springer-Verlag, New York, 1989.

[2] M. Newman, Integral Matrices, Academic Press, New York, 1972. Springer-Verlag, New York, 1984.

[3] O. T. O’Meara, Introduction to Quadratic Forms, Springer-Verlag, New York, 1973.

[4] I. Reiner, Maximal Orders, Academic Press, New York, 1975.

[5] K. Roggenkamp, Lattices over Orders I, Springer-Verlag, New York, 1970.

[6] M. Ronan, Lectures on Buildings, Perspectives in Mathematics, vol 7, Academic Press, San Diego, 1989.

[7] J-P. Serre, Trees, Springer-Verlag, Berlin, 1980.

[8] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, Princeton, 1971.

[9] M-F. Vignéras, Arithmétique des Algègres de Quaternions, Lecture Notes in Mathematics, 800, Springer-Verlag, New York, 1980.

Department of Mathematics, Fairfield University, Fairfield, Connecticut 06430

 E-mail address: aschwartz@fair1.fairfield.edu

Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755

 E-mail address: Thomas.Shemanske@dartmouth.edu