MING-HSUAN KANG AND WEN-CHING WINNIE LI
Abstract. In this paper we obtain a closed form expression of the zeta function Z(XΓ, u) of a finite
quotient XΓ= Γ\PGL3(F )/PGL3(OF) of the Bruhat-Tits building of PGL3over a nonarchimedean
local field F . Analogous to a graph zeta function, Z(XΓ, u) is a rational function and it satisfies
the Riemann hypothesis if and only if XΓ is a Ramanujan complex.
1. Introduction
First introduced by Ihara [Ih] for groups and later reformulated by Serre for regular graphs, the zeta function of a finite, connected, undirected graph X is defined as
Z(X, u) =Y
[C]
(1 − ul([C]))−1,
where the product is over equivalence classes [C] of backtrackless tailless primitive cycles C, and l([C]) is the length of a cycle in [C]. Taking the logarithmic derivative of Z(X, u), one gets
Z(X, u) = exp X n≥1 Nn n u n ,
where Nn counts the number of backtrackless and tailless cycles in X of length n.
Not only formally analogous to a curve zeta function, the graph zeta function is also a rational function. This can be seen in two ways. The first is the result of Ihara:
Theorem 1.0.1 (Ihara [Ih]). Suppose X = (V, E) with vertex set V and edge set E is (q + 1)-regular. Then its zeta function is a rational function of the form
Z(X, u) = (1 − u
2)χ(X)
det(I − Au + qu2I),
where χ(X) = #(V ) − #(E) is the Euler characteristic of X and A is the adjacency matrix of X.
2000 Mathematics Subject Classification. Primary: 22E35; Secondary: 11F70 . Key words and phrases. Bruhat-Tits building, zeta functions.
The research of both authors are supported in part by the DARPA grant HR0011-06-1-0012 and the NSF grants DMS-0457574 and DMS-0801096. Part of the research was performed while both authors were visiting the National Center for Theoretical Sciences, Mathematics Division, in Hsinchu, Taiwan. They would like to thank the Center for its support and hospitality.
If X is not regular, the same expression holds with qI replaced by the valency matrix of X minus the identity matrix. This was proved by Bass [Ba] and Hashimoto [Ha2]; Stark and Terras provided several proofs in [ST], while Hoffman [Ho] gave a cohomological interpretation. The reader is referred to [ST] and the references therein for the history and various zeta functions attached to a graph.
Endow two orientations on each edge of X. Define the neighbors of the directed edge u → v to be the edges v → w with w 6= u. The edge adjacency matrix Ae has its rows and columns indexed
by the directed edges e of X such that the ee0 entry is 1 if e0 is a neighbor of e, and 0 otherwise. Hashimoto [Ha] observed that Nn = TrAne so that
Z(X, u) = 1
det(I − Aeu)
.
This gives the second viewpoint of the rationality of the graph zeta function.
A (q + 1)-regular graph X is called Ramanujan if all eigenvalues λ of its adjacency matrix A other than ±(q + 1) satisfy |λ| ≤ 2√q (cf. [LPS]). The Ramanujan graphs are optimal expanders with extremal spectral property. It is easily checked that X is Ramanujan if and only if its zeta function Z(X, u) satisfies the Riemann hypothesis, that is, the poles of Z(X, u) other than ±1 and ±q−1, called nontrivial poles, all have absolute value q−1/2 (cf. [ST]).
When q is a prime power, the universal cover of a (q + 1)-regular graph can be identified with the (q + 1)-regular tree on PGL2(F )/PGL2(OF) for a nonarchimedean local field F with ring of
integers OF and q elements in its residue field. Let π be a uniformizer of F . The vertices of the
tree are PGL2(OF)-cosets and the directed edges are I-cosets, where I is the Iwahori subgroup
of PGL2(OF). Moreover, the (vertex) adjacency operator A on the tree is the Hecke operator
given by the double coset PGL2(OF)diag(1, π)PGL2(OF) and the edge adjacency operator Ae is
the Iwahori-Hecke operator given by the double coset Idiag(1, π)I. One obtains a (q + 1)-regular graph by taking a left quotient by a torsion-free discrete cocompact subgroup of PGL2(F ).
This set-up has a higher dimensional extension to the Bruhat-Tits building Bn associated to
PGLn(F )/PGLn(OF), which is a simply connected (q + 1)-regular (n − 1)-dimensional simplicial
complex. Its vertices are PGLn(OF)-cosets, naturally partitioned into n types, marked by Z/nZ.
There are n − 1 Hecke operators Ai, for 1 ≤ i ≤ n − 1, associated to PGLn(OF)-double cosets
represented by diag(1, ..., 1, π, ..., π) with determinant πi. A finite quotient X
Γ = Γ\Bn of Bn by
a torsion-free discrete cocompact subgroup Γ preserving the types of vertices is again a (q + 1)-regular finite complex. It is called a Ramanujan complex if all the nontrivial eigenvalues of Ai
explicit constructions of infinite families of Ramanujan complexes are given in Li [Li], Lubotzky-Samuels-Vishne [LSV1] and Sarveniazi [Sa], respectively, using deep results on the Ramanujan conjecture over function fields for automorphic representations of the multiplicative group of a division algebra by Laumon-Rapoport-Stuhler [LRS] and of GLn by Lafforgue [La]. Further, the
paper [LSV2] discusses what kind of Γ would fail to yield a Ramanujan complex.
To extend the results from graphs to complexes, one seeks a similarly defined zeta function of closed geodesics in XΓ with the following properties:
(1) it is a rational function with a closed form expression;
(2) it captures both topological and spectral information of XΓ; and
(3) it satisfies the Riemann hypothesis if and only if XΓ is a Ramanujan complex.
Questions of this sort were previously considered in Deitmar [De1], [De2], and Deitmar-Hoffman [DH], where partial results were obtained.
The purpose of this paper is to present a zeta function with the asserted properties for the case n = 3. In what follows, we fix a local field F with q elements in its residue field as before. Write G for PGL3(F ), K for its maximal compact subgroup PGL3(OF), and B for the Bruhat-Tits building
B3. Similar to a tree, the geometric objects in the building B can be parametrized algebraically.
More precisely, the vertices of B are the right K-cosets on which the group G acts transitively by left translation. A directed edge has type 1 or 2, and opposite edges have different types. Let σ = 11
π
. As the stabilizer of the type 1 edge K → σK, denoted by e0, is E := K ∩ σKσ−1, the
right E-cosets parametrize the type 1 edges of B. The Iwahori subgroup B := K ∩ σKσ−1∩ σ−1Kσ
stabilizes the three vertices K, σK and σ2K of the chamber C
0. Since the stabilizer of C0 in G is
B ∪ Bσ ∪ Bσ2 and the type 1 edges of C
0 are σie0 for 1 ≤ i ≤ 3, the right B-cosets parametrize
the directed chambers (C, e) of B, where e is a type 1 edge of the chamber C. Define the neighbors of a type 1 edge gK → g0K to be the type 1 edges g0K → g00K such that gK, g0K and g00K do not form a chamber. Further, the neighbors (C0, e0) of a directed chamber (C, e) with e the edge g1K → g2K and g3K the third vertex of C are defined as follows: C0 are the chambers other than
C which share the edge g2K → g3K and the type 1 edge e0 is from g3K to the third vertex of C0.
The zeta function of XΓ is defined as
Z(XΓ, u) =
Y
[C]
(1 − ulA([C]))−1,
where [C] runs through the equivalence classes of tailless primitive closed geodesics consisting of edges of the same type, and lA([C]) is the algebraic length of any geodesic in [C].
Main Theorem. Let Γ be a discrete cocompact torsion-free subgroup of G such that (I) ordπdet Γ ⊆ 3Z, and
(II) Γ is regular, namely, the stabilizer of any non-identity element of Γ in G is a torus. Then the zeta function of the (q + 1)-regular finite complex XΓ = Γ\B is a rational function
Z(XΓ, u) =
(1 − u3)χ(XΓ)
det(I − A1u + qA2u2− q3u3I) det(I + LBu)
, (1.1)
in which χ(XΓ) is the Euler characteristic of XΓ, and LB is the Iwahori-Hecke operator given by
the B-double coset Bt2σ2B, where t2 =
π−1
1 π
.
Similar to the graph zeta function, our complex zeta function can be expressed as Z(XΓ, u) = 1 det(I − LEu) det(I − (LE)tu2) = 1 det(I − LEu) det(I − LEu2) ,
where LE is the operator given by the double coset E(t2σ2)2E, which is also the adjacency matrix
of type 1 edges in XΓ. Since the opposite of the type 1 edges are the type 2 edges, the transpose
(LE)t is the adjacency matrix for type 2 edges. Likewise, LB may be viewed as the adjacency
matrix of directed chambers in XΓ. Consequently, the identity (1.1) can be expressed in terms of
operators on XΓ as (1 − u3)χ(XΓ) det(I − A1u + qA2u2− q3u3I) = det(I + LBu) det(I − LEu) det(I − (LE)tu2) , (1.2)
while the parallel identity of operators on a (q + 1)-regular graph X reads (1 − u2)χ(X)
det(I − Au + qu2I) =
1 det(I − Aeu)
.
The similarity is reminiscent of the zeta functions attached to a surface and a curve over a finite field. Since (1.2) is expressed in terms of the operators on the finite complex, it is likely to be the prototype of complex zeta functions in general.
Z(XΓ, u) clearly has properties (1) and (2). Now we discuss its connection with the Riemann
hypothesis. The trivial zeros of det(I − A1u + qA2u2− q3u3I) arise from the trivial eigenvalues of
A1 and A2 on XΓ; they are 1, q−1, q−2 and their multiples by cubic roots of unity. An equivalent
statement for XΓ being Ramanujan is that the nontrivial zeros of det(I − A1u + qA2u2− q3u3I)
all have absolute value q−1 (cf.[Li]), which is the Riemann hypothesis for Z(XΓ, u). The zeros of
each determinant in (1.2) are computed in [KLW], where equivalent statements are obtained. Theorem 1.0.2 ([KLW], Theorem 2). The following four statements on XΓ are equivalent.
(2) The nontrivial zeros of det(I − A1u + qA2u2− q3u3I) have absolute value q−1;
(3) The nontrivial zeros of det(I + LBu) have absolute values 1, q−1/2 and q−1/4; and
(4) The nontrivial zeros of det(I − LEu) have absolute values q−1 and q−1/2.
Thus the Riemann hypothesis for Z(XΓ, u) is actually a statement concerning the nontrivial zeros
of each determinant in (1.2), analogous to the Riemann hypothesis for the surface zeta function. A representation-theoretical proof of (1.2) is given in [KLW]. It should be pointed out that the right hand side of (1.2) is equal to Z(XΓ, u)/Z2(XΓ, −u), where Z2(XΓ, u) is the zeta function
of tailless type 1 closed galleries in XΓ. As shown in §10, this quotient also affords another
interpretation as the product of a geometric and an algebraic zeta functions: Z1(XΓ, u)Z−(Γ, u),
where Z1(XΓ, u) = 1/ det(I − LEu) involves type 1 geodesic cycles in XΓ, and Z−(Γ, u) involves
conjugacy classes in Γ of negative type. This interpretation gives an infinite product expression of the left hand side of (1.2) (cf. Theorem 10.3.2).
This paper is organized as follows. In §2 the types and lengths of elements in G and geodesics in B are introduced. Properties of elements in Γ and basic concepts of cycles in the finite complex XΓ are discussed in §3, while recursive relations of Hecke operators on XΓ are laid out in §4.
The based homotopy classes of closed geodesics in XΓ are partitioned into sets indexed by the
conjugacy classes [γ] of Γ, with each set consisting of based homotopy classes which are base-point free homotopic to the path from K to γK. Each set [γ] has a type, algebraic length and geometric length, defined in terms of those of the rational form of γ, which depends on γ up to conjugacy. Theorem 3.6.1 says that the lengths of the set [γ] are the minimal respective lengths of the homotopy cycles contained in the set. Cycles achieving minimal geometric (resp. algebraic) length in each [γ] are called tailless (resp. algebraically tailless). In other words, among the cycles base-point free homotopic to each other, the shortest ones are called tailless. This definition also applies to graphs. Algebraically tailless cycles afford an explicit algebraic characterization, as shown in §5 and §6 according as γ is split or rank-one split, and hence are more amenable to computation. We shall see in §5 and §6 that, for type 1 and type 2 cycles, there is no distinction between algebraic tailless and tailless (Corollaries 5.1.2 and 6.3.2).
While the zeta function only concerns tailless cycles of types 1 and 2, to find its closed form, we have to consider all cycles up to homotopy. Indeed, we shall compute the number of cycles, as well as those of type 1, in a set [γ] with given algebraic length. This is carried out in §5 and §6. As shown in §9, where the Main Theorem is proved, these numbers can be put together to show that the logarithmic derivative of the left hand side of (1.2) counts the number of type 1 tailless closed
geodesics in XΓ, namely, those from the logarithmic derivative of 1/ det(I − LEu), and some extra
terms arising from sets represented by rank-one split γ’s.
§7 and §8 are devoted to explaining these extra terms. In §7 we discuss type 1 tailless closed galleries and define chamber zeta function Z2(XΓ, u), while the zeta function on type 1 tailless
closed geodesics, Z1(XΓ, u), is discussed in §8. The boundary of a type 1 tailless closed gallery is
analyzed in §8.2, where it is shown that the boundary of an even/odd length gallery consists of two/one tailless type 1 cycle(s). The information on the boundary further leads to a criterion on the chambers occurring in a type 1 tailless closed gallery. This in turn allows us to compute the logarithmic derivative of det(I+LBu)
det(I−(LE)tu2) =
Z1(XΓ,u2)
Z2(XΓ,−u), which gives the extra terms.
In §10 the Ihara (group) zeta function Z(Γ, u) attached to Γ is introduced, analogous to the original definition in [Ih] for the case of PGL2(F ), as an infinite product over primitive conjugacy
classes in Γ. By separating these conjugacy classes into positive and negative types, we show that the product over those with negative type, denoted by Z−(Γ, u), accounts for the extra terms
alluded above, and thus provides a different interpretation of (1.2). Finally we remark that for PGL2, Ihara group zeta function coincides with the graph zeta function attached to the quotient
of the tree by the group, but this is no longer true for PGL3.
2. Edges and Geodesics in B
2.1. Hecke operators. The group G is the disjoint union of the K-double cosets Tn,m = K diag(1, πm, πm+n)K
as m, n run through all non-negative integers. We shall also regard Tn,m as the Hecke operator
acting on functions f ∈ L2(G/K) via
Tn,mf (gK) = X αK∈Tn,m/K f (gαK). In particular, A1 = T1,0 and A2 = T0,1.
2.2. Description of type 1 and type 2 edges. The vertices of B are parametrized by G/K. We identify a vertex gK with the equivalence class of the rank three OF-lattice L generated by the
three column vectors of g. Two vertices gK and g0K are adjacent if they can be represented by lattices L and L0, respectively, such that πL ⊂ L0 ⊂ L. Three mutually adjacent vertices form a 2-dimensional simplex, called a chamber. This structure makes the building B a simply connected 2-dimensional simplicial complex.
A vertex gK has a type τ (gK) defined by ordπdet g mod 3. Adjacent vertices do not have the
same type. The type of a directed edge gK → g0K is τ (g0K) − τ (gK) = i, which is 1 or 2. Out of each vertex there are q2 + q + 1 edges of a given type. The type 1 edges out of gK have terminal
vertices gαK, where αK are the K-cosets contained in the double coset
A1 = T1,0 = K 1 1 π K = [ a,b∈OF/πOF π a b 1 1 K [ c∈OF/πOF 1 π c 1 K[ 1 1 π K.
Similarly, the terminal vertices of type 2 edges out of gK can be described using the following q2 + q + 1 left K-coset representatives of A2 = T0,1:
π b π c 1 , π a 1 π and 1 π π , where a, b, c ∈ OF/πOF.
2.3. Geodesics and lengths in B. Since B is simply connected, all paths between two vertices are homotopic. By a geodesic between two vertices of B we mean a path with shortest length in the 1-skeleton of the building B, which is the (undirected) graph with vertex set G/K and adjacency operator A1+ A2.
It can be shown that all geodesics between two vertices g1K and g2K with g1−1g2 ∈ Tn,m lie in
the same apartment, and they use n type 1 edges and m type 2 edges. We say that they have type (n, m). When m = 0 (resp. n = 0), the path is called type 1 (resp. type 2) for short. Define lG(g1-1g2) := n + m to be the geometric length of the homotopy class in B of the geodesics from
g1K to g2K. The algebraic length of this homotopy class is lA(g1-1g2) := n + 2m. Note that the
same path traveled backwards has algebraic length m + 2n. Further, when the path has type 1 or 2, there is only one geodesic between the two vertices.
3. Finite quotients of B
3.1. The group Γ. Let Γ be a discrete cocompact torsion-free subgroup of G which acts on B by left translations. Then Γ intersects any compact subgroup of G trivially. In particular, Γ acts on B free of fixed points. Denote by XΓ = Γ\B the finite quotient, whose vertices are the double
cosets Γ\G/K.
Example 3.1.1. Let M be a number field or a function field such that F = Mv is the completion
of M at a nonarchimedean place v. Fix a place ∞ of M different from v, which is an archimedean place if M is a number field. Let H be a division algebra of dimension 9 over M which is unramified at v and ramified at ∞, and D = H×/center. Then for any congruence subgroup K of Q
finite places w6=v,∞ D(Ow), Γ = D(M ) ∩ K
Q
remaining places wD(Mw) is a discrete cocompact regular
subgroup of G. Since the torsion subgroup of Γ is finite, choosing a smaller congruence subgroup if necessary, we may assume that Γ is also torsion free.
3.2. Homotopy classes of closed paths in XΓ. The 1-skeleton of XΓ is an undirected graph
with the adjacency operator A1+ A2. We study cycles on this graph which are homotopic in XΓ.
A closed geodesic in XΓ starting at the vertex ΓgK can be lifted to a path in B starting at gK
and ending at γgK for some γ ∈ Γ. Denote by κγ(gK) the homotopy class of the geodesic paths
from gK to γgK in B. Note that, when projected to XΓ, κγ(gK) has shortest geometric length
among all cycles in its homotopy class in XΓ. By abuse of notation, we also use κγ(gK) to denote
its homotopy class in XΓ. Thus the fundamental group of XΓ based at ΓgK is
π1(XΓ, ΓgK) = {κγ(gK) : γ ∈ Γ}.
Since Γ has no fixed points, all κγ(gK) are distinct and π1(XΓ, ΓgK) is isomorphic to Γ.
When all base points are taken into account, the set of based homotopy classes of all closed geodesics in XΓ is parametrized by
Γ × Γ\G/K ∼= a
ΓgK∈Γ\G/K
π1(XΓ, ΓgK).
For each conjugacy class of Γ fix a representative γ and denote that class by hγiΓ. Let [Γ] = {γ}
be the set of representatives of conjugacy classes. Since the conjugacy class of γ in Γ corresponds bijectively to Γ modulo the centralizer CΓ(γ) of γ in Γ, we have Γ =`γ∈[Γ]hγiΓ ≈`γ∈[Γ]CΓ(γ)\Γ,
and hence Γ × Γ\G/K ∼= a γ∈[Γ] (CΓ(γ)\Γ) × (Γ\G/K) ∼= a γ∈[Γ] CΓ(γ)\G/K.
Letting, for each γ ∈ [Γ],
[γ] = {κγ(gK) | g ∈ CΓ(γ)\G/K},
(3.1)
we can express the set of all based homotopy classes of XΓ as the disjoint union of [γ] over γ ∈ [Γ].
Two based homotopy classes of XΓ are said to be base-point free homotopic if a cycle in one
(H1) Shifting the starting vertex to another vertex on the cycle;
(H2) Replacing the cycle by a homotopic cycle while holding the end points fixed. The set [γ] has the following geometric interpretation.
Proposition 3.2.1. Let γ ∈ [Γ]. The set [γ] defined by (3.1) has the following properties: (i) It is closed under base-point free homotopy;
(ii) Any two classes in [γ] are base-point free homotopic;
(iii) The set [γ] is independent of the choice of representative γ in the conjugacy class hγiΓ.
Consequently, [γ] consists of all based homotopic classes which are base-point free homotopic to κγ(K).
Proof. (i) Take any homotopy cycle κγ(gK) in [γ]. It is represented by the path gK → g2K →
· · · → γgK in B. If we shift the starting vertex to g2K, the resulting cycle in XΓ is represented
by the path g2K → · · · → γgK from κγ(gK) followed by γgK → γg2K in B, which is homotopic
to κγ(g2K). This shows that (H1) is satisfied. (H2) is obvious. This proves (i).
(ii) Let κγ(gK) : gK → · · · → γgK and κγ(hK) : hK → · · · → γhK be two homotopy cycles in
[γ]. Then κγ(gK) is homotopic in B and hence in XΓto the path C which is a path P (gK, hK) from
gK to hK followed by κγ(hK) then followed by P (γhK, γgK), the left translation of P (gK, hK)
by γ. Next, shifting the vertex on C from gK to hK, we obtain a new path C0, which is κγ(hK)
followed by P (γhK, γgK) and then by P (gK, hK). Note that on XΓ the last two paths are reverse
of each other so that C0 is homotopic to κγ(hK). This proves that κγ(gK) is base-point free
homotopic to κγ(hK), which establishes (ii).
(iii) This is because the classes in [γ] are base-point free homotopic to the cycles in B from K to g−1γgK, with ending vertices represented by all elements in the conjugacy class of γ in G. 3.3. Classification of elements in Γ. For an element h of H, denote by CH(h) the centralizer
of h in H. We investigate elements in Γ.
Proposition 3.3.1. Every element of Γ is diagonalizable over some finite extension of F .
Proof. If an element γ ∈ Γ is not diagonalizable, then there is some g in G such that, up to scalar, g−1γg is one of the following matrices:
1 a 1 a 1 , 1 1 a 1 , or 1 b a b
for some a, b ∈ F× with b 6= 1. For the first case, conjugation by wn = diag(π−n, 1, πn) gives wn-1g-1γgwn = w-1n 1 a 1 a 1 wn= 1 aπn 1 aπn 1 .
Choose n large so that aπn is integral, hence w-1
ng-1γgwn ∈ K, or equivalently γgwnK = gwnK.
This shows that the vertex gwnK is fixed by γ, which is not possible. For the second case, the
same argument with wn= diag(1, 1, πn) implies that gwnK is a fixed point of γ, again impossible.
For the third case, conjugation by wn= diag(1, 1, πn) yields
w-1ng-1γgwn= wn-1 1 b a b wn = 1 b aπn b .
When n is greater than ordπ b − ordπ a, we have w-1ng-1γgwn ∈ Kdiag(1, b, b)K. This means
that the homotopy class of paths from gwnK to γgwnK have the same geometric length for
all n ≥ ordπ b − ordπ a. If we show that no two gwnK lie in the same CΓ(γ)-orbit, then we
will have found infinitely many classes in [γ] having the same geometric length. This certainly contradicts the fact that XΓ is a finite complex. Suppose gwnK and gwmK lie in the same
CΓ(γ)-orbit, that is, CΓ(γ)gwnK = CΓ(γ)gwmK. Then there exists some h ∈ CΓ(γ) such that
g−1hg = h11 h22 h23 h22 and w-1ng-1hgwm = h11 h22 πmh23 πm−nh22 ∈ K. Then h22 and πm−nh22
are both units, which implies m = n.
For γ ∈ Γ, denote by F hγi the field extension of F generated by the eigenvalues of γ. This field is well-defined since the eigenvalues of γ are unique up to common multiples in F×. An element in Γ having three distinct eigenvalues in F is called split ; it is called rank-one split if it has exactly one eigenvalue in F . In the latter case we say it is unramified/ramified rank-one split if its eigenvalues generate an unramified/ramified quadratic extension of F . Note that Γ does not contain elements with no eigenvalue in F . Indeed, if γ is such an element, then its characteristic polynomial is irreducible over F . As ordπ(det γ) = 3m for some integer m, the eigenvalues of γ0 = π−mγ are
units in a cubic extension of F , which implies that γ0 lies in the intersection of Γ with a conjugate of K, and hence is the identity element. Together with the proposition above, we have shown
Theorem 3.3.2 (Classification of elements in Γ). Every element γ of Γ falls in one of the following types: 1) γ is the identity;
2) γ is split, that is, it has three distinct eigenvalues in F×;
3) γ is ramified/unramified rank-one split, that is, F hγi is a ramfield/unramified quadratic exten-sion of F ;
4) γ is irregular, that is, it has a repeated eigenvalue with multiplicity two.
The following conclusion on Γ shown in [KLW] results from the closed form expression of the zeta function identity of XΓ.
Proposition 3.3.3 ([KLW], Corollary 4). Γ contains rank-one split elements.
3.4. Rational form. Let γ be a non-identity element in Γ. If F hγi = F , we may assume that the eigenvalues are 1, a, b ∈ F× with ordπb ≥ ordπa ≥ 0. Then γ is conjugate to rγ := diag(1, a, b).
If γ is rank-one split, then its characteristic polynomial has the form (x − a)(x2 − b0x − c0) with
x2− b0x − c0 irreducible over F . The splitting field of x2− b0x − c0 is a quadratic extension L = F (λ)
of F . We fix the choice of λ so that it is a unit if L is unramified over F and it is a uniformizing element if L is ramified over F . Let x2− bx − c be the irreducible polynomial of λ over F and let ¯
λ be the Galois conjugate of λ. Then ordπc = 0 or 1 according as L is unramified or ramified over
F and ordπb ≥ 12ordπc. There are elements e, d ∈ F such that e + dλ and e + d¯λ are the roots of
x2− b0x − c0 in L. Consequently γ is conjugate to r γ := a e dc d e + db
. We shall assume that
all eigenvalues of rγ are minimally integral. In other words, a, e, d are in OF and at least one of
them is a unit. Call rγ the rational form of γ. Clearly it depends on the conjugacy class of γ.
We study centralizers of γ ∈ Γ.
Proposition 3.4.1. Let γ ∈ Γ be a non-identity element.
(1) If γ is rank-one split, then its centralizer CG(γ) ∼= F hγi× is a non-split torus, and CΓ(γ)
is a free abelian group of rank one.
(2) If γ is split, then its centralizer CG(γ) ∼= (F×)2 is a split torus, and CΓ(γ) is a free abelian
group of rank two.
(3) If γ is irregular, then its centralizer CG(γ) ∼= GL2(F ) is not a torus, and CΓ(γ) is
Proof. (1) Assume γ ∈ Γ is rank-one split. There is an element h ∈ G such that h-1γh = rγ = a e dc d e + db
. Up to scalars we may express
h-1CG(γ)h = 1 x cy y x + by | x, y ∈ F, not both 0 .
The map φ sending g = 1 x cy y x + by
to x+λy yields an isomorphism from h-1C
G(γ)h to F (γ)×
such that the norm of φ(g) is equal to det g. As a torsion-free discrete subgroup of CG(γ), CΓ(γ) is
a free abelian group. If CΓ(γ) has rank greater than one, then φ(h-1CΓ(γ)h) contains a nontrivial
unit u = x + λy. Thus x, y ∈ OF and the norm of u is in O×F. This means that g = φ−1(u)
has integral entries and det g is a unit in OF. In other words, g is a non-identity element in K.
Hence hgh-1 is a non-identity element in Γ ∩ hKh-1, and thus has finite order, contradicting the
torsion-free assumption of Γ. Therefore CΓ(γ) is a free abelian group of rank one.
(2) When γ is split, we have h-1γh = r
γ = diag(1, a, b) for some h ∈ G, and h-1CG(γ)h can
be expressed as {diag(1, x, y) | x, y ∈ F×}, which is isomorphic to F× × F× under the map
diag(1, x, y) 7→ (x, y). Since Γ intersects any compact subgroup of G trivially, CΓ(γ) can be
identified as a subgroup of CG(γ)/(CG(γ) ∩ K) ' (F×/O×F) × (F ×/O×
F) ' Z × Z, thus it has rank
at most 2. If CΓ(γ) has rank less than 2, then CΓ(γ)\CG(γ)K/K is infinite, which contradicts the
finiteness of XΓ. Therefore CΓ(γ) is a rank two abelian group.
(3) When γ is irregular, then h-1γh = r
γ = diag(1, a, a) for some h ∈ G, and h-1CG(γ)h is clearly
isomorphic to GL2(F ). Under this isomorphism, h-1CΓ(γ)h is mapped to a discrete co-compact
torsion-free subgroup of GL2(F ).
In what follows, we assume that Γ satisfies the two additional conditions below:
(I) ordπdet Γ ⊂ 3Z so that Γ identifies vertices of the same type, and consequently XΓ is a
finite connected (q + 1)-regular 2-dimensional simplicial complex.
(II) Γ is regular, that is, Γ does not contain irregular elements. Equivalently, the centralizer in G of any non-identity element in Γ is a torus.
Note that Γ arising from a division algebra of dimension 9 as in Example 3.1.1 contains no irregular elements.
Remark. The condition (II) is imposed to ease our computations. As shown in [KLW] using representation-theoretic approach, this assumption is not needed.
3.5. The type and lengths of a homotopy class. The type, geometric length and algebraic length of a homotopy class κγ(gK) of XΓare those of κγ(gK) in B. In other words, If g−1γg ∈ Tn,m,
then κγ(gK) has algebraic length lA(κγ(gK)) = n + 2m, geometric length lG(κγ(gK)) = n + m,
and type (n, m). By assumption, κγ(gK) has positive length if and only if γ is not identity.
3.6. The type and lengths of [γ]. Let γ ∈ [Γ] be non-identity, and let rγ be its rational form
as defined in §3.4. Fix a choice of Pγ ∈ G such that rγ = (Pγ)−1γPγ. As the centralizers of γ
and rγ are related by CG(γ) = PγCG(rγ)Pγ−1, we have CΓ(γ)Pγ = PγCP−1
γ ΓPγ(rγ), and [γ] may be
expressed in two ways:
[γ] = {κγ(gK) | g ∈ CΓ(γ)\G/K}
= {κγ(PγgK) | g ∈ CPγ−1ΓPγ(rγ)\G/K}.
(3.2)
The second expression will facilitate our computations later on.
Suppose rγ ∈ Tn,m. We say that [γ] has type (n, m), algebraic length lA([γ]) = n + 2m and
geometric length lG([γ]) = n + m. As before, call [γ] of type 1 or 2 according as m = 0 or n = 0.
We shall prove
Theorem 3.6.1. Let γ ∈ [Γ] and γ 6= id. Then
lA([γ]) = minκγ(gK)∈[γ] lA(κγ(gK)) and lG([γ]) = minκγ(gK)∈[γ] lG(κγ(gK)).
Moreover, for g ∈ CG(rγ), we have lA(κγ(PγgK)) = lA([γ]), lG(κγ(PγgK)) = lG([γ]) and the type
of κγ(PγgK) coincides with the type of [γ].
The second assertion is obvious since (Pγg)−1γPγg = g−1rγg = rγ for g ∈ CG(rγ). The proof of
the first assertion is contained in Theorem 5.1.1 for γ split and Theorem 6.3.1 for γ rank-one split. Note that lA(κγ(gK)) ≡ ordπdet γ (mod 3), hence lA(κγ(gK)) = lA([γ]) + 3m for some
3.7. Tailless cycles. In view of Theorem 3.6.1, a homotopy class κγ(gK) is called algebraically
tailless if its algebraic length agrees with lA([γ]). It is called tailless if its geometric length is
lG([γ]). By Proposition 3.2.1, a tailless based homotopy cycle has shortest geometric length among
all cycles base-point free homotopic to it.
3.8. The volume of [γ]. By Proposition 3.4.1 and assumption (II), CG(rγ) is a torus in G
con-taining the discrete cocompact subgroup CP−1
γ ΓPγ(rγ). Let
Ω = ∪g∈G g−1Kg.
Then the double coset CP−1
γ ΓPγ(rγ)\CG(rγ)/(CG(rγ) ∩ Ω) is finite. Its cardinality is the same as
that of CΓ(γ)\CG(γ)/CG(γ) ∩ Ω, called the volume of [γ]:
vol([γ]) = # CP−1
γ ΓPγ(rγ)\CG(rγ)/(CG(rγ) ∩ Ω) = # CΓ(γ)\CG(γ)/CG(γ) ∩ Ω.
(3.3)
Observe that
Lemma 3.8.1. For any g ∈ G, we have CG(rγ) ∩ gKg−1 = CG(rγ) ∩ gKg−1∩ K. Consequently,
CG(rγ) ∩ Ω = CG(rγ) ∩ K.
Proof. Let g ∈ G. It suffices to show CG(rγ) ∩ gKg−1 ⊂ K. Suppose h ∈ CG(rγ) ∩ gKg−1. Without
loss of generality, we may assume that the eigenvalues of h are roots of f (x), the characteristic polynomial of some element in GL3(OF). We distinguish two cases.
Case I. γ is split. Then CG(rγ) consists of diagonal matrices in G. Let α, µ, ν be the diagonal
entries of h with ordπα ≥ ordπµ ≥ ordπν. Then f (x) = (x − α)(x − µ)(x − ν) lies in OF[x]. The
constant term of f (x) is a unit in OF, which implies ordπα ≥ 0 ≥ ordπν. Thus if α, µ, ν have the
same order, then they are all units. If not, then ordπα > 0 and ordπν < 0. Since the coefficient of
x in f (x) is in OF, we have ordπµν ≥ 0, which contradicts ordπαµν = 0. Therefore h ∈ K.
Case II. γ is rank-one split. Then, as in the proof of Proposition 3.4.1,(1), h = α µ cν ν µ + bν with eigenvalues α, µ+νλ, µ+ν ¯λ. Here λ has minimal polynomial x2−bx−c over F , and λ is either
a unit of a uniformizer in the field F (λ). Thus β := α(µ2+µνb−ν2c) is a unit in F , and α +2µ+bν
and δ := α(2µ + bν) + µ2+ µνb − ν2c both lie in OF. If ordπα < 0, then ordπ(2µ + bν) = ordπα < 0
so that ordπα(2µ + bν) < 0 while ordπ(µ2 + µνb − ν2c) = ordπβ/α = −ordπα > 0, contradicting
We obtain the same contradiction. Thus α is a unit, and so are µ + νλ and µ + ν ¯λ. The choice of
λ implies µ, ν ∈ OF. Hence h ∈ K, as desired.
Thus we can express vol([γ]) as vol([γ]) = # CP−1
γ ΓPγ(rγ)\CG(rγ)/(CG(rγ) ∩ K).
(3.4)
Remark. For any integer m 6= 0, the eigenvalues of γm are the m-th power of those of γ, hence
rγm = (rγ)m up to a central element (due to normalization), and thus we may assume Pγm = Pγ.
Consequently, Pγ−1mΓPγm = Pγ−1ΓPγ for all m 6= 0. Clearly, CG(rγ) ⊆ CG(rγm). The reverse
containment follows from the argument in the proof of Proposition 3.4.1. Therefore CG(rγm) =
CG(rγ). This shows vol([γ]) = vol([γm]) for all m 6= 0.
4. Hecke operators on B and on XΓ
4.1. Recursive relations among Hecke operators. It is well-known that each Hecke operator is a polynomial in A1 and A2. Tamagawa [Ta] obtained a recursive relation on Hecke operators:
( X
n,m≥0
Tn,mun+2m)(I − A1u + qA2u2− q3u3I) = (1 − u3)I.
(4.1)
We prove a different recursive formula adapted for our needs. Theorem 4.1.1. q ∞ X k=1 Tk,0uk− (q − 1)( ∞ X k=1 X n+2m=k Tn,muk) 1 − q2u3 1 − u3 = u d dulog (1 − u3)rI I − A1u + A2qu2− q3u3I , (4.2) where r = (q+1)(q−1)3 2.
Proof. The algebra of Hecke operators is isomorphic to the polynomial ring C[z1, z2, z3]S3/hz1z2z3−
1i under the Satake isomorphism ψ (cf. [Sat]). To describe its values on {Tn,m}, let χ be the
quasi-character on the Borel subgroup P of G defined by
χ b1 ∗ ∗ b2 ∗ b3 = zordπ(b1) 1 z ordπ(b2) 2 z ordπ(b3) 3 ,
and regard it as a map from G/K to C[z1, z2, z3]/hz1z2z3− 1i. (The relation z1z2z3 = 1 follows
φ be the function on G given by
φ(bk) = χ(b)δP1/2(b) (b ∈ P, k ∈ K). Then the value of the Satake isomorphism at Tn,m is
ψ(Tn,m) =
Z
G
Tn,m(g)φ(g)dg (n ≥ 0, m ≥ 0),
where dg is the Haar measure on G so that K has volume 1. Direct computations give ψ(A1) =
q(z1 + z2+ z3), ψ(A2) = q(z1z2+ z2z3+ z3z1) and
ψ(I − A1u + qA2u2− q3u3I) = (1 − qz1u)(1 − qz2u)(1 − qz3u).
For k ≥ 1, let Tk = P n+2m=kTn,m, and set σk,1(z1, z2, z3) = z1k+ z k 2 + z k 3, σk,2(z1, z2, z3) = X 1≤a≤k−1
za1z2k−a+ z2az3k−a+ z3az1k−a,
and
σk,3(z1, z2, z3) =
X
a,b,c≥1,a+b+c=k
z1az2bz3c.
Our strategy is to show that the identity (4.2) holds after applying the Satake isomorphism ψ. For this, it suffices to compute the coefficient of za1
1 z a2
2 z a3
3 in ψ(Tk) with a1 ≥ a2 ≥ a3 ≥ 0 and
a1+ a2+ a3 = k, then use symmetry to determine ψ(Tk).
It is straightforward to check that the number of elements gK ∈ F
n+2m=kTn,m/K mapped to za1 1 z a2 2 z a3
3 by χ is equal to q2a1+a2 if a3 = 0, and (q3− 1)q2a1+a2−3 if a3 > 0. Moreover, for such gK
we have δP(gK)1/2= qa3−a1. Therefore the coefficient of z1a1z a2
2 z a3
3 in ψ(Tk) is equal to qa1+a2+a3 or
qa1+a2+a3−3(q3− 1) according to a
3 = 0 or a3 > 0. By symmetry, this yields
ψ(Tk) = qk(σk,1+ σk,2+ q3− 1 q3 σk,3). Noting that ∞ X k=1 σk,3uk = ((z1z2z3)u3+ (z1z2z3)2u6+ · · · ) ∞ X k=0 (1 + σk,1+ σk,2)uk= u3 1 − u3 ∞ X k=0 (1 + σk,1+ σk,2)uk, we obtain ψ( ∞ X k=1 Tkuk) = ∞ X k=1 (σk,1+ σk,2+ q3− 1 q3 σk,3)(qu) k = (q 3 − 1)u3 1 − q3u3 + 1 − u3 1 − q3u3 ∞ X k=1 (σk,1+ σk,2)(qu)k.
On the other hand, put G0 = F ∞
k=1Tk,0. One verifies that the number of elements in G0/K
mapped to za1 1 z a2 2 z a3 3 by χ is q2a1 if a2 = a3 = 0, (q −1)q2a1+a2−1 if a2 > a3 = 0, and (q −1)2q2a1+a2−2 if a2 ≥ a3 > 0. Therefore, ψ( ∞ X k=1 Tk,0uk) = ∞ X k=1 (σk,1+ q − 1 q σk,2+ (q − 1)2 q2 σk,3)(qu) k = q(q − 1) 2u3 1 − q3u3 + 1 + qu3− 2q2u3 1 − q3u3 ∞ X k=1 σk,1(qu)k+ (q − 1)(1 − q2u3) q(1 − q3u3) ∞ X k=1 σk,2(qu)k. Consequently, ψ q( ∞ X k=1 Tk,0uk) − (q − 1)( ∞ X k=1 Tkuk) 1 − q2u3 1 − u3 = ∞ X k=0 σk,1(qu)k+ (q − 1)(q2− 1)u3 1 − u3 = z1qu 1 − z1qu + z2qu 1 − z2qu + z2qu 1 − z2qu − 3ru 3 1 − u3 = u d dulog (1 − u3)r
(1 − z1qu)(1 − z2qu)(1 − z3qu)
= ψ u d dulog (1 − u3)r I − A1u + A2qu2− q3u3I . 4.2. Hecke operators on XΓ. The action of the Hecke operator Tn,m on L2(Γ\G/K) is
repre-sented by the matrix Bn,m, whose rows and columns are indexed by vertices of XΓ such that the
(ΓgK, Γg0K) entry records the number of homotopy classes of geodesic paths from ΓgK to Γg0K in XΓ of type (n, m). Alternatively, this is the number of γ ∈ Γ such that the homotopy classes
of the geodesics from gK to γg0K have type (n, m). The trace of Bn,m then gives the number of
geodesic cycles of type (n, m) up to homotopy. In other words, Tr(Bn,m) = #
κγ(gK) | γ ∈ [Γ], κγ(gK) ∈ [γ] has type (n, m)
. To facilitate our computations, form two kinds of formal power series:
(4.3) X n,m≥0 (n,m)6=(0,0) Tr(Bn,m)un+2m = X γ∈[Γ], γ6=id X κγ(gK)∈[γ] ulA(κγ(gK)), and (4.4) X n>0 Tr(Bn,0)un = X γ∈[Γ], γ6=id X κγ(gK)∈[γ] has type 1 ulA(κγ(gK)).
Proposition 4.2.1. u d dulog (1 − u3)χ(XΓ) det(I − A1u + A2qu2− q3u3I) (4.5) = q X n>0 Tr(Bn,0)un ! − (q − 1) X n,m≥0 (n,m)6=(0,0) Tr(Bn,m)un+2m 1 − q2u3 1 − u3 ,
where the operators are on L2(Γ\G/K), χ(X
Γ) = (q+1)(q−1)
2
3 V is the Euler characteristic of XΓ,
and V is the number of vertices in XΓ.
Proof. Note that Bn,m is Tn,m acting on the space L2(Γ\G/K), so (4.2) also holds with Tn,m
replaced by Bn,m. In other words,
u d duTr log (1 − u3)rI (I − A1u + A2qu2 − q3u3I) = q X n>0 Tr(Bn,0)un ! − (q − 1) X n,m≥0 (n,m)6=(0,0) Tr(Bn,m)un+2m 1 − q2u3 1 − u3 ,
where r = (q+1)(q−1)3 2. Recall that each vertex is incident to q2+ q + 1 type 1 edges and q2+ q + 1
type 2 edges so that the total number of undirected edges in XΓ is
2(q2+q+1)
2 V . Since each edge is
contained in q + 1 chambers, the number of chambers in XΓ is (q+1)3 (q2+ q + 1)V . Therefore the
Euler characteristic of XΓ is χ(XΓ) = V − (q2 + q + 1)V + (q + 1) 3 (q 2+ q + 1)V = (q − 1) 2(q + 1) 3 V = rV.
Using the identity
log(det A) = Tr(log A) for a V × V matrix A, we have
u d duTr log (1 − u3)rI (I − A1u + A2qu2− q3u3I) = u d dulog (1 − u3)χ(XΓ) det(I − A1u + A2qu2− q3u3I) ,
which proves the proposition.
To understand the combinatorial meaning of the right hand side of (4.5), we first determine the algebraic length of κγ(gK), then computePκγ(gK)∈[γ]ulA(κγ(gK))andPκγ(gK)∈[γ] has type 1ulA(κγ(gK)).
5. Homotopy cycles in [γ] for γ split
Let | | be the valuation on F such that |π| = q−1. In this section we fix a split γ ∈ [Γ] with rational form rγ = diag(1, a, b), where ordπb ≥ ordπa ≥ 0.
5.1. Minimal lengths of homotopy cycles in [γ]. We begin by proving the first assertion of Theorem 3.6.1 for the split case.
Theorem 5.1.1. Suppose γ ∈ Γ is split with rγ = diag(1, a, b), where ordπb ≥ ordπa ≥ 0. Then
(1) lA([γ]) = ordπa + ordπb = minκγ(gK)∈[γ] lA(κγ(gK)) and
(2) lG([γ]) = ordπb = minκγ(gK)∈[γ] lG(κγ(gK)).
Proof. The centralizer CG(rγ) consists of the diagonal matrices in G so that G = CG(rγ)U K, where
U = 1 x y 1 z 1 | x, y, z ∈ F/OF .
It suffices to consider the lengths of κγ(PγgK) with g ∈ U . Write g =
1 x y 1 z 1 . Then (Pγg)−1γPγg = g-1rγg = 1 x y 1 z 1 -1 1 a b 1 x y 1 z 1 = 1 x(1 − a) y(1 − b) + xz(b − a) a z(a − b) b ∈ K πe1 πe2 πe3 K
for some integers e1 ≤ e2 ≤ e3. In fact, for 1 ≤ i ≤ 3, e1 + · · · + ei = miny {ordπy} where y runs
through the determinant of all i × i minors of g−1rγg. Consequently,
e1 = min{0, ordπx(1 − a), ordπz(a − b), ordπ(y(1 − b) + xz(b − a))} ≤ 0,
(5.1)
e1+ e2 = min{ordπa, ordπ[x(1 − a)z(a − b) − a(y(1 − b) + xz(b − a))]} ≤ ordπa,
(5.2) and
e1 + e2+ e3 = ordπa + ordπb.
(5.3)
In particular, e3 ≥ ordπb from the last two inequalities. Moreover, we have, for any g ∈ G,
lA(κγ(PγgK)) = e3+ e2+ e1− 3e1 = ordπa + ordπb − 3e1 ≥ ordπa + ordπb = lA([γ])
and
lG(κγ(PγgK)) = e3− e1 ≥ ordπb − e1 ≥ ordπb = lG([γ]).
(5.5)
As noted before, the equalities in (5.4) and (5.5) hold for g ∈ CG(rγ). Therefore
lA([γ]) = min κγ(gK)∈[γ]
lA(κγ(gK)) and lG([γ]) = min κγ(gK)∈[γ]
lG(κγ(gK)).
This proves the theorem.
It follows from (5.5) that if lG(κγ(PγgK)) = lG([γ]) = ordπb, then e3 = ordπb and e1 = 0, which
in turn imply e2 = ordπa because e1+e2+e3 = ordπa+ordπb. Hence a tailless cycle κγ(PγgK) in [γ]
has the same type as [γ]. Further, by (5.4), the condition e1 = 0 implies lA(κγ(PγgK)) = lA([γ]),
so κγ(PγgK) is also algebraically tailless.
Conversely, suppose κγ(PγgK) is algebraically tailless. Then e1 = 0, that is, x(1 − a) ∈ OF,
z(a − b) ∈ OF and y(1 − b) + xz(b − a) ∈ OF. As seen above, κγ(PγgK) is tailless if the additional
condition e1 + e2 = ordπa is satisfied. By (5.2), this amounts to ordπx(1 − a)z(a − b) ≥ ordπa,
which obviously holds when ordπa = 0, i.e., γ has type 1. We record the discussion in
Corollary 5.1.2. Suppose γ ∈ [Γ] is split. Then all tailless cycles in [γ] are also algebraically tailless, and they have the same type as [γ]. Furthermore, if [γ] has type 1, then the algebraically tailless cycles in [γ] are tailless.
5.2. Counting homotopy cycles in [γ] in algebraic length. Let ∆A([γ]) = {gK ∈ G/K | lA(κγ(PγgK)) = lA([γ])}.
As noted before, ∆A([γ]) ⊃ CG(rγ)K/K and is invariant under left multiplication by CPγ−1ΓPγ(rγ).
So the number of algebraically tailless cycles in [γ] is the cardinality of CP−1
γ ΓPγ(rγ)\∆A([γ]).
The following theorem, stated in terms of a formal power series, gives the number of homotopy cycles of a given algebraic length in [γ].
Theorem 5.2.1. Suppose γ ∈ [Γ] is split with rγ = diag(1, a, b). Then
X κγ(gK)∈[γ] ulA(κγ(gK)) = #(C Pγ−1ΓPγ(rγ)\∆A([γ])) u lA([γ]) 1 − u 3 1 − q3u3 = vol([γ])(|1 − a||a − b||b − 1|)−1 ulA([γ]) 1 − u 3 1 − q3u3,
Proof. The group CG(rγ)∩K consists of diagonal matrices whose nonzero entries are units. In view
of Proposition 3.4.1, there are two generators s, t ∈ CG(rγ) such that CG(rγ) = hs, ti(CG(rγ) ∩ K)
and CP−1
γ ΓPγ(rγ) is a subgroup of hs, ti of index vol([γ]). We have (CG(rγ) ∩ K)U K = U K and
CP−1
γ ΓPγ(rγ)\G/K = CPγ−1ΓPγ(rγ)\hs, tiU K/K. Suppose h, h
0 ∈ hs, ti and v, v0 ∈ U are such that
CP−1
γ ΓPγ(rγ)hvK = CPγ−1ΓPγ(rγ)h
0v0K. Replacing h by a suitable multiple from C
Pγ−1ΓPγ(rγ) if
necessary, we may assume hvK = h0v0K, which is equivalent to v−1h−1h0v0 ∈ K. Since v and v0
are unipotent and h−1h0 is diagonal, v−1h−1h0v0 is an upper triangular matrix with diagonal entries being those of h−1h0. This implies that h−1h0 ∈ K and hence is equal to the identity matrix. It then follows from the definition of U that v = v0. This proves that the left hand side of the identity can be expressed as X κγ(PγgK)∈[γ] ulA(κγ(PγgK)) = vol([γ]) X v∈U ulA(κγ(PγvK)).
To proceed, we compute the sum on the right hand side.
Proposition 5.2.2. Let γ be split with rγ = diag(1, a, b). Then
X v∈U ulA(κγ(PγvK)) = u lA([γ]) |1 − a||a − b||b − 1| 1 − u3 1 − q3u3 .
Proof. Given v ∈ U , write v = 1 x y 1 z 1
. As computed in the proof of Theorem 5.1.1,
(Pγv)−1γPγv = v-1rγv = 1 x(1 − a) y(1 − b) + xz(b − a) a z(a − b) b = (vi,j).
For fixed m ≥ 0, we count the number of v’s such that lA(κγ(PγvK)) ≤ lA([γ]) + 3m. By (5.4),
the constraints are |vij| ≤ qm for all 1 ≤ i, j ≤ 3. In other words,
|x(1 − a)| ≤ qm, |z(a − b)| ≤ qm and |y(1 − b) + xz(b − a)| ≤ qm.
This implies
|x| ≤ qm|1 − a|−1
and |z| ≤ qm|a − b|−1
so that the numbers of x and z in F/OF are qm|1 − a|−1 and qm|a − b|−1, respectively. Further, for
chosen x and z, there are qm|1 − b|−1 choices of y satisfying the above constraint. We have shown
#v ∈ U
lA(κγ(PγvK)) = lA([γ]) = (|1 − a||a − b||b − 1|)−1 (5.6)
and, for m > 0, #v ∈ U
lA(κγ(PγvK)) = lA([γ]) + 3m = (q3m− q3m−3)(|1 − a||a − b||b − 1|)−1. (5.7)
Put together, this gives X v∈U ulA(κγ(PγvK)) = u lA([γ]) |1 − a||a − b||b − 1| 1 +X m≥1 (q3m− q3m−3)u3m = u lA([γ]) |1 − a||a − b||b − 1| 1 − u3 1 − q3u3 . The argument above shows that the number of algebraically tailless homotopy classes in [γ] is vol([γ]) times the number of elements in U with m = 0, which is given by (5.6). This proves Proposition 5.2.3. Suppose γ ∈ Γ is split with rγ = diag(1, a, b). Then
#(CP−1
γ ΓPγ(rγ)\∆A([γ])) = vol([γ])(|1 − a||a − b||b − 1|)
−1.
The proof of Theorem 5.2.1 is now complete.
5.3. Counting homotopy cycles of type 1 in [γ]. The theorem below gives the number of type 1 homotopy cycles in [γ] of given algebraic length. The result depends on the type of [γ].
Theorem 5.3.1. Suppose γ ∈ Γ is split with rγ = diag(1, a, b). The following assertions hold.
(i) If [γ] does not have type 1, then X
κγ(gK)∈[γ], type 1
ulA(κγ(gK)) = vol([γ])(|1 − a||a − b||b − 1|)−1 ulA([γ])(1 − q−1)(1 − q
2u3
1 − q3u3).
Moreover, no type 1 cycles in [γ] are tailless. (ii) If [γ] has type 1, then
X κγ(gK)∈[γ], type 1 ulA(κγ(gK)) = vol([γ])(|1 − a||a − b||b − 1|)−1 ulA([γ]) q−1+ (1 − q−1)(1 − q 2u3 1 − q3u3) .
Remark. The right hand side of the identities in Theorem 5.2.1 and Theorem 5.3.1 can be ex-pressed as vol([γ]) times the orbital integrals at the split element γ of suitably chosen spherical functions on G with fast decay.
Proof. Since rγ = diag(1, a, b), [γ] has type (ordπb − ordπa, ordπa) and lA([γ]) = ordπb + ordπa.
the difference is that we only need to consider those v ∈ U such that κγ(PγvK) has type 1. So we
count the number of
{v ∈ U | lG(κγ(PγvK)) = lA(κγ(PγvK)) = lA([γ]) + 3m = ordπ b + ordπ a + 3m}
for each m ≥ 0. As before, writing v as 1 x y 1 z 1
and following the proofs of Proposition 5.2.2
and Theorem 5.1.1, we arrive at the following constraints on x, y, z ∈ F/OF:
(1) min{0, ordπx(1 − a), ordπz(a − b), ordπ(y(1 − b) + xz(b − a))} = −m, and
(2) min{ordπa, ordπ[x(1 − a)z(a − b) − a(y(1 − b) + xz(b − a))]} = −2m.
For m > 0, the two constraints are equivalent to
(3) ordπx(1 − a) = −m = ordπz(a − b) and ordπ(y(1 − b) + xz(b − a)) ≥ −m.
Hence the number of x is (1 − q−1)qm|1 − a|−1, the number of z is (1 − q−1)qm|a − b|−1, and the
number of y is qm|1 − b|−1 so that the total number of v is (1 − q−1)2q3m(|1 − a||a − b||b − 1|)−1. For
m = 0 and ordπa > 0, the same constraint (3) holds. In this case the number of x is |1 − a|−1 = 1,
the number of y is |1 − b|−1 = 1 and the number of z is (1 − q−1)|a − b|−1 so that the total number of v is (1 − q−1)(|1 − a||a − b||b − 1|)−1. Finally, when m = ordπa = 0, the constraints (1) and (2)
are equivalent to
(4) ordπx(1 − a) ≥ 0, ordπz(a − b) ≥ 0 and ordπ(y(1 − b) + xz(b − a)) ≥ 0.
Hence the numbers of x, y and z are |1 − a|−1, |1 − b|−1 and |a − b|−1, respectively, so that the number of v is (|1 − a||a − b||b − 1|)−1.
Since vol([γ])(|1 − a||a − b||b − 1|)−1 is present in both cases, it suffices to compute 1
vol([γ])(|1 − a||a − b||b − 1|)−1
X
κγ(gK)∈[γ], type 1
ulA(κγ(gK)).
In case ordπ a > 0, this sum is equal to
ulA([γ])(1 − q−1+X
m≥1
(1 − q−1)2q3mu3m) = ulA([γ])(1 − q−1)(1 − q
2u3
1 − q3u3),
and in case ordπ a = 0, it is equal to
ulA([γ])(1 +X m≥1 (1 − q−1)2q3mu3m) = ulA([γ]) q−1+ (1 − q−1)(1 − q 2u3 1 − q3u3) .
This proves the theorem.
Corollary 5.3.2. Suppose γ ∈ Γ is split with rγ = diag(1, a, b). Assume that γ has type 1 with
a ∈ OF× . Let δ = δ([γ]) = ordπ(1 − a) and n = ordπb. Then
∆A([γ]) = {hvxK | h ∈ CG(rγ)/(CG(rγ) ∩ K), vx = 1 x 1 1 with x ∈ π−δOF/OF}
and for hvxK ∈ ∆A([γ]), the geodesic κγ(PγhvxK) in B is
PγhvxK → Pγhvxdiag(1, 1, π)K → · · · → Pγhvxdiag(1, 1, πn)K = γPγhvxK.
Here we used Pγhvxdiag(1, 1, πn)K = PγhvxrγK = PγrγhvaxK = γPγhvxK since vax−x∈ K.
6. Homotopy cycles in [γ] for γ rank-one split
In this section we fix a rank-one split γ ∈ [Γ] whose eigenvalues a, e+dλ, e+d¯λ, where a, e, d ∈ OF
and at least one of them is a unit, generate a quadratic extension L = F (λ) of F . Here λ is a unit or uniformizer in L according as L is unramified or ramified over F , i.e., γ is unramified or ramified
rank-one split. Let rγ =
a e dc d e + db
be the rational form of γ as in §3.4. Fix a matrix Pγ so
that Pγ−1γPγ = rγ.
6.1. The centralizers of rγ for γ rank-one split. Embed L× in GL2(F ) as the subgroup
u vc v u + vb
| u, v ∈ F, not both zero
, (6.1)
which is further imbedded in GL3(F ) as
1 u vc v u + vb
. Embed F× into GL3(F ) as the
diagonal matrices diag(F×, 1, 1). Note that rγ lies in F× × L×, and F× × L× modulo the
di-agonal embedding of F× in this product is the centralizer of rγ in G. Recall from (3.4) that
CP−1
γ ΓPγ(rγ)\CG(rγ)/(CG(rγ) ∩ K) has cardinality vol([γ]).
Observe that the group of units UL of L× is contained in K. If L is unramified over F , then
L× = hπiUL so that CG(rγ)K/K is represented by the vertices diag(πn, 1, 1)K, n ∈ Z, on a line
in B, and CP−1
γ ΓPγ(rγ)\CG(rγ)/(CG(rγ) ∩ K) by diag(π
n, 1, 1)K, n mod vol([γ]). If L is ramified
by a unit multiple. In this case CG(rγ)K/K is represented by the vertices diag(πn, 1, 1)K and
diag(πn, 1, 1)π
LK, n ∈ Z, lying on two lines in B. There are two possibilities for CPγ−1ΓPγ(rγ):
Case (i). The vertices in CP−1
γ ΓPγ(rγ)K/K are contained in the line diag(π
n, 1, 1)K, n ∈ Z.
Then vol([γ]) is even so that CP−1
γ ΓPγ(rγ)\CG(rγ)/(CG(rγ) ∩ K) is represented by the vertices
diag(πn, 1, 1)K and diag(πn, 1, 1)π
LK, n mod vol([γ])/2.
Case (ii). CP−1
γ ΓPγ(rγ)K/K contains a vertex on the line diag(π
n, 1, 1)π
LK, n ∈ Z. Let
y ∈ CP−1
γ ΓPγ(rγ) be such that yK = diag(π
N, 1, 1)π
LK has the least non-negative N . Then
y generates the group CP−1
γ ΓPγ(rγ), y
2K = diag(π2N −1, 1, 1)K, vol([γ]) = 2N − 1 is odd, and
CP−1
γ ΓPγ(rγ)\CG(rγ)/(CG(rγ) ∩ K) is represented by the vertices diag(π
n, 1, 1)K, 0 ≤ n ≤ N − 1 =
(vol([γ]) − 1)/2, and diag(πn, 1, 1)π
LK, 0 ≤ n ≤ N − 2 = (vol([γ]) − 3)/2.
6.2. Double coset representatives of CG(rγ)\G/K.
Proposition 6.2.1. The set
S = 1 x y 1 0 πn | x, y ∈ F/OF, n ≥ 0
represents the double coset CG(rγ)\G/K.
Proof. Write an element g ∈ G as wk for some upper triangular w and some k ∈ K. Since
CG(rγ) = F×× L×modulo the diagonal embedding of F×, we may assume that w =
1 x y 1 z πn ,
where x, y, z ∈ F/OF and n ∈ Z. We are reduced to proving
GL2(F ) = a n≥0 L× 1 πn GL2(OF), (6.2)
where L× is given by (6.1). The proof can be found in [Fl], Lemma 1 on p.30. 6.3. Minimal lengths of cycles in [γ]. First we discuss the type of [γ], which is defined in §4.4 to be (n, m) such that rγ ∈ Tn,m = Kdiag(1, πm, πn+m)K. Observe that ordπdet γ = ordπdet rγ =
ordπa(e + dλ)(e + d¯λ) ∈ 3Z by assumption (I) on Γ. Hence if e + dλ is a unit in L, then at least
one of e, d is a unit and a is not a unit. Consequently, [γ] has type (ordπa, 0). Next assume e + dλ
and d are non-units and a is a unit; in this case [γ] has type (0, min(ordπe, ordπd)). If L is ramified
over F (hence λ is a uniformizer of L), then there are two possibilities:
(i) ordπ(e + dλ)(e + d¯λ) = 1. This happens if and only if e is a non-unit, d is a unit, and
ordπa ≥ 2; in this case [γ] has type (ordπa − 1, 1).
(ii) ordπ(e + dλ)(e + d¯λ) > 1. This happens if and only if both e and d are non-units and a is a
unit; in this case [γ] has type (0, ordπe) if ordπe ≤ ordπd, and type (1, ordπd) if ordπe > ordπd.
This proves the first assertion of
Theorem 6.3.1. Let γ be a rank-one split element in [Γ] with rational form rγ =
a e dc d e + db .
Suppose that rγ ∈ Kdiag(1, πm, πm+n)K. Then
(1) The type (n, m) of [γ] is as follows.
(1.i) If ordπc = 0, then (n, m) = (ordπa, min{ordπe, ordπd}).
(1.ii) If ordπc = 1, then (n, m) = (ordπa, ordπe) provided that ordπe ≤ ordπd, otherwise
(n, m) = (max{ordπa − 1, 1}, max{ordπd, 1}).
(2) lA([γ]) = minκγ(gK)∈[γ]lA(κγ(gK)) = ordπa(e
2+ edb − cd2) = n + 2m.
(3) lG([γ]) = minκγ(gK)∈[γ]lG(κγ(gK)) = n + m.
This theorem combined with Theorem 5.1.1 completes the proof of Theorem 3.6.1.
Remark. If γ is ramified rank-one split and [γ] has type (n, 1), then [γ2] has type (2n + 1, 0).
Proof. It remains to show that the algebraic and geometric lengths of the cycles in [γ] are at least those of [γ] since, as observed before, the cycles κγ(PγgK) with g ∈ CG(rγ) have the same algebraic
and geometric lengths as [γ]. By Proposition 6.2.1, it suffices to compute (Pγg)−1γPγg = g−1rγg
for g ∈ S. Let g = 1 x y 1 0 πi
, where x, y ∈ F/OF and i ≥ 0. Then
g−1rγg = 1 −x −yπ−i 1 0 π−i a e dc d e + db 1 x y 1 0 πi =
a (a − e)x − dyπ−i (a − e − db)y − cdxπi
e dcπi dπ−i e + db ∈ K πe1 πe2 πe3 K.
Here e1 ≤ e2 ≤ e3, and as in the proof of Theorem 5.1.1, we have
e1 ≤ min{ordπa, −i + ordπd, ordπe} ≤ min{ordπa, ordπd, ordπe} = 0,
(6.3)
(6.4)
e1+ e2 ≤ min{ordπae, −i + ordπad, ordπ(e2+ bed − cd2)}
≤ min{ordπae, ordπad, ordπ(e2+ bed − cd2)} = m,
and
e1+ e2+ e3 = ordπa(e2+ bed − cd2) = n + 2m,
(6.5)
in which the last upper bound for e1 + e2 can be verified using the statement (1). Therefore
lA(κγ(PγgK)) = e1+ e2+ e3− 3e1 ≥ e1+ e2+ e3 = n + 2m = lA([γ]) since e1 ≤ 0. The inequalities
(6.4) and (6.5) together give the lower bounds e3 ≥ n + 2m − m = n + m, which in turn implies
lG(κγ(PγgK)) = e3− e1 ≥ n + m. This proves the theorem.
As shown in the above proof, an algebraically tailless cycle in [γ] satisfies the condition e1 = 0,
while a tailless cycle in [γ] should satisfy e1+ e2 = m and e1 = 0. This shows that a tailless cycle
is also algebraically tailless. Moreover, it also satisfies e2 = m, which shows that a tailless cycle
has the same type as [γ]. If furthermore, [γ] has type 1, then an algebraically tailless cycle in [γ] satisfies e1 = 0, which implies e1+ e2 ≥ 0 and hence e1 + e2 = 0 = m by (6.4) and e3 = n + m.
This shows that in this case an algebraically tailless cycle in [γ] is also tailless. We record this discussion in
Corollary 6.3.2. Suppose γ ∈ [Γ] is rank-one split. Then all tailless cycles in [γ] are also al-gebraically tailless, and they have the same type as [γ]. Moreover, if [γ] has type 1, then the algebraically tailless and tailless cycles in [γ] coincide.
We have shown that as long as [γ] has type 1, there is no distinction between algebraically tailless and tailless, regardless whether γ is split or rank-one split.
6.4. Counting the number of cycles in [γ] in algebraic length. As observed before, for all g ∈ CP−1
γ ΓPγ(rγ)\CG(rγ)sK/K, the cycles κγ(PγgK) have the same algebraic length. Since S
represents the double coset CG(rγ)\G/K, to count the number of cycles in [γ] of a given length, we
need to determine the cardinality of CP−1
γ ΓPγ(rγ)\CG(rγ)sK/K for s ∈ S. For this, we may take as
representatives the product of representatives of CP−1
γ ΓPγ(rγ)\CG(rγ)/(CG(rγ) ∩ K) (independent
of s) by the representatives of (CG(rγ) ∩ K)sK/K. The number of the former representatives is
It remains to compute the cardinality of the latter. Recall that L×∩ K consists of the units in L×, which we shall identify as the set of matrices
UL= u vc v u + vb | u, v ∈ OF, u2+ uvb − cv2 is a unit .
Denote by K0 the group GL2(OF). As analyzed in the proof of Proposition 6.2.1, we are reduced
to counting, for given m ≥ 0, the cardinality of UL
1 πm K0/K0. Proposition 6.4.1. #[UL 1 πm K 0 /K0] = 1 when m = 0,
qm when m ≥ 1 and ord
πc = 1,
qm+ qm−1 when m ≥ 1 and ordπc = 0.
Proof. It is clear that the cardinality is 1 when m = 0. Thus assume m ≥ 1. Case (I) ordπc = 1. Then any
u vc
v u + vb
∈ UL satisfies u ∈ OF×. For n ≥ 0, let
UL(n) = u vcπn vπn u + vbπn ∈ UL u, v ∈ O×F so that UL= UL(∞) ∪n≥0UL(n), where UL(∞) = u 0 0 u | u ∈ OF× . One verifies that
UL(n) 1 πm K 0 = [ u∈O×F/πm−nO F πm−n u πn K 0 for 0 ≤ n < m, and UL(n) 1 πm K 0 = 1 πm K 0
for n ≥ m and n = ∞. Therefore
#[UL 1 πm K 0 /K0] = 1 + X 0≤n<m (qm−n− qm−n−1) = qm.
Case (II) ordπc = 0. Let UL0 = u vc v u + vb ∈ UL u ∈ O×F and UL00= u vc v u + vb ∈ UL u ∈ πOF so that UL= UL0 ∪ U 00 L.
As in Case (I), we have
UL0 1 πm K 0 = [ m≥n≥0 u∈O×F/πm−nO F πm−n u πn K 0 .
One checks that
UL00 1 πm K 0 = [ z∈πOF/πmOF πm z 1 K 0 . Therefore #[UL 1 πm K 0 /K0] = qm+ qm−1 for m ≥ 1.
We summarize the above discussion in
Corollary 6.4.2. For each s = 1 x y 1 0 πn ∈ S, the cardinality of CP−1 γ ΓPγ(rγ)\CG(rγ)sK/K is vol([γ]) 1 when n = 0,
qn when n ≥ 1 and ordπc = 1,
qn+ qn−1 when n ≥ 1 and ord
πc = 0.
Now we are ready to state the main result of this section.
Theorem 6.4.3. Suppose γ ∈ [Γ] is rank-one split with rational form rγ =
a e dc d e + db . Set
(A) γ is unramified rank-one split. Then the following hold. (A1) X κγ(gK)∈[γ] ulA(κγ(gK)) = vol([γ])ulA([γ]) q δ+1+ qδ− 2 q − 1 + (q + 1)qδ+2u3 1 − q3u3 1 − u3 1 − q2u3 .
(A2) If [γ] does not have type 1, then X κγ(gK)∈[γ], type 1 ulA(κγ(gK))= vol([γ])ulA([γ]) qδ+ qδ−1+ (q 2− 1)qδ+1u3 1 − q3u3 .
(A3) If [γ] has type 1, then X κγ(gK)∈[γ], type 1 ulA(κγ(gK)) = vol([γ])ulA([γ]) q δ+1+ qδ− 2 q − 1 + (q2− 1)qδ+1u3 1 − q3u3 .
(B) γ is ramified rank-one split. Then the following hold. (B1) X κγ(gK)∈[γ] ulA(κγ(gK)) = vol([γ])qµulA([γ]) q δ+1− 1 q − 1 + qδ+3u3 1 − q3u3 1 − u3 1 − q2u3.
(B2) If [γ] does not have type 1, then X κγ(gK)∈[γ], type 1 ulA(κγ(gK))= vol([γ])ulA([γ]) qδ(qµ− µ) + (q − 1)q δ+µ+2u3 1 − q3u3 .
(B3) If [γ] has type 1, then X κγ(gK)∈[γ], type 1 ulA(κγ(gK)) = vol([γ])ulA([γ]) q δ+1− 1 q − 1 + (q − 1)qδ+2u3 1 − q3u3 .
Moreover, in each case, if [γ] does not have type 1, none of the type 1 cycles in [γ] are tailless. Remarks. 1. µ = 0 unless a, e, c are all nonunit, in which case it is 1 and δ = 0.
2. µ = 0 when [γ] has type one.
3. δ > 0 in case (A2), while δ may be zero in case (A3).
4. The right hand side of the identities (A1) - (B3) can be expressed as vol([γ]) times the orbital integrals at the rank-one split element γ of suitably chosen spherical functions on G with fast decay.
Proof. Recall that the algebraic length of a cycle in [γ] is equal to lA([γ]) + 3m for some m ≥ 0.
We shall follow the same notation and computation as in the proof of Theorem 6.3.1, letting g run through all elements in the double coset representatives S and computing, for each m ≥ 0,
the number of cycles κγ(PγgK) with lA(κγ(PγgK)) ≤ lA([γ]) + 3m using Corollary 6.4.2. As g = 1 x y 1 0 πi
, this amounts to computing the number of x, y ∈ F/OF and i ≥ 0 such that
e1 = min{ordπ((a − e)x − dπ−iy), ordπ(−cdπix + (a − e − db)y), −i + ordπd} ≥ −m.
This is equivalent to 0 ≤ i ≤ m + ordπd, (a − e)x − dπ−iy ∈ π−mOF and −cdπix + (a − e − db)y ∈
π−mOF. Denote ordπd by δ for short. So for each 0 ≤ i ≤ m + δ, we solve the following system of
linear equations α β = a − e −dπ−i −cdπi a − e − db x y = M x y (6.6)
for α, β ∈ π−mOF and count the distinct pairs (x, y) ∈ F/OF × F/OF. Recall that a, e, d are
integral, at least one of them is a unit, and a and e cannot be both units since ordπdet rγ > 0. Let
µ := ordπdet M = ordπ((a − e)2− db(a − e) − cd2),
which is 0 unless a, e and c are all nonunits, in which case it is 1. Put ε := min{ordπ(a − e), −i + δ, ordπ(a − e − bd)},
which is equal to −i + δ if δ ≤ i ≤ m + δ, and 0 if 0 ≤ i < δ. Then the coefficient matrix M = k1diag(πε, πµ−ε)k2 for some k1, k2 ∈ GL2(OF). Thus system (6.6) has the same number of
solutions as the system
α β = πε πµ−ε x y (6.7)
for α, β ∈ π−mOF and (x, y) ∈ F/OF × F/OF. We get the solutions x ∈ π−m−εOF/OF and
y ∈ π−m−µ+εOF/OF so that there are q2m+µ different pairs (x, y) for each 0 ≤ i ≤ m + δ. To
proceed, we distinguish two cases.
Case (A) ordπc = 0, that is, γ is unramified rank-one split. Then µ = 0. By Corollary 6.4.2, the
number of classes in [γ] with algebraic length at most lA([γ]) + 3m is
vol([γ])q2m(1 + X 1≤n≤m+δ qn+ qn−1) = vol([γ])q2m(q m+δ− 1 q − 1 + qm+δ+1− 1 q − 1 ) = vol([γ]) q − 1 (q 3m+δ+1 + q3m+δ− 2q2m).
Therefore X κγ(gK)∈[γ] ulA(κγ(gK)) = X κγ(PγgK)∈[γ] ulA(κγ(PγgK)) = vol([γ])ulA([γ]) 1 q − 1 qδ+1+ qδ− 2 + X m≥1 (q3m+δ+1+ q3m+δ− 2q2m− q3m+δ−2− q3m+δ−3+ 2q2m−2)u3m = vol([γ])ulA([γ]) 1 q − 1 qδ+1+ qδ 1 − q3u3 − 2 1 − q2u3 (1 − u3) = vol([γ])ulA([γ]) q δ+1+ qδ− 2 q − 1 + (q + 1)qδ+2u3 1 − q3u3 1 − u3 1 − q2u3 .
Among the cycles with lA(κγ(PγgK)) = lA([γ]) + 3m, we compute the number of those with
type 1. First consider the case m ≥ 1. In order that lA(κγ(PγgK)) = lA([γ]) + 3m and κγ(PγgK)
has type 1, two conditions must be satisfied:
e1 = min{ordπ((a − e)x − dπ−iy), ordπ(−cdπix + (a − e − db)y), −i + δ} = −m,
and
e1+ e2 = ordπ[((a − e)x − dπ−iy)(e + db) − dπ−i(−cdπix + (a − e − db)y)] = −2m.
These two conditions are equivalent to i = δ + m, ordπ(−cdπix + (a − e − db)y) = −m, and
ordπ((a − e)x − dπ−iy) ≥ −m. This amounts to solving system (6.6) with α ∈ π−mOF and
β ∈ π−mO×F, hence we obtain (q − 1)q2m−1distinct pairs (x, y). Combined with Corollary 6.4.2, we
see that the number of rank one cycles κγ(PγgK) with lA(κγ(PγgK)) = lA([γ]) + 3m is vol([γ])(q −
1)q2m−1(qδ+m+ qδ+m−1).
Next consider the case m = 0. Under the assumption ordπc = 0, we know from Theorem 6.3.1
that [γ] has type (ordπa, min{ordπe, ordπd}). Therefore it has type 1 if and only if ordπa > 0, in
which case all cycles in [γ] with algebraic length equal to lA([γ]) have type 1, and the number of
such cycles is vol([γ])qδ+1q−1+qδ−2, as computed above. If [γ] does not have type 1, then δ = ordπd > 0;
the condition e1 = e2 = 0 implies i = δ and only one solution (x, y) = (0, 0). In this case the
number of type 1 cycles in [γ] with algebraic length equal to lA([γ]) is qδ+ qδ−1 by Corollary 6.4.2.
