HARMONIC COCYCLES AND COHOMOLOGY OF

ARITHMETIC GROUPS (IN POSITIVE CHARACTERISTIC).

Marc Reversat

Laboratoire de Mathématiques Émile Picard, Unité Mixte de Recherche Université-C.N.R.S. 5580.

September 1999

Abstract. Let K be a global field of characteristic p > 0. We study the cohomology of arithmetic subgroups Γ of SL_{n+1}(K) (with respect to a fixed place of K), under the hypothesis that these groups have no p′-torsion (any arithmetic group possesses a normal subgroup of finite index without p′-torsion). We define the cohomology of Γ with compact supports and values in ℤ[1/p], and we relate it to spaces of harmonic cocycles, also with compact supports (§3). We give a description of the locus of these supports, in particular by introducing a notion of cusp in dimension n ≥ 1 (§4) and we calculate“geometrically” the Euler-Poincaré characteristic of this cohomology, up to torsion (§5).

1 Introduction.

(1.0.1) Let K be a global field of characteristic p > 0, i.e. the function field of a geometrically irreducible curve defined over a finite field of characteristic p. One sets G = SL_{n+1}, where n ≥ 1 is an integer. Let ∞ be a fixed place of K, K_{∞} be the completion of K at this place and ℑ be the the Bruhat-Tits building of G(K_{∞}) (see §2 for references and a brief description of this building); ℑ is a contractible simplicial complex of dimension n, where G(K_{∞}) acts simplicially. Let Γ be an arithmetic subgroup of G(K_{∞}) (see (2.4.1) for a precise definition), the G(K_{∞})-action induces a Γ-action on ℑ.

The purpose of this paper is to give information about the cohomology of Γ, more precisely, to relate this cohomology to geometric or simplicial properties of the building ℑ or of its quotient by Γ.

Links of such kind have been studied particularly for discrete groups Γ ⊂ G(K_{∞}) which are cocompact, i.e. such that Γ\ℑ is a finite complex, and, moreover, which are torsion free, and thus which act freely on ℑ. This has been carried out by A. Borel and J.-P. Serre in [Bo-Se 2], by G.A. Mustafin in [Mus],

Typeset by AMS-TEX

and, in their study of the Drinfeld symmetric spaces (which are rigid analytic realizations of the simplicial complexes ℑ), by P. Schneider and U. Stuhler ([Sc-St]). When the characteristic of the base field is zero, i.e. in the p-adic case or in the classical case, many authors have studied the cohomology of arithmetic, torsion free and cocompact groups acting on ℑ or on other symmetric spaces: see for instance the three papers mentioned above, in characteristic zero, and J.-P. Serre ([Se 1]), H. Garland ([GarH]), A. Borel and J.-P. Serre ([Bo-Se 1]) etc., and a recent paper of E. de Shalitt ([dS 2]). There is also, only in dimension 1 but in all characteristics, the study of Schottky groups by Mumford curves (see [Mum] and [Ger-vdP]).

In the situation studied here, the arithmetic groups are discrete, but they are not cocompact (the complex Γ\ℑ is not finite) and they have torsion elements (simplices of ℑ have non trivial stabilizers in Γ). When n = 1, i.e. in the dimension 1 case, this has been extensively studied, following the seminal work of Drinfeld [Dr] (see [Se 2] ch. 2, [Gek-Re] and [A-B]). To explain what we do here, we now recall somes properties of the dimension 1 case.

1.1 The dimension 1 case.

Let Γ be an arithmetic subgroup of G(K) = SL_{2}(K), where K is a global field of characteristic p > 0.

(1.1.1) In the dimension 1 case, the building ℑ is a tree and the quotient Γ\ℑ is a connected graph which is the union of a finite graph (Γ\ℑ)^{∘} without ends and of finitely many half lines; the graph (Γ\ℑ)^{∘} contains all the homology of Γ\ℑ and the half lines corresponds to the cusps of Γ (see [Se 2] ch. 2, [vdP], see also [Gek-Re] §2).

(1.1.2) The harmonic cocycles are functions f defined on the set of oriented edges of ℑ, with for instance values in ℤ, such that for any oriented edge e, f(−e) = −f(e), and such that for any vertex v of ℑ, ∑ f(e) = 0, where e runs in the set of edges beginning at v. An important fact is that the group \underline{H}_{!}(ℤ)^{Γ}, of ℤ-valued harmonic cocycles invariant under Γ and with compact (finite) supports modulo Γ, is canonically isomorphic with H^{1}(Γ, ℤ), under the hypothesis that Γ has no p′-torsion, i.e. that any torsion element of Γ is of order a power of p ([Gek-Re] prop. 3.4.5). This last hypothesis is indeed natural and not too restrictive, because any arithmetic group contains a normal subgroup of finite index and having this property (this is right in all dimensions, see 2.4).

The group H^{1}(Γ, ℤ) can also be interpreted as the cohomology group of the cochain complex (C^{i}(ℤ))_{i=0,1} (definitions are given in §3.1, see [Se 2] ch. 2, §2.8).

(1.1.3) The graph (Γ\ℑ)^{∘} has no ends; let \overline{T} be one of its maximal subtrees and T be a connected subtree of ℑ wich is a lift of \overline{T}. Let e_{1}, ⋯, e_{g}be edges of ℑ, beginning in T and equal modulo Γ to the edges of ((Γ\ℑ)^{∘} − \overline{T}). Let γ_{1}, ⋯, γ_{g} be elements of Γ such that the end of γ_{i}(e_{i}) is in T (1 ≤ i ≤ g). Then, the group G :=< γ_{1}, ⋯, γ_{g} > is free , it is canonically identified with the fundamental group of (Γ\ℑ)^{∘} and

G ⊂ Γ → Γ/Γ_{tors}

is an isomorphism, where Γ_{tors} is the subgroup of Γ generated by its torsion elements ([Re], compare with [Se 2] ch. 1, §3.3). As a consequence (but it can be seen more directly, see [Gek-Re] §3.2) one has for the Euler-Poincaré characteristic of Γ with values in ℚ

χ(Γ, ℚ) = 1 − g.

1.2 Summary of results.

The object of this article is to extend to the case of dimension n > 1 some of the properties reviewed in the preceding paragraph.

We suppose that the arithmetic group Γ ⊂ G(K) has no p′-torsion. Let R be a (commutative) ring such that multiplication by p is invertible. Let C^{·}(R) be the complex of R-valued cochains, i.e.

C^{q}(R) = Hom_{ℤ}(ℤ[ℑ_{q}^{⋆}], R) , 0 ≤ q ≤ n,

where ℑ_{q}^{⋆} is the set of oriented q- simplices of ℑ. We first prove that, canonically

H^{·}(Γ, R) ≃ H^{·}(C^{·}(R)^{Γ})

(proposition (3.1.4)), indeed this is a direct application of an argument of J.P. Serre: [Se 1], p.95; here, the hypothesises that Γ has no p′-torsion and that p is invertible in End_{ℤ}(R) are very useful. This formula leads to the definition of cohomology of Γ with compact supports, which is the part in H^{·}(Γ, R) corresponding in H^{·}(C^{·}(R)^{Γ}) to the cocycles having, up to coboundaries, finite supports modulo Γ (definition (3.1.5)). Afterwards, we introduce the notion of harmonic cocycle (definition (3.2.5)), as in the paper [Ga] of H. Garland^{1}. Let

\underline{H}_{!}^{q}(R)^{Γ} and H_{!}^{q}(Γ, R) , 0 ≤ q ≤ n

denote respectively the R-module of harmonic cocycles defined on the oriented q-simplices, invariant under Γ and with (finite) compact supports modulo Γ, and the q-th cohomological R-module of Γ with compact supports. Set R = ℤ[1/p] (indeed what we will do works for any subring of ℝ containing ℤ[1/p]). We prove, among other things, that the module of harmonic cocycles \underline{H}_{!}^{q}(R)^{Γ} is canonically isomorphic to the torsion free submodule of H_{!}^{q}(Γ, R) (theorem (3.3.1)). This extends to the higher dimensional case some aspects of the results recalled in (1.1.2); we will explain below (see (1.3.1)) why we obtain an isomorphism with the cohomology of Γ with compact supports, instead of the whole cohomology, as in the dimension 1 case.

1H. Garland writes that the idea of this notion goes back to Eckmann and Hodge. A variant, especially for the maximal rank (harmonic cocycles defined on pointed chambers), is used in [Sc-Te 1], [Sc-Te 2], [dS 1] and [dS 2].

In the 4-th paragraph, we introduce a notion of cusp for Γ. One knows that Γ\ℑ is the “skeleton” of a modular variety M_{Γ} ([Dr], see also [De-Hu]), which is affine; from that point of view, the cusps are skeletons of neighborhoods of the missing part, i.e. of the difference between M_{Γ} and its canonical completion (but we shall not say anything in this paper about modular varieties). It is not difficult to prove that there are only finitely many cusps modulo Γ (and that they are rational over K, as in the dimension 1 case, see [Se 2] ch. 2, §2.3). They are represented modulo Γ by finitely many sectors of ℑ, say S_{1}, ⋯, S_{d}, and the complementary complex in ℑ of

⋃_{1≤i≤d} ΓS_{i}

is finite modulo Γ (theorem (4.2.2)). Thus, this complementary complex and the cusps, modulo Γ, give respectively an analogy with (Γ\ℑ)^{∘} and the half lines (see (1.1.1)). The proof of theorem (4.2.2) is long, the argument is based on results of G. Harder (sätze 2.1.1 and 2.2.2 of [Ha 1]), concerning the locus of supports of cusp forms ([Ha 2] theorem 1.2.1).

The 5-th paragraph gives a partial generalization of (1.1.3), in calculating “geometrically” the Euler-Poincaré characteristic of Γ, for the cohomology with values in ℚ. This geometric calculation is a step towards notions analogous to the group G of (1.1.3); problems, going in that way, are suggested in (5.0.5).

1.3 Some comments. (1.3.1) Let Γ be an arithmetic group whithout p′-torsion. We have obtained a comparison theorem between the spaces \underline{H}_{!}^{·}(ℚ)^{Γ} of harmonic cocycles invariant under Γ, with finite supports modulo Γ and with values in ℚ (we take here just ℚ to be short) and the spaces H_{!}^{·}(Γ, ℚ) of cohomology with compact supports (theorem (3.3.1)). In the one dimensional case, H_{!}^{·}(Γ, ℚ) is replaced by the ordinary cohomology H^{·}(Γ, ℚ). The reason for this difference is that, in the dimension 1 case, the “simplicial complex at infinity” of Γ\ℑ (which comes from the spherical building of ℑ at infinity, but which is not simply the quotient of this spherical building by Γ), chambers of which can be viewed as the cusps, is a finite complex of dimension 0 (see (1.1.1)), then there is no complexity (no cycle) at infinity. In that situation, writing P for the inverse image in ℑ of the half lines of Γ\ℑ (the cusps, see (1.1.1)), a Mayer-Vietoris argument applied to the decomposition

ℑ = (ℑ − P) ∪ P

yields equality between the ordinary cohomology of Γ and the cohomology with compact supports. When the dimension n is greater than 1, the simplicial complex at infinity of Γ\ℑ is still finite, but it has dimension n − 1 > 0 and it may have non trivial homology; in this case the two cohomologies of Γ are not equal. In our opinion, to give a description of the homological nature of the simplicial complex at infinity of Γ\ℑ is a very important question, which seems to underly the conjecture (13.4.1) of [Lau]. It is related, as the next question, to calculations and interpretations of Eisenstein cohomologies (see [Ha 3] and op.cit.).

(1.3.2) In the dimension 1 case, harmonic cocycles invariant under Γ and with finite support modulo Γ can be interpreted canonically, for a suitable subgroup of the adèle group of G(K), as cusp forms that transform like the special representation ([Dr], [De-Hu], [Ge-Re], [A-B], etc.). This is not so clear in dimension greater than 1. This problem is mentioned in the introduction of [dS 2] and in [Sc-St], remark (c) p. 84. Our opinion is also that this is an interesting question.

(1.3.3) Problems can be formulated to determine objects, in dimension greater than 1, similar to G =< γ_{1}, ⋯, γ_{g} > (see (1.1.3)), they need notations and definitions, which are introduced in §5, then we have written this problem in (5.0.5).

2 The Bruhat-Tits building of SL_{n+1}(K_{∞}) (review). Notations.

2.1 Notations.

Let K denote a global field of positive characteristic p, i.e. the function field of a geometrically irreducible curve C defined over a finite field 𝔽 of characteristic p; we suppose that C is smooth and projective. Let ∞ be a fixed place of K, i.e. a closed point of C. One sets

(2.1.1)  A = H^{0}(C − {∞}, O_{C}),

A is the ring of elements of K regular outside ∞.

(2.1.2) Let 𝔙 be the set of places of K (the set of closed points of C) and let 𝔙_{f} = 𝔙 − {∞} the set of “finite places”^{2} of K.

Let ω be a place of K, we denote by K_{ω} the completion of K at ω, by O_{ω} its valuation ring, by 𝔽(ω) the residue field and by π_{ω} a uniformizing parameter at ω, we suppose that π_{ω} is in K.

Usually, we will denote by the same letter a place and the normalized valuation of K corresponding to it, except for ∞, this valuation being denoted ω_{∞}. If ω is a place of K, the corrresponding normalized valuation satisfies ω(π_{ω}) = [𝔽(ω) : 𝔽].

Let ω be a valuation of K, we denote by | |_{ω} (by | |_{∞} if ω = ∞) the corresponding normalized absolute value, i.e. the absolute value which satisfies | π_{ω} |_{ω}= ♯(𝔽(ω))^{−1}.

(2.1.3) Let n ≥ 1 be an integer, one sets

G = SL_{n+1} ,

let T be its standard torus (of diagonal matrices), P and U be respectively the standard Borel subgroup (of upper triangular matrices) and its unipotent radical (of matrices with in addition 1 on the diagonal). Let N be the normalizer of T in G. One sets \overline{W} = N(K_{∞})/T (K_{∞}) : the linear Weyl group , W = N(K_{∞})/T (O_{∞}) : the affine Weyl group.

2This terminology comes from an analogy with the classical case: A and K look like respectively ℤ and ℚ, the place ∞ like the archimedean place of ℚ...

Let B_{∞} be the inverse image of P (𝔽(∞)) by the natural map G(O_{∞}) ⟶ G(𝔽(∞)).

Θ Let 𝔸 (resp. 𝔸_{f}), be the ring of adèles (resp. finite adèles) of K, i.e. the set of elements (f_{ω}) ∈ ∏ K_{ω}, where ω runs in 𝔙 (resp. 𝔙_{f}), such that f_{ω} ∈ O_{ω} except for finitely many places ω. The groups of adèles and finite adèles of G(K) are denoted respectively G(𝔸) and G(𝔸_{f}).

(2.1.4) The groups of characters and of 1-parameter subgroups. Let

X^{⋆} = Hom_{gr}(T (K_{∞}), K_{∞}^{⋆}) and X_{⋆} = Hom_{gr}(K_{∞}^{⋆}, T (K_{∞}))

be the groups of characters and of 1-parameter subgroups of T (we do not distinguish between T, 𝔾_{m} and their K_{∞}-valued points because these groups are split). They are both free abelian groups of rank n. For any i, 0 ≤ i ≤ n, let χ_{i} ∈ X^{⋆} be the character which maps t = diag(t_{o}, ⋯, t_{n}) ∈ T (K_{∞}^{⋆}) to t_{i}, then

{χ_{i} / 0 ≤ i ≤ n}

generates X^{⋆} and any of its subsets with n elements is a basis. For any i, 0 ≤ i ≤ n − 1, let λ_{i} ∈ X_{⋆} be the 1-parameter subgroup which maps z ∈ K_{∞}^{⋆} to t = diag(t_{o}, ⋯, t_{n}) such that t_{j} = 1 if j ≠ i, i + 1, t_{i} = z and t_{i+1} = z^{−1}. Then

{λ_{i} / 0 ≤ i ≤ n − 1}

is a basis of X_{⋆}. There exists a perfect pairing

X^{⋆} × X_{⋆} → ℤ

which to (χ, λ) associates the integer m such that, for any z in K_{∞}^{⋆}, χ(λ(z)) = z^{m} (see for instance [Bo] §8).

Note that N(K_{∞}) acts on X^{⋆} and X_{⋆} via its action on T (K_{∞}) by conjugation.

2.2 The building ℑ.

Now we recall some basic facts about the Bruhat-Tits building ℑ of G(K_{∞}). For generalities on complexes and buildings, see [Ro], [Bro] and [GarP]. The building of G(K_{∞}) is completely described in [Bru-Ti], there is a good introduction to this topic in [Lan].

Let S be a simplicial complex. We denote by S_{q} the set of its simplices of dimension q. Let S′ be a subcomplex of S; recall that, by definifion, S′ is a union of closed simplexes, recall also that the complementary complex of S′ in S is the minimal subcomplex containing the complementary set of S′ in S, then, maybe, S′ and its complementary complex have common simplices.

(2.2.1) The fundamental apartment. Let

V_{0} = X_{⋆} ⊗_{ℤ} ℝ,

it is a real vector space with basis

{e_{i} = λ_{i} ⊗ 1 / 0 ≤ i ≤ n − 1}.

The pairing between X^{⋆} and X_{⋆}, introduced in (2.1.4), induces a pairing between V_{0} and X^{⋆} ⊗_{ℤ} ℝ, denoted by < ·, · >; it gives the identification

V_{0}^{⋆} = X^{⋆} ⊗_{ℤ} ℝ

where V_{0}^{⋆} is the dual space of V_{0}. We set

a_{i} = χ_{i} ⊗ 1 , 0 ≤ i ≤ n and a_{i,j} = a_{i} − a_{j} , 0 ≤ i, j ≤ n,

the a_{i,j}’s, i ≠ j, are the roots of G(K_{∞}) relative to T (K_{∞}) ([Bo] §8.17).

The vector space V_{0} is a Coxeter complex, the walls of the chambers being the hyperplanes

{< a_{i,j}, · > +k = 0} , 0 ≤ i ≤ n , k ∈ ℤ.

The torus T (K_{∞}) acts on V_{0} by translations: the translation corresponding to t = diag(t_{0}, ⋯, t_{n}) is given by the vector

∑_{0≤i≤n−1} m_{i}e_{i} where m_{i} = −(ω_{∞}(t_{0}) + ⋯ + ω_{∞}(t_{i})) ,

then, for any i, 0 ≤ i ≤ n − 1, and for any z ∈ K_{∞}^{⋆} , the translation in V_{0} corresponding to λ_{i}(z) is given by the vector −ω_{∞}(z)e_{i}. This action gives a map T (K_{∞})/T (O_{∞}) ⟶ V_{0} (because T (O_{∞}) acts trivially on V_{0}).

The normalizer N(K_{∞}) acts on V_{0}, this action comes from the action on X_{⋆}, for this action T (K_{∞}) acts trivially on V_{0}. It follows a map N(K_{∞})/T (K_{∞}) ⟶ GL(V_{0}).

Everything that we have said about the different actions on V_{0} can be summarized in the following diagram (compare with [Lan], proof of lemma 1.7)

0 ⟶ \frac{T (K_{∞})}{T (O_{∞})} ≃ ℤ^{n} ⟶ \frac{N(K_{∞})}{T (O_{∞})} = W ⟶ \frac{N(K_{∞})}{T (K_{∞})} = \overline{W} ⟶ 0
↓  ↓  ↓
0 ⟶  V_{0}  ⟶ GL(V_{0}) ∝ V_{0} ⟶ GL(V_{0}) ⟶ 0

where the two arrows are exact, the second vertical map is induced by the two others, which are explained above; GL(V_{0}) ∝ V_{0} is the affine transformation group of V_{0}.

The space V_{0} equipped with its simplicial structure and with the previous affine action of W (or of N(K_{∞})) is called the fundamental apartment of the building ℑ; one now defines this building.

(2.2.2) Definition of the building. Let i, j be integers such that 0 ≤ i ≠ j ≤ n and let U_{i,j} be the subgroups of u = (u_{k,l}) ∈ U(K_{∞}) (see (2.1.3)) such that u_{k,l} = 0 if k ≠ l and (k, l) ≠ (i, j). For any x ∈ V_{0} let

U_{i,j,x} = {u = (u_{k,l}) ∈ U_{i,j} / ω_{∞}(u_{i,j})+ < a_{i,j}, x >≥ 0}

and let U_{x} be the group generated by the U_{i,j,x}’s, 0 ≤ i ≠ j ≤ n.

On G(K_{∞}) × V_{0} one has the following equivalence relation ([Bru-Ti] 2.5 and 7.4, see also [Lan] §9): let (g, x) and (h, y) be in G(K_{∞}) × V_{0}, they are said to be equivalent (and one writes (g, x) ∼ (h, y)) if there exists n ∈ N(K_{∞}) such that y = nx and g^{−1}hn ∈ U_{x}. Let

ℑ = \frac{G(K_{∞}) × V_{0}}{∼} .

One has V_{0} ⊂ ℑ canonically, by x ↦ Class(1, x). The group G(K_{∞}) acts on ℑ (if g, h ∈ G(K_{∞}) and x ∈ V_{0}, the action of g on Class(h, x) gives Class(gh, x)) and this action extends the one of W on V_{0}. For any x ∈ V_{0} ⊂ ℑ, the stabilizer of x in G(K_{∞}) is the group

P_{x} := ⟨U_{x} , N_{x}⟩ where N_{x} = {n ∈ N(K_{∞}) / nx = x},

This is a bounded subgroup of G(K_{∞}), for the ∞-topology.

ℑ has a simplicial structure, which is induced by that of V_{0}, indeed ℑ is an affine building, its apartments are of the form gV_{0}, for g ∈ G(K_{∞}), the action of G(K_{∞}) on ℑ is simplicial. The building ℑ is the Bruhat-Tits building of G(K_{∞}).

(2.2.3) Another point of view. We now recall the description of the building ℑ by O_{∞}-submodules of K_{∞}^{n+1} (see [GarP] ch. 19).

We mentioned above that V_{0} is a Coxeter complex (see (2.2.1)). It follows from the definition of the walls of the chambers that a point v = ∑_{0≤i≤n−1} m_{i}e_{i} (with m_{i} ∈ ℝ) of V_{0} is a vertex if and only if for each i, j, 0 ≤ i, j ≤ n, < a_{i,j}, v >∈ ℤ.

On K_{∞}^{n+1} one considers the following G(K_{∞})-action: for any z ∈ K_{∞}^{n+1} and any g ∈ G(K_{∞}), the action of g on z is given by the product of matrices zg^{−1}, z being viewed as a one-line matrix. Note that, in [GarP] ch. 19, gz is chosen, with z viewed as a one-column matrix. Let {u_{0}, ⋯, u_{n}} be the canonical basis of K_{∞}^{n+1}.

Let v = ∑_{0≤i≤n−1} m_{i}e_{i} be a vertex of V_{0}. We set

M_{v} = ⨁_{0≤i≤n} π_{∞}^{<a_{i,n},v>}O_{∞}u_{i}.

This is an O_{∞}-submodule of K_{∞}^{n+1} of maximal rank n + 1. Note that for any t = diag(t_{0}, ⋯, t_{n}) ∈ T (K_{∞}), one has M_{tv} = t(M_{v}), where, on the left, tv is given by the action of t on V_{0} (see (2.2.1)) and, on the right, by the preceding action on K_{∞}^{n+1}.

Let ℑ′ be the building of homothety classes [M] of O_{∞}-submodule M of K_{∞}^{n+1} with rank n + 1 (see [GarP] ch. 19). Then, the map v ↦ [M_{v}] induces a simplicial isomorphism between V_{0} in ℑ and the apartement of ℑ′ with frame {O_{∞}u_{0}, ⋯, O_{∞}u_{n}} (op. cit.). With the G(K_{∞})-action described above, it follows that there is a G(K_{∞})-isomorphism ℑ ≃ ℑ′.

(2.2.4) The fundamental chamber. Let C_{0} be the set of x ∈ V_{0} such that

< a_{i,i+1}, x >> 0 , 0 ≤ i ≤ n − 1 and < a_{n,0}, x > +1 > 0.

By the isomorphism of (2.2.3), it corresponds in ℑ′ to the chamber having vertices [M_{0}], ⋯, [M_{n}] with

M_{0} = O_{∞}u_{0} ⊕ ⋯ ⊕ O_{∞}u_{n}
M_{i+1} = π_{∞}O_{∞}u_{0} ⊕ ⋯ ⊕ π_{∞}O_{∞}u_{i} ⊕ O_{∞}u_{i} ⊕ ⋯ O_{∞}u_{n} , 0 ≤ i ≤ n − 1.

More precisely, C_{0} is a chamber of ℑ having vertices v_{i}, 0 ≤ i ≤ n, such that v_{0} is the origin 0 of V_{0} and v_{i+1} is defined by

< a_{k,k+1}, v_{i+1} >= 0 for all k, k ≠ i, 0 ≤ k ≤ n − 1 and < a_{i,i+1}, v_{i+1} >= 1.

One has M_{v_{i}} = M_{i}, 0 ≤ i ≤ n. It follows from [GarP] ch. 19, §4, that the stabilizer of C_{0} in G(K_{∞}) is the parahoric subgroup B_{∞} (see (2.1.3)).

This chamber C_{0} is called the fundamental chamber, its vertex corresponding to the origin 0 ∈ V_{0} is called the fundamental vertex.

2.3 The spherical building at infinity.

Let ℑ_{∞} be the spherical building at infinity of ℑ. We shall say almost nothing about this building, for a complete description, see for instance [Bro] ch. VI, §9. As generality, one just recalls that ℑ_{∞} is constructed by attaching to each apartment of ℑ a sphere at infinity; a point of ℑ_{∞} is an equivalent class of half lines, for the relation of parallelism.

The group G(K_{∞}) acts on ℑ_{∞} and this action is compatible with that on ℑ. Let \overline{W} be the linear Weyl group (see (2.1.3)), one knows that \overline{W} is a finite reflexion group acting on V_{0}, this last one carrying the structure of Coxeter complex associated to \overline{W} ([Bro] ch. I and II).

(2.3.1) Let v be a vertex of ℑ, v ∈ V_{0}, a sector of ℑ in V_{0}, with vertex v, or beginning in v, is the (topological) closure of v + C, where C ⊂ V_{0} is a chamber for the simplicial stucture of V_{0} coming from \overline{W}. A sector of ℑ is the image under the G(K_{∞})-action of a sector of ℑ in V_{0}.

(2.3.2) Remark. (1) A sector of ℑ in V_{0} is a closed simplicial cone of V_{0} (viewed as an apartment of ℑ). There is the following geometric description: take a chamber C of ℑ in V_{0}, let v be a vertex of C and H_{1}, ⋯, H_{n} be the walls of C containing v. Let S be the closure of the connected component of

V_{0} − H_{1} ∪ ⋯ ∪ H_{n}

containing C. Then S is a sector of ℑ in V_{0}, with vertex v, and all these sectors have this form.

(2) Our definition implies that a sector of ℑ is a subcomplex of ℑ; this is the reason why we insist that the sectors be closed, which is not always the case in the literature (as in [Bro]).

(2.3.3) Let v be a vertex of ℑ and let Σ_{v} be the set of sectors of ℑ with vertex v. Then, there is a one to one map between Σ_{v} and the set of closed chambers of ℑ_{∞}, which to each S ∈ Σ_{v} associates the closed chamber \overline{Δ}_{∞} of ℑ_{∞} equal to the set of all equivalent classes of half lines beginning in v and contained in S ([Bro] ch. VI, §9, lemma 2). One says that the closed chamber \overline{Δ}_{∞} is defined by the sector S or that the sector S ends at \overline{Δ}_{∞}; if Δ_{∞} is the chamber of ℑ_{∞} with closure \overline{Δ}_{∞}, one also says that Δ_{∞} is defined by S or that S ends at Δ_{∞}.

(2.3.4) Let S and S′ be two sectors of ℑ (with different vertices) defining the same chamber Δ_{∞} of ℑ_{∞}, then there exists a sector S′′ contained in S and S′ and defining the same chamber Δ_{∞} of ℑ_{∞} (op.cit. lemma 4).

(2.3.5) The fundamental sector. Let H_{i} be the hyperplane of V_{0}, defined by < a_{i,i+1}, · >= 0, 0 ≤ i ≤ n − 1. Let S_{0} be the sector of V_{0} which contains the fundamental chamber and which is the closure of a connected component of

V_{0} − H_{0} ∪ ⋯ ∪ H_{n−1}.

Then, the vertex of S_{0} is the fundamental vertex v_{0} and one has

S_{0} = {x ∈ V_{0} / < a_{i,i+1}, x >≥ 0 , 0 ≤ i ≤ n − 1}.

This sector S_{0} is called the fundamental sector.

Let C_{∞} be the chamber of ℑ_{∞} defined by the fundamental sector S_{0}. One calls this the fundamental chamber at infinity.

(2.3.6 Let K_{∞}^{n+1} be equipped by the G(K_{∞})-action as before (see (2.2.3)). There exists a G(K_{∞})-isomorphism between the building ℑ_{∞} and the building of flags of K_{∞}^{n+1}, which is compatible with that described in (2.2.3) (arguments to prove that are closed to those of (2.2.3), see for instance [Bro] ch.VI, §9). Via this isomorphism, the fundamental chamber C_{∞} corresponds to the flag with vertices

K_{∞}u_{n−i} ⊕ ⋯ ⊕ K_{∞}u_{n} , 0 ≤ i ≤ n

({u_{0}, ⋯, u_{n}} is the canonical basis of K_{∞}^{n+1}). Then it is clear that the stabilizer of C_{∞} in G(K_{∞}) is P (K_{∞}). There is a canonical G(K_{∞})-equivariant bijection of the set of chambers of ℑ_{∞} with G(K_{∞})/P (K_{∞}).

2.4 Arithmetic groups.

A subgroup of G(K) is called arithmetic if it is commensurable with G(A) (see (2.1.1)). Such a group Γ is discrete in G(K_{∞}) for the ∞-topology and the stabilizer in Γ of any simplex of ℑ is finite.

For any arithmetic subgroup Γ of G(K), there exists an ideal I of A, I ≠ 0, such that

Γ(I) := ker(G(A) ⟶^{canonical} G(A/I))

is a subgroup of Γ with finite index. The groups of the form Γ(I) are whithout p’-torsion, it means that any torsion element of Γ(I) is of order a power of the characteristic p of the base field.

3 Harmonic cocycles and cohomology.

In all this paragraph, the letter Γ denotes a subgroup of G(K) which is arithmetic. We suppose that

(3.0.1) Γ has no p′-torsion, i.e. that the order of any torsion element of Γ is a power of p, where p is the characteristic of K.

Note that this hypothesis is not too restrictive, see 2.4.

3.1 Cohomology of Γ.

Let 0 ≤ q ≤ n be an integer and let ℑ_{q}^{∗} be the set of oriented q-simplices of ℑ; recall that a q-simplex is oriented if it is equipped with an equivalent class of total orderings of its vertices, two orderings being equivalent if they differ by an even permutation of the vertices. Let ℤ[ℑ_{q}^{∗}] be the ℤ-module generated by the oriented q-simplices of ℑ, with the relations, when q ≠ 0, σ_{1} + σ_{2} = 0, if σ_{1} and σ_{2} are the two oriented q-simplices corresponding to the same non oriented q-simplex. For 1 ≤ q ≤ n, let ∂_{q} : ℤ[ℑ_{q}^{∗}] → ℤ[ℑ_{q−1}^{∗}] be the morphism of ℤ-modules which maps the oriented simplex [v_{0}, ⋯, v_{q}] (v_{0}, ⋯, v_{q} are its vertices) to ∑_{0≤i≤q}(−1)^{i}[v_{0}, ⋯, v̂_{i}, ⋯, v_{q}], where the hat in v̂_{i} means that this term is omitted. Let ε : ℑ_{0}(M) ⟶ M be the augmentation map (which to each finite sum ∑_{σ} a_{σ}σ, σ ∈ ℑ_{0}(M) and a_{σ} ∈ ℤ, associates ∑_{σ} a_{σ}).

Note that Γ acts on each ℤ[ℑ_{q}^{∗}] (by the restriction of the action of G(K_{∞})) and that the maps ∂_{q} are morphisms of ℤ[Γ]-modules.

It is wellknown that the following sequence of ℤ[Γ]-modules

(3.1.1)  0 ⟶ ℤ[ℑ_{n}^{∗}] ⟶^{∂_{n}} ⋯ ⟶^{∂_{q+1}} ℤ[ℑ_{q}^{∗}] ⟶^{∂_{q}} ⋯ ⟶^{∂_{1}} ℤ[ℑ_{0}^{∗}] ⟶^{ε} ℤ ⟶ 0

is exact (because ℑ is contractible) and that it is split.

(3.1.2) Let R be a ℤ-module. We suppose that multiplication by the characteristic p of K is invertible in End_{ℤ}(R). This hypothesis will be necessary in (3.1.4). For 0 ≤ q ≤ n, we set C^{q}(R) = Hom_{ℤ}(ℤ[ℑ_{q}^{∗}], R); we suppose that Γ acts trivially on R, then C^{q}(R) is equiped with the usual action of Γ f ↦^{γ} f ∘ γ^{−1} (f in C^{q}(R), γ in Γ). Let d^{q} : C^{q}(R) ⟶ C^{q+1}(R) be the map defined by f ↦ f ∘ ∂_{q+1}, where f belongs to C^{q}(R) and 0 ≤ q ≤ n − 1; let e : R ⟶ C^{0}(R) be the map r ↦ rε, r ∈ R. They are maps of R[Γ]-modules.

(3.1.3) The following sequence of R[Γ]-modules is exact

0 ⟶ R ⟶^{e} C^{0}(R) ⟶^{d^{0}} ⋯ ⟶^{d^{q−1}} C^{q}(R) ⟶^{d^{q}} ⋯ ⟶^{d^{n−1}} C^{n}(R) ⟶ 0.

This is a direct consequence of the fact that (3.1.1) is exact and split.

(3.1.4) Proposition. One has natural isomorphisms of ℤ-modules

H^{·}(Γ, R) ≃ H^{·}(C^{·}(R)^{Γ}).

The left hand side term is the cohomology of Γ with value in the trivial Γ-module R, on the right this is the cohomology of the complex C^{·}(R)^{Γ} ({ }^{Γ} is the subset of { } of its elements invariant under Γ).

Proof. The argument of the proof is from J.P. Serre ([Se 1], p.95). Consider an injective resolution of C^{·}(R) in the category of complexes of R[Γ]-modules and apply the functor H^{0}(Γ, ). Consider the usual two spectral sequences which tend both to the cohomology of the total complex. It follows from (3.1.3) that the second spectral sequence degenerates in the E_{2} term, which implies that the cohomology of the total complex is H^{·}(Γ, R).

On the other hand, we have

C^{q}(R) = Hom_{ℤ}(ℤ[ℑ_{q}^{∗}], R) = Hom_{ℤ}(∐_{σ} ℤ[Γ/Γ_{σ}], R) = ∏_{σ} Hom_{ℤ}(ℤ[Γ/Γ_{σ}], R)

where σ runs in a representative set of ℑ_{q}^{∗} modulo Γ, Γ_{σ} is the stabilizer of σ in Γ and where ℤ[Γ/Γ_{σ}] is the free ℤ-module with basis Γ/Γ_{σ}. We also have

Hom_{ℤ}(ℤ[Γ/Γ_{σ}], R) ≃ Hom_{ℤ[Γ_{σ}]}(ℤ[Γ], R) = coind_{Γ_{σ}}^{Γ}(R)

this isomorphism coming from the fact that the action of Γ_{σ} on R is trivial. Then

H^{·}(Γ, C^{q}(R)) ≃ ∏_{σ} H^{·}(Γ, coind_{Γ_{σ}}^{Γ}(R)) ≃ ∏_{σ} H^{·}(Γ_{σ}, R)

the last isomorphism being given by the Shapiro’s lemma. As Γ is without p′-torsion and because multiplication by p is invertible in End_{ℤ}(R), one has H^{q}(Γ_{σ}, R) = 0 for all q > 0. It follows that the first spectral sequence degenerates also in the E_{2} term and then that the cohomology of the total complex is that of C^{·}(R)^{Γ}. □

Then, we are interested by the cohomology of the complex C^{q}(R)^{Γ}. The following definition precises an important part of this cohomology.

(3.1.5) Definition. Let R be a ℤ-module such that multiplication by p is invertible in End_{ℤ}(R). Let Γ be an arithmetic group. For any integer q, 0 ≤ q ≤ n, we denote by Z_{!}^{q}(R)^{Γ} the set of elements of Z^{q}(R)^{Γ} = (Kerd^{q})^{Γ} (see (3.1.3)) which have finite supports modulo Γ; let H_{!}^{q}(Γ, R) be its image in H^{q}(Γ, R) by

Z_{!}^{q}(R)^{Γ} ⊂ Z^{q}(R)^{Γ} ⟶^{can. surj.} H^{q}(C^{·}(R)^{Γ}) ≃ H^{q}(Γ, R)

(see (3.1.4)).We say that H_{!}^{q}(Γ, R) is the q-th R-module of cohomology with compact (finite) supports of the arithmetic group Γ.

(3.1.6) Remark. The cohomology groups H^{·}(Γ, R) are finitely generated R- modules. This is wellknown when n = 1 (see e.g. [Ge-Re]), when n > 1, it comes from the fact that the arithmetic groups are of finite type ([Be 1], see also [Be 2]).

3.2 Harmonic cocycles. (3.2.1) Let s and σ be two oriented simplices of ℑ.

One says that s is an oriented face of σ, and one writes s < σ, if, as non oriented simplices, s is a face of σ and if the orientation of s is the restriction of that of σ.

Write σ = [v_{0}, ⋯, v_{q}], where v_{0}, ⋯, v_{q} are the vertices of σ with a numbering which represents their equivalent class of orderings. Then, an oriented face s of σ of codimension one can be written s = [v_{0}, ⋯, v̂_{i}, ⋯, v_{q}]. We set η(s, σ) = (−1)^{i}, it does not depend on the choice in an equivalent class of the ordering of the vertices of σ.

As before, R is a ℤ-module.

(3.2.2) Definition. Let g be an element of C^{q}(R), with 1 ≤ q ≤ n, let s ∈ ℑ_{q−1}^{∗} be an oriented q − 1-simplex. We set

δ^{q}(g)(s) = ∑_{σ∈ℑ_{q}^{∗},σ>s} η(s, σ)g(σ)

the function δ^{q}(g) is an element of C^{q−1}(R) and δ^{q} is a R-linear and G(K_{∞})-linear map from C^{q}(R) to C^{q−1}(R), 1 ≤ q ≤ n.

We also set δ^{0} = 0 : C^{0}(R) ⟶ R.

(3.2.3) Remark. (i) Let g be an element of C^{q}(R), with 1 ≤ q ≤ n, let s ∈ ℑ_{q−1}^{∗} be an oriented (q − 1)-simplex. Let σ ∈ ℑ_{q}^{∗} and suppose that s is an oriented face of σ. If we write s = [v_{0}, ⋯, v_{q−1}], we have, up to equivalence of orderings of the vertices, two possibilities for σ, the vertex v of σ not in s being fixed, namely σ = σ_{1} = [v, v_{0}, ⋯, v_{q−1}] and σ = σ_{2} = [v_{0}, v, ⋯, v_{q−1}]. We have η(s, σ_{1})g(σ_{1}) = η(s, σ_{2})g(σ_{2}). Then we can write

1/2(η(s, σ_{1})g(σ_{1}) + η(s, σ_{2})g(σ_{2})) = η(s, σ_{1})g(σ_{1}) = η(s, σ_{2})g(σ_{2})

and

1/2 ∑_{σ∈ℑ_{q}^{∗},σ>s} η(s, σ)g(σ)

makes sense (see the previous definition), but, with this formula as definition of δ^{q}, the following remark is wrong.

(ii)Let f and g be in C^{q}(R), 1 ≤ q ≤ n, and suppose that

(f, g)_{q} =^{def.} ∑_{σ∈ℑ_{q}^{∗}} f(σ)g(σ)

makes sense, suppose moreover that q ≥ 1 and that f = d^{q−1}(h), with h ∈ C^{q−1}(R), suppose also that (h, δ^{q}(g))_{q−1} makes also sense. Then, it is not difficult to prove that

(d^{q−1}(h), g)_{q} = (h, δ^{q}(g))_{q−1}

then, we see that our next definitions of laplacians and harmonic cocycles are classical (see [GarH]).

(3.2.4) Definition. Let q be an integer, 1 ≤ q ≤ n, the q-th laplacian is the map

Δ^{q} =^{def} δ^{q+1} ∘ d^{q} + d^{q−1} ∘ δ^{q} : C^{q}(R) ⟶ C^{q}(R). We also set Δ^{0} = δ^{1} ∘ d^{0}.

(3.2.5) Definition. Let q be an integer, 0 ≤ q ≤ n, we say that an element f of C^{q}(R) is an harmonic cocycle of level q, or an harmonic q-cocycle, with values in R, if f is in the kernel of Δ^{q} . Let \underline{H}^{q}(R) be the set of harmonic cocycles of level q with values in R (then \underline{H}^{q}(R) = Ker(Δ^{q})).

Let Γ be a subgroup of G(K_{∞}). Let also \underline{H}^{q}(R)^{Γ} be the set of q-harmonic cocycles which are invariant under the Γ-action and \underline{H}_{!}^{q}(R)^{Γ} be the set of those which moreover have compact (finite) support modulo Γ.

(3.2.6) Remark. Suppose, as in (3.2.3), that all the expressions written with ( , )_{·} have sense. Take f ∈ C^{q}(R), we have

(Δ^{q}(f), f)_{q} = (δ^{q+1}∘d^{q}(f), f)_{q}+(d^{q−1}∘δ^{q}(f), f)_{q} = (d^{q}(f), d^{q}(f))_{q+1}+(δ^{q}(f), δ^{q}(f))_{q−1}

Then, if for instance R ⊂ ℝ, one has Δ^{q}(f) = 0 if and only if d^{q}(f) = 0 and δ^{q}(f) = 0.

The next proposition give an important case where this characterization is valid.

(3.2.7) Proposition. Let Γ be an arithmetic group and let R be a subring of ℝ. Let f ∈ C^{q}(R)^{Γ} (f is stable under Γ), with 0 ≤ q ≤ n. Suppose that the support of f is finite modulo Γ, then f is an harmonic cocycle if and only if d^{q}(f) = 0 and δ^{q}(f) = 0.

Proof. For any q, 1 ≤ q ≤ n, set C^{q}(Γ\ℑ^{⋆}, R) = Hom_{ℤ}(ℤ[Γ\ℑ_{q}^{⋆}], R). For any q, 1 ≤ q ≤ n, for any f ∈ C^{q}(Γ\ℑ^{⋆}, R) and any s ∈ Γ\ℑ_{q−1}^{⋆}, let

δ_{1}^{q}(f)(s) = ∑_{σ∈Γ\ℑ_{q}^{⋆},σ>s} η(s, σ)\frac{♯(Γ_{s})}{♯(Γ_{σ})}f(σ)

(see (3.2.1)), where Γ_{s} and Γ_{σ} are the stabilizers in Γ of lifts in ℑ, say s̃ and σ̃, of s and σ, where, if s̃ and σ̃ are chosen such that s̃ is a face of σ̃, one has writen η(s, σ) in place of η(s̃, σ̃) . We also set δ_{1}^{0} = 0.

For any q, 0 ≤ q ≤ n − 1, for any f ∈ C^{q}(Γ\ℑ^{⋆}, B) and any Σ ∈ Γ\ℑ_{q+1}^{⋆} let

d_{1}^{q}(f)(Σ) = ∑_{σ∈Γ\ℑ_{q}^{⋆},σ<Σ} η(σ, Σ)f(σ).

We also set d_{1}^{n} = 0.

A function f ∈ C^{q}(R)^{Γ} gives rise to one, say f_{1}, in C^{q}(Γ\ℑ^{⋆}, R) (if σ_{1} is the class mod. Γ of σ ∈ ℑ_{q}^{⋆}, we have f_{1}(σ_{1}) = f(σ)) and it is not difficult to see that

δ^{q}(f) = δ_{1}^{q}(f_{1}) , d^{q}(f) = d_{1}^{q}(f_{1}).

For f, g ∈ C^{q}(Γ\ℑ^{⋆}, R), f or g having a finite support, let

(f, g)_{q}^{Γ} = ∑_{σ∈Γ\ℑ_{q}^{⋆}} \frac{1}{♯(Γ_{σ})}f(σ)g(σ).

Let 1 ≤ q ≤ n, f ∈ C^{q−1}(Γ\ℑ^{⋆}, R) and g ∈ C^{q}(Γ\ℑ^{⋆}, R), f or g having a finite support, one has

(d_{1}^{q−1}(f), g)_{q}^{Γ} = (f, δ_{1}^{q}(g))_{q−1}^{Γ}.

It follows that, for any q, 1 ≤ q ≤ n, and any g ∈ C^{q}(Γ\ℑ^{⋆}, R)

(δ_{1}^{q+1} ∘ d_{1}^{q}(g) + d_{1}^{q−1} ∘ δ_{1}^{q}(g) , g)_{q}^{Γ} = (d_{1}^{q}(g), d_{1}^{q}(g))_{q+1}^{Γ} + (δ_{1}^{q}(g), δ_{1}^{q}(g))_{q−1}^{Γ} .

The proof for q = 0 uses the same arguments. □

3.3 Harmonic cocycles and cohomology of Γ.

The aim of this paragraph is to prove the theorem (3.3.1) about the cohomology with compact supports of an arithmetic group Γ, which is defined in (3.1.5). Recall that p is the characteristic of our base field K and that we say that a group has no p′-torsion if any of its torsion elements is of order a power of p.

(3.3.1) Theorem. Let Γ be an arithmetic group whithout p′-torsion; for any q, 0 ≤ q ≤ n, let H^{q}(Γ, ℤ[1/p])_{tors} be the torsion subgroup of H^{q}(Γ, ℤ[1/p]). Then, we have a canonical isomorphism of ℤ[1/p]-modules

H_{!}^{q}(Γ, ℤ[1/p]) ≃ \underline{H}_{!}^{q}(ℤ[1/p])^{Γ} ⊕ H_{!}^{q}(Γ, ℤ[1/p])_{tors}.

Moreover, these modules are finitely generated and \underline{H}_{!}^{q}(ℤ[1/p])^{Γ} is free (see (3.1.6)).

(3.3.2) Let R be a ring, equals to ℤ[1/p] or ℚ. For any q, 0 ≤ q ≤ n, let C_{!}^{q}(R)^{Γ} be the set of elements of C^{q}(R)^{Γ} which have finite supports modulo Γ. Note that the complex (C^{·}(R)^{Γ}, d^{·}) is canonically isomorphic to that defined in the proof of (3.2.7). Let B_{!}^{q}(R)^{Γ} and Z_{!}^{q}(R)^{Γ} be the intersections with C_{!}^{q}(R)^{Γ} of respectively B^{q}(R) and Z^{q}(R). Let f be in C^{q}(R)^{Γ}, note that for any simplex σ of Γ\ℑ^{⋆}, the expression f(σ) makes sense. One has (if q > 1)

δ^{q}(f)(s) = ∑_{σ∈Γ\ℑ_{q}^{⋆},σ>s} η(s, σ)\frac{♯(Γ_{s})}{♯(Γ_{σ})}f(σ).

Let moreover g be in C^{q}(R)^{Γ} and suppose that one of f or g is of finite support modulo Γ, we set (see the proof of (3.2.7))

(f, g)_{q}^{Γ} = ∑_{σ∈Γ\ℑ_{q}^{⋆}} \frac{1}{♯(Γ_{σ})}f(σ)g(σ).

Let 1 ≤ q ≤ n, f ∈ C^{q−1}(R)^{Γ} and g ∈ C^{q}(R)^{Γ}, f or g being of finite support modulo Γ; it is easy to see that

(d^{q−1}(f), g)_{q}^{Γ} = (f, δ^{q}(g))_{q−1}^{Γ}.

(3.3.3) Proposition. For any q, 1 ≤ q ≤ n, let B_{!}^{q}(ℤ[1/p])_{“tors”}^{Γ} be the set of f ∈ Z_{!}^{q}(ℤ[1/p])^{Γ} such that there exists a ∈ ℤ, a ≠ 0, with af ∈ B_{!}^{q}(ℤ[1/p])^{Γ} (i.e. the inverse image of H^{q}(Γ, ℤ[1/p])_{tors} by the canonical morphism Z_{!}^{q}(ℤ[1/p])^{Γ} ⟶ H^{q}(Γ, ℤ[1/p])). We have the direct sum of ℤ[1/p]-modules

Z_{!}^{q}(ℤ[1/p])^{Γ} = \underline{H}_{!}^{q}(ℤ[1/p])^{Γ} ⊕ B_{!}^{q}(ℤ[1/p])_{“tors”}^{Γ}.

Proof. One has \underline{H}_{!}^{q}(ℤ[1/p])^{Γ} ⊂ Z_{!}^{q}(ℤ[1/p])^{Γ} (see (3.2.7)).

(3.3.4) Lemma. (i) \underline{H}_{!}^{q}(ℤ[1/p])^{Γ} is orthogonal to B_{!}^{q}(ℤ[1/p])_{“tors”}^{Γ}, with respect to the scalar product ( , )_{q}^{Γ} (see (3.3.2));

(ii) \underline{H}_{!}^{q}(ℤ[1/p])^{Γ} is a free ℤ-module of finite rank.

Proof of the lemma. Let f ∈ \underline{H}_{!}^{q}(ℤ[1/p])^{Γ}, g ∈ B_{!}^{q}(ℤ[1/p])_{“tors”}^{Γ}, a ∈ ℤ − \{0\} such that ag ∈ B_{!}^{q}(ℤ[1/p])^{Γ} and let h be a q − 1-cochain such that d^{q−1}(h) = ag. One has

a(f, g)_{q}^{Γ} = (f, d^{q−1}(h))_{q}^{Γ} = (δ^{q}(f), h)_{q−1}^{Γ} = 0

because f ∈ Ker(δ^{q}) (see (3.2.7)). It follows also from (3.2.7) that \underline{H}_{!}^{q}(ℤ[1/p])^{Γ} is torsion free, which implies (ii). □

End of the proof of the proposition (3.3.3). Let G_{q} be a finite set of generators, over ℤ, of H^{q}(Γ, ℤ[1/p]), containing a set G′_{q} of generators of H^{q}(Γ, ℤ[1/p])_{tors} and containing a basis B_{q} of \underline{H}_{!}^{q}(ℤ[1/p])^{Γ}. Let G̃_{q} be a lift of G_{q} in Z_{!}^{q}(ℤ[1/p])^{Γ}, containing B_{q}. Let Z (resp. B) be the subset of elements of Z_{!}^{q}(ℤ[1/p])^{Γ} (resp. B_{!}^{q}(ℤ[1/p])_{“tors”}^{Γ}) generated by G_{q} (resp. G′_{q}); set H = \underline{H}_{!}^{q}(ℤ[1/p])^{Γ}. Let Z_{ℚ}, B_{ℚ} and H_{ℚ} be the same sets with ℚ in place of ℤ[1/p].

Note that the ℤ-modules Z and B are free of finite ranks and that H is a submodule of Z, thus, with the previous lemma, we see that the formula Z = H ⊕ B implies the proposition. Indeed, because H_{ℚ} ∩ Z = H, it remains to prove that Z_{ℚ} = H_{ℚ} ⊕ B_{ℚ}.

Let B_{ℚ}^{⊥} the subspace of Z_{ℚ} orthogonal to B_{ℚ} with respect to the pairing ( , )_{q}^{Γ}. We have Z_{ℚ} = B_{ℚ}^{⊥} ⊕ B_{ℚ}. Now we prove that H_{ℚ} = B_{ℚ}^{⊥}. Let f ∈ Z_{ℚ} and let f = u + d^{q−1}g be its decomposition with respect to the previous direct sum (u is in B_{ℚ}^{⊥} and g is a (q − 1)-cochain). Let h be a (q − 1)-cochain:

(δ^{q}(f − d^{q−1}g), h)_{q−1}^{Γ} = (f − d^{q−1}g, d^{q−1}h)_{q}^{Γ} = (u, d^{q−1}h)_{q}^{Γ} = 0,

then f −d^{q−1}g ∈ Ker(δ^{q}) and hence f −d^{q−1}g ∈ H_{ℚ} (see (3.2.7)). Then H_{ℚ} ⊃ B_{ℚ}^{⊥}, the other inclusion is given by (3.3.4). □

Proof of (3.3.1). For any q, 1 ≤ q ≤ n, propositions (3.1.4) and (3.3.3) imply the formula of the theorem, the case q = 0 is easy. The other sentence comes from (3.1.6).

(3.3.5) Corollary. Let Γ be an arithmetic group whithout p′-torsion; for any q, 0 ≤ q ≤ n, we have a canonical isomorphism of ℚ-vector spaces

H_{!}^{q}(Γ, ℚ) ≃ \underline{H}_{!}^{q}(ℚ)^{Γ}.

and these spaces are finite dimensional over ℚ.

4 The locus of supports of harmonic cocycles and of cohomology.

Let Γ be an arithmetic group whithout p′-torsion. The ℤ[1/p]-modules \underline{H}_{!}^{·}(ℤ[1/p])^{Γ} and H_{!}^{·}(Γ, ℤ[1/p]) are finitely generated (see (3.3.1)), then there exists a finite subcomplex of Γ\ℑ which contains all the supports modulo Γ of elements of these modules (the support of an element of these cohomology modules is defined by (3.1.4) and (3.1.5)). The purpose of this paragraph is to give a description of the locus of this subcomplex. Indeed, we first introduce a notion of cusps of an arithmetic group Γ, they are “the missing part of Γ\ℑ at infinity”, and, secondly, we prove that “neighborhoods of this missing part at infinity” are given modulo Γ by a union of finitely many sectors of ℑ, and that the complementary complexe of each of these neighborhoods is finite modulo Γ. The supports of the elements of \underline{H}_{!}^{·}(ℤ[1/p])^{Γ} and of H_{!}^{·}(Γ, ℤ[1/p]) are modulo Γ outside sufficiently small neighborhoods of the cusps. These properties are known in the case where n = 1 (i.e. when the Building is a tree, see [Se 2] ch. II, §2.3).

4.1. The cusps.

In all this paragraph, Γ is an arithmetic subgroup of G(K). Recall that ℑ_{∞} is the spherical building at infinity of ℑ (see §2.3).

(4.1.1) Definition. Let v_{0} be a fixed vertex of ℑ. For any chamber σ of ℑ_{∞}, let S_{σ} be a sector of ℑ beginning at v_{0} and ending at σ (see (2.3.1) and (2.3.3)). Let σ and σ′ be two chambers of ℑ_{∞}, we say that they are equivalent modulo Γ if

sup_{γ∈Γ} ♯((S_{σ} ∩ S_{γ(σ′)})_{n}) = ∞

i.e. if the number of chambers of S_{σ} ∩ S_{γ(σ′)} is unbounded, as γ goes through Γ. A Γ-equivalent class of chambers of ℑ_{∞} will be called a cusp of Γ.

This notion of cusp does not depend on the choice of the vertex v_{0}. This is an equivalent relation on the set of chambers of ℑ_{∞}.

(4.1.2) Proposition. The set of cusps of Γ is finite, it canonically one to one with Γ\G(K)/P (K).

Proof. The second part of the proposition implies the first, because it is wellknown that

(4.1.3) Lemma. The set Γ\G(K)/P (K) is finite.

(see for instance [Go]).

Let σ_{1},...,σ_{d} be chambers of ℑ_{∞} which represent Γ\G(K)/P (K) (see (2.3.6)), let S_{1},...,S_{d} be sectors of ℑ which end respectively at σ_{1},...,σ_{d} and which begin at the same vertex v_{0} of ℑ. Let S be a sector of ℑ, begining in v_{0}, and, for any l > 0, let S(l) be the set of chambers of S with distance at most l from v_{0}. For all l there exists i and γ , there exists a sector S_{i,γ} beginning in v_{0} and ending at γ(σ_{i}), such that S(l) ⊂ S_{i,γ}; the conclusion now is a direct consequence of the definition (4.1.1). □

(4.1.4) Remark. Indeed, the preceeding proof means that the cusps of Γ are “rationnal” over the global field K (this is known in dimension 1: [Se] ch. II, §2.3).

In the sequel, we will say equivalently that a cusp of Γ is a class of chamber of ℑ_{∞} or an element of Γ\G(K)/P (K).

4.2 On the structure of Γ\ℑ.

The following result gives the main property of our notion of cusp. First we introduce notations which will be used in the rest of this chapter.

(4.2.1) let Γ be an arithmetic subgroup of G(K). Let v_{0} be a fixed vertex of the building ℑ. Let σ_{1},...,σ_{d} be chambers of ℑ_{∞} which represent Γ\G(K)/P (K), i.e. which represent the cusps of Γ. Let S_{1},...,S_{d} be sectors of ℑ which end respectively at σ_{1},...,σ_{d} and which begin at v_{0}. For any integer l ≥ 0 and for any i, 1 ≤ i ≤ d, let S_{i}^{l} be the subcomplex of S_{i} such that for any chamber C of S_{i}^{l}, the (combinatorial) distance from C to the complementary complex S_{i}^{c} of S_{i} (in ℑ) is at least l, i.e. such that for any chamber C′ of S_{i}^{c}, any geodesic from C to C′ contains at least l chambers. Let

P (l) = ∪_{1≤i≤d} ∪_{γ∈Γ} γ(S_{i}^{l})

and D(l) be its complementary complex in ℑ.

P (l) is, modulo Γ, what one has called, in the introduction of this paragraph, a “neighborhood of this missing part at infinity”, this missing part being modulo Γ the union of the cusps. Note that Γ acts on P (l) and D(l).

(4.2.2) Theorem. For all l ≥ 0, the complex Γ\D(l) is finite.

This theorem implies that P (l) does not depend, up to a finite subcomplex modulo Γ, on the choices of the σ_{i}’s and of the choice of the vertex v_{0}. One has also the following remark.

(4.2.3) Suppose that Γ is whithout p′-torsion, then, for l sufficiently large, the supports of the elements of \underline{H}_{!}^{·}(ℤ[1/p])^{Γ} and H_{!}^{·}(Γ, ℤ[1/p]) are in D(l).

All the rest of this §4 is devoted to the proof of the theorem (4.2.2).

First we reduce to the case Γ = G(A). Let σ be a chamber at infinity which represents a cusp of G(A) and let S be a sector of ℑ ending at σ. Let g_{1}(σ), ⋯, g_{r}(σ), with g_{1}, ⋯, g_{r} ∈ G(A), be representatives of the cusps of Γ, which are equal modulo G(A) to the cusp σ of G(A). Let L_{1}, ⋯, L_{r} be sectors in ℑ ending respectively at g_{1}(σ), ⋯, g_{r}(σ). There exists a choice of these last sectors such that, for any integer l ≥ 0, the two complexes

⋃_{γ∈Γ, 1≤i≤r} γ(L_{i}^{l}) and ⋃_{γ∈Γ, 1≤i≤r} γg_{i}(S^{l})

(L_{i}^{l} and S^{l} are defined as in (4.2.1)) differ modulo Γ by finitely many chambers. This implies our assertion, because the number of cusps is finite.

(4.2.4) Then we suppose now that Γ = G(A). We denote by C_{0} and v_{0} respectively the fundamental chamber and the fundamental vertex (see (2.2.4)).

(4.2.5) Let Y be a subset of G(𝔸_{f}) which represents the set of double classes G(K)\G(𝔸_{f})/G(𝕆_{f}). It is wellknown that Y is finite (indeed one to one with PicA). For any \underline{y} ∈ Y one sets

Γ\underline{y} = \underline{y}G(𝕆_{f})\underline{y}^{−1} ∩ G(K).

These groups are arithmetic, we can suppose that \underline{1} = (1, ⋯, 1) ∈ G(𝔸_{f}) is in Y, then we have Γ\underline{1} = G(A).

It is also wellknown that there exists a one to one map

G(K)\G(𝔸)/(G(𝕆_{f}) × B_{∞}) → ∐_{\underline{y}∈Y} Γ\underline{y}\ℑ_{n}

(where ℑ_{n} is the set of chambers of the building ℑ) induces by

G(𝔸) = G(𝔸_{f}) × G(K_{∞}) → ∐_{\underline{y}∈Y} Γ\underline{y}\ℑ_{n}

which maps \underline{g} = (γ\underline{y}\underline{k}_{f}, g_{∞}) to γ^{−1}g_{∞} mod. Γ\underline{y} (where γ ∈ G(K), \underline{y} ∈ Y and \underline{k}_{f} ∈ G(𝕆_{f}); \underline{g} being given, \underline{y} is well defined).

This last remark explains why some arguments of the proof of theorem (4.2.2) will be of adelic nature. This is the object of the next section.

4.3 Proof of theorem (4.2.2): the adelic part.

(4.3.1) Let 𝔙 be the set of normalized valuations of K. If \underline{t} = (t_{ω})_{ω∈𝔙} is an idèle of K, i.e. is an element of GL_{1}(𝔸), we set

|\underline{t}| = ∏_{ω∈𝔙} |t_{ω}|_{ω}.

Let H be a subset of G(𝔸), let \underline{g} = (\underline{g}_{i,j}) be an element of H; for any integer i, 0 ≤ i ≤ n, we set |t_{i}(\underline{g})| = |\underline{g}_{i,i}|, if it makes sense, which is supposed in the following definitions.

Let c and c′ be two positive real numbers with c′ ≤ c, we set

H_{c′} = \{\underline{g} ∈ H / ∀i 0 ≤ i ≤ n − 1 c′ ≤ |t_{i}(\underline{g})/t_{i+1}(\underline{g})|\},

H_{c′}^{c} = \{\underline{g} ∈ H / ∀i 0 ≤ i ≤ n − 1 c′ ≤ |t_{i}(\underline{g})/t_{i+1}(\underline{g})| ≤ c\},

H_{(c′,c)} = \{\underline{g} ∈ H / ∀i c′ ≤ |t_{i}(\underline{g})/t_{i+1}(\underline{g})| , ∃i |t_{i}(\underline{g})/t_{i+1}(\underline{g})| > c , 0 ≤ i ≤ n − 1\}.

Let ω ∈ 𝔙. The preceeding definitions make sense for a subset H of G(K_{ω}), if one considers, as usual, G(K_{ω}) as a subgroup of G(𝔸). In that case, to be precise, one will write ^{ω}H_{c′}, ^{ω}H_{c′}^{c} and ^{ω}H_{(c′,c)} (the reason of that will be clear when one will consider subsets of G(K), G(K) being viewed as a subgroup of G(K_{ω})).

(4.3.2) Lemma. There exits two real numbers 0 < c′_{0} < c_{0} such that, for all c′ and c satisfying c′ ≤ c′_{0} < c_{0} ≤ c, the image in G(K)\G(𝔸)/G(𝕆)

1. of P (𝔸)_{c′} is onto,

2. of P (𝔸)_{c′}^{c} is finite.

One will suppose moreover (for technical reasons) that c′ ≤ 1.

Proof. This lemma is a direct consequence of deep results of G. Harder (apply g ↦ g^{−1} to Sätze 2.1.1 and 2.2.2 of [Ha 1], see also §1.1 of [Ha 2]). □

The next proposition is the key result of this section. Before to set it, we introduce a notation.

(4.3.3) For any \underline{y} ∈ Y (see (4.2.5)), let 𝔖\underline{y} be a subset of G(K) one to one with the set of double classes Γ\underline{y}\G(K)/P (K), i.e. a subset of G(K) which represents the cusps of the arithmetic group Γ\underline{y} (see (4.1.2)).

20

(4.3.4) Proposition. There exists a constant δ > 0 satisfying the following property: let c′ and c be such that c′ ≤ c′_{0} < c_{0} ≤ c and δc′ ≤ c′_{0} < c_{0} ≤ δc (see (4.3.2)), then there exists a finite subset F ⊂ U(K) such that

G(K)(P (𝔸)_{(c′,c)})(G(𝕆_{f}) × B_{∞}) ⊆ G(K)[ ∐_{\underline{y}∈Y} (\{\underline{y}\} × E\underline{y})](G(𝕆_{f}) × B_{∞}),

where for any \underline{y} ∈ Y, E\underline{y} = 𝔖\underline{y}F (^{∞}T (K)_{(δc′,δc)})\overline{W} (this is a subset of G(K_{∞})).

The constant δ does not depend on c′ and c, but F depends on them. The proof of this proposition needs many lemmata.

(4.3.5) Lemma. P (𝔸_{f}) = U(K)T (𝔸_{f})P (𝕆_{f}).

Proof. Let \underline{g} ∈ P (𝔸_{f}), \underline{g} = (\underline{g}_{i,j}). Let \underline{t} = diag(\underline{g}_{0,0}, ⋯, \underline{g}_{n,n}) ∈ T (𝔸_{f}). We want to find u = (u_{i,j}) ∈ U(K) such that \underline{w} := \underline{t}^{−1}u\underline{g} ∈ P (𝕆_{f}). This gives the relations for i ≤ j

\underline{w}_{i,j} = ∑_{1≤k≤j} \underline{g}_{i,i}^{−1}\underline{g}_{k,j}u_{i,k} ∈ P (𝕆_{f}).

u_{i,i}, ⋯, u_{i,j} having been chosen, because K is dense in 𝔸_{f}, we see easely that u_{i,j+1} exists. □

(4.3.6) Lemma. Let X be a set of representatives in GL_{1}(𝔸_{f}) of GL_{1}(K)\GL_{1}(𝔸)_{f}/GL_{1}(𝕆_{f}). There exists a constant δ_{1} > 0 satisfying the following property: let c and c′ be as in (4.3.2), for any \underline{g} ∈ P (𝔸)_{(c′,c)} there exists \underline{τ} ∈ T (X) and g_{∞} ∈ ^{∞}P (K_{∞})_{(δ_{1}c′,δ_{1}c)} such that

\underline{g} ≡ (\underline{τ}, g_{∞}) mod. (P (K) , P (𝕆_{f}) × \{1\})

(this formula means mod. P (K) on the left and mod. P (𝕆_{f}) × {1} on the right).

Proof. We write, following (4.3.5), \underline{g} = (u\underline{τ}\underline{ρ}, h_{∞}) with u ∈ U(K), \underline{τ} ∈ T (𝔸_{f}), \underline{ρ} ∈ P (𝕆_{f}) and h_{∞} ∈ P (𝕆_{∞}). We also write \underline{τ} = γ\underline{τ}_{1}\underline{ρ}_{1} with γ ∈ T (K), \underline{τ}_{1} ∈ T (X) (see the definition of X) and \underline{ρ}_{1} ∈ T (𝕆_{f}). We have

\underline{g} ≡ (τ_{1}, γ^{−1}u^{−1}h_{∞}) mod. (P (K) , P (𝕆_{f}) × \{1\})

and for all i = 0, ⋯, n (see (4.3.1))

| t_{i}(\underline{g}) |=| t_{i}(τ_{1}, γ^{−1}u^{−1}h_{∞}) |,

then for all i = 0, ⋯, n − 1

| t_{i}(1, γ^{−1}u^{−1}h_{∞})/t_{i+1}(1, γ^{−1}u^{−1}h_{∞}) |=
| t_{i}(τ_{1}, γ^{−1}u^{−1}h_{∞})/t_{i+1}(τ_{1}, γ^{−1}u^{−1}h_{∞}) | . | t_{i}(τ_{1}, 1)/t_{i+1}(τ_{1}, 1) |^{−1}=
| t_{i}(\underline{g})/t_{i+1}(\underline{g}) | . | t_{i}(τ_{1}, 1)/t_{i+1}(τ_{1}, 1) |^{−1}

We set δ_{1} = min_{0≤i≤n−1,τ_{1}∈T (X)} | t_{i}(τ_{1}, 1)/t_{i+1}(τ_{1}, 1) |^{−1}, which exists because X is finite. □

(4.3.7) Lemma. There exists a constant δ > 0 satisfying the following property: let c and c′ be as in (4.3.2), then for any \underline{g} ∈ P (𝔸)_{(c′,c)} there exists \underline{y} ∈ Y, σ ∈ 𝔖\underline{y} (see (4.3.3)) and g_{∞} ∈ (^{∞}P (K_{∞})_{(δc′,δc)}) such that

\underline{g} ≡ (\underline{y}, σg_{∞}) mod. (G(K) , G(𝕆_{f}) × \{1\}).

Proof. We have, following (4.3.6)

\underline{g} ≡ (\underline{τ}, g_{∞}) mod. (P (K) , P (𝕆_{f}) × \{1\})

with, in particular, g_{∞} ∈ ^{∞}P (K_{∞})_{(δ_{1}c′,δ_{1}c)}. There exists \underline{y} ∈ Y, γ ∈ G(K) and \underline{k} ∈ G(𝕆_{f}) such that \underline{τ} = γ\underline{y}\underline{k}. Note that γ runs in a finite set depending only on X, because τ is in T (X).

We set γ^{−1} = ασe with α ∈ Γ\underline{y}, σ ∈ 𝔖\underline{y} and e ∈ P (K) (see (4.3.3)). One has

\underline{g} ≡ (γ\underline{y}\underline{k}, g_{∞}) mod. (P (K) , P (𝕆_{f}) × \{1\})
\underline{g} ≡ (\underline{y}, γ^{−1}g_{∞}) mod. (G(K) , P (𝕆_{f}) × \{1\})
\underline{g} ≡ (α^{−1}\underline{y}, σeg_{∞}) mod. (G(K) , P (𝕆_{f}) × \{1\})

which implies, because α ∈ Γ\underline{y} ⊂ \underline{y}G(𝕆_{f})\underline{y}^{−1},

\underline{g} ≡ (\underline{y}, σeg_{∞}) mod. (G(K) , G(𝕆_{f}) × \{1\}).

As e runs in a finite set depending only on X (as the above γ’s) and Y, there exists δ_{2} > 0, which does not depend on c′ and c, such that eg_{∞} ∈ (^{∞}P (K_{∞})_{(δ_{1}δ_{2}c′,δ_{1}δ_{2}c)}). □

(4.3.8) Lemma. Let c and c′ be as in (4.3.2). Then there exists a finite subset F of U(K) satisfying the following property: for any \underline{y} ∈ Y and any σ ∈ 𝔖\underline{y} (see (4.3.3))

Γ\underline{y}σF (^{∞}T (K)_{(c′,c)})P (𝕆_{∞}) ⊇ σ(^{∞}P (K_{∞})_{(c′,c)}).

Proof. We prove a little better: let Γ be an arithmetic group, then there exists a finite subset F_{Γ} of U(K) such that

(4.3.9) (^{∞}P (K_{∞})_{(c′,c)}) = (Γ ∩ U(K))F_{Γ}(^{∞}T (K)_{(c′,c)})P (𝕆_{∞}).

This implies the lemma, for Γ = σ^{−1}Γ\underline{y}σ and because σ and \underline{y} run in finite sets.

Note that it is sufficient to prove (4.3.9) for Γ = G(A), because Γ ∩ U(K) and G(A) ∩ U(K) are commensurable.

Let R ⊂ K be a set of representatives of 𝔽(∞) (the residue field at ∞), with 0 ∈ R, let η_{0} be the integer defined by

−η_{0} = max{v_{∞}(a) / a ∈ A, v_{∞}(a) < 0},

let a_{0} ∈ A which realises −η_{0} and set (π_{∞} = π ∈ K is uniformising parameter at ∞)

R̃ = {ρπ^{−η}a_{0}^{l} / ρ ∈ R, 0 ≤ η < η_{0}, l ≥ 0, η + η_{0}l > 0},
Π = {π^{−η}a_{0}^{l} / 0 ≤ η < η_{0}, l ∈ ℤ}.

It is easy to see that

(^{∞}P (K_{∞})_{(c′,c)}) = (^{∞}T (K)_{(c′,c)})U(K_{∞})T (𝕆_{∞}),

indeed, with straightforward calculations one can prove that

(4.3.10) (^{∞}P (K_{∞})_{(c′,c)}) = (^{∞}T (Π)_{(c′,c)})U(R̃)P (𝕆_{∞}).

Let t = diag(t_{0}, ⋯, t_{n}) ∈ (^{∞}T (Π)_{(c′,c)}) and u ∈ U(R̃) and set tut^{−1} = (u_{i,j}). We have for i ≤ j and when u_{i,j} ≠ 0

| u_{i,j} |_{∞}≥| t_{i}/t_{j} |_{∞}≥ (c′)^{j−i} ≥ (c′)^{n}.

On the other hand, we know that there exists a real number ε satisfying the following property: for any λ in K there exists an a in A such that | λ − a |_{∞}≤ ε. These two remarks lead to prove that for any t ∈ (^{∞}T (Π)_{(c′,c)}), for any u ∈ U(R̃), there exists γ ∈ U(A) such that, if (v_{i,j}) denotes the matrix γ^{−1}tut^{−1}, if i < j and v_{i,j} ≠ 0, then

ε ≥| v_{i,j} |_{∞}≥ (c′)^{n}.

The set

F_{1} := {(v_{i,j}) / (v_{i,j}) = γtut^{−1}, γ ∈ U(A), t ∈ (^{∞}T (Π)_{(c′,c)}), u ∈ U(R̃), ε ≥| v_{i,j} |_{∞}≥ (c′)^{n}∀i, j}

is finite modulo P (𝕆_{∞}) on the right (note that R̃ and Π are discrete); let F be a finite subset of U(K) such that F_{1} ⊂ F P (𝕆_{∞}).

Let g ∈ (^{∞}P (K_{∞})_{(c′,c)}) and write following (4.3.10) g = tuh, with t ∈ (^{∞}T (Π)_{(c′,c)}), u ∈ U(R̃) and h ∈ P (𝕆_{∞}); we have

g = tuh = (tut^{−1})tuh = γ^{−1}(γtut^{−1})h

with γtut^{−1} ∈ F P (𝕆_{∞}). This proves (4.3.9). □

(4.3.11) Lemma. There exists a constant δ > 0 satisfying the following property: let c′ and c be such that c′ ≤ c′_{0} < c_{0} ≤ c and δc′ ≤ c′_{0} < c_{0} ≤ δc (see (4.3.2)), then there exists a finite subset F ⊂ U(K) such that

P (𝔸)_{(c′,c)} ⊆ G(K) ∐_{\underline{y}∈Y} [\{\underline{y}\} × (𝔖\underline{y}F (^{∞}T (K)_{(δc′,δc)}))](G(𝕆_{f}) × P (𝕆_{∞}))

Proof. Let δ be given by (4.3.7), let c and c′ be such that c′ ≤ c′_{0}, δc′ ≤ c′_{0} and c ≥ c_{0}, δc ≥ c_{0} (see (4.3.2)). Let \underline{g} ∈ P (𝔸)_{(c′,c)}. We write as in (4.3.7)

\underline{g} ≡ (\underline{y}, σg_{∞}) mod. (G(K) , G(𝕆_{f}) × \{1\})

with g_{∞} ∈ (^{∞}(P (K_{∞})_{(δc′,δc)}).

We have, following (4.3.8), σg_{∞} = γσetρ where t ∈ (^{∞}T (K)_{δc′,δc)}), e ∈ F, ρ ∈ P (𝕆_{∞}) and γ ∈ Γ\underline{y}, γ = \underline{y}\underline{k}\underline{y}^{−1} with \underline{k} ∈ G(𝕆_{f}). Thus we see that modulo (G(K) , G(𝕆_{f}) × P (𝕆_{∞}))

\underline{g} ≡ (\underline{y}, γσetρ) ≡ (γ^{−1}\underline{y}, σetρ) = (\underline{y}\underline{k}, σetρ) ≡ (\underline{y}, σet). □

Proof of proposition (4.3.4). There exists finitely many elements p_{i} of P (𝕆_{∞}), 1 ≤ i ≤ i_{0}, such that

G(𝕆_{∞}) = ⋃_{w∈\overline{W} ,1≤i≤i_{0}} p_{i}wB_{∞}

(because the existence of the canonical morphism G(𝕆_{∞})/B_{∞} ↪ G(𝔽(∞))/P (𝔽(∞))), then we have (as before, \underline{1} means (1, ⋯, 1) ∈ G(𝔸_{f}))

P (𝔸)_{(c′,c)}(\{\underline{1}\} × G(𝕆_{∞})) = P (𝔸)_{(c′,c)}(\{\underline{1}\} × (\overline{W} B_{∞}))

which gives with (4.3.11) the expected formula.

4.4 End of the proof of theorem (4.2.2).

To finish the proof, we need also the two following easy lemmata.

(4.4.1) Lemma. Let l > 0 be an integer and let T_{l} be the set of elements of T (K) which give on V_{0} (the fundamental appartement) translations by vectors of the form ∑_{0≤i≤n−1} m_{i}e_{i} with m_{i} ≥ l for all i (see (2.2.1)).

(1) Let S_{1} be the sector of V_{0} (see (4.2.1) and 2.3) beginning at v_{0} (the fundamental vertex) and ending at the fundamental chamber at infinity, then

(S_{1}^{l})_{n} ⊆ T_{l}\overline{W} C_{0} := \{twC_{0}/t ∈ T_{l}, w ∈ \overline{W} \} ⊆ (S_{1}^{l−1})_{n}

where S_{1}^{l} is defined in (4.2.1), (S_{1}^{l})_{n} is the set of chambers of S_{1}^{l}, C_{0} is the fundamental chamber, and \overline{W} the linear part of the affine Weyl group W.

(2) Let F a finite subset of U(K) (as that defined in (4.3.4)). Recall that 𝔖\underline{1} ⊂ G(K) is a set of representatives of the cusps of Γ\underline{1} = G(A), i.e. of G(A)\G(K)/P (K); we can suppose that 1 is in 𝔖\underline{1} and represents the fundamental chamber at infinity. Then there exists an integer κ ≥ 0 such that, for any l ≥ κ,

𝔖\underline{1}F T_{l}\overline{W} C_{0} ⊆ (∪_{1≤i≤d}S_{i}^{l−κ})_{n}.

Proof. Part (1) is a direct consequence of calculations which prove that the closed chamber \overline{C}_{0} is a fundamental domain for the action of G(K_{∞}) on ℑ (see [Bro] ch. I, §5.F anf the definition of ℑ, as in (2.2.2)).

Part (2) follows from part (1) and from the fact that U(K) stabilizes the fundamental chamber at infinity. □

(4.4.2) Lemma. Let l > 0 be an integer and let c′_{1} ≤ 1, c_{1} be real numbers such that

log(c_{1}) ≥ l(n + 1)log(♯𝔽(∞)) − \frac{(n − 1)(n + 1)^{2}}{4} log(c′_{1}).

Then, (^{∞}T (K)_{(c′_{1},c_{1})}) ⊂ T_{l}.

Remark. There exists such c′_{1} ≤ 1 and c_{1} which moreover satisfy (4.3.4).

Proof. Let t = diag(z_{0}π^{k_{0}}, ⋯, z_{n}π^{k_{n}}) be an element of T (K)_{(c′_{1},c_{1})}, with z_{i} in 𝕆_{∞}^{⋆} ∩K and k_{i} ∈ ℤ. This gives on V_{0} the translation by the vector ∑_{0≤i≤n−1} m_{i}e_{i} with, for all i, m_{i} = −(k_{0} + ⋯ + k_{i}) (recall that k_{0} + ⋯ + k_{n} = 0). Because t is in T (K)_{(c′_{1},c_{1})}, one has, for all i, k_{i} − k_{i+1} ≤ −(log(c′_{1}))/log(♯𝔽(∞)) and there exists i_{0} such that k_{i_{0}} − k_{i_{0}+1} ≤ −(log(c_{1}))/log(♯𝔽(∞)).

Let i be an integer, 0 ≤ i ≤ n − 1. For any integer j with 0 ≤ j ≤ i let α_{j,i} = (j+1)(n−i)/(n+1) and, when i < j ≤ n−1, let α_{j,i} = (i+1)(n−j)/(n+1). Then we have k_{0} + ⋯ + k_{i} = ∑_{0≤j≤n−1} α_{j,i}(k_{j} − k_{j+1}).

It follows from all these remarks that for all i

m_{i} = − ∑_{0≤j≤n−1} α_{j,i}(k_{j} − k_{j+1})

≥ \frac{1}{log(♯𝔽(∞))}((n − 1)(log(c′_{1}))max_{j}(a_{j,i}) + (log(c_{1}))min_{j}(α_{j,i}))

because c′_{1} ≤ 1 and c_{1} ≥ 1; this gives

m_{i} ≥ \frac{1}{(n + 1)log(♯𝔽(∞))}(\frac{(n − 1)(n + 1)^{2}}{4} log(c′_{1}) + log(c_{1})) ≥ l. □

Now we can finish the proof of the theorem. Let δ > 0 be the constant given by (4.3.4), let c′ and c be such that c′ ≤ c′_{0} < c_{0} ≤ c and c′_{1} = δc′ ≤ c′_{0} < c_{0} ≤ c_{1} = δc (see (4.3.2)). Let l > 0 be an integer. We suppose moreover that l, c′_{1} and c_{1} satisfy part (2) of (4.4.1) and (4.4.2). Let φ be the canonical map

φ : ℑ_{n} → Γ\underline{1}\ℑ_{n}

(recall that Γ\underline{1} = G(A)).

It follows from (4.4.1) that

φ(𝔖\underline{1}F T_{l}\overline{W} C_{0}) ⊆ φ(P (l − κ)_{n}).

One has also (see the choice of l)

(^{∞}T (K)_{(c′_{1},c_{1})}) ⊂ T_{l}.

Thus, it is suffiscient to prove that the complementary complex of φ(𝔖\underline{1}F (^{∞}T (K)_{(c′_{1},c_{1})})\overline{W} C_{0}), in Γ\underline{1}\ℑ_{n}, is finite. Let

G(𝔸) ⟶^{η} G(K)\G(𝔸)/(G(𝕆_{f}) × B_{∞}) ⟶^{ψ} ∐_{\underline{y}∈Y} Γ\underline{y}\ℑ_{n}

be the two maps of (4.2.5), the second, ψ, being one to one. It follows from (4.3.4) that ψ^{−1}∘φ(𝔖\underline{1}F (^{∞}T (K)_{(c′_{1},c_{1})})\overline{W} C_{0}) is in the complementary set of η(P (𝔸)_{(c′,c)}) and this last set is finite (see (4.3.2)). □

5 Euler-Poincaré characteristic.

Let Γ be an arithmetic group with no p′-torsion. In this paragraph we prove that a good choice of the locus of supports of elements of \underline{H}_{!}^{·}(ℚ)^{Γ} leads to determine “geometrically” the Euler-Poincaré characteristic of Γ for the cohomology with compact supports and with coefficients in ℚ (indeed our proof goes for all subfield of ℝ). This result is a generalization to dimension n > 1 of some aspects of [Se 2] ch. I §3.3, [Re], [Ge-Re] §3.2 (see §(1.1.3)).

(5.0.1) Let Γ be an arithmetic group. One denotes by χ_{!}(Γ, ℚ) its Euler-Poincaré characteristic for the cohomology with values in ℚ and with compact supports, i.e.

χ_{!}(Γ, ℚ) = ∑_{0≤q≤n}(−1)^{q}dim_{ℚ}H_{!}^{q}(Γ, ℚ).

(5.0.2) Let l be an integer such that any cocycle in C_{!}^{·}(ℚ)^{Γ} has, up to a coboundary, its support in D(l) (see (3.3.5), (4.2.2) and (4.2.3)).

Let D ⊂ ℑ such that

(i) as subspace of ℑ, D is contractible and Γ.D ⊇ D(l); (ii) D is a finite subcomplex of ℑ, (to be a subcomplex means that D contains all the faces of its simplices); (iii) any cocycle in C_{!}^{·}(ℚ)^{Γ} has, up to a coboundary, its support in Γ.D and the support of any element of \underline{H}_{!}^{·}(ℚ)^{Γ} is in Γ.D.

The aim of this paragraph is to prove the following

(5.0.3) Theorem. Let Γ be an arithmetic group without p′-torsion and let D be as in (5.0.2). For any q, 0 ≤ q ≤ n, let D_{q,Γ} be the set of non oriented q-simplices σ of D, satisfying the following property: there exists γ ∈ Γ such that γ(σ) is a q-simplex of D and γ(σ) ≠ σ. Let g_{q} = ♯(D_{q,Γ}). Then one has

χ_{!}(Γ, ℚ) = 1 + ∑_{0≤q≤n}(−1)^{q+1}g_{q}.

Proof. We use notations and definitions of §3. For any q, 0 ≤ q ≤ n, let C_{D}^{q}, Z_{D}^{q}, B_{D}^{q} be respectively the subspace of C_{!}^{q} := C_{!}^{q}(ℚ)^{Γ}, Z_{!}^{q} := Z_{!}^{q}(ℚ)^{Γ}, B_{!}^{q} := B_{!}^{q}(ℚ)^{Γ} (with B_{!}^{0} := B_{!}^{0}(ℚ)^{Γ} := 0) of elements with supports in Γ.D; one sets also H_{!}^{q} := \underline{H}_{!}^{q}(ℚ)^{Γ} (see (3.3.5)). We have (see the proof of (3.3.4)) Z_{D}^{q} = H_{D}^{q} ⊕ B_{D}^{q} for 0 ≤ q ≤ n.

Let q < n, the map δ^{q+1} : C_{D}^{q+1} ⟶ C_{D}^{q} is defined, its restriction to B_{D}^{q+1} is injective, indeed, if d^{q}(f) is in Kerδ^{q+1} (with f ∈ C_{!}^{q}), then (see (3.3.2))

0 = (δ^{q+1}d^{q}f, f)_{q}^{Γ} = (d^{q}f, d^{q}f)_{q+1}^{Γ}

thus d^{q}f = 0 (here we use that we are in ℝ). It follows then

C_{D}^{q} ≃ Z_{D}^{q} × B_{D}^{q+1} ≃ H_{!}^{q} × B_{D}^{q} × B_{D}^{q+1}

this formula being also right when q = 0, n if we set B_{D}^{0} = B_{D}^{n+1} = 0.

Let c_{q}, h_{q}, b_{q} be the dimensions over ℚ of respectively C_{D}^{q}, H_{!}^{q}, B_{D}^{q}. Let D_{q} be the set of q-simplexes of D. We have, 0 ≤ q ≤ n,

c_{q} = ♯(Γ\Γ.D_{q}) = ♯(D_{q}) − g_{q}

and, following the previous caculations,

∑_{0≤q≤n}(−1)^{q}c_{q} = ∑_{0≤q≤n}(−1)^{q}(h_{q} + b_{q} + b_{q+1}) = 1 + b_{0} + ∑_{0≤q≤n}(−1)^{q}h_{q},

and b_{0} = 0. We have found (see (3.3.5))

χ_{!}(Γ, ℚ) = 1 + ∑_{0≤q≤n}(−1)^{q+1}g_{q} + ( − 1 + ∑_{0≤q≤n}(−1)^{q}♯(D_{q})).

The following wellknown lemma finishes the proof.

(5.0.4) Lemma. Let D be a non empty subcomplex of ℑ which is, as a subspace, contractible. For any q, 0 ≤ q ≤ n, let D_{q} be the set of q-simplices of D. Then

∑_{0≤q≤n}(−1)^{q}♯(D_{q}) = 1.

(5.0.5) A problem. Notations are those of theorem (5.0.3). We can suppose that, for any chamber C of D and for any γ ∈ Γ, γ(C) is not in D (i.e. D_{n,Γ} is empty). Let q be an integer, 0 ≤ q ≤ n − 1, and Γ_{q} be the subset of γ ∈ Γ satisfying the following property: γ is in Γ_{q} if there exists a q-simplex σ of D such that γ(σ) is again in D. Let G_{0} be the subgroup of Γ generated by Γ_{0} and Γ_{tors} (Γ_{tors} is the group generated by the torsion elements of Γ) and, for any q, 1 ≤ q ≤ n − 1, let G_{q} be the normal subgroup of G_{q−1} generated by Γ_{q} and Γ_{tors}. Set G_{n} = {1}. Then we can expect

G_{0} = Γ

and

Hom_{gr}(G_{q}/G_{q+1}, ℤ[1/p]) ≃ Hom_{gr}(Π_{q+1}(Γ\ℑ, ·), ℤ[1/p]) , 0 ≤ q ≤ n − 1

(we consider homotopy groups for a fixed point of Γ\ℑ). Such formulae would generalize properties of arithmetic groups in dimension 1 (see (1.1.3).

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Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cédex 4. France

   E-mail address: reversat@picard.ups-tlse.fr