On the Zeta-Functions of Some Simple Shimura Varieties^{∗}

by

Robert Langlands

1. Introduction.

In an earlier paper [14] I have adumbrated a method for establishing that the zero-function of a Shimura variety associated to a quaternion algebra over a totally real field can be expressed as a product of L-functions associated to automorphic forms. Now I want to add some body to that sketch. The representation-theoretic and combinatorial aspects of the proof will be given in detail, but it will simply be assumed that the set of geometric points has the structure suggested in [13]. This is so at least when the algebra is totally indefinite, but it is proved by algebraic-geometric methods that are somewhat provisional in the context of Shimura varieties. However, contrary to the suggestion in [13] the general moduli problem has yet to be treated fully. There are unresolved difficulties, but they do not arise for the problem attached to a totally indefinite quaternion algebra, which is discussed in detail in [17].

It does not add to the essential difficulties if we enlarge our perspective a little and consider not only the zeta-function defined by the constant sheaf but also that defined by the sheaves associated to finite dimensional representatives of the group defining the variety, and we might even dissipate some of the current misconceptions about the nature of these sheaves. Their existence is a formal consequence of Shimura’s conjecture. We should moreover not confine ourselves to the multiplicative group of the quaternion algebra, but should in addition consider subgroups lying between the full multiplicative group and the kernel of the norm, for then we can see the effect of L-indistinguishability [7] in the place where it was first noticed.

In this introduction the results of [7], to which [22] is meant to serve as a kind of exegesis, are used in conjunction with facts about continuous cohomology to arrive at an assertion about the zeta-function which the remainder of the paper is devoted to proving. Some readers will find that I have given too free rein to a lamentable tendency to argue from the general to the particular, and have obfuscated them by interjecting unfamiliar concepts of representation theory into what could be a purely geometric discussion. My intention is not that, but rather to equip myself, and perhaps them as well, for a serious study of the Shimura varieties in higher dimensions. We are in a forest whose trees will not fall with a few timid hatchet blows. We have to take up the double-bitted axe and the cross-cut saw, and hope that our muscles are equal to them.

The method of proof has already been described in [14]. It is ultimately combinatoric. The Bruhat-Tits buildings, which arise naturally in the study of orbital integrals and Shimura varieties, are used systematically. However the automorphic L-functions used to express the zeta-functions of the varieties are unusual and most of §2 is taken up with the attempt to understand them and express their coefficients in manageable, elementary terms. The appearance of L-indistinguishability complicates the task considerably.

The meaning of the conjectures of [13] is also obscure, even to their author, and considerable effort is necessary before it is revealed sufficiently that a concrete expression for the coefficients of the zeta-functions is obtained. Once this is done, in §3 and the appendix, the equality to be proved is reduced to elementary assertions which are proved by combinatorial arguments in §4.

A connected reductive group G over Q and a weight μ of the associate group ^{L}G^{0} are the principal data specifying a Shimura variety. The conditions they must satisfy are described in [4]. If A_{f} is the group of finite adèles one needs an open compact subgroup K of G(A_{f}) as well. μ is the weight of ^{L}G^{0} defined by the co-weight h_{0} of [13]. The primary datum is h_{0}, rather than μ. To completely define S(K) one needs h_{0}. The variety will be denoted by S(K) and only K will appear explicitly, for G and μ are usually fixed. The group ^{L}G^{0} comes provided with a Borel subgroup ^{L}B^{0} and a Cartan subgroup ^{L}T^{0} in ^{L}B^{0}. We may

∗ Appeared in Can. J. Math., Vol. XXXI, No. 6, 1979, pp. 1121–1216. Included by permission of Canadian Mathematical Society. suppose that μ is a positive weight of ^{L}T^{0}. Moreover if L is a large Galois extension of Q then 𝔊(L/Q) acts on ^{L}G^{0}, fixing the subgroups ^{L}B^{0} and ^{L}T^{0}. If 𝔊(L/E) is the stabilizer of μ then the Shimura conjecture, which has been proved for the groups we shall consider, states that S(K) has a model over E characterized by the arithmetic structure of its special points [4]. We will always use this model. The set of complex points on S(K) is a double coset space

(1.1)  G(Q)\G(A)/K_{∞}K .

Here K_{∞} ⊆ G(R) and G(R)/K_{∞} is a finite union of Hermitian symmetric spaces.

Let Z_{0} be the intersection of the kernels of the rational characters over Q of the center Z of G and let ξ be a representation of G on the vector space V which is trivial on Z_{0}, both ξ and V being defined over Q. If K is sufficiently small, as we assume, then

V (Z) ×_{(G(Q),ξ)} G(A)/K_{∞}K → S(K)

defines a locally constant sheaf F_{ξ}^{K} or F_{ξ} on S(K). Using the étale coverings S(K′) → S(K), K ⊆ K′, which are defined over E, and the formalism of [12], we may defined the sheaves F_{ξ} as l-adic sheaves in the étale topology.

For the groups that occur in this paper the quotient (1.1) is compact and the varieties S(K) are proper, and we introduce the cohomology groups

H^{i}(S(K), F_{ξ}) .

They can be taken over Q or over Q_{l}, according to the exigencies of the context. If g ∈ G(A_{f}) the formalism of [12], which is the usual formalism, associates to g a linear transformation

T^{i}(g): H^{i}(S(K), F_{ξ}) → H^{i}(S(K), F_{ξ}) .

The T^{i}(g) act to the right and commute with 𝔊(\overline{E}/E) which acts to the left. I recall that in the theory of Shimura varieties E is given as a subfield of C.

We shall be concerned with the zeta-function of F_{ξ} as a formal rather than as an analytic object, and so we shall only be interested in the individual local factors, and these only at the primes 𝔭 of E for which the suggestions of [13] apply. If Φ_{𝔭} is the Frobenius at 𝔭 and τ^{i} the representation of 𝔊(\overline{E}/E) on H^{i}(S(K), F_{ξ}) then the logarithm of the zeta-function is given by

log Z_{𝔭}(s, S(K), F_{ξ}) = ∑_{n=1}^{∞} n^{−1} ∑_{i}(−1)^{t}trace τ^{i}(Φ_{𝔭}^{n})|ϖ_{𝔭}|^{ns} .

If q is the number of elements in the residue field then

|ϖ_{𝔭}| = q^{−1} .

We shall be more interested in

Z_{𝔭}(s, S(K), F_{ξ}) = ∏_{𝔭|p} Z_{𝔭}(s, S(K), F_{ξ}) .

We want to show that the zeta-function can be expressed in terms of the L-functions associated to automorphic forms. Considerations that will be explained shortly suggest an elegant conjecture. It is false and, in general, meaningless, but it is meaningless for interesting reasons, stemming from L-indistinguishability, and for the groups treated in this paper we will be able to modify and correct it, by taking the results of [7] into account. I will present it in a slicker form than it at first appeared, even though its genesis is thereby somewhat obscured.

Some auxiliary objects must be introduced. To simplify our considerations we suppose that the restriction of ξ to the center Z of G is of the form

ξ(z) = ν(z)I

where ν is a rational character. This does not affect the generality, since ξ can be decomposed into a direct sum of representations satisfying this condition.

If T is a Cartan subalgebra of G over R, γ is a regular element of T (R), δ belongs to 𝔇(T/R) (the explanation of this and other notation used below will be found in [16]), and f is a smooth function in G(R) which is compactly supported modulo Z(R) and satisfies

f(zg) = ν(z)f(g) z ∈ Z(R)

we may introduce

Φ^{δ}(γ, f) = ∫_{T^{h}(R)\G(R)} f(g^{−1}h^{−1}γhg)dg

as in [7]. Here h in 𝔄(T) represents δ. We may also introduce the stable orbital integrals [7]

Φ^{T/1}(γ, f) = ∑_{𝔇(T/R)} Φ^{δ}(γ, f) .

Choose f = f_{ξ} so that

Φ^{T/1}(γ, f_{ξ}) = 0

unless T (R) is fundamental, that is, unless Z(R)\T (R) is compact, and so that

Φ^{T/1}(γ, f_{ξ}) = \frac{trace ξ(γ)}{meas Z(R)\T (R)}

if T (R) is fundamental. It has to be shown that f_{ξ} exists, but for the groups we shall ultimately consider this has been done (cf. [15], §4).

If π_{∞} is a representation of G(R) set 𝔪(π_{∞}) = 𝔪(π_{∞}, ξ) equal to 0 unless

π_{∞}(z) = ν^{−1}(z)I z ∈ Z(R) .

Otherwise let 𝔪(π_{∞}) be the trace of

π_{∞}(f_{ξ}) = ∫_{Z(R)\G(R)} f_{ξ}(g)π_{∞}(g)dg .

We temporarily disregard the circumstance that 𝔪(π_{∞}) is in fact not well-defined. If π_{f} is a representation of G(A_{f}) let 𝔪(π_{f}) = 𝔪(π_{f}, K) be the multiplicity with which the trivial representation of K occurs in π_{f}.

The numbers 𝔪(π_{∞}) and 𝔪(π_{f}) will occur as exponents in the expression of Z(s, S(K), F_{ξ}) as a product of L-functions associated to automorphic representations π = π_{∞} ⊗ π_{f}, but to specify an L-function one needs a representation of ^{L}G as well. Let r^{0} be the representation of ^{L}G^{0} with highest weight μ. The element μ, or rather its restriction to the derived group, is a poids minuscule in the terminology of Bourbaki. Thus the weights of r^{0} are the ωμ with ω in the Weyl group Ω(^{L}T^{0},^{L}G^{0}) of ^{L}T^{0} in ^{L}G^{0}. Let ^{L}M^{0} be the group generated by ^{L}T^{0} and the coroots α^{∨} orthogonal to μ. The stabilizer of μ is Ω(^{L}T^{0},^{L}M^{0}) and the dimension of r^{0} is

[Ω(^{L}T^{0},^{L}G^{0}): Ω(^{L}T^{0},^{L}M^{0})] .

Let x be a non-zero vector transforming according to the weight μ. There is exactly one way on extending r^{0} to a representation, again denoted r^{0}, of

LG^{0} × 𝔊(L/E)

on the same space so that

r^{0}(σ)x = x σ ∈ 𝔊(L/E) .

L is here just some large Galois extension of Q, and could be taken to be \overline{Q}. Let

r = Ind(^{L}G,^{L}G^{0} × 𝔊(L/E), r^{0}) .

The group ^{L}G is a semi-direct product

LG^{0} × 𝔊(L/Q) .

Let q be the dimension of S(K). If the phenomenon of L-indistinguishability did not manifest itself, one might suspect that

(1.2)  Z(s, S(K), ξ) = ∏ L(s − q/2, π, r)^{𝔪(π)𝔪(π_{∞})𝔪(π_{f})} .

The zeta-function on the left is the product of the local zeta-functions, including a factor from the infinite places. The product on the right is over all automorphic representations of G(A) and 𝔪(π) is the multiplicity with which π occurs in the space of automorphic forms. The grounds for the suspicion are initially flimsy, but I shall try to explain them. If the conjectures of Weil and Ramanujan are compatible the shift in the variable form from s to s − q/2 must be present.

Let N be the number of absolutely irreducible components of ξ, counted with multiplicity, and let λ^{1}, …, λ^{k} be the highest weights of the distinct components ξ̃^{1}, …, ξ̃^{k} of the contragredient representation ξ̃ with respect to some order on the roots of a fundamental Cartan subgroup T (R) in G(R). Let g be one-half the sum of the positive roots with respect to this order, and for every ω in the complex Weil group Ω(T (C), G(C)) set

Λ_{ω}^{i} = ω(λ^{i} + g) .

For each Λ_{ω}^{i} there is a discrete series representation π(Λ_{ω}^{i}). We set

Π(ξ) = {π(Λ_{ω}^{i})|1 ≤ i ≤ k, ω ∈ Ω(T (C), G(C))} .

It is a union of the L-indistinguishable classes [11]

Π(ξ^{i}) = {π(Λ_{ω}^{i})|ω ∈ Ω(T (C), G(C))} .

f_{ξ} has been so chosen that

∑_{π_{∞}∈Π(ξ)} 𝔪(π_{∞}) ,

which is well-defined, is equal to (−1)^{1}dN if

d = dimension r^{0} .

In the notation of [2]

⊕_{π_{∞}∈Π(ξ)}H^{i}(𝔤, 𝔨_{∞}, π_{∞} ⊗ ξ)

is 0 unless i = q when its dimension is dN, for by the results of those notes

⊕_{π_{∞}∈Π(ξ^{j})}H^{i}(𝔤, 𝔨_{∞}, π_{∞} ⊗ ξ^{j}′)

is 0 unless j = j′ and i = q when its dimension is d. To see this one has to observe, among other things, that Ω(T (C), G(C)) is isomorphic to Ω(^{L}T^{0},^{L}G^{0}), that Ω(^{L}T^{0},^{L}M^{0}) is isomorphic to Ω(T (R), K_{∞}), and that the restriction of

⊕_{π_{∞}∈Π(ξ^{j})}π_{∞}

to the connected component of G(R) is therefore the direct sum of d irreducible representations, namely the discrete series representations with the same infinitesimal character as ξ̃^{j}.

If, as occasionally happens ([12], [14]), Π(ξ) consisted of a single element π_{∞}, then each time that

π = π_{∞} ⊗ π_{f}

occurred in the space of automorphic forms it would contribute a subspace to the cohomology group H^{q}(S(K), F_{ξ}) over C of dimension d𝔪(π_{f}). A corresponding L-function should appear as a factor of the zeta-function. The degree of the factors appearing in its expression as an Euler product should be

[E: Q]d𝔪(π_{f}) = 𝔪(π_{f}) dimension r

for almost all 𝔭. The L-function should appear in the numerator or denominator according as q is even or odd. We are led to guess that it is

L(s − q/2, π, r)^{(−1)^{q}m(π_{f})} = L(s − 1/2, π, r)^{𝔪(π_{∞})𝔪(π_{f})} .

Occam’s razor and the ordinary Eichler-Shimura theory then suggest (1.2). Unfortunately ∏(ξ) generally consists of several elements.

We might still be able to hold onto (1.2) if whenever π_{∞} and π′_{∞} were two L-indistinguishable representations of G(R) the representations

π = π_{∞} ⊗ π_{f} and π′ = π′_{∞} ⊗ π_{f}

occurred in the space of automorphic forms with the same multiplicity. We would just have to choose a representative from each L-indistinguishable class and agree that the product in (1.2) was to be taken over those π = π_{∞} ⊗ π_{f} for which π_{∞} belonged to our set of representatives. But we would have to search for another definition of the exponent 𝔪(π_{∞}), because, as it stands, different choices of f_{ξ} lead to different values for trace π_{∞}(f_{ξ}). It does not matter, for π and π′ do not always occur with the same multiplicity [7]. One may occur while the other does not. This clearly means that the degrees of the Euler products L(s − q/2, π, r) are then too large. We must seek Euler products of smaller degree.

L-indistinguishability appears when the sets 𝔇(T)of [16] have more than one element. If, as we may assume, the center of ^{L}G^{0} is connected then we may use the definitions of [16] to introduce groups H over Q and homomorphisms ψ: ^{L}H → ^{L}G. Suppose the principle of functoriality applies and the L-indistinguishable class of π is the image under ψ_{∗} of that of π′. Then

L(s, π, r) = L(s, π′, r ∘ ψ) .

In general r is irreducible but r ∘ ψ is often reducible

r ∘ ψ = ⊕r_{i} .

We must expect that the functions L(s − q/2, π′, r_{i}) will appear in the ultimate, correct form of (1.2). The definition of H leads naturally to such a decomposition of r ∘ψ. The constituents r_{i} might not be irreducible, but they seem nonetheless to yield the L-functions necessary for an analysis of Z(s, S(K), F_{ξ}).

Without repeating the definition of the groups H, we recall that each of them is attached to a triple (T, κ, g_{1}). If T_{sc} is the Cartan subgroup of the simply connected form of G defined by T the term κ is a homomorphism of its lattice of coweights X_{∗}(T_{sc}) into C^{×}. Moreover g_{1} allows an identification of X_{∗}(T_{sc}) with the lattice X^{∗}(^{L}T_{sc}^{0}) of rational characters of the Cartan subgroup of the L-group ^{L}G_{sc}^{0} and κ can therefore be transported ot a homomorphism κ′ of ^{∗}(^{L}T_{sc}^{0}) into C^{×}. We may extend κ′ to a 𝔊(K/Q)- invariant homomorphism

ϵ: X^{∗}(^{L}T^{0}) → C^{×} .

Since

LT^{0} = Hom(X^{∗}(^{L}T^{0}), C^{×}) ,

ϵ is an element of ^{L}T^{0} and lies in the center of ψ(^{L}H). The representation r ∘ ψ is the direct sum of its restrictions r_{i} to the eigenspaces of r(ϵ), but these may not be irreducible. Since any two choices of the extension ϵ differ by a central element in ^{L}G, these subspaces are well-defined.

We now try to modify (1.2) by including the L(s−q/2, π′, r_{i}) in such a way that at least the local factors at infinity of the hypothetically equal Euler products, Z(s, S(K), F_{ξ}) on the left and some combination of the L(s, π′, r_{i}) on the right, are likely to be the same. It seems to be sufficient to consider H defined by a T for which T (R) is fundamental.

The representation r^{0} may be regarded as a subrepresentation of the restriction of r to ^{L}G^{0} ⋊ 𝔊(\overline{Q}/E). It is clear that the eigenspaces of r(ϵ) also decompose r^{0} into

⊕r_{i}^{0}

and that

r_{i} = Ind(^{L}G,^{L}G^{0} ⋊ 𝔊(\overline{Q}/E), r_{i}^{0}) .

To define L(s, π_{∞}, r_{i}) we need only know the restriction of r_{i} to the local associate group at infinity, ^{L}G_{∞} = LG^{0} × 𝔊(C/R).

Implicit in the definition of E is an imbedding \overline{Q} ↪ C, and hence E is a subfield of C and 𝔊(C/R) is a subgroup of 𝔊(\overline{Q}/Q). The double cosets

𝔊(C/R)\𝔊(\overline{Q}/Q)/𝔊(\overline{Q}/E)

parametrize the infinite places v of E, the coset represented by r defining the valuation

x → |τ(x)| .

The decomposition group 𝔊_{v} at v = v(τ) is

𝔊(\overline{Q}/E) ∩ τ^{−1}𝔊(C/R)τ

and the restriction of r_{i} to ^{L}G_{∞} is

⊕_{v} Ind(^{L}G_{∞},^{L}G^{0} × 𝔤_{v}, r_{i}^{0}|τ𝔊_{v}τ^{−1}) = ⊕r_{i}(v) .

Consequently

L(s, π′_{∞}, r_{i}) = ∏_{v} L(s, π′_{∞}, r_{i}(v)) ,

and we attempt to arrange that the contributions from the L(s − q/2, π′_{∞}, r_{i}(v)) for a given v yield the local factor of Z(s, S(K), F_{ξ}) at the same place.

However, our primary interest in this paper is not with equality of the Euler factors at infinity, but with equality at almost all finite places, and we are only using the infinite places as a guide to the correct statement. The one place given by the imbedding E ⊆ \overline{Q} ↪ C provided by the definition of E will serve.

The weights of r^{0} on ^{L}T^{0} are the elements of

{ωμ|ω ∈ Ω(^{L}T^{0},^{L}G^{0})}

and the differences of any two μ_{1}, μ_{2} of these weights is an integral linear combination of roots of ^{L}T^{0}. Thus

μ_{1}(ϵ)/μ_{2}(ϵ) = κ′(μ_{1} − μ_{2}) = ±1 ,

because T (R) is fundamental and κ′ therefore of order one or two. Thus there are one or two r_{i}^{0} and, at the cost of adding a second of dimension zero, we suppose there are two, r_{1}^{0} and r_{2}^{0}. We are also going to decompose the set Π(ξ) into two subsets Π^{1}(ξ) and Π^{2}(ξ), with r_{i}^{0} and Π^{i}(ξ) matched. We first see how to distinguish between r_{1}^{0} and r_{2}^{0}.

The L-indistinguishable class Π(ξ^{j}) is equal to Π_{φ_{j}}, where

φ_{j}: W_{C/R} → ^{L}G .

The notation is that of [11]. Suppose φ_{j} = ψ ∘ φ′_{j} where

φ′_{j}: W_{C/R} → ^{L}H

and that π′_{∞} lies in Π_{φ′_{j}}. Then

L(s, r ∘ φ_{j}) = L(s, π_{∞}, r) = L(s, π′_{∞}, r ∘ ψ) = L(s, r ∘ ψ ∘ φ′_{j}) .
If T (R) is fundamental we may suppose [11] that φ′_{j} takes C^{×} ⊆ W_{C/R} to ^{L}T^{0} and then
φ′_{j}(z) = z^{Λ′} \overline{z}^{σΛ′}, z ∈ C^{×} .

Here Λ′ is one of the Λ_{ω}^{j}, at least if we use the identification of X^{∗}(T) with X_{∗}(^{L}T^{0}) provided by g_{1}, and σ is the non-trivial element of 𝔊(C/R). Λ′ is non-singular and there is exactly one weight μ′ of r^{0} which lies in the closure of the Weil chamber opposite to that containing Λ′. Since any other weight μ′′ is of the form ωμ′,

⟨Λ′, μ′′⟩ = ⟨Λ′, μ′′ − μ′⟩ + ⟨Λ′, μ′⟩ > ⟨Λ′, μ′⟩

if μ′′ ≠ μ′. Given π′_{∞} we take r_{1}^{0} to be the representation with μ′ as a weight, and r_{2}^{0} to be the other. Observe that the labeling depends on the L-indistinguishability class of π′_{∞}, and hence on the class in Φ(H) represented by φ′_{j}, but not directly on Λ′.

We now decompose ∏(ξ) into two subsets ∏^{1}(ξ), ∏^{2}(ξ), matching ∏^{i}(ξ) with r_{i}. Since T (R) is taken to be fundamental μ is defined by a coweight

μ^{∨} = h′_{0}

of T. Here h′_{0} is conjugate under G(R) to h_{0}, and if we use g_{1} as in [16] to introduce an isomorphism

X_{∗}(T)⟶^{∼}X^{∗}(^{L}T^{0})

then μ lies in the orbit of μ^{∨} under the Weyl group. For each j let Λ^{j} be an element of {Λ_{ω}^{j}|ω ∈ Ω(T (C), G(C))} which is such that it and μ^{∨} lie in opposing closed Weyl chambers. Since any two choices of Λ^{j} lie in the same orbit under Ω(T (R), G(R)), the representation π(Λ^{j}) = π^{j}(μ^{∨}) is well-defined and independent of the choice of Λ^{j}. Every element w of the normalizer of T (C) in G(C) lies in 𝔄(T/R) [21] and

w → {a_{τ} = τ(w)w^{−1}|τ ∈ 𝔊(C/R)}

yields an injection

Ω(T (C), G(C))/Ω(T (R), G(R)) ↪ 𝔈(T/R) .

The image is 𝔇(T/R), but that does not matter. If ω is represented by w we put π(ω^{−1}Λ^{j}) in ∏^{1}(ξ) or in ∏^{2}(ξ) according as κ({a_{τ} }) is 1 or −1. The assignment does depend on the choice of Λ^{j}, but that may be inevitable. I observe that it is not difficult to see [21] that under the isomorphism

H^{−1}(𝔊(C/R), X_{∗}(T))⟶^{∼}H^{1}(𝔊(C/R), T (C))

the cocycle {a_{τ} } corresponds to ωμ^{∨} − μ^{∨}.

One point on which we have insisted when assigning the elements of ∏(ξ) to the two sets ∏^{1}(ξ) and ∏^{2}(ξ) is that π(Λ^{j}) lie in ∏^{1}(ξ) for each j. In order to justify this we recall the way in which the complex structure on S(K) is introduced as well as the form suggested by Serre [20] for the local factors Z_{v}(s, S(K), F_{ξ}) in the case of trivial ξ.

If K′_{∞} is the centralizer of h′_{0} then

G(R)/K′_{∞} ≃ G(R)/K_{∞}

and the complex structure defining that on S(K) is obtained from an imbedding

G(R)/K′_{∞} ↪ G(C)/P (C)

if P is the parabolic subgroup whose Lie algebra is spanned by those X for which

μ^{∨}(z)X ≡ X or μ^{∨}(z)X = z^{−1}X

for z ∈ GL(1).

We choose the order on the roots of T with respect to which Λ^{j} lies in the positive Weyl chamber. Suppose U is a subspace of L^{2}(G(Q)\G(A)/K) transforming under G(R) according to π(Λ^{j}). It is explained in [12] (cf. also [2]) how to pass from an element of

Hom_{K′_{∞}}(Λ^{q}𝔤/𝔨′_{∞}, U ⊗ ξ) = ⊕_{j}Hom_{K′_{∞}}(Λ^{q}𝔤/𝔨′_{∞}, U ⊗ ξ^{j})^{N_{j}}

to a q-form on S(K) with values in F_{ξ}. Here N_{j} is the multiplicity with which ξ^{j} occurs in ξ. It is easier to work with

Hom_{K′_{∞}}(Λ^{q}𝔤/𝔨′_{∞} ⊗ ξ̃^{j}, U) .

Clearly

𝔤/𝔨′_{∞} = 𝔭^{+} ⊕ 𝔭^{−}

where 𝔭^{+} is spanned by the image of the root vectors associated to noncompact positive roots. The elements of 𝔭^{−} yield holomorphic tangent vectors; those of 𝔭^{+} yield anti-holomorphic tangent vectors. We have

Hom_{K′_{∞}}(Λ^{q}𝔭^{+} ⊗ ξ̃^{j}, U) ⊆ Hom_{K′_{∞}}(Λ^{q}𝔤/𝔨′_{∞} ⊗ ξ̃^{j}, U) ,

and it is the elements of the first space which yield forms of Hodge type (q, 0).

Let ρ_{P} be one-half the sum of the non-compact positive roots and ρ_{K} one-half the sum of the compact positive roots. The space Λ^{q}𝔭^{+} is one-dimensional and transforms under K′_{∞} according to the weight 2ρ_{P}. The highest weight of ξ̃^{j} is Λ^{j} − ρ_{P} − ρ_{K} and that of Λ^{q}𝔭^{+} ⊗ ξ̃^{j} is therefore Λ^{j} + ρ_{P} − ρ_{K}. However, it is a fundamental fact [5] that the restriction of π(Λ^{j}) to K′_{∞} has an irreducible component with highest weight Λ^{j} + ρ_{P} − ρ_{K}. Thus

Hom(Λ^{q}𝔭^{+} ⊗ ξ̃^{j}, U) ≠ 0

and U or π(Λ^{j}) contributes cohomology of type (0, q). No other element of ∏(ξ^{j}) does so.

On the other hand, we have written the restriction of φ′_{j} to C^{×} ⊆ W_{C/R} as

z → z^{Λ′}z^{σΛ′}

and we have taken μ′ and Λ′ to lie in opposing Weyl chambers. If ξ is trivial and g′ is one-half the sum of the roots α for which ⟨Λ′, α⟩ < 0, then Λ′ = −g′ and

μ′(φ′_{j}(z)) = z^{−⟨g′,μ′⟩}\overline{z}^{−⟨σg′,μ′⟩} = \overline{z}^{2⟨g′,μ′⟩}(z\overline{z})^{−⟨g′,μ′⟩} .

If q is the dimension of S(K) then

⟨g′, μ′⟩ = q/2 .

[ Added in proof (November, 1979). It appears that the local zeta-functions at 𝔭 calculated in this paper are those associated to ξ̃ and not to ξ. The correct definitions would entail replacing r by its contragredient r̃ and r(v) by

r̃(v) = r̃_{1}(v) + r̃_{2}(v) ,

if v is the place of E defined by E ⊆ Q ⊆ C. If α is the composite of

W_{C/R} → W_{R/R} = R^{×}

with the absolute value, then

L(s − q/2, π′_{∞}, r̃(v) ∘ φ) = L(s, α^{−q/2} ⊗ (r̃(v) ∘ φ_{j})) .

The one-dimensional subspace corresponding to the weight −μ′ transforms under the restriction of α^{−q/2} ⊗ (r̂(v) ∘ φ_{j})) to C^{×}, ξ being trivial, according to the character

\overline{z}^{−2⟨g′,μ′⟩} = \overline{z}^{−1} ]

We have been led to our labeling by the principle that the π(Λ^{j}), as the representations giving rise to forms of type (0, q), should be matched with the weight −μ′, which yields, when ξ is trivial, the character z → \overline{z}^{−q}. One is led to this principle by a suggestion of Serre [20].

If S is a variety over the number field E the factor at infinity of its zeta-function can of course be expressed in terms of Γ-functions and is a product over the infinite places of E of local zeta-functions. For each place v we take a corresponding imbedding E ↪ C and introduce the set S(C) of complex points on

S ⊗_{E} C .

The local zeta-function at the place v may be expressed as

∏_{i} L(s, ρ_{i})^{(−1)i}

where ρ_{i} is a representation of the Weil group W_{C/E_{v}} on the cohomology group

H^{i}(S(C)) = ⊕_{p+q=i}H^{p,q}(S(C)) .

Here, in order to conform to custom, q loses temporarily its significance as the dimension of S(K). The restriction of ρ_{i} to C^{×} is defined by demanding that ρ_{i}(z) act on H^{p,q}(S(C)) as z^{−p} \overline{z}^{−q}. If E_{v} is complex there is nothing more to be said. If it is real we have to define ρ_{i}(w) if w is an element of W_{C/E_{v}} which maps to the complex conjugation in 𝔊(C/E_{v}) and has square −1. When E_{v} is real, complex conjugation defines an involution ι of S(C) and an associated map ι^{∗} on cohomology. We let ρ_{i}(w) act on H^{p,q}(S(C)) as (−1)^{p}ι^{∗}.

Now we must bring these puzzling divagations to bear upon some simple examples. Suppose F is a totally real field and G̃ is the multiplicative group of a quaternion algebra D over F. The split algebra is excluded. Let A be a connected subgroup of G_{1} = Res_{F/Q}GL(1) which is defined over Q and let G be the

inverse image of A in G̃_{1} = Res_{F/Q}G̃ with respect to the norm.

The group ^{L}G is a quotient of

(∏_{𝔊(\overline{Q}/F)\𝔊(\overline{Q}/Q)} GL(2, C)) × 𝔊(K/Q) = ^{L}G^{0} × 𝔊(K/Q)

by a subgroup of the center of ^{L}G^{0}, namely by the set of (z_{σ}) for which

∏_{𝔊(K/F)\𝔊(K/Q)} λ_{σ}(z_{σ}) = 1, (λ_{σ}) ∈ Y_{∗},

with Y_{∗} defined as in §6 of [7]. ^{L}T^{0} is the image of the diagonal matrices and X^{∗}(^{L}T^{0}) is

{(a_{σ}, b_{σ})|a_{σ}, b_{σ} ∈ Z, (λ_{σ}) ∈ Y_{∗} if λ_{σ} = a_{σ} + b_{σ}} .

We have supposed that F ⊆ \overline{Q} ⊆ C. Then μ will be (μ_{σ}) and (μ_{σ}) will be 0 unless the imbedding x → σ(x) of F in C splits D, when it is (1, 0). Observe that (λ_{σ}) with λ_{σ} equal to 1 if the quaternion algebra splits at σ and 0 otherwise must belong to Y_{∗}. This is to be treated as a condition on A, because μ is determined by D alone.

For such a G and μ we want to present a correct and verifiable expression for Z(s, S(K), F_{ξ}) as a product of L-functions associated to automorphic forms. Two representations π = ⊗π_{w} and π′ = ⊗π′_{w} will be said to be L-indistinguishable if π_{w} and π′_{w} are L-indistinguishable for all w and equivalent for almost all w. Our expression for Z(s, S(K), F_{ξ}) will be given as a product over L-indistinguishable classes. We must describe the contribution from each class.

Suppose that 𝔪(π′), the multiplicity with which π′ occurs in the space of automorphic forms, is constant within the L-indistinguishability class Π of π. If Π_{w} is the L-indistinguishability class of π_{w} then

ϖ = ⊗_{w} (⊕_{Π_{w}}π′_{w}) = ⊗_{w}ϖ_{w} = ϖ_{∞} ⊗ ϖ_{f}

is a representation which contains each π′ in Π exactly once. We set

𝔪 (Π_{∞}) = ∑_{π_{∞}∈Π_{∞}} trace π_{∞}(f_{ξ})

provided

π_{∞}(z) = ν^{−1}(z)I, z ∈ Z(R),

for one, and hence all elements of Π_{∞}. Otherwise 𝔪(Π_{∞}) is to be 0. 𝔪(Π_{∞}) is well-defined, and is easily seen to be 0, or −1 when ξ is absolutely irreducible. Let 𝔪(Π_{f}) be the multiplicity with which the trivial representation of K occurs in ϖ_{f}; and let 𝔪(Π) be 𝔪(π). The contribution of the class Π to the zeta-function is

L(s − q/2, π, r)^{𝔪(Π)𝔪(Π_{∞})𝔪(Π_{f})} .

If 𝔪(π′) is not constant within Π there is a Cartan subgroup T of G with

(1.3)  [𝔈(T/A), 𝔈(T/F)] = 2

and a character θ of T (Q)\T (A) such that Π = Π(θ) [7]. Let γ → \overline{γ} be the automorphism of T (A) deduced from conjugation on the corresponding quadratic field, and define θ̄ by

θ̄(\overline{γ}) = θ(γ) .

We suppose not only that T satisfies (1.3) but also that T (R) is fundamental and that θ ≠ θ̄ for otherwise 𝔪(Π_{∞}) = 0 and the L-indistinguishability class Π does not contribute to the zeta-function.

We introduce S^{0} ⊆ S as in §8 of [7], or as in [22]. Then S^{0}\S is of order two. Let ϵ represent the non-trivial element. As we know, ϵ is associated to

κ′: X_{∗}(T_{sc}) → C^{×} .

Let ⟨ϵ, π_{∞}⟩ be the pairing of [7]. It is easily seen that there is a constant η = ±1 such that

(−1)^{i−1} = η ⟨ϵ, π_{∞}⟩ , π_{∞} ∈ ∏^{i}(ξ) .

The reason can be briefly given. In SL(2) over R we take

T = \{( \begin{matrix}a & b \\ −b & a \end{matrix} )| a^{2} + b^{2} = 1\} .

The non-trivial element of the normalizer of T is represented by

( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} )

in SL(2, C) and by

( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} )

in GL(2, R). The cocycle associated to the first matrix is

a_{1} = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , a_{σ} = ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) ,

and the determinant of the second is −1. Thus ⟨ϵ, π_{∞}⟩ is constant on each Π_{∞}^{i}.

Π_{f} is of course the set of π_{f} for which π_{∞} ⊗ π_{f} lies in Π for some π_{∞}. If π_{f} = ⊗π_{w} set

⟨ϵ, π_{f}⟩ = ∏_{w} ⟨ϵ, π_{w}⟩ .

Let

Π_{f}^{i} = {π_{f} ∈ Π_{f}| ⟨ϵ, π_{∞}⟩ ⟨ϵ, π_{f}⟩ = 1 for π_{∞} ∈ Π_{∞}^{i}}

and let 𝔪^{i}(Π_{f}) be the multiplicity with which the trivial representation of K occurs in

ϖ_{f}^{i} = ⊕_{Π_{f} ∈Π_{f}^{i}} π_{f} .

Let 𝔪′(θ) be 0 unless Π(θ_{∞}) is contained in a Π(ξ^{j}), when it is to be (−1)^{q}N^{j}. Recall that N^{j} is the multiplicity with which ξ^{j} occurs in ξ. On the set of π ∈ Π for which 𝔪(π) > 0, 𝔪(π) is constant. Denote this constant value by 𝔪(Π). When 𝔪(Π) is not constant on all of Π the contribution of Π = Π(θ) to the zeta-function Z(s, S(K), F_{ξ}) is

(1.4)  ∏_{π} L(s − q/2, θ, r_{i})^{𝔪′(θ_{∞})𝔪^{i}(Π_{f})𝔪(Π)} .

We should recall that although the collection {r_{i}} is the same for all θ_{∞} the labeling may vary. Moreover

𝔪′(θ_{∞}) = 𝔪(Π_{∞}) .

If i𝔪(π) is not constant on Π then the stable multiplicity 𝔫(π), which we also write as 𝔫(Π), introduced in [7] is 𝔪(Π)/2 and (1.4) may be written as

L(s − q/2, π, r)^{𝔫(Π)𝔪(Π_{∞})𝔪(Π_{f})} \{\frac{L(s − q/2, θ, r_{1})}{L(s − q/2, θ, r_{2})}\}^{(𝔪(Π_{∞})/2)(𝔪^{1}(Π_{f})−𝔪^{2}(Π_{f}))𝔪(Π)}

if

𝔪(Π_{f}) = 𝔪^{1}(Π_{f}) + 𝔪^{2}(Π_{f})

If, perchance, there is only one r_{i} we must as above introduce a second of dimension 0 in order to employ this notation. When 𝔪(π) is constant on Π we define

𝔪^{1}(Π_{f}) − 𝔪^{2}(Π_{2}) = 0 .

If 𝔪(π) is constant on Π then 𝔪(Π) = 𝔫(Π). Thus we are asserting that the zeta-function Z(s, S(K), F_{ξ}) can be presented as the product of a stable part

(1.5)  ∏_{Π} L(s − q/2, π, r)^{𝔫(Π)𝔫(Π_{∞})𝔪(Π_{f})}

and a labile part. The labile part is itself a product over the stable conjugacy classes of Cartan subgroups T with [𝔈(T/A): Im 𝔈(T/F)] = 2. We want to represent the labile contribution from T as a product over characters of T (Q)\T (A), but we must rememvber that two different θ̄ can yield the same L-indistinguishable class Π(θ). Since 𝔪(Π_{∞}) will be 0 if θ is not of type (a) in the sense of [7] and since 𝔪^{1}(Π_{f}) − 𝔪^{2}(Π_{f}) will be 0 if θ_{v} = θ̄_{v} for some finite v, we may apply Lemmas 6.7 and 7.1 of [7]. They allow us to write, in the notation of that paper,

𝔪(Π) = \frac{e(π)}{2} μ(T) .

Since e(π) is hte number of different θ yielding the same Π(θ), the contribution from T is

(1.6)  Π_{θ} (\frac{L(s − q/2, θ, r_{1})}{L(s − 1/2, θ, r_{2})})^{(𝔪(Π_{∞})/4)(𝔪^{1}(Π_{f} (θ))−𝔪^{2}(Π_{f} (θ)))μ(T)}

To prove the assertion one proves, in particular, that if we substitute L_{𝔭}(s−q/2, π, r) for L(s−q/2, π, r) in (1.5) and L_{𝔭}(s − q/2, θ, r_{i}) for L(s − q/2, θ, r_{i}) in (1.6) and take the product with the same exponents then the result is Z_{𝔭}(s, S(K), F_{ξ}). We shall take K sufficiently small, and prove that this is so for almost all 𝔭. The restriction on K is ultimately of no consequence.

Before beginning I take this opportunity to mention that while studying the problems arising from Shimura varieties I have frequently been instructed by Deligne’s conversation and correspondence., His comments on the structure of the set of geometric points over \overline{F}_{𝔭} on a Shimura variety associated to a quaternion algebra over a real quadratic field were invaluable.

2. The trace formula.

The procedure to be followed is that of [12] and [14]. We apply the trace formula to calculate the coefficients of the logarithms of the products in (1.5) and (1.6), and then compare with the coefficients of the logarithm of the zeta-function, which are obtained from the explicit description of the sets S(κ̄_{𝔭}).

We first turn our attention to the problem of expressing the logarithms of (1.5) and (1.6) in a form suitable for the final comparison. The expressions both are defined for all ξ, whether defined over Q or not. Since they are multiplicative in ξ, we may as well suppose that ξ is absolutely reducible, for that will simplify some of our considerations. The logarithm of (1.6) demands the most modification, and we begin with it. We are really interested in the logarithm of the product of the local L-functions at 𝔭, and it is

(2.1)  μ(T) ∑_{θ} \frac{𝔪(∏_{∞}(θ))}{4} (𝔪^{1}(Π_{f}(θ)) − 𝔪^{2}(Π_{f}(θ))) × log L_{p}(s − q/2, θ, r_{1} − r_{2}) .

The first step in the transformation of this expression leads to a clumsy, intermediate result, which we will be able to put in a useful form only after applying the trace formula to the group T.

In its role as carrier of θ, T is appearing as one of the groups H of [16]. This means in particular that the element g_{1} of that note, which allows us to identify X^{∗}(T) with X_{∗}(^{L}T^{0}), is fixed. Moreover it is fixed in a manner consonant with the remarks at the end of that paper. Then, as in the discussion at the end of §6 of [7], the given g_{1} leads to

ψ: ^{L}T → ^{L}G .

The map ψ was denoted ξ in [16].

The following lemma has been implicit in the earlier discussion, and may seem to be a matter of definition, but so far as I can see it needs a proof.

Lemma 2.1. Suppose φ′: W_{C/R} → ^{L}T, φ is ψ ∘ φ′, and θ_{∞} ∈ Π_{φ′}. Then Π_{φ′} = {θ_{∞}} and

Π_{φ} = Π(θ_{∞}) .

That Π_{φ′} consists of the single element θ_{∞} does indeed follow from the definitions of [11], which show in addition that is enough to verify the equality Π_{φ} = Π(θ_{∞}) when T is not split and G = GL(2). In this case, one must compare the definitions of [5] and [11].

In general the explicit description of ^{L}G in [7] gives ^{L}T^{0} as a quotient of the group matrices

( \begin{matrix}α_{1} & 0 \\ 0 & β_{1} \end{matrix} ) × … × ( \begin{matrix}α_{n} & 0 \\ 0 & β_{n} \end{matrix} ) , α_{i}, β_{i} ∈ C^{×} .

The co-weights are given by n pairs of integers, (a_{i}, b_{i}), and the co-weight is positive if and only if a_{i} ≥ b_{i} for all i.

Taking T to be non-split and G to be GL(2), we let

φ′: z → ( \begin{matrix}z^{a} \overline{z}^{b} & 0 \\ 0 & z^{b} \overline{z}^{a} \end{matrix} ) , z ∈ C^{×} .

It is easily verified that the class of φ′ is determined by its restriction to C^{×}. Let γ_{1} and γ_{2} be the co-weights (1, 0) and (0, 1) of ^{L}T^{0}. They are also weights of T. The definition of [11] gives

θ_{∞}: t → γ_{1}(t)^{a}\overline{γ_{1}(t)}^{b}, t ∈ T (R),

as the unique element of Π_{φ′}. Note that γ_{1}(t) = \overline{γ}_{2}(t) when t ∈ T (R).

Π_{φ} also consists of a single element. If a = b then φ factors through W_{C/R} → W_{R/R} → R^{×} and, in terms of R^{×},

φ: x → ( \begin{matrix}|x|^{a} & 0 \\ 0 & (sgn x)|x|^{a} \end{matrix} ) .

The unique element of Π_{φ} is the representation of the principal series corresponding to the two characters of R^{×} appearing here. According to the remark preceding Corollary 5.14 of [6], this is also π(θ_{∞}).

If a ≠ b then Π_{φ} consists of a representation π in the discrete series. It does not change if a and b are interchanged. It will be simpler to be explicit if we assume a > b. According to the definitions of §3 of [11], the character of π on T (R), viewed now as a subset of G(R), is

− \frac{\{γ_{1}(t)^{a−1/2}γ_{1}(t)^{b+1/2} − γ_{2}(t)^{a−1/2}\overline{γ}_{2}(t)^{b+1/2}γ_{2}(t)γ_{1}^{−1}(t)\}}{1 − γ_{2}(t)γ_{2}^{−1}(t)}.

This may be written as

− \frac{γ_{2}(t)}{|γ_{2}(t)|} \frac{∑_{ω∈Ω(T (C),G(C))} sgn ωθ_{∞}^{0}(ω(t))}{1 − γ_{2}(t)γ_{1}^{−1}(t)} (2.2)

It follows from Corollary 5.14 of [6] that if μ_{1} and μ_{2} are the two characters of R^{×} defined by

μ_{1}(x) = |x|^{a}, μ_{2}(x) = |x|^{b}(sgn x)^{a−b+1} ,

then the character of π(θ_{∞}) on T (R) is the negative of the character of the finite-dimensional representation π(μ_{1}, π_{2}). Lemma 5.7 of [6] allows one to compute easily the character of π(μ_{1}, μ_{2}) on T (R), and it is seen to equal the negative of (2.2).

For the purposes of the following corollary and lemma we choose an order on roots of T with respect to which μ^{∨} lies in the closed negative Weyl chamber. Otherwise we use the customary parameters to represent roots and weights. For example, let ζ be the weight, dominant with respect to this order, represented by

ζ = (1, 0) × … × (1, 0) .

Let λ be the highest weight of ξ and set

θ_{∞}^{0}(t) = λ(t)(ζ(t)/|ζ(t)|) .

Corollary 2.2. Π(θ_{∞}) is Π(ξ) if and only if θ_{∞} is conjugate under Ω(T (C), G(C)) to θ_{∞}^{0}.

Once again it is enough to verify the assertion for G = GL(2). Then ∏(ξ) consists of a single element π and on T (R) the character of π is

t → −trace ξ̃(t) .

If λ = (a, b), a > b, this is

− \frac{γ_{1}(t)^{a}γ_{2}(t)^{b} − γ_{1}(t)^{b}γ_{2}(t)^{a}γ_{2}(t)γ_{1}^{−1}(t)}{1 − γ_{2}(t)γ_{1}^{−1}(t)} .

Since γ_{1}(t) = \overline{γ_{2}}(t), this is equal to

− \frac{γ_{2}(t)}{|γ_{2}(t)|} \frac{∑_{ω} sgn ωθ_{∞}^{0}(ω(t))}{1 − γ_{2}(t)γ_{1}^{−1}(t)} .

Comparing this with (2.2), we obtain the corollary.

It will be useful to have the following lemma on record.

Lemma 2.3. Let Λ ∈ {Λ_{ω}} be λ + g, wehre g is one-half the sum of the positive roots. If φ′: W_{C/R} → ^{L}T and its restriction to C^{×} is

z → z^{Λ} \overline{z}^{σΛ}

then

∏_{φ′} = {θ_{∞}^{0}} .

We may write an element of T (R) as t = e^{H} with σ(H) = H. The unique element of Π_{φ′} takes t to

e^{Λ(H)} = e^{λ(H)}e^{g(H)} .

Since

λ(t) = e^{λ(H)}

we need only verify that

e^{g(H)} = ζ(t)/|ζ(t)| .

It is enough to do this when G = GL(2). If

γ_{1}(H) = z , γ_{2}(H) = \overline{z}

then

γ_{1}(t) = e^{z} , γ_{2}(t) = e\overline{z}

and

e^{g(H)} = e^{z/2−\overline{z}/2} = e^{z}/|e^{z}| = ζ(t)/|ζ(t)| .

We know that T is associated to a quadratic extension L of F. if L is not totally imaginary then 𝔪(Π(θ_{∞})) is 0 for all characters of T (Q)\T (A); hence the product (1.6) is equal to 1, and of no interest. Suppose that L is totally imaginary, and let κ be the associated character of I_{F}. It is also a character of Z(A) and Z(R). We set 𝔪(θ_{∞}) equal to 0 unless the restriction of θ_{∞} to Z(R) is

z → ν^{−1}(z)κ(z) ,

and then we take

𝔪(θ_{∞}) = ∫_{Z(R)/T (R)} θ_{∞}(t)f_{ξ}^{T} (t)dt

with

f_{ξ}^{T} (t) = (−1)^{q} \frac{∑_{Ω(T (C),G(C))} sgn ωθ_{∞}^{0}(ω(t^{−1}))}{meas Z(R)\T (R)} .

Lemma 2.4. The number 𝔪(θ_{∞}) is 0 if and only if 𝔪(Π(θ_{∞})) is 0, and 𝔪(Π(θ_{∞})) is not 0 if and only if Π(θ_{∞}) is Π(ξ). If θ_{∞}(t) ≡ θ_{∞}^{0}(ω(t)) then

𝔪(θ_{∞}) = sgn ω𝔪(Π(θ_{∞})) .

The first assertion is a consequence of the Weyl integration formula, and the explicit formula for the restriction of

∑ ; π∈∏(θ_{∞}) χ_{π}

to T (R). The same formulae also show that 𝔪(Π(ξ)) = (−1)^{q}. It is clear that 𝔪(θ_{∞}) has been so defined that

𝔪(θ_{∞}) = (−1)^{q}sgn ω

when

θ_{∞}(t) ≡ θ_{∞}^{0}(ω(t)) .

If 𝔪(Π(θ_{∞})) is 0 the corresponding factor of (1.6) is 1, and there is nothing to be said. Suppose therefore that Π(θ_{∞}) = Π(ξ). Then Π_{∞}^{i} = ∏^{i}(ξ) may be introduced. We have agreed that Π_{∞}^{1} will contain the representation π(Λ_{1}). Let ϕ be the characteristic function of K ⊆ G(A_{f}) divided by the measure of Z_{K}\K with

Z_{K} = Z(A_{f}) ∩ K .

We are supposing that

[𝔈(T/A)Im 𝔈(T/Q)] = 2 ,

and we regard κ as the non-trivial character of this quotient. Following §2 of [7], we set Φ^{T/κ}(t, ϕ) equal to

\{∏_{v}(λ(L_{v}/F_{v}, ψ_{v})κ_{v}(\frac{t_{1} − t_{2}}{t_{1}^{0} − t_{2}^{0}} ) \frac{|(t_{1} − t_{2})^{2}|v^{1/2}}{|t_{1}t_{2}|_{v}^{1/2}} )\} × \{ ∑_{ℭ(T/A_{f})} κ(δ)Φ^{δ}(t, ϕ)\} .

Here the product is over all finite places of F. If δ is represented by h in 𝔄(T/A_{f}) = ∏_{w} 𝔄(T/A_{w}), the product now being taken over all finite places of Q, then

Φ^{δ}(t, ϕ) = ∫_{T^{h}(A_{f})G\(A_{f})} ϕ(g^{−1}t^{h}g)dg .

It is a consequence of the definitions and principles of [7] that if π_{∞} ∈ Π_{∞}^{1} then

⟨ϵ, π_{∞}⟩ {𝔪^{1}(Π_{f}(θ)) − 𝔪^{2}(Π_{f}(θ))}

is equal to

∫_{Z_{K}\T (A_{f})} θ_{f}(t)Φ^{T/κ}(t, ϕ)dt = ⟨θ_{f}, Φ^{T/κ}( · , ϕ)⟩ .

It should not be forgotten that the pairing ⟨ϵ, π_{∞}⟩ depends on θ_{∞}. The representations r_{1} and r_{2} do also. In order to have a pair of representations that do not depend on θ_{∞}, we let r^{+} be that r_{i} for which X_{i} contains μ^{∨}, and r^{−} the other. If θ_{∞} = θ_{∞}^{0}, then Lemma 2.3 implies that

r_{1} − r_{2} = r^{+} − r^{−} .

In general if θ_{∞}(t) = θ_{∞}^{0}(ω^{−1}(t)) then {θ_{∞}} = Π_{φ′} if φ′ restricted to C^{×} is

z → z^{ωΛ} \overline{z}^{σωΛ} .

Since

⟨ωΛ, ωμ^{∨}⟩ = ⟨Λ, μ^{∨}⟩ ,

the labeling must be such that r_{1}^{0} contains the weight ωμ^{∨}. Thus

r_{1} − r_{2} = κ′(ωμ^{∨} − μ^{∨})(r^{+} − r^{−}) .

In order to stress its dependence on θ_{∞}, rather than on ω, we denote the coefficient κ′(ωμ − μ) appearing here by η(θ_{∞}).

Lemma 2.5. If we take θ_{∞}(t) = θ_{∞}^{0}(ω(t)) then

α = ⟨ϵ, π_{∞}⟩ η(θ_{∞})sgn ω ,

with π_{∞} ∈ ∏_{∞}^{1}, is independent of ω.

Before verifying the lemma, we observe that the pairing ⟨ϵ, π_{∞}⟩ depends not only on θ_{∞} but also on the choice of an additive character and a regular element in T (R). Eventually we will be forced to recognize this, but not yet.

To prove the lemma we show that α does not change if ω is replaced by ω′ω, where ω′ is the reflection defined by a simple root. For each real place of F there is one such reflection.

a) If the division algebra defining G is not split at the place then, according to the definitions of [7], replacing ω by ω′ω changes the sign of ⟨ϵ, π_{∞}⟩. It does not affect η(θ_{∞}).

b) If the division algebra splits at the place then ⟨ϵ, π_{∞}⟩ remains the same but η(θ_{∞}) changes sign.

Putting all these lemmas together, we conclude that the sum (2.1) is equal to

(2.3)  \frac{1}{4} αμ(T) ∑_{θ} 𝔪(θ_{∞}) ⟨θ_{f}, Φ^{T/κ}( · , ϕ)⟩ log L_{𝔭}(s − q/2, θ, r^{+} − r^{−}) .

The sum here is taken over those θ for which

\begin{matrix}θ(z) = ν^{−1}(z)κ(z), & z ∈ Z(R), \\ θ(z) = κ(z), & z ∈ Z_{K} . \end{matrix}

If we are to put (2.3) in a form to which the trace formula can be applied, we must view the Hecke algebra in the manner of [9]. We are now going to assume that K = K^{𝔭}K_{𝔭}, where K^{𝔭} ⊆ G(A_{f}^{𝔭}) and K_{𝔭} is a special maximal compact of G(A_{𝔭}). According to §2 of [7]

Φ^{T/κ}(ϕ) = 0

unless L is unramified at every place of F dividing 𝔭. This we may as well assume. We also assume that F is itself unramified over 𝔭. Then

⟨θ_{f}, Φ^{T/κ}( · , ϕ)⟩ = 0

unless θ_{𝔭}, the restriction of θ to T (Q_{𝔭}), is unramified. To such a θ_{𝔭} is associated a conjugacy class {t(θ_{𝔭})} in ^{L}T, and

log L_{𝔭}(s − q/2, θ, r^{+} − r^{−}) = ∑_{j=1}^{∞}(|ϖ^{j}|^{s−q/2}/j) {trace r^{+}(t^{j}(θ_{𝔭})) − trace r^{−}(t^{j}(θ_{𝔭}))}

if ϖ is a uniformizing parameter for Q_{𝔭}. In addition there is an element f_{𝔭}^{j} of the Hecke algebra of T (Q_{𝔭}) such that

|ϖ^{j}|^{−q/2} {trace r^{+}(t^{j}(θ_{𝔭})) − trace r^{−}(t^{j}(θ_{𝔭}))} = ∫_{Z_{𝔭}T \(Q_{𝔭})} θ_{𝔭}(t)f_{𝔭}^{j}(t)dt

for all unramified θ_{𝔭}. Here

Z_{𝔭} = Z(Q_{𝔭}) ∩ K_{𝔭} .
If t = (t^{𝔭}, t_{𝔭}) ∈ T (A_{f}) = T (A_{f}^{𝔭})T (Q_{𝔭}), we write
ϕ(t) = ϕ^{𝔭}(t^{𝔭})ϕ_{𝔭}(t_{𝔭}) ,

where, for example, ϕ_{𝔭} is the characteristic function of K_{𝔭} divided by the measure of Z_{𝔭}\K_{𝔭}. We may also write

Φ^{T/κ}(t, ϕ) = Φ^{T/κ}(t^{𝔭}, ϕ^{𝔭})Φ^{T/κ}(t_{𝔭}, ϕ_{𝔭}) .

For brevity we denote the second factor by φ_{𝔭}(t_{𝔭}). It is +1 times the characteristic function of the maximal compact U_{𝔭} of T (Q_{𝔭}) divided by the measure of Z_{𝔭}\U_{𝔭}.

Applying the trace formula on Z(R)Z_{K}T (Q)\T (A) to the function

t = (t_{∞}, t^{𝔭}, t_{𝔭}) → f_{ξ}^{T} (t_{∞})ϕ^{T/κ}(t^{𝔭}, ϕ^{𝔭})φ_{𝔭}^{∗}f_{𝔭}^{j}(t_{𝔭}) ,

we see that the coefficient of |ϖ^{j}|^{s}/j in the expansion of (2.3) is

(2.4)  \frac{αμ(T)}{2} meas(Z(R)Z_{K}T (Q)\T (A)) ∑ f_{ξ}^{T} (t)Φ^{T/κ}(t, ϕ^{𝔭})φ_{𝔭}^{∗}f_{𝔭}^{j}(t) .

The sum is over T (Q) ∩ Z(R)Z_{K}\T (Q).

The first thing to observe is that the term corresponding to a t in Z(Q) is 0 because f_{ξ}^{T} vanishes on Z(R). If t is not central we write Φ^{T/κ}(T, ϕ^{𝔭}) as the product of

Φ_{0}^{T/κ}(t, ϕ^{𝔭}) = ∑_{𝔈(A_{f}^{𝔭})} κ(δ)Φ^{δ}(t, ϕ)

and

∏ λ(L_{v}/F_{v}, ϕ_{v})κ_{v} (\frac{t_{1} − t_{2}}{t_{1}^{0} − t_{2}^{0}} ) \frac{|(t_{1} − t_{2})^{2}|_{v}^{1/2}}{|t_{1}t_{2}|_{v}^{1/2}} .

The product is over all finite places of F which do not divide 𝔭. Using various product formulae, we may replace it by the inverse of the same product taken over the infinite primes and the primes dividing 𝔭.

The expression

(2.5)  α\{∏ λ(L_{v}/F_{v}, ψ_{v})κ_{v} (\frac{t_{1} − t_{2}}{t_{1}^{0} − t_{2}^{0}} ) \frac{|(t_{1} − t_{2})^{2}|_{v}^{1/2}}{|t_{1}t_{2}|_{v}^{1/2}} \}^{−1}f_{ξ}^{T} (t) ,

in which the product is taken over the infinite places, depends on t and on μ^{∨}, but it does not depend on the choice of t^{0} or of the ψ_{v}.

Lemma 2.6. If t lies in T (R) and is regular then the expression (2.5) is equal to

\frac{trace ξ(t)}{meas Z(R)\T (R)} .

It is clear that we could define the expression (2.5) for any group lying between

∏_{v} S_{v}(R) ,

and

{(g_{v}) ∈ ∏_{v} G_{v}(R)| ∏_{v} Nm g_{v} > 0} .

Here the product is over the infinite places of F. G_{v}(R) is GL(2, R) if the quaternion algebra defining G splits at v and the multiplicative group of a quaternion algebra of it does not. S_{v}(R) consists of the elements of norm 1 in G_{v}(R). It is certainly enough to prove the lemma for the largest of these groups. Since the group

{(g_{v}) ∈ ∏_{v} G_{v}(R)|det g_{v} > 0 for all v}

contains T (R) and supports the character of ψ_{∞}, we may work with it instead. This yields a situation that factors, and we may finally suppose that there is only one place.

We take the additive character ψ_{v} to be x → e^{2πix} and define t^{0} by

t_{1}^{0} = γ_{1}(t^{0}) = i .

We choose γ_{1}, γ_{2} so that γ_{1} − γ_{2} and μ^{∨} lie in opposite Weyl chambers, and set

t_{1} = γ_{1}(t) = re^{iθ} .

According to [10],

λ(E_{v}/F_{v}, ψ_{v}) = i .

Thus

λ(E_{v}/F_{v}, ψ_{v})κ (\frac{t_{1} − t_{2}}{t_{1}^{0} − t_{2}^{0}} ) \frac{|(t_{1} − t_{2})^{2}|_{v}^{1/2}}{|t_{1}t_{2}|^{1/2}} = e^{iθ} − e^{−tθ} .

If D does not split and ω is the non-trivial element of the Weyl group,

\frac{θ_{∞}^{0}(t) − θ_{∞}^{0}(ω(t))}{e^{iθ} − e^{−iθ}} = \frac{λ(t)e^{iθ} − λ(ω(t))e^{−iθ}}{e^{iθ} − e^{−iθ}} = trace π_{∞}(t) .

Consequently the value of ⟨ϵ, π_{∞}⟩, defined with respect to θ_{∞}^{0}, is −1. Since η(θ_{∞}) is clearly also 1, the value of α is 1. Since π_{∞} is ξ̃ in this case, the assertion of the lemma now follows from the definition of f_{ξ}^{T}.

Suppose D splits at v. There is a unique element π_{∞} in Π^{1}(ξ) and it contains a lowest weight of T (R) with respect to the order making γ_{1} − γ_{2} positive. The character of π_{∞} is easily calculated and is found to be

−θ_{∞}^{0}(t)/(e^{iθ} − e^{−iθ}) .

If π̃_{∞} is the corresponding element of the holomorphic discrete series, then

χ_{π_{∞}}(t) − χπ̃_{∞}(t) = −\frac{θ_{∞}^{0}(t) + θ_{∞}^{0}(ω(t))}{e^{iθ} − e^{−iθ}} .

As a consequence

⟨ϵ, π_{∞}⟩ = 1 .

Since η(θ_{∞}^{0}) = 1, the number α is −1. Once again, the assertion of the lemma follows from the definition of f_{ξ}^{T}.

The expression

(2.6)  \{∏_{v|𝔭} λ(L_{v}/F_{v}, ψ_{v})κ_{v} (\frac{t_{1} − t_{2}}{t_{1}^{0} − t_{2}^{0}} ) \frac{|(t_{1} − t_{2})^{2}|_{v}^{1/2}}{|t_{1}t_{2}|_{v}^{1/2}} \}^{−1}φ_{𝔭}^{∗}f_{𝔭}^{j}(t)

depends on the regular element t in T (Q_{𝔭}) and on the order on the roots of T provided by the identification of X_{∗}(T) and X^{∗}(^{L}T^{0}). It is easily seen that r^{0}, the restriction to ^{L}T^{0} × 𝔊(\overline{Q}/E) of the representation of LG^{0} × 𝔊(\overline{Q})/E) that appeared in the construction of r, is the direct sum of two representations r_{0}^{+} and r_{0}^{−} such that

r^{+} = Ind(^{L}T,^{L}T^{0} × 𝔊(\overline{Q}/E), r_{0}^{+})
r^{−} = Ind(^{L}T,^{L}T^{0} × 𝔊(\overline{Q}/E), r_{0}^{−}) .

The restriction of r^{+} or r^{−} to the local associate group ^{L}T_{𝔭} = ^{L}T^{0} × 𝔊(\overline{Q}_{𝔭}/Q_{𝔭}) is therefore the direct sum of induced representations parametrized by the double coset space

𝔊(\overline{Q}/E)\𝔊(\overline{Q}/Q)/𝔊(\overline{Q}_{𝔭}/Q_{𝔭}) .

We had fixed E ⊆ \overline{Q} ⊆ C and we have now fixed \overline{Q} → \overline{Q}_{𝔭} as well. This double coset space also indexes the primes 𝔭 of E dividing 𝔭, the coset containing σ yielding the valuation defined by the imbedding x → σ^{−1}(x) of E in \overline{Q}_{𝔭}. We write

r^{+} = ⊕r_{𝔭}^{+} r^{−} = ⊕r_{𝔭}^{−}

and define f_{𝔭}^{j} in the Hecke algebra by

|ϖ^{j}|^{−q/2}{trace r_{𝔭}^{+}(t^{j}(θ_{𝔭})) − trace r_{𝔭}^{−}(t^{j}(θ_{𝔭}))} = ∫_{Z_{𝔭}/T (Q_{𝔭})} θ_{𝔭}(t)f_{𝔭}^{j}(t)dt .

Then

f_{𝔭}^{j} = ∑_{𝔭} f_{𝔭}^{j} .

The representation of f_{𝔭}^{j} as a sum yields a representation of (2.6) as a sum, the terms being obtained by replacing f_{𝔭}^{j} with f_{𝔭}^{j}. Although we should consider each of them, there is no loss of generality in fixing our attention on the prime 𝔭 defined by the imbedding E → \overline{Q} → \overline{Q}_{𝔭}.

The group G(F) is a subgroup of G̃_{1}(F), if G̃_{1} = Res_{F/Q}GL(2), and ^{L}G^{0} is a quotient of ^{L}G̃_{1}^{0}. Let T̃_{1} be the centralizer of T in G̃_{1}, and then ^{L}T^{0} is a quotient of ^{L}T̃_{1}^{0}. If Ũ_{𝔭} and U_{𝔭} are the maximal compact subgroups of T̃_{1}(Q_{𝔭}) and of T (Q_{𝔭}) we may define an imbedding f → f′ of the Hecke algebra ℋ_{𝔭}(T) of T (Q_{𝔭}) into ℋ_{𝔭}(T̃_{1}). The value of f′ at t is 0 unless t = su with u ∈ Ũ_{𝔭} and s ∈ T (Q_{𝔭}) and then

f′(t) = \frac{meas U_{𝔭}}{meas Ũ_{𝔭}} f(s) .

If we take ^{L}T and ^{L}T̃_{1} to be ^{L}T^{0} × 𝔊(Q_{𝔭}^{un}/Q_{𝔭}) and ^{L}T̃_{1}^{0} × 𝔊Q_{𝔭}^{un}/(Q_{𝔭}), then ℋ_{𝔭}(T) and ℋ_{𝔭}(T̃_{1}) may be regarded as algebras of funcitons on ^{L}T^{0} × Φ_{𝔭} and ^{L}T̃_{1}^{0} × Φ_{𝔭}, with Φ_{𝔭} being the Frobenius. The map f → f′ may also be obtained by pulling back functions by means of

LT̃_{1}^{0} × Φ_{𝔭} → ^{L}T^{0} × Φ_{𝔭} .

The representations r_{𝔭}^{+} and r_{𝔭}^{−} may be lifted to ^{L}T, and it will be advantageous for us to regard f_{𝔭}^{j} as an element of ℋ_{𝔭}(T̃_{1}).

We have supposed that the totally real field used to define G is imbedded in \overline{Q} and hence in C and \overline{Q}_{𝔭}. The set

Q = 𝔊(\overline{Q}/F)\𝔊(\overline{Q}/Q)

parametrizes the imbeddings of F in R ⊆ C and in \overline{Q}_{𝔭}. We represent it as a set of crosses and circles, the crosses denoting the infinite places at which the quaternion algebra defining G splits.

◯ × …× ◯ …×

We decompose the set into orbits under Φ_{𝔭}, and suppose that the action of Φ_{𝔭} on each orbit is by a cyclic

shift to the right \underbrace{× … ◯}_{Q_{v_{1}}}  \underbrace{◯× …×}_{Q_{v_{2}}} …

Let 𝔫_{v} be the number of elements in the v^{th} orbit. Then

∑ 𝔫_{v} = 𝔫 = [F : Q] .

We let 𝔪_{v} be the number of points in the v^{th} orbit which are marked by a cross. The orbits also parametrize the places of F dividing ρ, and so we may label an orbit by the place it defines. That is why we have chosen the symbol v.

Over Q_{𝔭}

T̃_{1} ≃ ∏_{v|𝔭} T_{v}

with

T_{v}(Q_{𝔭}) = L_{v}^{×}

if

L_{v} = L ⊗_{F} F_{v} .

Thus ℋ_{𝔭}(T̃_{1}) ≃ ⊗_{v}ℋ_{𝔭}(T_{v}).

Since t = t(θ_{𝔭}) ∈ ^{L}T^{0} × Φ_{𝔭} and r_{𝔭}^{+} and r_{𝔭}^{−} are induced,

trace r_{𝔭}^{+}(t^{j}) − trace r_{𝔭}^{−}(t^{j}) = 0

if j is not divisible by k = [E_{𝔭} : Q_{𝔭}]. Consequently f_{𝔭}^{j} is then to be 0. Let t = a × Φ_{𝔭} with a ∈ ^{L}T^{0}. Then

t^{k} = aΦ_{𝔭}(a) … Φ_{𝔭}^{k−1}(a) × Φ_{𝔭}^{k}

and

Φ_{𝔭}t^{k}Φ_{𝔭}^{−1} = a^{−1}t^{k}a

is conjugate under ^{L}G^{0} to g^{k}. Hence if k|j

trace r_{𝔭}^{+}(t^{j}) − trace r_{𝔭}^{−}(t^{j}) = [E_{𝔭} : Q_{𝔭}]{trace r_{0}^{+}(t^{j}) − trace r_{0}^{−}(t^{j})} .

The representations r_{0}^{+} and r_{0}^{−} are, for the present purposes, to be treated as representations of ^{L}T^{0} × 𝔊(Q_{𝔭}^{un}/E_{𝔭}) or of ^{L}T̃_{1}^{0} × 𝔊(Q_{𝔭}^{un}/E_{𝔭}). For each i in Q, let γ_{1}^{i}, γ_{2}^{i} be the weights of ^{L}T̃_{1}^{0} or of ^{L}T given by

\begin{aligned}γ_{1}^{i} & = (0, 0) × … × (0, 0) × (1, 0) × (0, 0) × … × (0, 0), \\ γ_{2}^{i} & = (0, 0) × … × (0, 0) × (0, 1) × (0, 0) × … × (0, 0) . \end{aligned}

There is one non-zero factor and it is at the i^{th} place. For the moment we will not be too concerned about which order makes the roots γ_{1}^{i} − γ_{2}^{i} positive. We take it to be that coming from the identification of X_{∗}(T) and X^{∗}(^{L}T^{0}).

Let \overline{Q} ⊆ Q be the set of marked spots. It is easily seen that

(2.7)  r_{0}^{+} − r_{0}^{−} = + ⊗_{i∈\overline{Q}} (γ_{1}^{i} − γ_{2}^{i})

on ^{L}T̃_{1}^{0}. To examine this difference on ^{L}T̃_{1}^{0} × 𝔊(Q_{𝔭}^{un}/Q_{𝔭}) we must not only describe r_{0} on ^{L}G^{0} × 𝔊(Q/E) but also explicitly describe the lifting of ψ: ^{L}T → ^{L}G to ψ̄_{1}: ^{L}T̃_{1} → ^{L}G̃_{1}.

We lift r_{0} and regard it as a representation of ^{L}G̃_{1}^{0} × 𝔊(Q/E). The group ^{L}G̃_{1}^{0} is a product

∏_{i∈Q} GL(2, C) .

Let r^{i} be the representation obtained by projecting on the i^{th} factor and then taking the standard representation of GL(2, C) on the space X^{i} of column vectors of length two. The restriction of r_{0} to ^{L}G̃_{1}^{0} is

⊗_{i∈Q} r^{i} ,

acting on

⊗_{i∈Q} X^{i} .

Recall that Q is a homogeneous space on which 𝔊(\overline{Q}/Q) acts to the right. By its definition 𝔊(\overline{Q}/E) consists of those elements of 𝔊(\overline{Q}/Q) that leave E invariant. If σ ∈ 𝔊(\overline{Q}/E) then

r_{0}(σ): ⊗_{i∈\overline{Q}} x^{i} → ⊗_{i∈\overline{Q}} x^{i^{σ}} .

On ^{L}T̃_{1}^{0} the homomorphism ψ̄_{1} is easily described. It takes

t → ∏_{i∈Q} ( \begin{matrix}γ_{1}^{i}(t) & 0 \\ 0 & γ_{2}^{i}(t) \end{matrix} ) .

To define it explicitly on ^{L}T̃_{1} we need to choose a set of representatives z_{i} for the cosets in Q. If we examine the constructions in [16], we see that this entails choosing g_{1} correctly, but we are in fact allowed to choose g_{1} anew continually. We let

τ_{i}σ = d_{τ_{i}}(σ)τ_{j}, d_{τ_{i}}(σ) ∈ 𝔊(\overline{Q}/F) .

Set

a(ρ) = \{ \begin{matrix}( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , ρ ∈ 𝔊(\overline{Q}/L), \\ ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , ρ ∈ 𝔊(\overline{Q}/F), ρ ∉ 𝔊(\overline{Q}/L) . \end{matrix}

Then

ψ̄_{1}(σ) = ∏_{i} a(d_{i}(σ)) × σ .

Lemma 2.7. The function f_{𝔭}^{j} is 0 if for some v_{0} the algebra L_{v_{0}} = L ⊗_{F} F_{v_{0}} is a field and Q_{v_{0}} has marked points.

We may suppose that [E_{𝔭} : Q_{𝔭}] divides j. Let l_{v} be the greatest common divisor of 𝔫_{v} and j. Up to equivalence the representation r_{0} and the representations r_{0}^{+} and r_{0}^{−}, as representations of ^{L}T̃_{1}, do not depend on the choice of coset representatives. In an orbit under ρ = Φ_{𝔭}^{j} we take the representatives to be

τ, τρ, τρ^{2}, …, τρ^{e_{v}−1}, e_{v} = 𝔫_{v}/l_{v} .

Then

d_{τρ^{i}}(ρ) = \{ \begin{matrix}1, & 0 ≦ i < e_{v} − 1, \\ τρ^{e_{v}}τ^{−1}, & i = e_{v} − 1 . \end{matrix}

Certainly

τρ^{e_{v}}τ^{−1} = τΦ_{𝔭}^{n_{v}j/l_{v}}τ^{−1}

is the Frobenius over F_{v} to the power j/l_{v}. If L_{v} is a field, then it is quadratic over F_{v} and, with the assumption under which we are working at present, unramified. Thus τρ^{e_{v}}τ^{−1} lies in 𝔊(\overline{Q}/L) if and only if 2|j/l_{v}.

Let {x_{1}^{i}, x_{2}^{i}} be the standard basis of X^{i}. The collection
⊗_{i∈\overline{Q}} x_{j(i)}^{i}, j(i) = 1, 2

is a basis for the space in which r_{0} acts. The representation r_{0}^{+} acts on the span of those elements for which

(−1)^{∑_{i∈\overline{Q}} j(i)} = ±(−1)^{|\overline{Q}|} ,

and r_{0}^{−} on the span of the other elements. The sign is determined by (2.7). It is clear that r_{0}(Φ_{𝔭}^{j}) permutes the basis elements amongst themselves, and that the element indexed by {j(i)} is fixed if and only if j(i) is constant on orbits of Φ_{𝔭}^{j} and

a(τρ^{e_{v}}τ^{−1}) = ( \begin{matrix}1 & 0 \\ 0 & 1 . \end{matrix} )

Since each basis element is an eigenvector for ^{L}T̃_{1}^{0}, we have

trace r_{0}^{+}(t^{j}) = trace r_{0}^{−}(t^{j}) = 0,

and

trace r_{0}^{+}(t^{j}) − trace r_{0}^{−}(t^{j}) = 0,

if 2l_{v_{0}} does not divide j and t = a × Φ_{𝔭}, a ∈ ^{L}T̃_{1}^{0}. Thus in this case at least, f_{𝔭}^{j} = 0.

We now suppose that 2l_{v_{0}} divides j, and make a different choice of coset representatives. We take the representatives of the cosets in Q_{v} to be of the form

τ, τΦ_{𝔭}, …, τΦ_{𝔭}^{n_{v}−1}.

If ρ = Φ_{𝔭} then

d_{τρ^{i}}(ρ) = \{ \begin{matrix}1, & 0 ≦ i < 𝔫_{v} − 1, \\ τρ^{𝔫_{v}}τ^{−1}, & i = 𝔫_{v} − 1 . \end{matrix}

It helps to picture Q_{v} as

××◯ …×^{l_{v}}××◯ …×◯ …

There are l_{v} orbits under Φ_{𝔭}^{j} and each orbit consists entirely of marked or entirely of unmarked points. If k_{v} is the number of marked orbits then

𝔪_{v}/k_{v} = 𝔫_{v}/l_{v} .

We may write

r_{0}^{+}(t^{j}) − r_{0}^{−}(t^{j}) = ± ⊗_{z} (r_{z}^{+}(t^{j}) − r_{z}^{−}(t^{j})) .

The tensor product is taken over the marked orbits and the meaning of r_{z}^{+}(t^{j}) and r_{z}^{−}(t^{j}) is, I hope, clear. We show that

trace r_{z}^{+}(t^{j}) = trace r_{z}^{−}(t^{j})

if z ∈ Q_{v_{0}}.

All our calculations will be within Q_{v_{0}}. So we may as well take the indices of the points in this set to be 1, …, 𝔫_{v_{0}}, agreeing that indices in the formulae to follow are to be read modulo 𝔫_{v_{0}}. If t^{j} = b × Φ_{𝔭}^{j}, then

γ_{e}^{i}(b) = γ_{e}^{i}(a)γ_{e}^{i+1}(a) … γ_{e}^{𝔫_{v_{0}}} (a)γ_{e′}^{𝔫_{v_{0}} +1}(a) … γ_{e′}^{2𝔫_{v_{0}}} (a)γ_{e}^{2𝔫_{v_{0}} +1}(a) …

Here e′ ≠ e, and the subscripts, except perhaps at the beginning and end, appear in blocks of length 𝔫_{v_{0}}. There are exactly two vectors of the form

⊗_{i∈z}x_{j(i)}^{i}

fixed by Φ_{𝔭}^{j}, but since we have chosen a new set of coset representations the subscripts j(i) are no longer constant. At all events,

trace r_{z}^{+}(t^{j}) − trace r_{z}^{−}(t^{j}) = (∏_{i∈z} γ_{j(i)}^{i}(b) − ∏_{i∈z} γ_{j′(i)}^{i}(b)) .

Since

Φ_{𝔭}(b) = a^{−1}bΦ_{𝔭}^{j}(a),

the character

a → ∏_{i∈z} γ_{j(i)}^{i}(b)

is invariant under Φ_{𝔭}. It is therefore a power of

a → γ_{1}^{1}(a) … γ_{1}^{n_{v_{0}}} (a)γ_{2}^{1}(a) … γ_{2}^{n_{v_{0}}} (a) ,

and the power must clearly be

\frac{j𝔫_{v_{0}}}{2𝔫_{v_{0}}l_{v_{0}}} = \frac{j}{2l_{v_{0}}} .

We conclude that

∏_{i∈z} γ_{j(i)}^{i}(b) = ∏_{i∈z} γ_{j′(t)}^{i}(b) .

The lemma follows.

Our purpose has been to find an explicit expression for (2.6), with f_{𝔭}^{j} replacing f_{𝔭}^{j}, when every prime of F dividing 𝔭 is unramified in L. We are still not finished, but we have shown that it is 0 unless v splits in L whenever Q_{v} contains a marked point. If we pass from ℋ_{𝔭}(T) to ℋ_{𝔭}(T̃_{1}) we replace φ_{𝔭} by φ′_{𝔭} and φ′_{𝔭} is a product

φ′_{𝔭}(t) = ∏_{v} φ_{v}(t^{v})

if t = (t^{v}), t^{v} ∈ T_{v}(Q_{𝔭}). The expression

(2.8)  λ(L_{v}/F_{v}, ψ_{v})κ_{v} (\frac{t_{1}^{v} − t_{2}^{v}}{t_{1}^{0} − t_{2}^{0}} ) φ_{v}(t^{v})

does not depend on the choice of t^{0} or of ψ_{v}. For lack of space, the image of the global element t^{0} in T_{v}(Q_{𝔭}) is also denoted t^{0}.

Lemma 2.8. Let U_{v} be the maximal compact subgroup of T_{v}(Q_{𝔭}) and let δ_{v} be the characteristic function of U_{v} divided by its measure. If v splits in L then (2.8) is equal to δ_{v}(t^{v}). If v does not split and U_{v} and K_{𝔭} are contained in a common maximal subgroup of G̃_{1}(Q_{𝔭}), then (2.8) is equal to δ_{v}(t^{v}) if the order of t_{1}^{v} − t_{2}^{v} in L_{v} is even and to −δ_{v}(t^{v}) if the order of t_{1}^{v} − t_{2}^{v} in L_{v} is odd.

The assertion pertaining to split v is clear. If v is not split we take ψ_{v} to be such that the largest ideal on which it is trivial is the ring of integers of F_{v} and we take t^{0} to be such that t_{1}^{0} − t_{2}^{0} is a unit in L_{v}. Then

λ(L_{v}/F, ψ_{v}) = 1

and

κ_{v} (\frac{t_{1}^{v} − t_{2}^{v}}{t_{1}^{0} − t_{2}^{0}} )

is 1 when the order of t_{1}^{v} − t_{2}^{v} is even and −1 when it is odd. However, it is observed after the proof of Lemma 2.2 of [7] that with these choices of ψ_{v} and t^{0}, and the assumption that U_{v} and K_{𝔭} are contained in a common maximal compact subgroup K̃_{𝔭} of G̃_{1}(Q_{𝔭}), the function

φ_{v}(t) = Φ^{T_{v}/κ_{v}}(t, ϕ_{v})

is the characteristic function of U_{v} divided by its measure. Here we write

K̃_{𝔭} = ∏_{v} K_{v}

and take ϕ_{v} to be the characteristic function of K_{v} divided by its measure.

We return to the functions f_{𝔭}^{j}, which we regard as elements of

ℋ_{𝔭}(T̃_{1}) ≃ ⊗_{v}ℋ_{𝔭}(T_{v}) .

We suppose now that Q_{v} contains no marked points if v does not split in L. If we regard r_{0}^{+} and r_{0}^{−} as representations of the associate group of T̃_{1} over Q_{𝔭} then we may factor r_{0}^{+} and r_{0}^{−} as a tensor product

⊗_{v|𝔭}(r_{v}^{+} − r_{v}^{−}) .

In order to specify r_{v}^{+} and r_{v}^{−} conveniently we choose the γ_{1}^{i} and γ_{2}^{i} in such a manner that all of the γ_{1}^{i} − γ_{2}^{i} are positive with respect to an order that puts μ^{∨} in the closed negative Weyl chamber. This is a choice that refers only to T and not to the identification of X_{∗}(T̃_{1}) and X^{∗}(^{L}T̃_{1}).

If Q_{v} contains no marked points then r_{v}^{+} is the trivial one-dimensional representation and r_{v}^{−} is zerodimensional. If v splits in L then r_{v}^{+} ⊕ r_{v}^{−} acts on the span of

⊗_{i∈\overline{Q}_{v}}x_{j(i)}^{i} .

The action of Φ_{𝔭} = σ sends

⊗_{i∈\overline{Q}_{v}}x_{j(i)}^{i} → ⊗_{i∈\overline{Q}_{v}}x_{j(i^{σ})}^{i}

and a ∈ ^{L}T̃_{1}^{0} acts as

⊗x_{j(i)}^{i} → (∏_{i∈\overline{Q}_{v}} γ_{j(i)}^{i}(a)) (⊗x_{j(i)}^{i}) .

If we write

μ^{∨} = ∑ μ_{v}^{∨},

then

μ_{v} = ∑_{i∈\overline{Q}_{v}} γ_{2}^{i} .

The vector ⊗x_{j(i)}^{i} lies in the space of r_{v}^{+} or r_{v}^{−} according as (−1)^{∑ j(i)} is 1 or −1.

The following lemma is in any case clear.

Lemma 2.9. The function f_{𝔭}^{j} is a product ∏ f_{𝔭,v}^{j} with f_{𝔭,v}^{j} in ℋ_{𝔭}(T_{v}).

In order to save on subscripts we suppose for the purposes of describing the f_{𝔭,v}^{j} that 𝔭 remains prime in F and that F splits in L. This allows us to drop the v when we choose.

Suppose μ is a weight of r_{0}^{+} + r_{0}^{−}. Then there is a partition of \overline{Q} into two disjoint subsets \overline{Q}′ and \overline{Q}′′ with 𝔪′ and 𝔪′′ elements respectively, and

μ = ∑_{i∈\overline{Q}′} γ_{2}^{i} + ∑_{i∈\overline{Q}′′} γ_{1}^{i} .

Moreover 𝔪′ + 𝔪′′ = 𝔪 (= 𝔪_{v}). Let

δ_{e} = ∑_{i∈Q} γ_{e}^{i}

and set

ν = 𝔪′δ_{2} + 𝔪′′δ_{1} .

If k_{𝔭} is a large unramified Galois extension of Q_{𝔭} then

Nm_{k_{𝔭}/Q_{𝔭}}μ = [k_{𝔭} : Q_{𝔭}]ν/𝔫 .

We set

β(ν) = κ(μ − μ^{∨}) = (−1)^{k′′} .

Lemma 2.10. Suppose there is only one prime v of F dividing 𝔭 and that it splits in L. Let 𝔫 = 𝔫_{v}, l = l_{v}, k = k_{v}. If j|[E_{𝔭} : Q_{𝔭}] and t = a × Φ_{𝔭} then

trace r_{0}^{+}(t^{j}) − trace r_{0}^{−}(t^{j})

is equal to

∑_{𝔪/k|𝔪′} \frac{k′}{k′!k′′!} β(ν)ν(a)^{j/𝔫} .

Here

k′ = \frac{𝔪′}{𝔪} k k′′ = \frac{𝔪′′}{𝔪} k .

Notice that

\frac{j}{𝔫} ν = \frac{j}{l} (\frac{𝔫}{l} ν) = \frac{j}{l} (\frac{k}{𝔪} ν)

is a weight if 𝔪/k|𝔪′, and so the terms of the sum appearing in the lemma are well-defined. The argument used to prove Lemma 2.7 shows that

trace r_{0}^{+}(t^{j}) − trace r_{0}^{−}(t^{j}) = ∏_{z}(trace r_{z}^{+}(t^{j}) − trace r_{z}^{−}(t^{j})) .

The basis vectors ⊗_{i∈z}x_{j(i)}^{i} for z_{z}^{+} ⊕ r_{z}^{−} are permuted amongst themselves by Φ_{𝔭}^{j}. The only fixed vectors are

⊗_{i∈z} x_{j}^{i}, j = 1, 2 .

Thus

trace r_{z}^{+}(t^{j}) − trace r_{z}^{−}(t^{j}) = δ_{2}(a)^{j/l} + (−1)^{𝔫/l}δ_{1}(a)^{j/l}

Since there are k orbits, we must raise the right side t the k^{th} power. Expanding by the binomial theorem and recalling that

𝔪/k = 𝔫/l ,

we obtain the lemma.

If ϖ is a local uniformizing parameter for Q_{𝔭} then the element of the Hecke algebra corresponding to the function

a × Φ_{𝔭} → ν(a)^{j/𝔫}

is the characteristic function of

ϖ^{jν/𝔫}Ũ_{𝔭}

divided by its measure. We denote this function by θ_{jν/𝔫}.

Before summarizing our results on the sum (2.1) we observe that since Z(R)\T (R) is compact the image of T (Q) in

T (R)Z_{K}\T (A) ≃ Z_{K}\T (A_{f})

is discrete. Clearly

meas(Z_{K}T (Q)\T (A_{f})) = \frac{meas(Z(R)Z_{K}T (Q)\T (A))}{meas Z(R)\T (R)} .

Summary of the discussion of the sum (2.1).

(a) It can be expressed as a sum over the primes 𝔭 of E dividing 𝔭.

(b) If 𝔭 is such a prime let ϖ_{𝔭} be a uniformizing parameter for E_{𝔭}. The term of the sum corresponding to 𝔭, which is now our only object of interest, may be expanded in powers of

|ϖ_{𝔭}|^{s} = |ϖ|^{se}

if e = [E_{𝔭} : Q_{𝔭}].

(c) The coefficient of j^{−1}|ϖ_{𝔭}|^{js} is 0 if, for some v|𝔭, v does not split in L and there is a marked place in Q_{v}. It contains the factor

\frac{μ(T)}{2[𝔈(T/A) : Im 𝔈(T/F)]} meas(Z_{K}T (Q)\T (A_{f})) .

(d) To obtain the coefficient this factor has to be multiplied by a sum over the non-central elements of

T (Q) ∩ Z(R)Z_{K}\T (Q) = T (Q) ∩ Z_{K}\T (Q)

and over the possible ν arising from collections {(𝔪′_{v}, 𝔪′′_{v})|𝔪′_{v} +𝔪′′_{v} = 𝔪_{v}}. In the quotient on the right, T (Q) is regarded as a subgroup of T (A_{f}). The terms of the sum are themselves the product of three factors. The first is trace ξ(t), and depends only on ξ and t. The second is Φ_{0}T/K(t, ϕ^{𝔭}), and depends only on the image of t in T (A_{f}^{𝔭}).

(e) The third factor may be represented by meas Ũ_{𝔭}/meas U_{𝔭} times a product over the places v of F dividing 𝔭 of further factors, each depending only on the image of t in T_{v}(Q_{𝔭}), times β(ν) = κ(μ − μ^{∨}). Here μ is any weight of r_{0} such that, for some large Galois extension k_{𝔭} of Q_{𝔭},

Nm_{k_{𝔭}/Q_{𝔭}}μ

is a multiple of ν.

(f) The factor corresponding to a given v is 0 unless v splits in F or there are no marked places in Q_{v}.

(g) If there are no marked places in Q_{v} and v does not split in F then the corresponding factor is

\frac{±1}{meas U_{v}} · \frac{|t_{1}t_{2}|_{v}^{1/2}}{|(t_{1} − t_{2})^{2}|_{v}^{1/2}} .

The sign ±1 may be expressed as a product of two factors. The first is (−1)^{ord(t_{1}−t_{2})}. To obtain the second we write

G̃_{1}(Q_{𝔭}) = ∏_{v|𝔭} G_{v}(Q_{𝔭}) ,

with

G_{v} = Res_{F_{v}/Q}G̃ ,

and let K̃_{𝔭} = ∏ K_{v} be a maximal compact subgroup of G̃_{1}(Q_{𝔭}) containing K_{𝔭}. If g ∈ G_{v}(Q_{𝔭}) and g^{−1}U_{v}g ⊆ K_{v} then the second factor is (−1)^{ord Nm^{g}}.

(h) We write ν = ∑ ν_{v}. If v splits in F, the corresponding factor is 0 unless k_{v}ν_{v}/𝔪_{v} is a weight. If it is, the corresponding factor is to be

\frac{k_{v}!}{k′_{v}!k′′_{v}!} p^{je𝔪_{v}/2}θ_{ν′_{v}}(t) \frac{|t_{1}t_{2}|_{v}^{1/2}}{|(t_{1} − t_{2})^{2}|_{v}^{1/2}} .

Here

k′_{v} = 𝔪′_{v}k_{v}/𝔪_{v}  k′′_{v} = 𝔪′′_{v}k_{v}/𝔪_{v}  ν′_{v} = jeν_{v}/𝔫_{v} .

Notice that l_{v} must now be taken to be the greatest common divisor of 𝔫_{v} and je. Moreover if 𝔪′_{v} + 𝔪′′_{v}, θ_{ν′_{v}}(t) ≠ 0, and q_{v} is the smallest of 𝔪′_{v} and 𝔪′′_{v} then

𝔭^{jem_{v}/2} \frac{|t_{1}t_{2}|_{v}^{1/2}}{|(t_{1} − t_{2})^{2}|_{v}^{1/2}} = p^{jeq_{v}} .

The logarithm of (1.5) has to be subjected to a similar treatment. Fortunately, we can handle it with more dispatch, aprtly because we can rely to some extent on our discussion of (1.6), and partly because (1.5) has been so set up that the trace formula, in its stabilized form [7], is immediately applicable and quickly leads to the expressions needed for the comparison.

The logarithm of the local factor at 𝔭 of (1.5) is

(2.9)  ∑∏ 𝔫(Π)𝔪(Π_{∞})𝔪(Π_{f}) log L_{𝔭}(s − q/2, π, r) ,

where π ∈ Π. The number 𝔫(Π) is the common value of 𝔫(π), π ∈ Π. Moreover we may as well agree that the sum is to be taken only over those Π such that

\begin{matrix}π_{∞}(z) = ν^{−1}(z)I, & z ∈ Z(R), \\ π_{f}(z) = I, & z ∈ Z_{K}, \end{matrix}

if π = π_{∞} ⊗ π_{f} ∈ ∏, for otherwise either 𝔪(Π_{∞}) or 𝔪(Π_{f}) is 0. This said, we may replace 𝔪(Π_{∞}) by its value

∑_{∏_{∞}∈Π_{∞}} trace π_{∞}(f_{ξ}) .

We also may write

𝔪(Π_{f}) = ∑_{π_{f} ∈Π_{f}} 𝔪(π_{f}) .

Then 𝔪(π_{f}) is non-zero only if π_{𝔭} contains the trivial representation of K_{𝔭}. Thus we may replace

log L_{𝔭}(s − q/2, π, r)

by

∑_{𝔭|𝔭} ∑_{j=1}^{∞} \frac{|ϖ|^{js}}{j} trace π_{𝔭}(h_{𝔭}^{j}) ,

where h_{𝔭}^{j}k is an appropriate element of the Hecke algebra of G(Q_{𝔭}). It is defined by

trace π_{𝔭}(h_{𝔭}^{j}) = |ϖ|^{−jq/2}trace r_{𝔭}(g(π_{𝔭})) ,

whenever π_{𝔭} is an irreducible representation of G(Q_{𝔭}) containing the trivial representation of K_{𝔭}. Here r_{𝔭} = r_{𝔭}^{+} ⊕ r_{𝔭}^{−} and g(π_{𝔭}) is an element of the conjugacy class in ^{L}G^{0} × Φ_{𝔭} associated to π_{𝔭}. We conclude that (2.9) may be written as the sum over 𝔭|𝔭 and the sum of j from 1 to ∞ of |ϖ|^{js}/j times

(2.10)  ∑_{π} 𝔫(π)trace π_{∞}(f_{ξ})trace π^{𝔭}(ϕ^{𝔭})trace π_{𝔭}(h_{𝔭}^{j}) .

Recall that ϕ^{𝔭} is the characteristic function of K^{𝔭} ⊆ G(A_{f}^{𝔭}) divided by its measure.

We may apply the stabilized trace formula for G(Q)Z(R)Z_{K}\G(A) to the function

g → f_{ξ}(g_{∞})ϕ^{𝔭}(g^{𝔭})h_{𝔭}^{j}(g_{𝔭})

to obtain the form for (2.10) that we need.

In order to write down the contribution from the scalars we need to draw on our knowledge of harmonic analysis on real groups, namely on the limit formula of Harish-Chandra, to see that

f_{ξ}(z) = (−1)^{q}trace ξ(z)/meas Z(R)\G′(R), z ∈ Z(R) .

Here G′ is a form of G over R, the one for which Z(R)\G′(R) is compact, and the measure on Z\G′ is obtained from that on Z\G by transporting invariant forms of highest degree as in §15 of [6]. To tell the truth, I am unable to supply a reference to the appropriate computation. The method to be used is described in [8], and the reader can verify for himself that the constant is correct. Appealing to §7 of [7], we see that the contribution of the scalar matrices to (2.10) is

(2.11)  ∑_{Z(Q)∩Z(R)Z_{K}\Z(Q)} \frac{meas(Z(R)Z_{K}G(Q)\G(A))}{meas(Z(R)\G(R)} × (−1)^{q}trace ξ(z)ϕ^{𝔭}(z)h_{𝔭}^{j}(z) .

It will remove some complication from the discussion of the remaining terms of the trace formula if we take advantage of the possibility we have allowed ourselves of only working with sufficiently small K. It is convenient so to arrange matters that the equation

r^{−1}tg = zt

with g ∈ G(Q), t ∈ G(Q), and z ∈ Z(Q) ∩ Z(R)Z_{K} implies that z = 1. The equation certainly implies that z lies in the center of the derived group. Since this is finite, its intersection with Z(R)Z_{K} will be {1} when K is small.

The remaining contribution to the stabilized trace formula is a sum over a set of representatives T for the stable conjugacy classes of Cartan subgroups of

\frac{1}{2} ∑′_{T (Q)∩Z(R)Z_{K}\T (Q)} \frac{meas(T (Q)Z(R)Z_{K}\T (A))μ(T)}{[𝔈(T/A) : Im 𝔈(T/F)]} × Φ^{T/1}(t, f_{ξ})Φ^{T/1}(t, ϕ^{𝔭})Φ^{T/1}(t, h_{𝔭}^{j}) .

The prime indicates that scalars are excluded from the sum. Since Φ^{T/1}(t, f_{ξ}) is 0 unless T (R) is fundamental, we agree to sum only over such T. Then

Φ^{T/1}(t, f_{ξ}) = trace ξ(t)/(Z(R)\T (R)) .

By appealing to our earlier discussion and to the formalism of Hecke algebras, we easily see that the following assertions are valid.

(i) h_{𝔭}^{j} = 0 if e = [E_{𝔭} · Q_{𝔭}] does not divide j.

(ii) If

G_{v} = Res_{F_{v}/Q_{𝔭}} G̃

there is a natural imbedding of ℋ_{𝔭} into ⊗_{v/𝔭}ℋ_{𝔭}(G_{v}).

(iii) The image of h_{𝔭}^{j} under this imbedding is a product of ⊗h_{𝔭,v}^{j}.

(iv) If T_{v} is a split Cartan subgroup of G_{v} and t_{1}, t_{2} the eigenvalues of t ∈ T_{v}(Q_{𝔭}) ≃ L_{v}^{×}, then, when e|j,

∫_{T_{v}(Q_{𝔭})\G_{v}(Q_{c}urp)} h_{𝔭,v}^{j}(g^{−1}tg)dg

is equal to

\frac{|t_{1}t_{2}|_{v}^{1/2}}{|t_{1} − t_{2}|_{v}} ∑_{𝔪_{v}/k_{v}|ν_{v}} \frac{k_{v}!}{k′_{v}!k′′_{v}!} θ_{ν′_{v}}(t) (ν′_{v} = \frac{j}{𝔫_{v}} ν_{v}) .

The last assertion is, of course, a consequence of the definition of the Satake homomorphism, and the relation of the Satake homomorphism with the formalism of the associate group. Here ν_{v} is defined by a pair of non-negative integers 𝔪′_{v}, 𝔪′′_{v} with 𝔪′_{v} + 𝔪′′_{v} = 𝔪_{v}, and the sum is over those for which

k′_{v} = k_{v}𝔪′_{v}/𝔪_{v} k′′_{v} = k_{v}𝔪′′_{v}/𝔪_{v}

are integral.

For any T let U_{𝔭} be, as before, the maximal compact subgroup of T (Q_{𝔭}), and Ũ_{𝔭} = ∏ U_{v} the maximal compact subgroup of T̃_{1}(Q_{𝔭}). Let K̃_{𝔭} = ∏ K_{v} be the maximal compact subgroup of G̃_{1}(Q_{𝔭}) containing K_{𝔭}. Replacing T̃_{1} by a conjugate if necessary, suppose Ũ_{𝔭} ⊆ K̃_{𝔭} and define

𝔲(T) = [K̃_{𝔭} : K_{𝔭}Ũ_{𝔭}] .

Then

𝔲(T)meas U_{𝔭}Φ^{T/1}(t, h_{𝔭}^{j})

is equal to

∏_{v} meas U_{v} ∫_{T_{v}(Q_{𝔭})\G_{v}(Q_{𝔭})} h_{𝔭,v}^{j}(g^{−1}tg)dg . (2.12)

To verify this we observe that both sides are independent of the measures chosen, and that it therefore suffices to work with one convenient choice of measure. Let h be (meas K_{𝔭})h_{𝔭}^{j}, where h_{𝔭}^{j} is taken in ℋ_{𝔭}(G) and let h′ be (meas K̃_{𝔭})h_{𝔭}^{j}, with h_{𝔭}^{j} now regarded as an element of ℋ_{𝔭}(G̃_{1}) = ⊗_{ι}ℋ_{𝔭}(G). Then h is the restriction of h′ to G(Q_{𝔭}).

Φ^{T/1}(t, h) = ∑_{T̃(Q_{𝔭})G(Q_{𝔭})\G̃_{1}(Q_{𝔭})} ∫_{T (Q_{𝔭})\G(Q_{𝔭})} h(g^{−1}tg)dg .

If we take the measure on T (Q_{𝔭})\G(Q_{𝔭}) to be the same as that on its image T̃_{1}(Q_{𝔭})\ T̃_{1}(Q_{𝔭})G(Q_{𝔭}) in T̃_{1}(Q_{𝔭})\G̃_{1}(Q_{𝔭}), the right side is equal to

∫_{T̃_{1}(Q_{𝔭})/G̃_{1}(Q_{𝔭})} h′(g^{−1}tg)dg .

To check the assertion completely we need to show that with this choice

𝔲(T)meas J_{𝔭}\K_{𝔭} = meas Ũ_{𝔭}\K̃_{𝔭} .

However

meas U_{𝔭}\K_{𝔭} = meas Ũ_{𝔭}\Ũ_{𝔭}K_{𝔭}

and

meas Ũ_{𝔭}\K̃_{𝔭} = 𝔲(T)meas Ũ_{𝔭}\Ũ_{𝔭}K_{𝔭} .

We have only to evaluate the individual integrals on the right of (2.12). They have been given in §6 of [12], albeit for the group GL(2, Q_{𝔭}) and not for G_{v}(Q_{𝔭}) = GL(2, F_{v}), but the formulae and the proofs are valid without change. In order to disencumber ourselves of the subscript v when stating the results, we suppose that 𝔭 remains prime in F.

We start from G_{v}(Q_{𝔭}) = GL(2, F_{v}) and a

ν = a′δ_{2} + a′′δ_{1} .

Let

ν̃ = a′δ_{1} + a′′δ_{2}

and let h be a spherical function on GL(2, F_{v}) for which

(|t_{1} − t_{2}|_{v}/|t_{1}t_{2}|_{v}^{1/2}) ∫_{A(F_{v})\GL(2,F_{v})} h(g^{−1}tg)dg = \frac{1}{2}\{θ_{ν}(t) + θν̃(t)\}

if A is a split torus in GL(2). If a′ = a′′ and K_{v} the maximal compact subgroup containing K_{𝔭}, then h is the characteristic function of

( \begin{matrix}ϖ_{v}^{a′} & 0 \\ 0 & ϖ_{v}^{a′′} \end{matrix} ) K_{v}

divided by the measure of K_{v}. There will be no need for an explicit evaluation of the other orbital integrals in this case.

In general set

Φ^{T_{v}}(t, h) = ∫_{T_{v}(Q_{𝔭})\G_{v}(Q_{𝔭})} h(g^{−1}th) .

If a′ ≠ a′′ these integrals can be computed directly, along the lines of §3 of [15] or of §7 of [6], but I prefer to apply the method of §6 of [12]. Lemma 6.4 of that paper and the Weyl integration formula imply that

\frac{1}{2} ∑′_{T_{v}} ∫_{T_{v}(Q_{𝔭})} χ_{π}(t)Φ^{T_{v}}(t, h)Δ^{2}(t)dt = 0

if π is an absolutely cuspidal representation of G_{v}(Q_{𝔭}). The sum is over a set of representatives for the non-split Cartan subgroups and

Δ(t) = |(t_{1} − t_{2})^{2}|_{v}^{1/2}/|t_{1}t_{2}|_{v}^{1/2} .

In addition Φ^{T_{v}}(t, h) is 0 unless

|det t|_{v} = |ϖ_{v}|^{a} a = a′ + a′′ .

Let G′_{v}(Q_{𝔭}) be the multiplicative group of the quaternion algebra over F_{v}. Exploiting the relation between characters of G_{v}(Q_{𝔭}) and characters of G′_{v}(Q_{𝔭}) provided in [6], and using as well the Weyl integration formula for G′_{v}(Q_{𝔭}) and orthogonality of characters, we deduce that there is a function φ(x) on F^{×} such that

Φ^{T_{v}}(t, h) = meas(T_{v}(Q_{𝔭})\G′_{v}(Q_{𝔭}))φ(det t)

if T_{v} is not split. The measure on G′_{v}(Q_{𝔭}) may be assumed to be obtained by transferring that of G_{v}(Q_{𝔭}).

Let χ be the representation g → χ(det g). If we use the Satake homomorphism to compute the trace of χ(h) we obtain

0  χ ramified
\frac{1}{2} (|ϖ_{v}|^{a′−a′′/2} + |ϖ_{v}|^{a′′−a′/2}) χ(ϖ_{v})^{a} χ unramified .

We may also compute it by using the Weyl integration formula. This yields the sum of

\frac{1}{2} ∑′_{T_{v}} meas(T_{v}(Q_{𝔭})\G′_{v}(Q_{𝔭})) ∫_{T_{v}(Q_{𝔭})} χ(det t)φ(det t)Δ^{2}(t)dt

and

\frac{1}{2} ∫_{A_{v}(Q_{𝔭})} χ(det t)θ_{ν}(t)Δ(t)dt .

The second expression is 0 unless χ is unramified, but then it equals

\frac{1}{2} |ϖ_{v}|^{−|a′−a′′|/2}χ(ϖ_{v})^{a} .

This yields a value for the first expression. Applying the orthogonality relations and the Weyl integration formula to the characters of the one-dimensional representations of G′_{v}(Q_{𝔭}), we conclude that φ(x) is 0 unless |x| = |ϖ_{v}|^{a} and that then

φ(x) = \frac{|ϖ_{v}|^{|a′−a′′|/2}}{2 meas K′_{v}}

if K′_{v} is the maximal compact subgroup of G′_{v}(Q_{𝔭}). In order to have a convenient notation we let Ξ_{a} be 0 unless |det t| = |ϖ|^{a} when Ξ_{a}(t) is to be 1.

Finally we deduce from Lemma 4.1 of [15] that if z is a scalar then h(z) = 0 unless |det z| = |ϖ_{v}|^{a} but then

h(z) = −\frac{|ϖ_{v}|^{|a′−a′′|/2}}{2 meas K′_{v}} .

All the information we need is now at our disposal. It will be a help to review it once again, and display it in a form convenient for reference.

Summary of the discussion of the sum (2.9).

(a) It can be expressed as a sum over the primes 𝔭 of E dividing 𝔭 and over the possible ν, defined by {(𝔪′_{v}, 𝔪′′_{v})|𝔪′_{v} + 𝔪′′_{v} = 𝔪_{v}}. The 𝔪′_{v}, 𝔪′′_{v} are to be integral except perhaps when 𝔪′_{v} = 𝔪′′_{v} = 𝔪_{v}/2, and then we allow them to be half-integral.

(b) The term corresponding to 𝔭 and ν may itself be expanded in powers of |ϖ_{𝔭}|^{s} = |ϖ_{𝔭}|^{se}, e = [E_{𝔭} : Q_{𝔭}]. We now fix a 𝔭 and a ν and consider the coefficients of |ϖ_{𝔭}|^{j}/j.

(c) We treat the case that 𝔪′_{v} ≠ 𝔪′′_{v} for at least one v first. Then the coefficient is a double sum, over the stable conjugacy classes of Cartan subgroups T fundamental at infinity and split at every v for which 𝔪′_{v} ≠ 𝔪′′_{v}, and over non-scalar t in T (Q) ∩ Z(R)Z_{K}\T (Q). The individual terms are given as products. One factor is

\frac{μ(T)}{2[𝔈(T/A) : Im 𝔈(T/F)]} meas(T (Q)Z_{K}\T (A_{f})) .

A second is

Φ^{T/1}(t, ϕ^{𝔭})trace ξ(t) · u(T) \frac{meas Ũ_{𝔭}}{meas U_{𝔭}} .

The third may again be written as the product over the places of v dividing 𝔭 of factors depending only on the image of t in T_{v}(Q_{𝔭}). If T_{v} is split, the factor is 0 unless 𝔪_{v}/k_{v} divides 𝔪′_{v} and 𝔪′′_{v}. Otherwise it is

𝔪^{je𝔪_{v}/2} \frac{k_{v}!}{k′_{v}!k′′_{v}!} θ_{ν′_{v}}(t) \frac{|t_{1}t_{2}|_{v}^{1/2}}{|t_{1} − t_{2}|_{v}} .

If T_{v} is not split the expression is more complicated. It is the sum of two terms. The first is

\{∑_{0 ≦ i<k_{v}/2} p^{je𝔪_{v}i/k_{v}} \binom{k_{v}}{i}\} \frac{meas T_{v}(Q_{𝔭})\G′_{v}(Q_{𝔭})}{meas K′_{v}} Ξ_{je𝔪_{v}/𝔫_{v}}(t) .

The second is 0 unless k_{v} is even, but it is then

p^{je𝔪_{v}/2}\binom{k_{v}}{k_{v}/2}\frac{1}{meas K_{v}} ∫_{T_{v}(Q_{𝔭})\G_{v}(Q_{𝔭})} φ_{𝔪_{v}}(g^{−1}tg)dg

if φ_{𝔪_{v}} is the characteristic function of

( \begin{matrix}ϖ^{𝔪_{v}/2} & 0 \\ 0 & ϖ^{𝔪_{v}/2} \end{matrix} ) K_{v} .

The point is that for split T_{v} there are several possible ν_{v}, and we can decompose the contributions from the orbital integrals of h_{𝔭}^{j} into parts labeled by them; but for a T_{v} that is not split there is only one reasonable ν_{v} and we have to take the orbital integral in one piece.

(d) Whatever is not included in the terms gathered in (c) must now be put together and credited to that ν for which 𝔪′_{v} = 𝔪′′_{v} for all v. There are two contributions; the first is a sum over z in Z(Q)∩Z(R)Z_{K}\Z(Q). Each term of the sum is a product, the first factor being

\frac{meas Z(R)Z_{K}G(Q)\G(A)}{meas Z(R)\G′(R)} trace ξ(z)ϕ^{𝔭}(z) \frac{meas K̃_{𝔭}}{meas K_{𝔭}} .

The second factor is itself a product over the places v dividing 𝔭 of terms depending only on the image of z in the group G_{v}(Q_{𝔭}). It will be easier to write them down if we first observe that a simple calculation, which can be left to the reade4r (cf. §15 of [6]), shows that

meas K_{v} = (|ϖ_{v}|^{−1} − 1) meas K′_{v} .

The factor at v is itself a sum of two terms. The first is

−(−1)^{𝔪_{v}} \frac{(|ϖ_{v}|^{−1} − 1)}{meas K_{v}} Ξ_{je𝔪_{v}/𝔫_{v}}(z) \{∑_{0 ≦ i<k_{v}/2} \binom{k_{v}}{i}𝔭^{je𝔪_{v}i/k_{v}}\} .

The second is 0 unless k_{v} is even, when it is

\frac{(−1)^{𝔪_{v}}}{meas K_{v}} Ξ_{je𝔪_{v}/𝔫_{v}}(z)\binom{k_{v}}{k_{v}/2}𝔭^{je𝔪_{v}/2} .

The second contribution is a sum over the stable conjugacy classes of Cartan subgroups T with T (R) fundamental of a sum over the non-scalar elements in T (Q) ∩ Z(R)Z_{K}\T (Q). The terms of the sum are themselves products. The first factor of the product is

\frac{μ(T)}{2[𝔈(T/A) : Im 𝔈(T/F)]} meas(T (Q)Z_{K}\T (A_{f}))Φ^{T/1}(t, ϕ^{𝔭}) × trace ξ(t) · u(T)\frac{meas Ũ_{𝔭}}{meas U_{𝔭}} .

The second is a product over v dividing 𝔭 of terms that once again depend only on the image of t in T_{v}(Q_{𝔭}). If T_{v} is split the first factor is 0 unless k_{v} is even, but then it is

𝔭^{je𝔪_{v}/2}\binom{k_{v}}{k_{v}/2}Θ_{ν′_{v}}(t)\frac{|t_{1}t_{2}|_{v}^{1/2}}{|t_{1} − t_{2}|_{v}} , ν′_{v} = \frac{je}{𝔫_{v}} ν_{v} .

If T_{v} is not split, the factor is the same as in (c).

3. The zeta-function.

The next step in the proof is to use the structure of the set of geometric oints on S(K) over the algebraic closure κ̃_{𝔭} of the residue field κ_{𝔭} of E at 𝔭 to obtain a formula for

log Z_{𝔭}(s, S(K), F_{ξ})

which can be compared with the formulae of the preceding paragraph. This logarithm has a power series expansion

∑_{j=1}^{∞} M(j)\frac{|ϖ_{𝔭}|^{js}}{j}

with

M(j) = ∑_{i}(−1)^{i}trace τ^{i}(Φ) ,

and we shall be concerned only with the coefficients M(j).

According to [13] the set S(K, κ̃_{𝔭}) is a union of subsets indexed by equivalence classes of Frobenius pairs (γ, h^{0}). These must be described. Recall that we have supposed that the totally real field F used to define G is imbedded in \overline{Q} ⊆ C. The set of imbeddings of F in \overline{Q} or C is then represented by the finite homogeneous space

Q = 𝔊(\overline{Q}/F)\𝔊(\overline{Q}/Q) .

To nourish our intuition we wrote this set as

×× …×◯ … ◯× .

The infinite places at which the algebra splits are marked by a cross; the others are not marked, but represented by circles. We assumed that F is unramified at 𝔭. Hence the Frobenius Φ_{𝔭} acts, and we decomposed the set into its orbits.

\underbrace{× … ◯ …×}_{Q_{v_{1}}} ; \underbrace{◯ …× … ◯}_{Q_{v_{2}}} ;

let 𝔫_{v} be the number of elements in the orbit labelled by v. Then

∑ 𝔫_{v} = 𝔫 = [F : Q] .

To be definite we viewed the Frobenius as acting on each orbit by a cyclic shift to the right. We also let 𝔪_{v} be the number of marked points in the orbit, and observed that the orbits or the indices v also label the primes of F dividing 𝔭.

There are two kinds of Frobenius pairs, or rather of equivalence classes of such pairs. To describe one of the first type we start from a totally imaginary quadratic extension L of F which splits at least one prime of F dividing 𝔭. We choose a set of representatives for the isomorphism classes over F of such fields. Each L determines a stable conjugacy class of Cartan subgroups of G over Q. Choose a representative T_{L} for each such class. T_{L} is contained in a unique Cartan subgroup T̃_{L} of G̃_{1}. it will be less taxing on my powers of abstraction if I fix an isomorphism of T̃_{L} with the algebraic group over Q defined by the multiplicative group of L. In particular T̃_{L}(Q) will be identified with L^{×}, and every imbedding of L in \overline{Q} defines a rational character of T̃_{L} and as such it is a basis of the lattice of all rational characters. The dual basis of the lattice of coweights is also indexed by P.

We fix, to have a point of reference, h_{L}: R → T_{L} which is conjugate under G(R) to the h defining the Shimura variety. Suppose g ∈ 𝔄(T/Q). Let

T = T_{L}^{g} = g^{−1}T g .

An h^{0}: R → T must be of the form

h_{L}^{x}: r → x^{−1}h_{L}(r)x, x ∈ G(R) .

The diagram

is commutative. Since 𝔇(T/F) → 𝔇(T/R) is surjective we may by a suitable choice of x and g arrange that h_{L}^{xg^{−1}} is any element in the orbit of h_{L} under the Weyl group of T_{L}(C) in G(C). However, h_{L} itself can only vary within an orbit of the Weyl group of T (R) in G(R).

As in [13] h^{0} defines a coweight μ^{∨} of T. Since g has been fixed for now, we may pull μ^{∨} back to a coweight of T_{L} and hence of T̃_{L}, which I again denote by μ^{∨}. It may be written in terms of the dual basis we have chosen. The coefficients will be 0 or 1. It will help to have a pictorial way of representing these coefficients.

Let P_{v} be the inverse image of Q_{v} in P. The group 𝔊(\overline{Q}_{𝔭}/Q_{𝔭}) acts on P_{v}. If v splits in L then P_{v} falls into two orbits and we represent the map P_{v} → Q_{v} by the diagram:

P_{v} \{ \begin{matrix}× & ◯ & … & ◯ & … & × \\ ◯ & × & … & ◯ & … & × \end{matrix}

↓
Q_{v} ×× … ◯ …× .

Each horizontal line represents an orbit. A point is marked by an × if the coefficient of the corresponding element of the dual basis is 1 and is left unmarked if the corresponding coefficient is 0. Above a marked point of Q_{v} there is one marked and one unmarked point, and above an unmarked point there is no marked point. let 𝔪′_{v} and 𝔪′′_{v} be the number of marked points in the two orbits. Then

𝔪′_{v} + 𝔪′′_{v} = 𝔪_{v} .

If v does not split in L we may still represent P_{v} → Q_{v} as

\{ \begin{matrix}◯ & … & × & … & ◯ \\ ◯ & … & ◯ & … & × \end{matrix}
↓
◯ …× …× .

However there is only one orbit in P_{v} and no significance is to be attached to the two horizontal rows inits pictorial representation.

Starting from L, g, and h^{0} we set about defining Frobenius pairs (γ, h^{0}) of the first kind. We must suppose that 𝔪′_{v} ≠ 𝔪′′_{v} for at least one v with v split. If k_{𝔭} is a sufficiently large but finite Galois extension of Q_{𝔭} we set

Nm_{k_{𝔭}/Q_{𝔭}}μ^{∨} = ν^{∨}

and write ν^{∨} as an integer linear combination of the elements of the dual basis. If l = [k_{𝔭} : Q_{𝔭}] the coefficient of a coweight in P_{v} is l𝔪_{v}/2𝔫_{v} if v does not split in L. If v splits the coefficient is l𝔪′_{v}/𝔫_{v} in the first row and l𝔪′′_{v}/𝔫_{v} in the second.

We first construct an appropriate γ_{1} in T_{L}(Q) ⊆ L^{×} and then take γ = g^{−1}γ_{1}g. If v splits let 𝔞_{v} and 𝔟_{v} be the two prime ideals in L dividing it. If it does not let 𝔤_{v} be the ideal of L generated by the prime ideal in F defined by the valuation v. Set

𝔞 = (∏ 𝔤_{v}^{𝔪_{v}}) (∏ 𝔞_{v}^{2𝔪′_{v}}𝔟_{w}^{2𝔪′′_{v}}) .

Some power 𝔞^{s} of 𝔞 is principal. Let

𝔞^{s} = (δ_{1}) .

δ_{1} lies in T̃_{L}(Q). Recall that G was defined as the inverse image of A in Res_{F/Q}GL(1). If B is the quotient of these two groups we have injections

T_{L}(Q)\T̃_{L}(Q) ↪ B(Q) A(Q)\F^{×} ↪ B(Q) .

Let δ̄_{1} be the image of δ_{1}. Since μ^{∨} and ν^{∨} are coweights of T_{L}, χ(δ̄_{1}) is a unit for all weights of B and some power δ̄_{1}^{t} of δ̄_{1} is the image of a unit β in F^{×}. Set

γ_{1} = δ_{1}^{2t}/β .

The pair (γ, h^{0}) is then of Frobenius type.

Suppose \overline{γ} lies in T and (\overline{γ}, h^{0}) is also of Frobenius type. Let

\overline{γ} = g^{−1}\overline{γ}_{1}g .

By the definition of pairs of Frobenius type there are positive integers c and d for which

|\overline{γ}_{1}^{d}|_{v} = |γ_{1}^{c}|_{v}

at every finite place v of L. Thus γ_{1}^{−c} \overline{γ}_{1}^{d} is a unit. Since L is a totally imaginary extension of F we may even arrange, by multiplying c and d by a common factor, that γ_{1}^{−c} \overline{γ}_{1}^{d} is a unit in F. Consequently (γ, h^{0}) and (\overline{γ}, h^{0}) are equivalent, and the critical data for the construction of a pair of the first kind are L, g, and h^{0}, or, if one prefers,h_{L}^{xg^{−1}}. Actually, h_{L}^{xg^{−1}} is determined by μ^{∨}, regarded as a coweight of T_{L}, and we use μ^{∨} rather than h^{0}. It is also clear that the class of (γ, h^{0}) depend sonly on the image δ of g in 𝔇(T/Q), and it is finally most convenient to take L, δ, and μ^{∨} as the fundamental data. I note in passing that, because 𝔪′_{v} ≠ 𝔪′′_{v} for some i, no power of γ is central and the group H^{0} of [13], now denoted by I^{0}, is T.

The use of the symbol H in [13] conflicts with its use in [7]. Since the construction of those two papers appear simultaneously in the study of Shimura varieties, it will be best to use I^{0} and I for the groups denoted H^{0} and H in [13]. It will also be best to denote the groups \overline{G}^{0} and \overline{G} of that paper by J^{0} and J, and not overburden the letter G.

Not every pair δ, μ^{∨} can arise, and it may save us some confusion if we describe now the relation between δ and μ^{∨} that must be satisfied. Observe first that μ^{∨} must lie in the orbit of μ_{L}^{∨} under the Weyl group if μ_{L}^{∨} is the coweight associated to h_{L}. Since the image of T_{L} is anisotropic, μ_{L}^{∨} − μ^{∨} defines an element of H^{−1}(𝔊(C/R), X_{∗}(T_{L})) and hence, by the Tate-Nakayama theory, an element α_{∞}(μ^{∨}) of H^{1}(R, T_{L}). On the other hand δ lies in 𝔇(T_{L}/Q) which may be mapped to 𝔇(T_{L}/R) ⊆ H^{1}(R, T_{L}). The condition is that the image of δ is α_{∞}(μ^{∨}). To verify this we observe that the image of δ is the class of H^{1}(R, T_{L}) corresponding to

μ_{L}^{∨} − gx^{−1}(μ_{L}^{∨}) = μ_{L}^{∨} − μ^{∨} .

Lemma 3.1. Suppose L, T_{L}, and h_{L} are given. Then L, δ, μ^{∨} are possible data for the construction of a Frobenius pair if and only if α_{∞}(μ^{∨}) is the image of δ in 𝔇(T/R).

The necessity has just been verified. On the other hand if g represents δ, the image of δ is α_{∞}(μ^{∨}), and μ^{∨} = ω(μ_{L}^{∨}), with ω in the Weyl group over C, then there is an x in G(R) such that gx^{−1} normalizes T and represents ω. We then define h^{0} by

h^{0}(r) = x^{−1}h_{L}(r)x .

We next describe sufficient conditions for the class of pairs associated to L, δ, and μ^{∨} to be the same as that associated to L, δ̄, and μ̄^{∨}. We continue to assume that the pairs are of the first kind. Let Ω(T_{L}, G; Q) be the quotient of the intersection of 𝔄(T/Q) with the normalizer of T by T (\overline{F}). It is a group and may be properly larger than Ω(T_{L}(Q), G(Q)), the Weyl group over Q. It acts on 𝔇(T_{L}/Q) to the right, and in the present circumstances consists of two elements, corresponding to the two automorphisms of L over F.

Lemma 3.2. Suppose δ, μ^{∨} and δ̄, μ̄^{∨} are given, and yield classes of Frobenius pairs of the first kind. These classes are the same if the following conditions are satisfied:

(i) There is an ω in Ω(T_{L}, G; Q) such that

Nm_{k_{𝔭}/Q_{𝔭}}(μ̄^{∨} − ω(μ^{∨})) = 0

if k_{𝔭} is any sufficiently large Galois extension of Q_{𝔭}.

(ii) If (i) is satisfied then μ̄^{∨} − ω(μ^{∨}) defines an element of H^{−1}(𝔊(k_{𝔭}/Q_{𝔭}), X_{∗}(T_{L})) and hence, by the Tate-Nakayama theory, an element α_{𝔭}(μ̄^{∨}, ω(μ^{∨})) of H^{1}(Q_{𝔭}, T_{)}. The equation

ωδα_{𝔭}(μ̄^{∨}, ω(μ^{∨})) = δ̄

is to obtain.

(iii) If l is a finite prime different from 𝔭 then ωδ and δ̄ have the same image in 𝔇(T/Q_{l}).

Let δ and δ̄ be represented by g and \overline{g}. Because of the first condition we may take

γ = g^{−1}γ_{1}g, \overline{g}^{−1}ω(γ_{1})\overline{g},

with a common γ_{1} in T_{L}(Q). If ω is represented by w the third condition implies that for l ≠ 𝔭

\overline{g} = twgu, t ∈ T_{L}(\overline{Q}_{l}), u ∈ G(Q_{l}),

and, consequently, that

\overline{γ} = u^{−1}γu .

There is one more condition to be verified if equivalence is to be established. It is a condition on b, the element associated to (γ, h^{0}) in [13] and discussed at length in the appendix of this paper, and on \overline{b}, the element associated in the same way to (\overline{γ}, h̄^{0}). The elements γ and \overline{γ} are conjugate over G(\overline{Q}) and hence, by the corollary on p. 170 of [18] over G(Q_{𝔭}^{un}). Let

\overline{γ} = u^{−1}γu u ∈ G(Q_{𝔭}^{un}) .

In order to establish equivalence we must show that there is a t in T (𝔨) for which

\overline{b} = t^{−1}u^{−1}bσ(u)σ(t)

if σ is the Frobenius on Q_{𝔭}^{un} or 𝔨.

Since no power of γ_{1} is central

\overline{g} = swgu, s ∈ T_{L}(Q_{𝔭}^{un}) .

Thus δ̄ is represented by the cocycle

\{τ(w)τ(g)τ(u)u^{−1}g^{−1}w^{−1}\}, τ ∈ 𝔊(\overline{Q}_{𝔭}/Q_{𝔭}),

and

δ̄ = ωδβ

in 𝔇(T/Q_{𝔭}) if

β = wgτ(u)u^{−1}g^{−1}w^{−1} .

Notice that β is represented by the inflation of a cocycle of 𝔊(Q_{𝔭}^{un}/Q_{𝔭}) and is completely determined by

wgσ(u)u^{−1}g^{−1}w^{−1}a_{σ} .

It is a consequence of the definitions and the corollary on p. 170 of [18] that, after multiplying b and \overline{b} by yσ(y^{−1}) and \overline{y}σ(\overline{y}^{−1}), y ∈ T (𝔨), \overline{y} ∈ \overline{T} (𝔨), we may suppose that α_{𝔭}(μ̄^{∨}, ω(μ^{∨})) is represented by the inflation of a cocycle {α_{τ} } of 𝔊(Q_{𝔭}^{un}/Q_{𝔭}) and that

\overline{g}\overline{b}\overline{g}^{−1} = wgbg^{−1}w^{−1}a_{σ} .

Thus

u\overline{b}u^{−1} = bg^{−1}w^{−1}a_{σ}wg .

The second assumption is that

β ∼ α_{𝔭}(μ̄^{∨}, ω(μ^{∨})) .

Thus we may even assume that

g^{−1}w^{−1}a_{σ}wg = σ(u)u^{−1} .

Then

\overline{b} = u^{−1}bσ(u) .

Lemma 3.3. Suppose the classes of Frobenius pairs associated to L, δ, μ^{∨} and L̄, δ̄, μ̄^{∨} are of the first kind and the same. Then L = L̄ and the conditions of the previous lemma are fulfilled.

If the classes are the same then L ⊗ F_{v} and L̄ ⊗ F_{v} are isomorphic for all finite places v of F which do not divide 𝔭. Consequently L and L̄ are isomorphic. Since we are working within a set of representatives for the isomorphism classes of quadratic extensions, L = L̄ and T_{L} = TL̄.

Let δ and δ̄ be represented by g and \overline{g} and let (γ, h^{0}) and (\overline{γ}, h^{0}) be corresponding Frobenius pairs. We suppose they are equivalent. Replacing γ and \overline{γ} by appropriate powers of themselves and perhaps multiplying by an appropriate central element as well, we may suppose they are conjugate in G(Q_{l}) for l ≠ 𝔭 and hence in G(Q). As usual let

γ = g^{−1}γ_{1}g \overline{γ} = \overline{g}^{−1}\overline{γ}_{1}\overline{g} .

Then γ_{1} and \overline{γ}_{1} are also conjugate in G(Q) and there is an ω in Ω(T_{L}, G, Q) such that

\overline{γ}_{1} = ω(γ_{1}) .

Recall that if λ is a rational character of T_{L} then

|λ(γ_{1})| = |ϖ|^{τ⟨λ,ν^{∨}⟩}

and

|λ(γ_{2})| = |ϖ|^{τ⟨λ,ν̄^{∨}⟩} .

Here r and \overline{r} are two positive rational numbers and

ν^{∨} = Nm_{k_{𝔭}/Q_{𝔭}}μ^{∨},  ν̄^{∨} = Nm_{k_{𝔭}/Q_{𝔭}} μ̄^{∨} .

Consequently

rν̄^{∨} = \overline{r}ω(ν^{∨}) .

Since the sum of the coefficients in the expression of ν^{∨} as a linear combination of the elements in the dual basis is the same as the sum of the coefficients for ν̄^{∨}, r = \overline{r} and

Nm_{k_{𝔭}/Q_{𝔭}} μ̄^{∨} = Nm_{k_{𝔭}/Q_{𝔭}}ω(μ^{∨}) .

This is the first condition of the previous lemma. To verify the second we have only to observe that if ω is represented by w then

γ = g^{−1}γ_{1}g = g^{−1}w^{−1}\overline{γ}_{1}wg

and

\overline{γ} = \overline{g}^{−1}\overline{γ}_{1}\overline{g}

are conjugate in G(Q_{l}), and hence

wg = t\overline{g}u, t ∈ T (\overline{Q}_{l}), u ∈ G(Q_{l}) .

As in the proof of the previous lemma

\overline{g} = swgu, s ∈ T_{L}(Q_{𝔭}^{un}), u ∈ G(Q_{𝔭}^{un}),

and

\overline{γ} = u^{−1}γu .

Since γ is not central, u is determined modulo T (Q_{𝔭}^{un}) and the class of the cocycle in T_{L}

{wgτ(u)u^{−1}g^{−1}w^{−1} | τ ∈ 𝔊(Q_{𝔭}^{un}/Q_{𝔭})}

is well-defined. As we observed before

δ̄ = ωδβ .

Moreover we may once again suppose that α_{𝔭}(μ̄^{∨}, ω(μ^{∨})) is obtained by inflating a cocycle {α_{τ} } of 𝔊(Q_{𝔭}^{un}/Q_{𝔭}) and that

u\overline{b}u^{−1} = bg^{−1}w^{−1}a_{σ}wg .

Since we are assuming the two Frobenius pairs equivalent, there is a t in \overline{T} (𝔨) for which

\overline{b} = t^{−1}u^{−1}bσ(u)σ(t)

and

(ut^{−1}u^{−1})bσ(u)u^{−1}(uσ(t)u^{−1}) = bg^{−1}w^{−1}a_{σ}wg .

Canceling b and taking

z = wgutu^{−1}g^{−1}w^{−1},

we obtain

zwgσ(u)u^{−1}g^{−1}w^{−1}σ(z^{−1}) = a_{σ} .

Since {a_{τ} } and {τ(u)u^{−1}} are both continuous cocycles, it follows readily from this equation that z ∈ T_{L}(Q_{𝔭}^{un}) and that

β = α_{𝔭}(μ̄^{∨}, ω(μ^{∨})) .

We introduce next another type of Frobenius pair, which we will say is of the second kind. We again start from the totally imaginary quadratic extension L of F, the Cartan subgroup T_{L}, an element g in 𝔄(T/Q), and an h^{0}, but suppose that 𝔪′_{v} = 𝔪′′_{v} for every place v of F dividing 𝔭 and splitting in L. Such data exist, for we may so choose L that no place dividing 𝔭 splits in it. We construct γ as before. The ideal 𝔞 is now an ideal in F, and so some power of γ is central.

Lemma 3.4. Any two Frobenius pairs of the second kind are equivalent.

Suppose (γ, h^{0}) and (\overline{γ}, h̄^{0}) are two pairs of the second kind. We take γ and \overline{γ} central. Choosing k_{𝔭} so large that it splits both T_{L} and TL̄ and noting that

𝔪′_{v} = 𝔪′′_{v} = 𝔪̄′_{v} = 𝔪̄′′_{v} = 𝔪_{v}/2

we see easily that there are two positive integers c and d and a unit ξ in F^{×} for which

ξγ^{c} = \overline{γ}^{d} .

Replacing γ by ξγ^{c} and \overline{γ} by \overline{γ}^{d} we may suppose γ = \overline{γ}. The equivalence now follows from the fact that b is well-defined (cf. Lemma A.2).

It is clear that a pair of the second kind cannot be equivalent to a pair of the first kind. With the following lemma, which is really a matter of definition, our classification of Frobenius pairs is complete.

Lemma 3.5. Every Frobenius pair is equivalent to one of the first or second kind.

The definition of (γ, h^{0}) involves the introduction of a subgroup I^{0} of G [13], and the choice of a Cartan subgroup T of I^{0} which is defined over Q and through which h^{0} factors. Replacing γ by a conjugate over Q if necessary, we may suppose that, for one of the representatives L and some g in 𝔄(T/Q),

T = g^{−1}T_{L}g .

If we also suppose, as we may, that the image of T in I_{ad}^{0} is anisotropic over Q_{𝔭} then, by the very definitions, (γ, h^{0}) must be a Frobenius pair associated to L, g, and h^{0}.

In addition to the group I^{0} there is a group I over Q and groups J^{0} and J over Q_{𝔭} attached to a Frobenius pair (recall that J^{0} and J are denoted by \overline{G}^{0} and \overline{G} in [13]). I is an inner twisting of I^{0}. For a pair of the first kind, I^{0} is T and the twisting is trivial. For a pair of the second kind I^{0} is G. An inner twisting of G is obtained from an inner twisting of Res_{F/Q}G̃ or a twisting of the quaternion algebra D defining G̃. Thus for pairs of the second kind, I will be defined by the same subgroup A of Res_{F/Q}GL(1) and a new quaternion algebra D′. According to the prescription for passing from I^{0} to I we are not to twist away from infinity and 𝔭, but D′ must be ramified at every infinite place. Since the number of infinite places at which D splits is

𝔪 = ∑_{v} 𝔪_{v}

the invariant will be changed at 𝔪 infinite places. Once we decipher the prescription given in [13] for the twisting at 𝔭, we will see that the invariant of D at v is to be changed if and only if 𝔪_{v} is odd. In particular, the total number of places at which the invariant is to be changed is even, and the prescription can actually be carried out.

Over Q_{𝔭}

Res_{F/Q}G̃ = ∏_{v} Res_{F_{v}/Q_{𝔭}} G̃ = ∏_{v} G^{v} .

The centralizer of T in this group is a product ∏ T^{v}, and if we regard μ^{∨} as a coweight of the product it may be factored as

∏ μv^{∨} (multiplicative notation) .

The cocycle which defines the twisting is also a product and the v^{th} factor lifts to the cochain

σ → a_{σ}^{v} = ∏_{τ∈𝔊(k_{𝔭}/Q_{𝔭})} a_{σ,τ}^{στμ^{∨}}

in G^{∨}. Here {a_{σ,τ} } is a representative of the fundamental class of the extension k_{𝔭}/Q_{𝔭}.

A straightforward calculation shows that

H^{1}(Q_{𝔭}, G_{ad}^{v}) ≃ H^{1}(F_{v}, G̃_{ad}) .

The isomorphism is obtained by writing G_{ad}^{v} over \overline{Q}_{𝔭} as a product

∏_{𝔊(\overline{Q}_{𝔭}/Q_{𝔭})/𝔊(\overline{Q}_{𝔭}/F_{v})} ^{σ}G̃_{ad},

and then restricting a cocycle of 𝔊(\overline{Q}_{𝔭}/Q_{𝔭}) with values in G_{ad}^{v} to 𝔊(Q_{𝔭}/F_{v}) and projecting to the factor 1G̃_{ad}. We also have the familiar imbedding

H^{1}(F_{v}, G̃_{ad}) ↪ H^{2}(𝔊(Q_{𝔭}/F_{v}), Q_{𝔭}) .

We have to show that, starting from the given element in H^{1}(Q_{𝔭}, G_{ad}^{v}), we finish with the element of H^{2}(𝔊(\overline{Q}_{𝔭}/F_{v}), \overline{Q}_{𝔭}) with invariant 𝔪_{v}/2. We may restrict and project before or after taking the co-boundary of {a_{σ}^{v}}, and it is convenient to take the co-boundary first. We obtain

{∏_{τ} a_{ρ,τ}^{ρτμ_{v}^{∨}}} {∏_{τ} ρ(a_{σ,τ})^{ρστμ_{v}^{∨}} ∏_{τ} a_{ρσ,τ}^{−ρστμ_{v}^{∨}}} .

The products run over τ ∈ 𝔊(k_{𝔭}/Q_{𝔭}). substituting στ for τ in the first factor and using the co-cycle relation

a_{ρ,σ} = ρ(a_{σ,τ})a_{ρ,στ} a_{ρσ,τ}^{−1}

we see that this co-boundary equals

{a_{ρ,σ}^{ν_{v}^{∨}}}

with

ν_{v}^{∨} = Nm_{k_{𝔭}/Q_{𝔭}}μ_{v}^{∨} .

The invariant of the restriction of {a_{ρ,σ}} to 𝔊(k_{𝔭}/F_{v}) is

𝔫_{v}/[k_{𝔭} : Q_{𝔭}] ,

and the invariant of the image of the composite homomorphism is therefore

\frac{𝔪_{v}[k_{𝔭} : Q_{𝔭}]}{2𝔫_{v}} · \frac{𝔫_{v}}{[k_{𝔭} : Q_{𝔭}]} = \frac{𝔪_{v}}{2} .

For the pairs of the second kind J^{0} = I^{0} and J = I. For pairs of the first kind J^{0}, a group over Q_{𝔭} is the inverse image of A in

∏ J^{v}

where J^{v} is T^{v} if v splits in L and 𝔪′_{v} ≠ 𝔪′′_{v} and is G^{v} otherwise. J is obtained in the same manner with D′ replacing D.

There is also a space X and a multiplicity d to associate to a pair (γ, h^{0}) or, rather, to a class of such pairs. For the moment we ignore the multiplicity and forego a detailed analysis of the space X. The group K is taken to be a product K^{𝔭}K_{𝔭} with K^{𝔭} ⊆ G(A_{f}^{𝔭}) and K_{𝔭} a special compact subgroup of G(Q_{𝔭}), that is, the stabilizer of a special vertex in the Bruhat-Tits building. X depends on K_{𝔭}. The set of points in S(κ̄_{𝔭}) corresponding to the class of (γ, h^{0}) is formally d copies of

Y_{K} = H(Q)\G(A_{v}^{𝔭}) × X/K^{𝔭} .

The group K^{𝔭} acts on the right through its action on G(A_{f}^{𝔭}). I(Q) acts on both factors, and the Frobenius Φ_{𝔭} acts on Y through its action on X.

As in [12] we use the Lefschetz fixed point formula to compute the alternating sum of the traces of Φ_{𝔭}^{j}, j > 0, on the cohomology of F_{ξ}. We take the sum over the fixed points of the traces on the fibers. The fibers over Q_{l} are obtained by first taking the fibers of the sheaves over Z/l^{k}Z, then letting k → ∞ to obtain fibers over Z_{l}, and then tensoring with Q_{l}. Since the fixed points lie in S(κ̄_{𝔭}) = S_{K}(κ̄_{𝔭}), the only thing that really matters is the resulting sheaf over S(κ̄_{𝔭}), a set with the discrete topology, and the action of Φ_{𝔭} on it.

We first look at the points in Y_{K}, and find a formula for their contribution to the alternating sum. If \overline{K} = \overline{K}^{𝔭}K_{𝔭} with \overline{K}^{𝔭} ⊆ K^{𝔭} then the inverse image of Y_{K} in S\overline{K}(κ̄_{𝔭}) is

Y\overline{K} = H(Q)\G(A_{v}^{𝔭}) × X/ \overline{K}^{𝔭} .

The map is the obvious one. Consequently [12] the sheaf over Y_{K} is

(H(Q)\G(A_{f}^{𝔭}) × X) ×_{K^{𝔭}} V (Q_{l}) .

Here K^{𝔭} acts on V (Q_{l}) through its projection on G(Q_{l}). The action of Φ_{𝔭} on the sheaf is obtained by letting it act on X.

The point y ∈ Y_{K} represented by (g, x) is fixed by Φ_{𝔭}^{j} if

(g, Φ_{𝔭}^{j}x) = (hgk, hx), k ∈ K^{𝔭}, h ∈ I(Q) .

There are really two equations here

Φ_{𝔭}^{j}x = hx  and  k = g^{−1}h^{−1}g .

The map from the fiber at Φ_{𝔭}^{j} to the fiber at y is

(g, Φ_{𝔭}^{j}x) × v → (g, x) × v .

At a fixed point,

(g, Φ_{𝔭}^{j}x) × v = (hgk, hx) × v = (g, x) × kv

and the trace is

trace ξ(k^{−1}) = trace ξ(h) .

There is a lemma to be proved before we can find an expression for the contribution of the points on Y_{K} to the alternating sum of the traces. If h_{1}, h_{2} lie in I(Q) and k_{1}, k_{2} lie in K^{𝔭}, the equation

(h_{1}gk_{1}, h_{1}x) = (h_{2}gk_{2}, h_{2}x)

is equivalent to the two equations

h_{1}^{−1}h_{2} = gk_{1}k_{2}^{−1}g^{−1}  and  h_{1}^{−1}h_{2}x = x .

Since the center Z of G is contained in I^{0} and I is obtained from I^{0} by an inner twisting, Z is also a subgroup of I.

Lemma 3.6 There is an open compact subgroup K_{0} of G(A_{f}) such that if K ⊆ K_{0} then for any Frobenius pair the equations

h = gkg^{−1}, hx = x,

with h ∈ I(Q), g ∈ G(A_{f}^{𝔭}), k ∈ K^{𝔭}, x ∈ X imply that h lies in Z.

We may as well divide by Z, and consequently suppose that Z is {1}. Since I(R) is compact, h is semisimple. Let it lie in the torus T over Q. I claim that if λ is a rational character of T and v any valuation of \overline{Q}, then

|λ(h)|_{v} = 1 .

If v is archimedean, this is a consequence of the compactness of I(R). If v is non-archimedean but prime to 𝔭, it is a consequence of the first of the assumed equations. If v divides 𝔭, it is a consequence of the second assumed equation and the definition of X [13]. We conclude that λ(h) is a root of unity. Since λ(h) lies in a Galois extension whose degree is at most the product of the order of the Weyl group of G with the order of the group of automorphisms of the Dynkin diagram, it is one of a fixed finite set of roots of unity. We have merely to take K_{0} sufficiently small that the ensuing congruence conditions force it to be 1.

We assume henceforth that K ⊆ K_{0}. There is another vexatious possibility with which I would prefer not to have to deal, simply because it encumbers the notation. Suppose h ∈ I(Q), g ∈ I(Q) and g^{−1}hg = zh with z ∈ (Q) ∩ K. Then z certainly must lie in the center of the derived group which is finite. I take K_{0} so small that this equation implies z = 1.

If z ∈ Z(Q) ∩ K then (zg, zx) = (gz, x). If

(g, Φ_{𝔭}^{j}x) = (hgk, hx)

and h_{1} ∈ I(Q), k_{1} ∈ K′′, then (h_{1}gk_{1}, Φ_{𝔭}^{j}h_{1}x) = ((h_{1}hh_{1}^{−1})h_{1}gk_{1}(k_{1}^{−1}kk_{1}), (h_{1}hh_{1}^{−1})h_{1}x) .

Thus to each fixed point in Y_{K} is associated a conjugacy class {h} in Z(Q) ∩ K\I(Q), and if N^{j}(h) is the number of fixed points yielding the conjugacy class {h} then the total contribution of Y_{K} to the alternating sum of the traces is

∑_{(h)} N^{j}(h)trace ξ(h) .
If x ∈ X and j > 0, set
T_{x}^{j} = {g ∈ J(Q_{𝔭}) | Φ_{𝔭}^{j}x = gx} .

Lemma 3.7. If h lies in I(Q) and in T_{x}^{j}, j > 0, then the centralizer I(h, Q_{𝔭}) of h in I(Q_{𝔭}) is the same as its centralizer J(h, Q)_{𝔭}) in J(Q_{𝔭}).

The proof of this lemma will have to be postponed until we have examined the sets T_{z}^{j} more closely. Let ψ_{x}^{j} be the characteristic function of T_{z}^{j}, and if {x} is a set of representatives for the orbits of J(Q_{𝔭})

in X, set

φ^{j}(h) = ∑_{\{x\}} \frac{1}{meas J(x)} ∫_{J(h,Q_{𝔭})\J(Q_{𝔭})} ψ_{x}^{j}(g^{−1}hg)dg .

Here h is an arbitrary element of J(Q_{𝔭}) and J(x) is the stabilizer of x in J(Q_{𝔭}). We shall eventually see that the integrals are finite and that, for each j, all but finitely many of the ψ_{x}^{j} are identically zero.

Lemma 3.8. Suppose ϕ^{𝔭} is the characteristic function of K^{𝔭} divided by its measure, and let

Z_{K} = Z(A_{f}) ∩ K .

Then N^{j}(h) is equal to

\frac{meas(Z_{K}I(h, Q)\I(h, A_{f}))φ^{j}(h)}{meas Z_{K}} ∫_{I(h,A_{f}^{𝔭})\G(A_{f}^{𝔭})} ϕ^{𝔭}(g^{−1}hg)dg .

This lemma is a more general form of some of the lemmas in §5 of [12]. The number N^{j}(h) is equal to the sum over i of

∑_{h_{1}} ∑_{(g,\overline{g})} meas K^{𝔭}ϕ(g^{−1}h_{1}g)ψ_{x_{i}}^{j}(\overline{g}^{−1}h_{1}\overline{g}) .

Here h_{1} runs over the conjugates of h in I(Q) modulo Z(Q) ∩ K and (g, \overline{g}) runs over

I(Q)\G(A_{f}^{𝔭}) × J(Q_{𝔭})/K^{𝔭} × J(x_{i}) .

We may drop the sum of h_{1} if we divide on the left, not by I(Q), but by I(h, Q). Since

Z_{K}I(Q) ∩ (gK^{𝔭}g^{−1} × \overline{g}J(x_{i})\overline{g}^{−1}) = Z_{K},

we may also replace the sum by an integral over Z_{K}I(h, Q)\G(A_{f}^{𝔭}) × J(Q_{𝔭}) provided we multiply by

meas Z_{K}/(meas K^{𝔭})(meas J(x_{i})) .

The integration may be taken first over Z_{K}I(h, Q)\I(h, A_{f}) and then over

I(h, A_{f}^{𝔭})\G(A_{f}^{𝔭}) × I(h, Q_{𝔭})\J(Q_{𝔭}) .

Appealing to Lemma 3.7, we replace the denominator in the second factor by J(h, Q_{𝔭}). The first integration simply yields a factor

meas(Z_{K}I(h, Q)\I(h, A_{f})) .

The lemma follows.

We must next consider the multiplicity d attached to a Frobenius pair. I have first to confess that the multiplicity suggested in [13] is not quite correct. It was suggested that it could be incorrect because insufficiently many examples had been studied, and that is still a possibility, but the error to be mentioned now arises from a difference source, a misinterpretation of my calculations for the special cases. Such mistakes — another is correction [14] — must be annoying to anyone who is seriously attempting to understand this sequence of papers. I can only apologize and assure him that they do not seem serious. I hope to have a fairly thorough discussion of the conjectures and the examples available sooner or later. It appears to be safe for now to take d to be the number of elements in H^{1}(Q, I) which become trivial in H^{1}(Q_{v}, I) for all places v of Q and which have trivial image in H^{1}(Q, G_{der}\G). In [13] triviality in H^{1}(Q_{𝔭}, I) was not demanded. I observe as well that in [13] a rather eccentric notation was employed. The set H^{1}(𝔊(\overline{Q}/Q), I(\overline{Q})) was denoted by H^{1}(\overline{Q}, I) and not by H^{1}(Q, I).

With this definition of d, we have the following simple lemma.

Lemma 3.9 (a) For Frobenius pairs of the second kind d = 1.

(b) For Frobenius pairs of the first kind d is equal to μ(T), if T = I^{0}, a Cartan subgroup, and μ(T) is the order of the kernel of 𝔈(T/Q) → 𝔈(T/A).

For a pair of the second kind,

G_{der}\G ≃ I_{der}\I

and I_{der} is a simply-connected group. Thus it follows from the Hasse principle or, more directly, from the fact that H^{1}(Q, I_{der}) = 1 that d = 1. For pairs of the first kind, T = I^{0} = I and the kernel of

H^{1}(Q, T) → H^{1}(Q, G_{der}\G)

is 𝔈(T/Q). The lemma is verified.

In order to compare the alternating sum of the traces with the results of §2, we need to express it as the sum of a stable and labile part. We begin with the contribution from the Frobenius pairs of the first kind attached to a given totally imaginary quadratic extension of F.

Earlier we fixed T_{L} and μL^{∨} and, when g ∈ 𝔄(T/Q) and h^{0} were given, regarded μ^{∨} as a coweight of T_{L}. There is more than one possibility for

ν^{∨} = Nm_{k_{𝔭}/Q_{𝔭}}μ^{∨} .

Let ν1^{∨}, ν2^{∨}, … be the finitely many possibilities. For each of them we choose a μj^{∨} with norm νj^{∨} and a g_{j} ∈ 𝔄(T_{L}/Q) with image δ_{j} in 𝔇(T_{L}/Q) so that δ_{j} and μj^{∨} satisfy the condition of Lemma 3.1. It follows from the density of D(Q) in D(R) that g_{j} exists. Set T_{j} = g_{j}^{−1}T_{L}g_{j} and regard μj^{∨} now as a coweight of T_{j}. Rather than working with T_{L} and μL^{∨} we prefer now to work with T_{j},g ∈ 𝔄(T_{j}/Q), and those μ^{∨} for which

Nm_{k_{𝔭}/Q_{𝔭}}μ^{∨} = Nm_{k_{𝔭}/Q_{𝔭}}μj^{∨} .

However, when we come to assemble the contributions from the various T_{j}, we must divide by 2, the order of the group Ω(T_{L}, G; Q), because, by Lemma 3.2, νi^{∨} and νj^{∨} yield the same classes of Frobenius pairs if

νj^{∨} = ω(νi^{∨}), ω ∈ Ω(T_{j}, G; Q) .

If

Nm_{k_{𝔭}/Q_{𝔭}}μ^{∨} = Nm_{k_{𝔭}/Q_{𝔭}}μi^{∨}

then by the Tate-Nakayama theory μ^{∨} − μi^{∨} defines elements α_{𝔭}(μ^{∨}, μi^{∨}) and α_{∞}(μ^{∨}, μi^{∨}) in H^{1}(Q_{𝔭}, T_{i}) and H^{1}(Q_{∞}, T_{i}). By Lemma A.9 there is an element α(μ^{∨}, μi^{∨}) in H^{1}(Q, T_{i}) whose image in H^{1}(Q_{v}, T_{i}) is α_{∞}^{−1}(μ^{∨}, μi^{∨}) if v = ∞, α_{𝔭}(μ^{∨}, μi^{∨}) if v = 𝔭, and 1 otherwise. If g and μ^{∨} actually define a Frobenius pair then δα^{−1}(μ^{∨}, μi^{∨}) is trivial at ∞, and, by Lemma 3.2, the class of the Frobenius pair is determined by its local behavior at the finite places. Conversely Lemma 3.3 shows that, for a given i, the cohomology class of δα^{−1}(μ^{∨}, μi^{∨}) is determined locally by the Frobenius pair. To free ourselves of any ambiguity we first of all agree to take μ^{∨} = μi^{∨}, and then to choose g from a set of representatives {g} for those elements 𝔄(T_{i}/Q) which are trivial at ∞ modulo those which are trivial everywhere.

For such g set T_{i}^{g} = g^{−1}T_{i}g. A datum such as a measure may be transported from T_{i} toT_{i}^{g}. The part of the contribution to the alternating sum of the traces corresponding to T_{i} is the product of

(d/2) meas(Z_{K}T_{i}(Q)\T_{i}(A_{f}))/meas Z_{K}

and

∑_{{g}} ∑_{h∈T_{i}(Q)} φ_{g}^{j}(h) ∫_{T^{g}(A_{f}^{𝔭})\G(A_{f}^{𝔭})} ϕ(x^{−1}h^{g}x)dx . (3.1)

Here φ_{g}^{j} is φ^{j}, but for the Frobenius pair attached to g. I have not stressed this before, but it is understood that any term of this sum is zero if the first factor is zero, even when the second factor is infinite. It will eventually be clear that if the first factor is not zero, then the second factor is finite.

If 𝔈(T_{i}/A_{f}) is the set of elements in 𝔈(T_{i}/A) which are trivial at ∞, then

𝔈(T_{i}/A_{f})/𝔈(T_{i}/A_{f}) ∩ Im 𝔈(T_{i}/Q) ≃ 𝔈(T_{k}/A)/Im 𝔈(T_{i}/Q) .

Thus it should be possible to write (3.1) as a sum over the characters κ of 𝔈(T/A)/Im 𝔈(T/Q) of

[𝔈(T_{i}/A) : Im 𝔈(T_{i}/Q)]^{−1} ∑_{h} ∑_{{g}} κ(δ)φ_{g}^{j}(h^{g}) × ∫_{T_{i}^{g}(A_{f}^{𝔭})\G(A_{f}^{𝔭})} ϕ(x^{−1}h^{g}x)dx .

Here g runs over a set of representatives for 𝔈(T_{i}/A_{f}) and δ in 𝔈(T_{i}/A_{f}) is the image of g. However we do have to observe that in the definition of X and J it was not essential that g lie in 𝔄(T_{i}/Q). It need only lie in 𝔄(T_{i}/Q_{𝔭}). Thus φ_{g}^{j} is defined for g ∈ ∏_{v} 𝔄(T/Q_{v}) by its coordinate in 𝔄(T_{i}/Q_{𝔭}). We set

φ_{g}^{j}(h^{g}) = φ_{δ}^{j}(h) ,

sometimes taking δ in 𝔇(T_{i}/Q) and sometimes in 𝔇(T_{i}/Q_{𝔭}).

The inner sum may be written as the product of

Φ^{T_{i}/κ}(h) = ∑_{𝔈(T_{i}/A_{f}^{𝔭})} κ(δ)Φ^{δ}(h, f)

and

∑_{𝔈(T_{i}/Q_{𝔭})} κ(δ)φ_{δ}^{j}(h) .

The first of these two factors we have met before, and there is little to be said about it. It is the second which must be studied carefully.

Let 𝔨 be the completion of the maximal unramified extension of Q_{𝔭}. If b_{i} is the element of T_{i}(𝔨) associated to μi^{∨} by the procedure of the appendix, then the element b of T^{g}(𝔨) associated to μi^{∨} by the same procedure is b_{g} = g^{−1}b_{t}g. We recall the manner in which b_{g} is used to construct the space X [13]. To stress that it depends on g, I write X_{g} instead of X.

The group G(𝔨) is contained in G̃_{1}(𝔨), and if Q = 𝔊(\overline{Q}/F)\𝔊(\overline{Q}/Q) then

G̃_{1}(𝔨) = ∏_{Q} GL(2, 𝔨) .

Since F is unramified at 𝔭 the action of 𝔊(\overline{Q}_{𝔭}/Q_{𝔭}) on Q factors through 𝔊(Q_{𝔭}^{un}/Q_{𝔭}), which is also the group of continuous automorphisms of 𝔨 and contains σ as Φ_{𝔭}. To obtain the action of σ on G̃_{1}(𝔨) we choose a set of representatives \{τ\} for the cosets of Q and write τσ = d_{τ} (σ)τ′ with d_{τ} (σ) ∈ 𝔊(\overline{Q}/F). Then

σ : (g_{τ}) → (g′_{τ})

with g′_{τ} = d_{τ} (σ)(g′_{τ}). Observe that in fact d_{τ} (σ) is determined only modulo the inertial group of F ⊆ \overline{Q}_{𝔭}, but that does not matter, for d_{τ} (σ) is acting on g′_{τ} ∈ GL(2, 𝔨) in the usual way.

Let 𝔬_{𝔨} be the ring of integers in 𝔨. The element σ acts in the same way on the collection of (M_{i}), i ∈ Q, where, for each i, M_{i} is an 𝔬_{𝔨}-lattice in the space of column vectors of length two over 𝔨. Let (M_{i}^{0}) be a fixed point of σ and let K̃_{𝔭}(𝔨) be the stabilizer of (M_{i}^{0}) in G̃_{1}(𝔨). We may so choose (M_{i}^{0}) that if

K̃_{𝔭} = K̃_{𝔭}(𝔨) ∩ G̃_{1}(Q_{𝔭})

then

K_{𝔭} = K̃_{𝔭} ∩ G(Q_{𝔭}) .

Observe that G̃_{1}(Q_{𝔭}) is taken to be the set of points in G̃_{1}(𝔨) fixed by σ.

We introduce the set

𝔛 = G(𝔨)/K_{𝔭}(𝔨) ⊆ 𝔛̃ = G̃_{1}(𝔨)/K̃_{𝔭}(𝔨)

with

K_{𝔭}(𝔨) = K̃_{𝔭}(𝔨) ∩ G(𝔨) .

𝔛̃ is just the set of (M_{i}). the action of σ on 𝔛 is given by its action on G(𝔨) or by its action on the sequences (M_{i}). We introduce the transformation F_{v} of 𝔛 which sends 𝔠 to b_{g}σ(𝔠).

Let 𝔵 in 𝔛 be (M_{i}) and let 𝔜 = F_{g}𝔵 be (N_{i}). According to the definition of [13], supplemented by the correction in [14], the point 𝔵 lies in X_{g} if and only if the following two conditions are satisfied.

(i) M_{i} = N_{i} if i is an unmarked point. (ii) M_{i} ⊋ N_{i} ⊋ 𝔭M_{i} if i is a marked point.

It will be easier to make the comparison of the following paragraph if we can express

(3.2)  ∑_{𝔈(T_{i}/Q_{𝔭})} κ(δ)φ_{δ}^{j}(h)

entirely in terms of the set 𝔛̃. By the corollary on p. 170 of [18]

H^{1}(\overline{Q}_{𝔭}/Q_{𝔭}^{un}, T_{i}) = 1 .

Thus we may choose the g to lie in G(Q_{𝔭}^{un}) and then write

g = tu,  t ∈ T̃_{i}(Q_{𝔭}^{un}),  u ∈ G̃_{1}(Q_{𝔭}) .

Here T̃_{i} is the centralizer of T_{i} in G̃_{1}. Thus

T_{i}^{g} = u^{−1}T_{i}u

and

b_{g} = u^{−1}b_{i}u .

We let F_{i} be the operator on 𝔛̃ which takes the point represented by x in G̃_{1}(𝔨) to the point represented by b_{i}σ(x), and define X̃_{i}P in the same way that we defined X_{g} except that F_{i} replaces F_{g}. Then

uX_{g} = u𝔛 ∩ X̃_{i} .

As in [7], 𝔈(T_{i}/Q_{𝔭}) may be identified with

G̃_{1}(Q_{𝔭})/T̃_{i}(Q_{𝔭})G(Q_{𝔭}) ≃ ∏_{v|𝔭} F_{v}^{×}A(Q_{𝔭}) ∏_{v|𝔭} Nm L_{v}^{×}

with

L_{v} = L ⊗_{F} F_{v} .

We may therefore regard κ as a character of G̃_{1}(Q_{𝔭}). If

J(Q_{𝔭}) = \{\overline{g} ∈ G(Q_{𝔭}) | b_{σ}(\overline{g})b^{−1} = \overline{g}\}

the sum (3.2) is equal to

(3.3) ∑_{u∈G̃_{1}(Q_{𝔭})/T_{i}(Q_{𝔭})G(Q_{𝔭})} ∑_{\{x\}} \frac{κ(u)}{meas J(X)} ∫_{T_{i}(Q_{𝔭})\(Q_{𝔭})} ψ_{x}^{j}(g^{−1}hg)dg .

Here {x} is a set of representatives for the orbits of J(Q_{𝔭}) in u𝔛 ∩ X̃_{i} and ψ_{x}^{j} is the characteristic function of

{g ∈ J(Q_{𝔭}) | Φ_{𝔭}^{j}x = gx}

with Φ_{𝔭} = F_{i}^{e} if e = [E_{𝔭} : Q_{𝔭}]. I observe that, by the definition of E, the operator Φ_{𝔭} must take the set of marked points to itself, and does in fact operate on X̃_{i}.

It is manifest that G̃_{1}(Q_{𝔭}) ∩ K̃_{𝔭}(𝔨) takes u𝔛 to itself for all u, and hence that (3.3) is 0 unless κ is unramified. We assume then that κ is unramified. For a given T_{i} there are at most two possibilities for κ. If it is not trivial, it can be unramified only if L is unramified at every place of F dividing 𝔭.

We want to transform the expression (3.3), and in order to do so we need the following lemma.

Lemma 3.10. The set X̃_{i} is contained in G̃_{1}(Q_{𝔭})𝔛.

If G_{1} = Res_{F/Q}GL(1) then
Nm : G̃_{1}(Q_{𝔭}) → G_{1}(Q_{𝔭})

is surjective. Let g ∈ G̃_{1}(𝔨) and let a = Nm g. We want to show that if the image of g in 𝔛̃ lies in X̃ then

g ∈ G̃_{1}(Q_{𝔭})G(𝔨)K̃_{𝔭}(𝔨) .

Since G is the inverse image of A in G̃_{1}, all we need do is show that

a = a_{1}a_{2}a_{3}

with a_{1} ∈ G_{1}(Q_{𝔭}), a_{2} ∈ A(𝔨), a_{3} ∈ G_{1}(𝔨), and |λ(a_{3})| = 1 for every rational character of G_{1}.

The composition of μi^{∨} with T_{i} → A is a coweight μ̄i^{∨} of A and thus of G_{1}. Both A and G_{1} are split over Q_{𝔭}^{un} because F is assumed to be unramified at 𝔭. Consequently ϖ^{μ′_{i}} lies in A(Q_{𝔭}^{un}) if ϖ is a uniformizing parameter of Q_{𝔭}. Let \overline{b} = Nm b_{i}. The condition for the image of g to lie in X̃ which was added in [14] and used to deduce conditions (i) and (ii) above is that

aσ(a^{−1}) = \overline{b} ϖ^{−μ_{i}^{∨}}b_{3}

with |λ(b_{3})| = 1 for every rational character of G_{1}. A simple variant of Lemma A.7 allows us to establish that b_{3} = a_{3}σ(a_{3}^{−1}) with a_{3} of the desired form. For simplicity we replace a by aa_{3}^{−1} and suppose that b_{3} = 1.

The considerations of the appendix apply to the group A and the coweight μ̄i^{∨}. There are two ways to construct an element b_{A}, that is, the element b of the appendix when A replaces G. On one hand, we can take \overline{b}. On the other, we can observe that A splits over an unramified extension l_{𝔭} of Q_{𝔭} with Galois group generated by σ, and that the fundamental cocycle of l_{𝔭}/Q_{𝔭} may be taken to be

a_{σ^{j},σ^{k}} = \{ \begin{matrix}1, & 0 ≦ j, k < [l_{𝔭} : Q_{𝔭}], j + k < [l_{𝔭} : Q_{𝔭}], \\ ϖ, & 0 ≦ j, k < [l_{𝔭} : Q_{𝔭}], j + k ≧ [l_{𝔭} : Q_{𝔭}] \end{matrix}

This leads to ϖ^{μ̄_{i}^{∨}}. The considerations of the appendix show that

\overline{b}ϖ^{−μ_{i}^{∨}} = a_{2}σ(a_{2}^{−1}), a_{2} ∈ A(𝔨) .

Thus

a_{1} = aa_{2}^{−1} ∈ G_{1}(Q_{𝔭}) .

We may now regard x as a function on X̃_{i}, defined by

κ(x) = κ(u)

if x ∈ u𝔛. It is constant on orbits of T̃_{i}(Q_{𝔭}). Let

J̃_{1}(Q_{𝔭}) = {g ∈ G̃_{1}(Q_{𝔭}) | bσ(g)b^{−1} = g} .

This group contains T̃_{i}(Q_{𝔭}) and the usual bijection

T_{i}(Q_{𝔭})\J(Q_{𝔭}) ≃ T̃_{i}(Q_{𝔭})\(Q_{)}J(Q_{𝔭}) ⊆ T̃_{i}(Q_{𝔭})\J̃_{1}(Q_{𝔭})

is defined. We choose measures on T̃_{i}(Q_{𝔭}) and J̃_{1}(Q_{𝔭}) in such a way that the restriction of the quotient measure to the image of the arrow corresponds by transport of structure to the measure on T_{i}(Q_{𝔭})\J(Q_{𝔭}) appearing in (3.3).

Recall the definition of the number u(T_{i}) (p. 1156). Replacing T_{i} by a conjugate if necessary, we suppose that K_{𝔭} ⊆ K̃_{𝔭} ⊆ Ũ_{𝔭} where K̃_{𝔭} is a maximal compact subgroup of G̃_{1}(Q_{𝔭}) and Ũ_{𝔭} the maximal compact subgroup of T̃_{i}(Q_{𝔭}). Let J̃_{1}(x) be the stabilizer of x in J̃_{1}(Q_{𝔭}). We want to show now that the expression (3.3) is equal to

u(T_{i}) \frac{meas Ũ_{𝔭}}{meas U_{𝔭}}

times

∑ \frac{κ(x)}{meas J_{1}(x)} ∫_{T̃_{i}(Q_{𝔭})\J̃_{1}(Q_{𝔭})} ψ_{x}^{j}(g^{−1}hg)κ(g)dg . (3.4)

The sum is over a set of representatives for the orbits of J̃_{1}(Q_{𝔭}) in X̃_{i}, and U_{𝔭}, Ũ_{𝔭} are the maximal compact subgroups of T_{i}(Q_{𝔭}) and T̃_{i}(Q_{𝔭}).

We described J(Q_{𝔭}) explicitly above. It is clear that J̃_{1}(Q_{𝔭}) admits a similar description. It follows from these descriptions that

Norm J(Q_{𝔭}) = Norm G(Q_{𝔭}),
Norm J̃_{1}(Q_{𝔭}) = Norm G̃_{1}(Q_{𝔭}) .

Consequently the outer sum in (3.3) may be taken over

T̃_{1}(Q_{𝔭})J(Q_{𝔭})\J̃_{1}(Q_{𝔭})

and the index in the inner sum may be taken to be ux where x runs over a set of representatives for the orbits of J(Q_{𝔭}) in 𝔛 ∩ X̃_{i}. Changing the order of summation and combining the new inner sum and the integral into a single integral, we obtain

∑_{\{x\}} \frac{κ(x)}{meas J(x)} ∫_{T̃_{1}(Q_{𝔭})\J̃_{1}(Q_{𝔭})} ψ_{x}^{j}(g^{−1}hg)κ(g)dg .

The sum is over a set of representatives for the orbits.

Every orbit of J̃_{1}(Q_{𝔭}) in X̃ meets 𝔛, and so the set {x} does meet every orbit of J̃_{1}(Q_{𝔭}). The difficulty with which we have to contend is that it may contain several points from the same orbit. Given x in {x}, choose a maximal compact subgroup C̃ of J̃_{1}(Q_{𝔭}) containing J̃_{1}(x) and let C = C̃ ∩ J(Q_{𝔭}). If g ∈ J̃_{1}(Q_{𝔭}) then gx ∈ 𝔛 if and only if g ∈ J(Q_{𝔭})C̃. The number of orbits of J(Q_{𝔭}) in

J̃_{1}(Q_{𝔭})x ∩ 𝔛

is

[J(Q_{𝔭})C̃ : J(Q_{𝔭})J̃_{1}(x)] = [C̃ : CJ̃_{1}(x)] .

Thus we are free to sum over a set of representatives for the orbits of J̃_{1}(Q_{𝔭}), provided we incorporate the factor

[C̃ : CJ̃_{1}(x)]/meas J(x) .

What we must do is show that this equals

u(T_{i})meas Ũ_{𝔭}/meas U_{𝔭}meas J̃_{1}(x) .
Let T̃_{i}(x) and T_{i}(x) be the stabilizers of x in T̃_{i}(Q_{𝔭}) and T_{i}(Q_{𝔭}). Then

\frac{[C̃ : CJ̃_{1}(x)]}{meas J(x)} = \frac{[C̃ : CT̃_{i}(x)]}{[J̃_{1}(x) : J(x)T̃_{i}(x)]} · \frac{1}{meas T_{i}(x)\J(x)} · \frac{1}{meas T_{i}(x)} .

The middle factor on the right combines with the denominator of the first factor to give

(meas T̃_{i}(x)\J̃_{1}(x))^{−1} = meas T̃_{i}(x)/meas J̃_{1}(x) .

On the other hand,

[C̃ : CT̃_{i}(x)] = u(T_{i})[Ũ_{𝔭} : U_{𝔭}T̃_{i}(x)]

and

[Ũ_{𝔭} : U_{𝔭}T̃_{i}(x)] = \frac{[Ũ_{𝔭} : T̃_{i}(x)]}{[U_{𝔭} : T_{i}(x)]} = \frac{meas Ũ_{𝔭}}{meas U_{𝔭}} · \frac{meas T_{i}(x)}{meas T̃_{i}(x)} .

The desired equality follows.

We started with a particular quadratic extension L, chose T_{L}, and then ν1^{∨}, ν2^{∨}, …, and g_{1}, g_{2}, …, as well as T_{1}, T_{2}, …. On the other hand, in the previous paragraph we expressed at leaast part of the contribution to the sums (2.1) and (2.9) as a sum over T and ν. We may as well assume the T are the T_{L}. Moreover the νi^{∨} and the ν are essentially the same. If

νi^{∨} = ∑_{v} νv^{∨}  and  ν = ∑ ν_{v}

then

νv^{∨} = \frac{𝔫_{v}}{[k_{𝔭} : Q_{𝔭}]} ν_{v} .

Since we are dealing with Frobenius pairs of the first kind, 𝔪′_{v} ≠ 𝔪′′_{v} for at least one v.

Let δ_{i} be the image of g_{i} in 𝔇(T/Q)(T = T_{L}) and δ_{∞}^{i} and δ_{𝔭}^{i} the images of δ_{i} in 𝔇(T/R) and 𝔇(T/Q_{𝔭}). Then

Φ^{T/κ}(t, ϕ^{𝔭}) = κ(δ_{∞}^{i})κ(δ_{𝔭}^{i})Φ^{T_{i}/κ}(t, ϕ^{𝔭}) .

The term κ(δ_{∞}^{i}) is 1 if κ is trivial and cancels the term β(ν) if κ is not trivial. The term κ(δ_{𝔭}^{i}) if 1 if κ is trivial. Otherwise it is just what is needed to allow us to refer the definition of the sign appearing in part (g) of our summary of the discussion of (2.1) to T_{i} rather than T.

Observe that u(T) is 1 when every place v of F dividing 𝔭 is unramified, that the space X̃_{i}, together with the action of Φ_{𝔭}, may be represented as a Cartesian product over the v dividing 𝔭, and that the expression is then also a product. A comparison of the summaries in the previous paragraph with our results for Frobenius pairs of the first kind shows that, in order to have perfect cancellation of the contributions of these pairs with the contributions from the Selberg trace formula parametrized by the ν with 𝔪′_{v} ≠ 𝔪′′_{v} for some v, we need only verify the combinatorial facts to be reviewed below.

Combinatorial facts to be proved. Before describing these facts, we recapitulate the relevant definitions in the form and with the notation that is now appropriate. F_{v} ⊆ Q_{𝔭}^{un} ⊆ \overline{Q}_{𝔭} is an unramified extension of Q_{𝔭} and we take G to be Res_{F_{v}/Q_{𝔭}}GL(2). The imbeddings of F_{v} in \overline{Q}_{𝔭} are indexed by

𝔊(Q_{𝔭}^{un}/F_{v})\𝔊(Q_{𝔭}^{un}/Q)

and the imbedding x → Φ_{𝔭}^{−i}(x) will also be labelled by the integer i. Let M^{0} be a lattice in the space of column vectors of length two over O_{v} invariant under 𝔊(F_{v}/Q_{𝔭}) and let K ⊆ GL(2, F_{v}) = G(Q_{𝔭}) be the stabilizer of M^{0}. K is a maximal compact subgroup. Let 𝔨 be be completion of Q_{𝔭}^{un}. Then

G(𝔨) = {(g_{i}) | 1 ≦ i ≦ n, g_{1} ∈ GL(2, 𝔨)} .

Let M^{0} also denote M^{0} ⊗ 𝔬_{𝔨} and let

K(𝔨) = {(g_{i}) ∈ G(𝔨) | g_{i}M^{0} = M^{0} for all i} .

A point 𝔵 in

𝔛 = G(𝔨)/K(𝔨)

is a sequence (M_{i}), 1 ≦ i ≦ n, where M_{i} is an 𝔬_{𝔨}-lattice in the space of column vectors of length two over 𝔨. The action of σ, the Frobenius of 𝔨, on (M_{i}) is σ : (M_{i}) → (M′_{i}) with M′_{i} = M_{i=1}, 2 ≦ i ≦ n, and M′_{1} = σ^{n}(M^{n}), if in the last equation σ^{n} denotes the usual action of σ^{n} on 𝔬_{𝔨}-lattices.

Suppose J^{0} is either a split Cartan subgroup of G or G itself, and T is a Cartan subgroup of J^{0} whose image in J_{ad}^{0} is anisotropic. Let μ be a coweight of T of the form

∑_{i=1}^{n} a_{i}γ_{j(i)}^{i} ,

with j(i) equal to 1 or 2 and a_{i} equal to 0 or 1. The set {γ_{j}^{i}} is the standard basis of the lattice of coweights. We suppose that

Nm_{k_{𝔭}/Q_{𝔭}}μ

factors through the center of J^{0}, but if J^{0} ≠ G not through the enter of G, and use μ as in the appendix to define b, and then let F be the operator 𝔵 → b_{σ}(𝔵).

The set X is a subset of 𝔛. If 𝔵 = (M_{i}) and 𝔜 = F𝔵 = (N_{i}), then 𝔵 lies in X if and only if the following two conditions are satisfied:

(i) M_{8} = N_{i} if i is an unmarked point.
(ii) M_{i} ⊋ N_{i} ⊋ 𝔭M_{i} if i is a marked point.

Observe in particular that if (M_{i}) and (M′_{i}) with M′_{i} = g_{i}M_{i} lie in X then ord(det g_{i}) is independent of i. If we fix an index i_{0}, 1 ≦ i_{0} ≦ 𝔫, we may define a function κ on X by

κ(𝔵) = ϵ^{ord(det g_{i_{0}} )},

if 𝔵 = (M_{i}) and M_{i} = g_{i}M_{i}^{0}. Here ϵ is 1 if J^{0} is a split Cartan subgroup and ±1 if J^{0} is G. If 𝔤 = g𝔵 with g = (g_{i}) in J(Q_{𝔭}) and u = ord(det g_{i}), then

κ(𝔤) = ϵ^{u}κ(𝔵) .

If there are no marked places κ is independent of i_{0} even when ϵ = −1. If there are marked places a different choice of i_{0} may change its sign, but that does not matter.

If e = [E_{𝔭} : Q_{𝔭}] then Φ_{𝔭} = F^{e} maps X to itself and we set

T_{x}^{j} = {g ∈ J(Q_{𝔭}) | Φ_{𝔭}^{j}x = gx} .

J is obtained from J^{0} by the usual inner twisting. Finally, if {x} is a set of representatives for the orbits of J(Q_{𝔭}) in X and ψ_{x}^{j} is the characteristic function of T_{x}^{j}, we set

φ_{κ}^{j}(h) = ∑_{\{x\}} \frac{κ(x)}{meas J(x)} ∫_{J(h,Q_{𝔭})\J(Q_{𝔭})} ψ_{x}^{j}(g^{−1}hg)κ(g)dg

with κ(g) = ϵ^{ord(Norm g)}. We must assume that g → x(g) is trivial on J(h, Q_{𝔭}).

Here is what we must establish.

(a) Suppose J^{0} is a split Cartan subgroup and ϵ = 1. Let

Nm_{k_{𝔭}/Q_{𝔭}}μ = \frac{[k_{𝔭} : Q_{𝔭}]}{𝔫} (𝔪′′δ_{1} + 𝔪′δ_{2})

with

δ_{c} = ∑_{i=1}^{n} γ_{c}^{i}, c = 1, 2 ,

𝔪′ ≠ 𝔪′′, and 𝔪′ + 𝔪′′ = 𝔪, the number of marked places. Let l be the number of orbits under Φ_{𝔭}^{j} and k the number of marked orbits. Then φ_{κ}^{j}(h) is 0 unless 𝔪/k divides 𝔪′ and 𝔪′′. However, if k′ = 𝔪′/k𝔪 and k′′ = 𝔪′′k/𝔪 are integral, then

φ_{κ}^{j}(h) = \frac{k!}{k′!k′′!} 𝔭^{jeq}θ_{v}(h)

with q equal to the smallest of 𝔪′ and 𝔪′′ and ν = jcn^{−1}(𝔪′′δ_{1} + 𝔪′δ_{2})

(b) Suppose 𝔪′ = 𝔪′′ and they are both integral. Then J^{0} = J = G. Let t lie in the split Cartan subgroup T and have distinct eigenvalues t_{1}, t_{2}. Let ϵ be 1. If k′ = 𝔪′k/𝔪 = k/2 and k′′ = 𝔪′′/𝔪 = k/2 are not integral, then φ_{κ}^{j}(t) is 0. Otherwise it is

\binom{k}{k/2}^{p^{jem}/2} \frac{|t_{1}t_{2}|^{1/2}}{|t_{1} − t_{2}|^{1/2}} θ_{ν}(t)

with

ν = jen^{−1}(𝔪′δ_{2} + 𝔪′′δ_{1}) .

Suppose t lies in a Cartan subgroup T that is not split. If ϵ = −1, only the case of a T associated to an unramified quadratic extension matters. If 𝔪 ≠ 0, then φ_{κ}^{j}(t) must be 0. If 𝔪 = 0, then b = 1 and

φ_{κ}^{j}(t) = \frac{±1}{meas U} \frac{|t_{1}t_{2}|^{1/2}}{|(t_{1} − t_{2})^{2}|^{1/2}} .

Here U is the maximal compact subgroup of T (Q_{𝔭}), and the sign is to be determined by the rule given in part (g) of the summary of our discussion of (2.1).

If ϵ = 1 but the eigenvalues are still distinct, then φ_{κ}^{j}(t) is a sum of two terms. The first is

\{∑_{0 ≦ i<k/2} p^{jem/k}\binom{k}{i}\} \frac{meas T (Q_{𝔭})\G′(Q_{𝔭})}{meas K′} Ξ_{jem/𝔫}(t) .

Here G′ is the group obtained from the quaternion algebra over F_{v}, and K′ is a maximal compact subgroup of G′(Q_{𝔭}). The second term is 0 unless k/2 is integral, when it is

𝔭^{jem/2}\binom{k}{k/2}\frac{1}{meas K} ∫_{T (Q_{𝔭})\G(Q_{𝔭})} φ_{ejm/𝔫}(g^{−1}tg)dg

if φ_{ejm/𝔫} is the characteristic function of

( \begin{matrix}ϖ^{ejm/2𝔫} & 0 \\ 0 & ϖ^{ejm/2𝔫} \end{matrix} ) K .

(c) The final case to consider is that 𝔪′ = 𝔪′′, and they are both half-integral. The formulae are to be the same as before. Notice, however, that split T no longer come into question, and that k/2 can no longer be integral.

Notice also that Lemma 3.7 is a consequence of part (a).

Finally, we have to examine the contribution of Frobenius pairs of the second kind to the alternating sum of the traces and compare it with the results of §2. We take the sum over conjugacy classes {h} in I(Q) of the contributions N^{j}(h) given by Lemma 3.8 and decompose it into stable and labile parts. The group I is now G′, defined in the same manner as G, but in terms of a different quaternion algebra D′. The Hasse invariants of D′ are the same as those of D at all finite places that do not divide 𝔭, but they are \frac{1}{2} at every infinite place, and the invariant at a place v dividing 𝔭 is 𝔪_{v}/2.

The contribution of the scalar matrices is already stable. It is a sum over Z(Q) ∩ K\Z(Q). The terms of the sum may be represented as products. The first factor is

\frac{meas(Z_{K}G′(Q)\G′(A_{f}))}{meas Z_{k}} ϕ^{𝔭}(h) .

In order to make the comparison with the results of §2, we must recall a simple property of Tamagawa numbers, viz.,

meas(Z_{K}Z(R)G(Q)\G(A)) = meas(Z_{K}Z(R)G′(Q)\G′(A)) .

The measure on the right is equal to meas(Z(R)\G′(R))meas(Z_{K}G′(Q)\G′(A_{f}))

The second factor is φ^{j}(h), defined with respect to the space X associated to G. Since it is X̃, the space associated to G̃_{1}, that factors as a Cartesian product, we want to replace φ^{j}(h) by φ̃^{j}(h). Lemma 3.10 is still valid, and hence every orbit of J̃_{1}(Q_{𝔭}) = G̃′_{1}(Q_{𝔭}) in X̃ meets X. The number of orbits of J(Q_{𝔭}) = G′(Q_{𝔭}) contained in the J̃_{1}(Q_{𝔭})-orbit of x is

[C̃ : CJ̃_{1}(x)] .

Here C̃ is again a maximal compact subgroup of J̃_{1}(Q_{𝔭}) containing J̃_{1}(x). Since

[C̃ : CJ̃_{1}(x)] = \frac{[C̃ : J̃_{1}(x)]}{[C : J_{1}(x)]} = \frac{meas C̃}{meas C} \frac{meas J_{1}(x)}{meas J̃_{1}(x)}

and

\frac{meas C̃}{meas C} = \frac{meas K̃_{𝔭}}{meas K_{𝔭}} ,

we may replace φ^{j}(h) by φ̃^{j}(h), provided that we multiply by the quotient

meas K̃_{𝔭}/meas K_{𝔭} .

If we observe that ∑ 𝔪_{v} is the dimension of the Shimura variety, we see that, to establish that the contribution of the scalar matrices is equal to their contribution to (2.9), we need only the following:

Additional combinatorial facts. We must revert to the notations ued when describing the other combinatorial facts to be proved.

(d) Let κ be identically 1 and let z lie in the center of G′(Q_{𝔭}). If k is odd, then φ_{κ}^{j}(h) is equal to

− \frac{(|ϖ_{v}|^{−1} − 1)}{meas K} Ξ_{jem/𝔫}(z) \{∑_{0 ≦ i<k/2} \binom{k}{i} p^{jemi/k}\} .

If k is even, it is the sum of this and

\frac{1}{meas K} Ξ_{jem/𝔫}(z)\binom{k}{k/2} p^{jem/2} .

The contribution of the non-scalar elements in G′(Q) remains to be treated, but for it no new combinatorial facts are needed. The treatment is by now routine. We obtain a double sum, over a set of representatives T for the stable conjugacy classes of Cartan subgroups, and over the characters κ of

𝔈(T/A)/Im 𝔈(T/Q) = 𝔈(T/A_{f})/\𝔈(T/A_{f}) ∩ Im 𝔈(T/Q) .

Each term is itself a sum over h in Z_{K}T (Q)\T (Q) of

\frac{μ(T)}{2[𝔈(T/A) : Im 𝔈(T/F)]} trace ξ(y)Φ^{T/κ}(t, ϕ^{𝔭})

times

∑_{δ∈𝔈(T/Q_{𝔭})} κ(δ)φ^{j}(h^{δ}) .

This sum is basically the same as (3.2), except that φ^{j}(h^{δ}) is here defined with respect to a fixed X rather than with respect to varying X_{g}. It may, however, be treated in exactly the same way, with the same conclusions.

This is the reason that no additional combinatorial facts are needed. Now 𝔪′_{v} = 𝔪′′_{v} for all v, whereas earlier this happened only for some v. Since we factored the set into a product over v, every possibility for the individual factors has had to be taken into account already.

4. Combinatorics.

The preparation over, we come now, with sighs of relief from reader and author, to the amusing part of the paper. The combinatorial facts to be verified turn out to be statements about a simple type of tree, the Bruhats-Tits buildings for SL(2). They may well be familiar to combinatorialists, but a cursory glance at the standard texts yielded nothing of help.

The notation will now be that used when stating the facts to be proved. Once a few preliminary remarks are out of the way, we will be able to dismiss most of the preceding discussion from our minds, and indulge ourselves in a little elementary mathematics.

We have agreed to take b = 1 when μ = 0. Otherwise, we have not made any particular choice of the element b used to define X. It will be convenient to do so for the calculations of this paragraph. b lies in J^{0}(𝔨), but we are free to modify it to cbσ(c^{−1}) with c in J^{0}(𝔨). Thus we can suppose it is of the form b = (b_{i}) with

b_{2} = … = b_{𝔫} = 1  and  b_{1} = B .

If J^{0} is a split torus, which we take to be the group of diagonal matrices, we may take

B = ( \begin{matrix}p^{𝔪′′} & 0 \\ 0 & p^{𝔪′} \end{matrix} ) .

If J^{0} is G we may take

B = ( \begin{matrix}p^{𝔪/2} & 0 \\ 0 & p^{𝔪/2} \end{matrix} )

when 𝔪 is even and

B = p^{𝔪−1/2} ( \begin{matrix}0 & 1 \\ p & 0 \end{matrix} )

when 𝔪 is odd. There is no real need for a specific choice of B, but it does no harm.

A point 𝔵 ∈ 𝔛 is represented by a sequence {M_{i} | 1 ≦ i ≦ n}. Define M_{i} for all i ∈ Z by the periodicity condition

Bσ^{n}(M_{i+n}) = M_{i} .

In addition, extend the notion of marked or unmarked point by periodicity. Then {M_{i}} defines a point of X if and only if the following conditions are satisfied.

(o) Bσ^{n}(M_{i+n}) = M_{i};

(i) At an unmarked point i, M_{i} = M_{i−1} ;

(ii) At a marked point i, M_{i} ⊋ M_{i−1} ⊋ 𝔭M_{i}.

The supplementary condition is absorbed into (i) and (ii). Because of the special form of b, it states simply that, if M_{i} = g_{i}M_{i}^{0}, then

ord(det(g_{i}g_{i−1}^{−1})) = \{ \begin{matrix}0, & i unmarked, \\ −1, & i marked . \end{matrix}
The operation of Φ_{𝔭}^{j} takes (M_{i}) to (M′_{i}) with
M′_{i} = M_{i−ej} .

The point h = (h_{i}), 1 ≦ i ≦ n, lies in J(Q_{𝔭}) if and only if h_{i} is independent of i and

Bσ^{n}(h_{i})B^{1} = h_{i}

for all i. It will be convenient to change the notation slightly and to write h = (h, …, h), that is, we identify h with any one of its coordinates. If x = (M_{i}), then h ∈ T_{x}^{j} if and only if

M_{i−ej} = hM_{i}

for all i.

If M and M′ are two-dimensional lattices over 𝔬_{𝔨}, they are said to be homothetic if M′ = λM with λ ∈ 𝔨^{x}. Let \overline{M} be the class of lattices homothetic to M. The Bruhat-Tits building for SL(2, 𝔨) is a tree whose vertices consist of the homothety classes of lattices, the classes \overline{M} and \overline{N} being joined by an edge if, for some λ ∈ 𝔨^{x},

M ⊋ λN ⊋ 𝔭M .

σ acts on this building and the fixed point set of σ^{j} is the Bruhat-Tits building for SL(2, k) if k is the unramified extension of Q_{𝔭} of degree j.

The point x in X determines an infinite path in the building

with vertices \overline{M}_{i}. If x and x′ = (M′_{i}) determine the same path then, because of conditions (i) and (ii), there is a z ∈ Z(Q_{𝔭}) for which x′ = zx, that is, M′_{i} = zM_{i} for all i. Since J(Q_{𝔭}) contains Z(Q_{𝔭}), x and x′ lie in the same orbit. Observe also that if we have a sequence (\overline{M}_{i}) satisfying the following three conditions, then it may be lifted to a point of X.

(o) B\overline{M}_{i+n} = σ^{−n}(\overline{M}_{i});

(i) \overline{M}_{i} = \overline{M}_{i−1} if i is unmarked;

(ii) \overline{M}_{i} and \overline{M}_{i−1} are joined by an edge if i is marked.

We may as well dispose immediately of the case that μ = 0 and B = 1. Then J(Q_{𝔭}) = G(Q_{𝔭}) = GL(2, F_{v}), and M_{i} = M is independent of i. Since BM_{i+n} = σ^{−n}(M_{i}), L is actually a lattice over F_{v}. There is only one orbit and we may take it to be the point with M = M^{0} or, if we were being extremely precise, M^{0} ⊗ 𝔬_{𝔨}, but at present it is best not to distinguish between a lattice over 𝔬_{v} and the lattice over 𝔬_{𝔨} it determines. If ψ is the characteristic function of K, then

φ_{κ}^{j}(h) = (meas K)^{−1} ∫_{G(h,Q_{𝔭})\G(Q_{𝔭})} κ(g)ψ(g^{−1}hg)dg .

That φ_{κ}^{j}(h) has the desired value when h is regular and lies in a split torus follows form the properties of the Satake homomorphism. That it has the desired value otherwise is immediate for κ trivial, and follows from an observation in §2 of [7] when κ is not trivial.

It is also easy to show that, if there are marked points and κ is not trivial, then φ_{κ}^{j}(h) is 0. Suppose x = (M_{i}) is a point of X. We may define another point x′ = (M′_{i}) by demanding that M′_{i} = M_{i′} whenever i is marked and i′ is the first marked point following i. If h ∈ J(Q_{𝔭}) and y = hx, then y′ = hx′. Moreover

κ(x′) = −κ(x) .

Since φ_{κ}^{j}(h) can be calculated by a sum over {x′} rather than a sum over {x}, we conclude that φ_{κ}^{j}(h) = −φ_{κ}^{j}(h).

We suppose henceforth not only that there are marked points, but also that κ is trivial, and fix our attention on a specific h in J(Q_{𝔭}). We observe first of all that φ_{κ}^{j}(h) is the sum over the orbits of J(h, Q_{𝔭}), the centralizer of h in J(Q_{𝔭}), of

κ(x)ψ_{x}^{j}(h)/meas J(h, x) .

Here J(h, x) is the stabilizer of x in J(h, Q_{𝔭}). To see this start from the definition of φ_{κ}^{j}(h) as

∑_{\{x\}} \frac{κ(x)}{meas J(x)} ∫_{J(h,Q_{𝔭})\J(Q_{𝔭})} κ(g)ψ_{x}^{j}(g^{−1}hg)dg .

This expression is equal to

∑_{\{x\}} \frac{κ(x)}{meas J(x)} ∑_{J(h,Q_{𝔭})\J(Q_{𝔭})/J(x)} κ(g)ψ_{gx}^{j}(h) × meas(J(h, Q_{𝔭})\J(h, Q_{𝔭})gJ(x)) .

Since

κ(g)κ(x) = κ(gx)

and

\frac{meas(J(h, Q_{𝔭})\J(h, Q_{𝔭})gJ(x))}{meas J(x)} = \frac{1}{meas J(h, gx)} ,

the assertion follows.

We are therefore interested only in the set U of those x for which h ∈ T_{x}^{j}. Suppose we have a subset U′ of U and an open subgroup J_{0} of J(h, Q_{𝔭}) satisfying the following three conditions.

(i) Every orbit of J(h, Q_{𝔭}) in U meets U′.

(ii) If x and y in U′ lie in the same orbit of J(h, Q_{𝔭}), then x = gy with g ∈ J_{0}.

(iii) For all x ∈ U′, J(h, x) is a subgroup of J_{0}.

It is then clear that

(4.1)  φ_{κ}^{j}(h) = meas J_{0} ∑_{x∈U′} κ(x) .

We also want to reformulate the two conditions of periodicity:

(a) σ^{n}(BM_{i}) = M_{i−n};

(b) hM_{i} = M_{i−ej}.

Let uej + vn = l. I claim that (a) and (b) are equivalent to the conditions

(c) σ^{ejn/l}(M_{i}) = B^{−ej/l}h^{n/l}M_{i};

(d) h^{u}B^{v}σ^{vn}(M_{i}) = M_{i−l}.

From (b)

h^{n/l}M_{i} = M_{i−enj/l} .

Since σ(B) = (B), the relation (a) implies that

M_{i−enj/l} = σ^{enj/l}B^{ej/l}M_{i} ,

and we deduce (c). Applying (a) and (b) again, we have

M_{i−l} = M_{i−vn−uej} = h^{u}M_{i−vn} = h^{u}B^{v}e^{vn}(M_{i}) .

Conversely, if we assume (d), we may write

M_{i−n} = M_{i−ln/l} = h^{un/l}B^{vn/l}σ^{vn^{2}/l}(M_{i})

because h ∈ J(Q_{𝔭}) and

Bσ^{n}(h) = hB .

We then apply (c) to infer that the right side of

h^{un/l}B^{vn/l}σ^{vn^{2}/l}(M_{i}) = h^{un/l}BB^{−uej/l}σ^{n−uejn/l}(M_{i})

is equal to

Bσ^{n}(M_{i}) = σ^{n}(BM_{i}) .

This is (a). To deduce (b) from (c) and (d), we write

M_{i−ej} = M_{i−lej/l} = h^{uej/l}B^{vej/l}σ^{vejn/l}(M_{i}) ,

and replace the right side by

h^{−vn/l}B^{vej/l}σ^{vejn/l}(M_{i}) = M_{i} .

The conditions (c) and (d) will be more useful than (a) and (b), but there is still one useful consequence to be drawn from (a). Consider the set Ā of points \overline{M} in the Bruhat-Tits building for which dist(\overline{M}, σ^{n}(B\overline{M})) is a minimum. It is clear that Ā is invariant under \overline{M} → σ^{n}(B\overline{M}). Moreover, Ā is convex, in the sense that every vertex on the path joining two points of Ā is again in Ā. To prove this statement, we proceed by induction on the length of the path. Thus it is enough to show that if \overline{M} and \overline{N} lie in Ā but no other point on the path joining \overline{M} and \overline{N} does, then \overline{M} and \overline{N} are adjacent.

We examine first the path connecting \overline{M} and σ^{n}(B\overline{M})

if \overline{M} ≠ σ^{n}(B\overline{M}) and L̄ is the point in the path succeeding \overline{M} then

dist(L̄, σ^{n}(BL̄)) ≦ dist(\overline{M}, σ^{n}(B\overline{M})) .

Since we must have equality, σ^{n}(B, \overline{M}) is on the path joining L̄ to σ^{n}(BI)

Observe that L̄ lies in Ā.

Returning to \overline{M} and \overline{N}, we notice that we can construct a cycle of the form:

If \overline{M} and \overline{N} are not adjacent, the sides can have no edge in common with the top or bottom, and the cycle is not trivial. This is impossible, because the building is a tree.

Suppose x = (M_{i}) lies in X. Let \overline{M}_{i_{0}} be such that

dist(\overline{M}_{i_{0}}, Ā) ≦ dist(\overline{M}_{i}, Ā)

for all i. I claim that \overline{M}_{i_{0}} ∈ Ā. Otherwise, we could again construct a nontrivial cycle

Here \overline{M} is the point in Ā closest to \overline{M}_{i_{0}}.

The skeleton S(x) of x = (M_{i}) in X will either be the set of integers i for which \overline{M}_{i} ∈ Ā or the path formed by joining these vertices in succession. Suppose, for example, that Ā is

and \overline{M} → σ^{n}(B\overline{M}) is a shift to the left by two. Let 𝔫 be 8. Then the skeleton could be To form the full path we have to add subpaths which issue from Ā. Thus the full path, or at least a representative part, could be

The sets A are easily described explicitly. If J(Q_{𝔭}) is the split Cartan subgroup, then A consists of the lattices M(u′, u′′) formed by the set of

( \begin{matrix}ϖ^{u′}x \\ ϖ^{u′′}y \end{matrix} )

with x, y in 𝔬_{𝔨}. Here u′ and u′′ are any two integers. The set Ā is an infinite line

and \overline{M} → σ^{n}(B\overline{M}) is a shift of size |𝔪′ − 𝔪′′|. We take it to be to the right.

If J(Q_{𝔭}) is GL(2, F_{v}), then A is the set of lattices defined over F_{v} and \overline{m} → σ^{n}(B\overline{M}) acts trivially on Ā. If J(Q_{𝔭}) is G′(F_{v}), then Ā consists of two points, the image of the lattice formed by

\binom{x}{y}, x, y ∈ 𝔬_{𝔨},

together with the image of the lattice formed by

\binom{x}{ϖy}, , x, y ∈ 𝔬_{𝔨} .

The map \overline{M} → σ^{n}(B\overline{M}) interchanges the two points in Ā.

The sets A and Ā are invariant under J(Q_{𝔭}), and J(Q_{𝔭}) acts transitively on Ā. I now introduce a convex subset D̄ of Ā, as well as D, the set of lattices M for which \overline{M} ⊆ D̄. D̄ will be invariant under J(h, Q_{𝔭}). If h is central in J(Q_{𝔭}), then D̄ will be Ā. Otherwise, J(h, Q_{𝔭}) is a Cartan subgroup of J(Q_{𝔭}). We can pass to an extension b_{v}, at most quadratic, of F_{v} that splits J(h). The Bruhat-Tits building over F_{v} becomes, perhaps after taking the first barycentric subdivision, a subset of the Bruhat-Tits building over L_{v} (cf. §3 of [15]). Over L_{v} we may associate an apartment to J(h, Q_{𝔭}). D̄ is to consist of the points in Ā at a minimum distance from this apartment. It is geometrically clear that D̄ can contain at most two points.

It is also easily seen (cf. §3 of [15]) that in all cases D̄ is a connected tree and that the same number q_{0} + 1 of edges issue from each vertex. We tabulate the possibilities and give, in addition, the number f of orbits in D̄ or D under the action of J(h, Q_{𝔭}).

(a) J^{0} = J is a split torus. Then q_{0} = 1 and f = 1.

(b) J^{0} = G and h is central. Then q_{0} = 𝔭^{n} and f = 1.

(c) J^{0} = G and the centralizer of h is a split torus. Then q_{0} = 1 and f = 1.

(d) J^{0} = J = G and the centralizer of h is not a split torus. If the extension splitting J(h) is unramified, then q_{0} = −1 and f = 1. If it is ramified, then q_{0} = 0 and f = 1.

(e) J^{0} = G but J = G′, and the centralizer of h is not a split torus. If the extension splitting J(h) is unramified, then q_{0} = 0 and f = 2. If it is ramified, then q_{0} = 0 and f = 1.

The reduced skeleton RS(x) will either be the set of i on which dist(L̄_{i}, D̄) attains its minimum or the path obtained by joining these points in order. The reduced skeleton is contained in the skeleton. We choose a set of representatives D_{1} for the orbits of J(h, Q_{𝔭}) in D. Then D̄_{1} is a set of representatives for the orbits of J(h, Q_{𝔭}) in D̄. In all cases but one, D_{1} consists of a single element. All points of D_{1} have the same stabilizer J_{0} and J_{0} is a maximal compact subgroup of J(h, Q_{𝔭}).

We may now define the set U′. For each possible reduced skeleton RS, we choose an integer i(RS) in it. Then U′ consists of those x = (M_{i}) which are such that if RS = RS(x) and i = i(RS), then the minimum distance from the path ( …, \overline{M}_{−1}, \overline{M}_{0}, \overline{M}_{1}, …) to D̄ is equal to dist(\overline{M}_{i}, \overline{N}) with N ∈ D_{1} and M_{i} = gN with

order(det g) = dist(\overline{M}_{i}, \overline{N}) .

To verify that U′ satisfies the three conditions imposed, we have only to observe that if g is any element of J(h, Q_{𝔭}) and, for the same M_{i} and N,

dist(g\overline{M}_{i}, \overline{N}) = dist(\overline{M}_{i}, \overline{N}) ,

then g\overline{N} = \overline{N}.

There are some remarks to be made about the periodicity condition (c) before we examine the set U′ more closely. Since that condition is to be valid for all i, it implies in particular that if φ_{κ}^{j}(h) ≠ 0, the transformation

M → h^{−n/l}B^{ej/l}σ^{ejn/l}(M)

has a fixed point in D, because it fixes a point in the reduced skeleton and therefore the closest point in D̄ to the reduced skeleton. Recalling that in all but a single case there is only one orbit in D, and that in the exceptional case D̄ consists of two points, we conclude that the transformation fixes every point of D. So does σ^{ejn/l}. Thus, h^{−n/l}B^{ej/l} fixes every point of D.

Examining the various cases, we come to the following conclusions about the nature of those h for which φ_{κ}^{j}(h) ≠ 0.

(a) If J^{0} = J is a split torus, then

h = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} )

with

|α| = |ϖ^{ej𝔪′′/𝔫}|, |β| = |ϖ^{ej𝔪′/𝔫}| .

In particular, 𝔫 must divide ej𝔪′ and ej𝔪′′, and consequently 𝔫/l = 𝔪/k must divide 𝔪′ and 𝔪′′.

(b) If J = G and the centralizer of h is a split Cartan subgroup, then the eigenvalues α and β of h have equal absolute values and

|α| = |β| = |ϖ|^{ej𝔪/2𝔫} .

(d) J^{0} = G and the centralizer of h is not split, then

|Norm h| = |ϖ|^{ej𝔪/𝔫} .

In all cases, the order of the determinant of h^{u}B^{v} is

uej𝔪/𝔫 + v𝔪 = 𝔪l/𝔫 = k .

The conditions described here are also sufficient for the transformation to fix every element of D. Moreover, that φ_{κ}^{j}(h) is 0 when they are nto satisfied is a part of the combinatorial facts to be proved. We suppose henceforth that the conditions are satisfied.

Suppose

M = h^{−𝔫/l}B^{ej/l}σ^{ej𝔫/l}(M) .

The set of points in the Bruhat-Tits building over 𝔨 that can be joined to \overline{M} by an edge may be viewed as the projective line over the algebraic closure of the finite field F_{𝔭}. The transformation

\overline{N} → h^{−𝔫/l}B^{ej/l}σ^{ej𝔫/l}(\overline{N})

allows us to put on it the structure of a projective line over F_{𝔭^{d}}, d = ej𝔫/l. According to Lang’s theorem, there is only one such structure. Thus the set of \overline{N} that are fixed by the transformation and can be joined to \overline{M} by an edge contains 𝔭^{ej𝔫/l} + 1 elements.

If the minimum distance from \{ …, \overline{M}_{−1}, \overline{M}_{0}, \overline{M}_{1}, … \} to D̄ is positive, then the path of the reduced skeleton consists of a single point, say \overline{M}_{i}. Since D̄ is invariant under Bσ^{n} and under h, the periodicity condition

\overline{M}_{i−1} = h^{u}B^{v}σ^{v𝔫}(\overline{M}_{i}) = h^{u}\overline{M}_{i}

implies that \overline{M}_{i−1} also lies in the reduced skeleton. Hence \overline{M}_{i−1} = \overline{M}_{i} when the minimum distance is positive, as we now assume. Since u and 𝔫/l are relatively prime, we deduce from the two equations

h^{u}\overline{M}_{i} = \overline{M}_{i}  and  h^{−𝔫/l}\overline{M}_{i} = \overline{M}_{i}

that

h\overline{M}_{i} = \overline{M}_{i} .

Since there are k marked points in an interval of length l, the path from \overline{M}_{i} to \overline{M}_{i−1} = \overline{M}_{i} must have k edges. Consequently, points of the above type can exist only for k even.

It is now an easy matter to construct the points of U′ for which the minimum distance to D̄ is positive. Given a possible reduced skeleton RS and i = i(RS), we choose any \overline{M}_{i} and Ā which is fixed by h, and construct a path \overline{M}_{i}, \overline{M}_{i−1}, …, \overline{M}_{i−l}, from \overline{M}_{i} to \overline{M}_{i−l}. To construct it we must suppose that k is even. Then \overline{M}_{i′−1} = \overline{M}_{i′} if i′ is unmarked and one of the 𝔭^{ej𝔫/l} + 1 elements that can be joined to \overline{M}_{i′} by an edge and satisfy

h^{𝔫/l}\overline{M}_{i′−1} = b^{ej/l}σ^{ej𝔫/l}(\overline{M}_{i′−1})

if i′ is marked. We say that the edge from \overline{M}_{i′} to \overline{M}_{i′−1} is progressive or retrogressive according as \overline{M}_{i′−1} is at a greater or a lesser distance from D̄ than \overline{M}_{i′}. We must be careful that at any stage we have added at least as many progressive as retrogressive edges, for otherwise we would approach too closely to D̄, and we must ultimately take as many retrogressive as progressive steps, in order that we arrive back at \overline{M}_{i} = \overline{M}_{i−l}. Indeed we must be careful to return to the initial \overline{M}_{i} at any i′ that lies in the given rational skeleton. Finally, we have to choose \overline{M}_{i} so that the closest point to it in D̄ is an \overline{N} with N in D_{1}. We define \overline{M}_{i′} in general by the periodicity condition (d), and we lift the full path to a point x = (M_{i′}) in X with M_{i} = gN and

order(det g) = dist(\overline{M}_{i}, \overline{N}) .

Then x lies in U′.

There are some observations to be made. First of all, the number W of points that are yielded by a given choice of \overline{M}_{i} and all possible choices of the reduced skeleton and all possible paths from \overline{M}_{i} to \overline{M}_{i−l} is independent of \overline{M}_{i}. We shall show that

(4.2)  W = \binom{k}{k/2}𝔭^{ej𝔪/2} .

The minimum distance can be positive only when D ≠ A, that is, only when J(Q_{𝔭}) = G(Q_{𝔭}) and J(h, Q_{𝔭}) is a Cartan subgroup. Then A is just the Bruhat-Tits building over F_{v}. The argument that led to (4.1) shows that if T (Q_{𝔭}) = J(h, Q_{𝔭}) and |det h| = |ϖ|^{ej𝔪/𝔫} then

(4.3)  (meas K_{𝔭})^{−1} ∫_{T (Q_{𝔭})\G(Q_{𝔭})} φ_{ej𝔪/𝔫}(g^{−1}hg)dg

is (meas U_{𝔭})^{−1} times the number of points \overline{M} in the Bruhat-Tits building over F_{v} that are fixed by h and satisfy

dist(\overline{M}, D̄_{1}) ≦ dist(\overline{M}, D̄) .

However, \binom{k}{k/2}𝔭^{ej𝔪/2} times (4.3) is the second term of the desired formula for φ_{κ}^{j}(h) when κ is trivial. Therefore, the contribution to (4.3) of those \overline{M} for which

0 ≦ dist(\overline{M}, D̄_{1}) ≦ dist(\overline{M}, D̄)

is yielded by those x for which the minimum distance from\{ …, \overline{M}_{−1}, \overline{M}_{0}, \overline{M}_{1}, … \} to D̄ is positive.

The equality (4.2) is a consequence of the next lemma, which will be proved towards the end of the paragraph. Let T_{1} be a connected tree and suppose that from every vertex of T_{1} there issue q_{1} + 1 edges, with q_{1} ≧ 0. Let k ≧ 0 be an even integer, and for each nonempty subset S of Z with period l choose i = i(S) in S. Let P be a point of T_{1} and L an edge containing it.

Lemma 4.1. Let 𝔨 be the set of pairs (x, S) where x = (P_{i}, P_{i−1}, …, P_{i−k}), with i = i(S), is a path from P to P in T_{1} with no edge in L and with P_{i′} = P if and only if i′ ∈ S. The number of pairs in 𝔨 is then

\binom{k}{k/2}q_{1}^{k/2} .

The equality (4.2) follows upon taking q_{1} = 𝔭^{ej𝔫/l}. This lemma will be proved at the same time as another lemma from which we can deduce the combinatorial facts needed but not yet proved. Suppose T_{0} ⊆ T_{1} is another connected tree and that there issue q_{0} + 1 points from every vertex of T_{0}. Let P and P′ be two points in T_{0} a distance d apart and let r = (k − d)/2 be a non-negative integer.

Lemma 4.2. Let 𝔔 be the set of pairs (x, S) where x = (P_{i}, P_{i−1}, …, P_{i−k}) is a path from P_{i} = P to P_{i−l} = P′ and P_{i′} ∈ T_{0} if and only if i′ ∈ S. The number of points in 𝔔 is

− ∑_{0 ≦ a<r}(q_{0} − 1)\binom{k}{a}q_{1}^{a} + \binom{k}{r}q_{1}^{r} .

This lemma will enable us to count the number of paths in U′ for which the minimum distance to D̄ is 0. The reduced skeleton is then a path in D̄. The distance

d = dist(L̄, h^{u}B^{v}σ^{v𝔫}(L̄))

is constant on D̄. If J^{0} = G then d is 0 if k is even and is 1 if k is odd. If J^{0} is a split Cartan subgroup, then

h^{u}B^{v} = ( \begin{matrix}α^{u}ϖ^{v𝔪′′} & 0 \\ 0 & β^{u}ϖ^{v𝔪′} \end{matrix} )

and

|\frac{α^{u}ϖ^{v𝔪′′}}{β^{u}ϖ^{v𝔪′}} | = |ϖ|^{v(𝔪′′−𝔪′)+uej(𝔪′′−𝔪′)/𝔫} = |ϖ|^{(k′′−k′)} .

Consequently, d = [k′′ − k′] = k − 2r if r is the minimum of k′ and k′′.

To construct the set U′_{0} of points of U′ whose reduced skeleton lies in D̄, we have merely to construct the associated path in the Bruhat-Tits building over 𝔨, for we can uniquely lift the path to a point in U′. If RS is the reduced skeleton, we have only to construct that part of the path lying between i(RS) and i(RS)−l, for we may complete the part of the path to the full infinite path by invoking the periodicity condition (d). Moreover, the unmarked points are irrelevant, for at them we just mark time, and so we might as well discard them and obtain the new period k. The reduced skeleton is also a path in D̄ labeled by the points of RS. That part of it between i(RS) and i(RS) − k joins \overline{M}_{i(RS)} = \overline{M} in D̄_{1} to \overline{M}_{i(RS)−k} = h^{u}B^{v}σ^{v𝔫}(\overline{M}). We complete the reduced skeleton to the complete path segment from \overline{M}_{i(RS)} to \overline{M}_{i(RS)−k} by adding flagella which project into the ambient Bruhat-Tits building. Thus, if n = 8 and i = i(RS), the reduced skeleton could be

and the full path

Applying the lemma with q_{1} = 𝔭^{ej𝔫/l}, we see that the number of points in U′ for which the reduced skeleton lies in D̄ is

(4.4)  f \{\binom{k}{r}𝔭^{ej𝔫r/l} − (q_{0} − 1) ∑_{0 ≦ a<r} \binom{k}{a}𝔭^{ej𝔫a/l}\} ,

if f is the number of orbits of J(h, Q_{𝔭}) in D̄.

To see that this gives us all the combinatorial facts we needed, we just have to run through the various possibilities, adding the occasional simple comment. If J(h, Q_{𝔭})is a split Cartan subgroup, then q_{0} = 1, q_{0} − 1 = 0, and f = 1. Moreover, 𝔫r/l = q is the smallest of 𝔪′ and 𝔪′′. Thus

\frac{|U′_{0}|}{meas J_{0}} = \frac{1}{meas U_{𝔭}} \binom{k}{k′}𝔭^{ejq}

is the anticipated value of φ_{κ}^{j}(h) if k′ ≠ k′′. If k′ = k′′, it is exactly what is required to supplement the contribution to φ_{κ}^{j}(h) from U′ − U′_{0}.

Suppose J(h, Q_{𝔭}) is a Cartan subgroup of J(Q_{𝔭}) but is not split. Then −f(q_{0} − 1) is 2 if the corresponding extension is unramified and 1 if it is ramified. If T (Q_{𝔭}) = J(h, Q_{𝔭}), then

\frac{meas T (Q_{𝔭})\G′(Q_{𝔭})}{meas K′} = \frac{−f(q_{0} − 1)}{meas U_{𝔭}} .

U_{𝔭} is the maximal compact subgroup of T (Q_{𝔭}).

If k is odd, then U′ = U′_{0}. Moreover, the extension can be unramified only if 𝔪 is odd; but then J(Q_{𝔭}) = G′(Q_{𝔭}) and f = 2. The value of φ_{κ}^{j}(h) is seen to be exactly that stated in the list of combinatorial facts to be proved. If k is even, the value of φ_{κ}^{j}(h) is given there as a sum of two terms, the first a sum over 0 ≦ i < k/2. This term is yielded by the second part of (4.4), a sum over 0 ≦ a < r = k/2. The second part of φ_{κ}^{j}(h) was expressed in terms of the integral (4.3). Most of it was accounted for by the contribution from U′ − U′_{0}. The remainder is taken care of by the first term of (4.4). Notice that f must be 1 if k is even.

Suppose, finally, that h is central. Then U′ = U′_{0} and q_{0} = |ϖ_{v}|^{−1}. It is consequently manifest that φ_{κ}^{j}(h) has the anticipated value.

We have still to prove Lemmas 4.1 and 4.2. Since there is nothing else to be done, all symbols apart from those entering the statements of these two lemmas are free. The skeleton of a pair (x, S) in 𝔓 or 𝔔 is S. If (x, S) is in 𝔓, an edge will be called progressive if it is moving away from P. If it is in 𝔔, an edge will be called progressive if it is moving away from T_{0}. An edge that is not progressive will be called retrogressive. The sense of the path is from P_{i} to P_{i−l}.

We represent that part of the skeleton lying between i = i(S) and i − l as

×_{i} ◯ × × ◯ ◯ ◯ × … ×_{i−l}

The points in S are unmarked. If there is to be any pair with S as skeleton, the gaps must all be of even length. Let there be 𝔪 = 𝔪(S) gaps of length 2s_{1}, …, 2s_{𝔪}. Given a pair (x, S), we add to S the integers j′ for which the edge from L_{j′} to L_{j′−1} is progressive if i ≧ j ≧ i − l and j ≡ j′(mod k). The result will be called the frame F. We can recover S from F. To do this, let ϵ_{j} be +1 or −1, according as j is or is not in F. If j_{2} ≦ j_{1}, set

N_{j_{1},j_{2}} = ∑_{j=j_{2}}^{j_{1}} ϵ_{j} .

Then j_{1} ∈ S if and only if N_{j_{1},j_{2}} ≧ 0 for all j_{2}. There are k − s points in a period of F and k − s ≧ k/2.

Conversely, suppose we start from a subset F of Z which is periodic of period l and contains k −s ≧ k/2 points in each period. Define ϵ_{j} and N_{j_{1},j_{2}} as before, and let S be the set of j_{1} for which N_{j_{1},j_{2}} ≧ 0 whenever j_{2} ≦ j_{1}. We verify by induction that S is not empty. Choose j_{1} ∈ F. If j_{1} ∈ S, there is nothing to prove. Otherwise, choose the largest j_{2} ≦ j_{1} for which N_{j_{1},j_{2}} < 0. Then j_{2} < j_{1} because j_{1} ∈ F and N_{j_{1},j_{2}} = 1. Moreover, j_{2} > j_{1} − k because

N_{j_{1},j_{2}−l} = (k − 2s) + N_{j_{1},j_{2}} .

Finally, the set {j_{2} + 1, …, j_{1}} must contain an even number of elements because N_{j_{1},j_{2}+1} = 0. Discard this set and all sets congruent to it modulo k, and pull the remaining points together to obtain a new set F′ of period k − (j_{1} − j_{2}). Since exactly half of the points {j_{2} + 1, …, j_{1}} lie in F, the set F′ contains

k − s − \frac{1}{2} (j_{1} − j_{2})

points within a period and

k − s − \frac{1}{2} (j_{1} − j_{2}) ≧ \frac{1}{2} (k − (j_{1} − j_{2})) .

It is clear that F and F′ have the same skeleton. The induction assumption guarantees that the skeleton of F′ is not empty.

The points in S immediately preceding the gaps, as one moves toward smaller integers, will be called extremities. The integer j_{1} is an extremity if and only if N_{j_{1},j_{2}} = 0 for some j_{2} ≦ j_{1}. To a frame we can attach:

(i) the skeleton;

(ii) the number 𝔪 of extremities within a period;

(iii) to each extremity i_{α}, 1 ≦ α ≦ 𝔪, i − k < i_{α} ≦ i the length 2s_{α} of the succeeding gap;

(iv) the spine, which is obtained from the skeleton by discarding the extremities. If s = ∑ s_{α}, the spine has k − 2s elements in a period and may therefore be empty.

We want to treat the two lemmas in a uniform manner, and to this purpose we introduce, when dealing with Lemma 4.1, the tree T_{0} consisting of P alone. We take P′ to be P, and let q = 0, q_{0} = −1. When dealing with Lemma 4.2, we take q to be q_{0}. Let N(c) be the number of paths of length c in T_{0} joining P to P′. The number of elements in 𝔓 or 𝔔 is

(4.5)  ∑(1 − q/q_{1})^{𝔪}q_{1}^{s}N(k − 2s) ,

the sum being taken over all possible frames. To see this we observe that to construct a path x = (P_{i}, …, P_{i−k}), we first take one of the N(k − 2s) paths from P to P′ in T_{0}, with points labeled by the integers between i and i − l lying in the spine, and then at each extremity add one of the q_{1} − q possible edges from T_{0} into T_{1}, and finally, at all other points of the frame, add one of the q_{1} possible progressive edges.

The expression (4.5) is equal to

(4.6)  ∑_{l=0}^{r} \{∑_{𝔪 ≧ l} \binom{m}{l}(−q)^{l}N(k − 2s)\} q_{1}^{s−1} .

The inner sum is taken over all frames with 𝔪 ≧ l. It can be put in a more manageable form.

If we have a frame F with k − s elements and 𝔪 ≧ l, then s ≧ l. We construct \binom{𝔪}{l} new frames with k − (s − l) elements. Take any subset E of the extremities with l elements (within a period) and, for each element of the subset, add to F the last element of the gap in the skeleton following it. Since the added elements do not lie in F, the result is a frame with k − (s − l) elements. The added elements will not be extremities of F′, because the skeleton S of F is contained in the skeleton S′ of F′. The extremities of F′ are the extremities of F that do not lie in E.

The procedure yields not only F′ but also a subset E′ of its spine. E′ consists of the added elements, and any two elements of E′ are separated by a point in the spine of F′. Conversely, suppose we start with F′ and a set E′ of l separated points in its spine. Remove these l points from F′. The result is still a frame, and the skeleton S of F is contained in the skeleton S′ of F′. The points of E′ lie in the gaps of S. I claim that they are the last points of the gaps in which they lie. Since we may argue by induction, we have only to show that if j is a point of the spine, j′ is the smallest point of the spine with j < j′, j′ < j + l, and we remove j from F′ to obtain F, then the skeleton of F is obtained by removing all j_{1} with j ≦ j_{1} < j′ from the skeleton of F′.

We must certainly remove j. Let j_{1} with j < j_{1} < j + l lie in the skeleton of F′. Suppose first that 0 < j_{1} < j′. Then j_{1} must be an extremity, and so N′_{j_{1},j_{2}} = 0 for some j_{2}. If j_{2} ≦ j, then

N′_{j_{1},j_{2}} = N′_{j_{1},j+1} + N′_{j,j_{2}} ≧ N′_{j,j_{2}} > 0 ,

because j lies in the spine. This is impossible and j < j_{2} < j_{1}. If j_{3} < j_{2}, then

0 ≦ N′_{j_{1},j_{3}} = N′_{j_{1},j_{2}} + N′_{j_{2}−1,j_{3}} = N′_{j_{2}−1,j_{3}} .

Thus, j_{2} − 1 also lies in the skeleton. Iterating, we conclude that

N′_{j_{1},j+1} = 0 .

Hence, N′_{j_{1},j} = −1, and j_{1} is not in the skeleton of F.

Now suppose that j′ ≦ j_{1} < j + l. If j_{1} ≧ j_{2} > j, then

N_{j_{1},j_{2}} = N′_{j_{1},j_{2}} ≧ 0 .

If j − l < j_{2} ≦ j, then

N_{j_{1},j_{2}} = N′_{j_{1},j_{2}} − 1 = N′_{j_{1},j′+1} + (N_{j′,j+1} − 1) + (N_{j,j_{2}} − 1) .

Since j and j′ lie in the spine, all summands on the right are positive or zero.

Let S_{𝔫,l} be the number of ways of choosing l separated points from a cyclic set with 𝔫 elements. We now regard the inner sum in (4.6) as taken over all F′ with k − (s − l) elements, and all possible choices of E′. Since the spine of F′ has k − 2(s − l) elements and there are \binom{k}{s−l} possible choices for F′, the sum (4.6) is equal to

∑_{l=0}^{r} ∑_{s=l}^{r} \binom{k}{s − l}(−q)^{l}N(k − 2s)S_{k−2(s−l),l}q_{1}^{s−l} .

We reverse the order of summation and replace s − l by l to obtain

∑_{s=0}^{r} ∑_{l=0}^{s} \binom{k}{l}(−q)^{s−l}N(k − 2s)S_{k−2l,s−l}q_{1}^{l} .

We then change the order of summation once again, and consider

∑_{s=l}^{r}(−q)^{s−l}N(k − 2s)S_{k−2l,s−l} .

Here we substitute s for s − l to obtain

(4.7)  ∑_{s=0}^{r′}(−q)^{s}N(k′ − 2s)S_{k′,s}

with r′ = r − l, k′ = k − 2l.

To establish the lemmas, we must show that the sum (4.7) has the following values.

(i) When q = 0 and q_{0} = −1, it is 0 unless r′ = 0, and then it is 1.

(ii) When q = q_{0}, it is 1 if r′ = 0 and −(q_{0} − 1) if r′ > 0.

Observe that in the first case d = 0 and k′ = 2r′. If q = 0, the sum reduces to

N(k′)S_{k′,0} .

If q_{0} = −1, then N(k′) = 0 unless k′ = 0 and N(0) = S_{0,0} = 1. If q = q_{0} = −1, the k′ = 2r′ and the sum reduces to

N(0)S_{2r′,r′} = S_{2r′,r′} .

Since S_{2r′,∩′} is 2 if r′ > 0, the value of the sum is again correct. If q = q_{0} = 0, the sum reduces to

N(k′)S_{k′,0} .

Since N(k′) is always 1 when q_{0} = 0 and since S_{k′,0} = 1 for all k′ ≧ 0, the value is again correct.

Suppose that q = q_{0} > 0. We regard S_{k′,s} as the number of ways of choosing a set of s points from 1, …, k′ which is cyclically separated. Let S_{k′,s}^{0} be the number of ways of choosing such X which do not contain k′. We first show that

(4.8)  ∑_{s=0}^{r′}(−q_{0})^{s}N(k′ − 2s)S_{k′,s}^{0} = 1 .

Suppose that X doe snot contain k′. Remove from {1, …, k′} the points of X and their immediate successors. This leaves k′ − 2s points which can be used ot label the edges of a path of length k′ − 2s from P to P′. Starting from this path and the labeling, we construct q_{0}^{s} new paths of length k′. Choose any edge emanating from P and cal it the exceptional edge. The exceptional edge at any other point will be the edge leading toward P. If i ∈ X and there is no i′ < i not in X, then we add to the path an adge issuing from P and then the same edge in the opposite direction. We are allowed to take any but the exceptional edge. These new edges are labeled by i and i + 1. If there is such an i we take the largest and add an adge and its opposite in just the same way, except that it must issue form the final point of the edge labeled by i − 1, and it must not be the exceptional edge. Carrying this out for each i ∈ X, we obtain a path of length k′ from P to P′ labeled by 1, …, k′.

The sum (4.8) is a sum over all paths from P to P′ of length k′ of the sum over s of (−1)^{s} times the multiplicity with which it is obtained by the above construction. The sum over s is easily evaluated. Given a path of length k′, let 𝔫 be the number of subpaths of length two which consist of a move out from a point along an edge which is not exceptional and a return. Then the sum is

∑_{s=0}^{𝔫}(−1)^{s}\binom{𝔫}{s} .

It is 0 unless there are no such subpaths, and then it is 1. A little reflection convinces one that there are no such subpaths in only one case, that of the path which moves out from P along the exceptional edge and returns r′ times, and then proceeds directly to P′. This establishes (4.8).

Let S_{k′,s}^{1} be the number of ways of choosing X so that it does contain k′. To complete the proof of the lemma, we have only to show that

∑_{s=1}^{r′}(−q)^{s−1}N(k′ − 2s)S_{k′,s}^{1} = 1

if r′ ≧ 1 and k′ ≧ 2. A separated subset of {1, …, k′} that contains k′ yields, upon removal of k′, a separated subset of {2, …, k′ − 1} that does not contain k′ − 1. Conversely, a separated subset of {2, …, k′ − 1} that does not contain k′ − 1 yields, upon addition of k′, a separated subset of {1, …, k′}. Thus

S_{k′,s}^{1} = S_{k′−2,s−1}^{0} ,

and our sum is equal to

∑_{s=0}^{r′−1}(−q_{0})^{s}N(k′ − 2 − 2s)S_{k′−2,s−1}^{0} ,

which is have already seen to equal 1.

Appendix.

Suppose (γ, h^{0}) is a Frobenius pair [13]. We choose a Cartan subgroup T over Q such that T_{ad}, the image of T in I_{ad}^{0}, is anisotropic at infinity and 𝔭, and let k_{𝔭} be a finite Galois extension of Q_{𝔭} which splits T. We suppose h^{0} factors through T and let μ^{∨} be the coweight h_{0}^{0} of T. We set

ν^{∨} = ∑_{τ∈𝔊(k_{𝔭}/Q_{𝔭})} τμ^{∨} .

Let {a_{σ,τ} } be a fundamental 2-cocycle of the extension k_{𝔭}/Q_{𝔭}. We define the Weil group W_{k_{𝔭}/Q_{𝔭}} as the set of pairs (x, σ) with x ∈ k_{𝔭}^{×}, and σ ∈ 𝔊(k_{𝔭}/Q_{𝔭}), multiplication being defined by

(x, σ)(y, τ) = (xσ(y)a_{σ,τ} , στ) .

If σ belongs to 𝔊(k_{𝔭}/Q_{𝔭}), we set

a_{σ} = ∏_{τ∈𝔊(k_{𝔭}/Q_{𝔭})} a_{σ,τ}^{στμ^{∨}} .

It lies in

T (k_{𝔭}) ≃ X_{∗}(T) ⊗ k_{𝔭}^{×} .

If w = (x, σ), we set

b_{w} = x^{ν^{∨}}a_{σ} .

Lemma A.1. The 1-cochain w → b_{w} is a cocycle.

It must be verified that

a_{ρ}ρ(a_{σ})a_{ρσ}^{−1} = a_{ρ,σ}^{ν^{∨}} .

The left side is

{∏_{τ} a_{ρ,τ}^{ρτμ^{∨}}} {∏_{τ} ρ(a_{σ,τ})^{ρστμ^{∨}}} {∏ a_{ρσ,τ}^{−ρστμ^{∨}}} .

Replace τ by στ in the first product and use the relation

a_{ρσ,τ} ρ(a_{σ,.τ})a_{ρσ,τ} = a_{ρ,σ}

to obtain

∏_{τ} a_{ρ,σ}^{ρστμ^{∨}} = a_{ρ,σ}^{ν^{∨}} .

Suppose we replace a_{ρ,σ} by

\overline{a}_{ρ,σ} = c_{ρ}ρ(c_{σ})c_{ρσ}^{−1}a_{ρ,σ} .

Then W_{k_{𝔭}/Q_{𝔭}} is replaced by \overline{W}_{k_{𝔭}/Q_{𝔭}}, but

(x, σ) → (xc_{σ}^{−1}, σ)

is an isomorphism from W_{k_{𝔭}/Q_{𝔭}} to \overline{W}_{k_{𝔭}/Q_{𝔭}}. The 1-cochain \{a_{σ}\} is replaced by

σ → \overline{a}_{σ} = a_{σ} ∏_{τ} (c_{σ}^{στμ^{∨}}σ(c_{τ})^{στμ^{∨}}c_{στ}^{−στμ^{∨}})
= a_{σ}c_{σ}^{ν^{∨}}σ(d)d^{−1}

if

d = ∏_{τ} c_{τ}^{τμ^{∨}} .

If we pull back the cocycle \{\overline{b}_{w}\} from \overline{W}_{k_{𝔭}/Q_{𝔭}} to W_{k_{𝔭}/Q_{𝔭}}, we obtain

w → σ(d)b_{w}d^{−1} = w(d)b_{w}d^{−1} ,

because w acts on T (k_{𝔭}) through its projection to σ. At all events, we obtain a cocycle in the same class, and so our constructions do not depend on the choice of a fundamental 2-cocycle.

We choose another Cartan subgroup \overline{T} for which \overline{T}_{ad} is still anisotropic at infinity and 𝔭 and which still splits over k_{𝔭}. We suppose that h̄^{0} : R → I^{0} is conjugate under I(R) to h^{0} and factors through \overline{T}. Let μ̄^{∨} be the coweight h̄_{0}^{0} of \overline{T}. We use it to define the 1-cocycle \{\overline{b}_{w}\}.

Lemma A.2. There is a c in I^{0}(k_{𝔭}), such that

\overline{b}_{w} = cb_{w}w(c^{−1})

for all w.

Since T_{ad} is anisotropic at 𝔭, the coweight ν^{∨} is actually a coweight of the center of I^{0}, and hence b_{w} and a_{σ} have the same image a′_{σ} in I_{ad}^{0}(k_{𝔭}). We use the cocycle {a′_{σ}} to twist I^{0} and obtain a group I over Q_{𝔭}. Then

d_{w} = \overline{b}_{w}b_{w}^{−1}

is a cocycle of W_{k_{𝔭}/Q_{𝔭}} with values in I(k_{𝔭}) and we have to show that it bounds.

The first step is to verify that it takes values in the derived group I_{der} of I and factors through 𝔊(k_{𝔭}/Q_{𝔭}). The difference between μ^{∨} and μ̄^{∨}, and hence that between ν^{∨} and ν̄^{∨}, is a sum of coroots. Since ν^{∨} and ν̄^{∨} are coweights of the center, ν^{∨} = ν̄^{∨}. Thus

d_{w} = a_{σ}\overline{a}_{σ}^{−1}

and factors through 𝔊(k_{𝔭}/Q_{𝔭}). I write d_{σ} instead of d_{w}. If λ is a rational character of I, then λ is orthogonal to coroots and

⟨στμ^{∨}, λ⟩ = ⟨στμ̄^{∨}, λ⟩ .

Consequently, λ(d_{σ}) = 1 and d_{σ} lies in I_{der}.

If I_{sc} is the simply-connected covering group of I_{der}, then

H^{1}𝔊((k_{𝔭}/Q_{𝔭}), I_{sc}(k_{𝔭})) = H^{1}(𝔊(\overline{Q}_{𝔭}/Q_{𝔭}), I_{sc}(\overline{Q}_{𝔭})) = 1 .

If C is the kernel of

I_{sc} → I_{der} ,

then the composition

H^{1}(𝔊(k_{𝔭}/Q_{𝔭}), I_{der}(k_{𝔭})) → H^{1}(𝔊(\overline{Q}_{𝔭}/Q_{𝔭}), I_{der}(\overline{Q}_{𝔭})) → H^{2}(𝔊(\overline{Q}_{𝔭}/Q_{𝔭}), C(\overline{Q}_{𝔭}))

is injective. We show that the image of d_{σ} is trivial.

Choose an integer m so that

mμ^{∨} = μ1^{∨} + μ2^{∨}

where μ1^{∨} is a coweight of T_{sc} and μ2^{∨} is a coweight of the center Z. Then

mμ̄^{∨} = μ̄1^{∨} + μ̄2^{∨} .

For each ρ, σ let b_{ρ,σ} be an m^{th} root of a_{ρ,σ}. Then

∏_{τ} b_{σ,τ}^{στ(μ_{1}^{∨}+μ_{2}^{∨})} (1)

and ∏_{τ} b_{σ,τ}^{στ(μ̄_{1}^{∨}+μ_{2}^{∨})} (2)

are liftings of a_{σ} and \overline{a}_{σ} to I_{sc}(\overline{Q}_{𝔭}) × Z(\overline{Q}_{𝔭}). Moreover

{∏_{τ} b_{σ,τ}^{ρτ(μ̄_{1}^{∨}+μ_{2}^{∨})}} {∏_{τ} b_{σ,τ}^{στ(μ_{1}^{∨}+μ_{2}^{∨})}}^{−1} = {∏_{τ} b_{σ,τ}^{στμ̄_{1}^{∨}}} {∏_{τ} b_{σ,τ}^{στμ_{1}^{∨}}}^{−1} (3)

is a lifting of d_{σ} to I_{dc}(\overline{Q}_{𝔭}).

Let {c_{ρ,σ,τ} } be the boundary of {b_{ρ,σ}}. The boundary of (1) is
{∏_{τ} b_{ρ,τ}^{ρτ(μ_{1}^{∨}+μ_{2}^{∨})}} {∏_{τ} ρb_{σ,τ}^{ρστ(μ_{1}^{∨}+μ_{2}^{∨})}} {∏_{τ} b_{ρσ,τ}^{ρσ,τ(μ_{1}^{∨}+μ_{2}^{∨})}}^{−1} ,

which equals

b_{ρ,σ}^{mν^{∨}} ∏_{τ} c_{ρ,σ,τ}^{ρστ(μ_{1}^{∨}+μ_{2}^{∨})}

It must of course lie in the center of I_{sc} × Z. There is a similar formula for the boundary of (2). Taking account of the Galois action on I(\overline{Q}_{𝔭}), one readily concludes that the boundary of (3) is

ρ_{ρ,σ} = {∏_{τ} c_{ρ,σ,τ}^{ρστ(μ̄_{1}^{∨}+μ_{2}^{∨})}} {∏_{τ} c_{ρ,σ,τ}^{ρστ(μ_{1}^{∨}+μ_{2}^{∨})}}^{−1} .

We should perhaps remind ourselves that μ1^{∨} and μ̄1^{∨} are coweights of different groups, namely, T_{sc} and \overline{T}_{sc}. We may also write this boundary as

e_{ρ,σ} = {∏_{τ} c_{ρ,σ,τ}^{ρστμ̄_{1}^{∨}}} {∏_{τ} c_{ρ,σ,τ}^{ρστμ_{1}^{∨}}}^{−1} .
Let X_{∗}(T_{der}) and X_{∗}(T_{sc}) be the lattices of coweights of T_{der} and T_{sc} and let
Y_{∗} = X_{∗}(T_{der})/X_{∗}(T_{sc}) .

If X^{∗}(T_{der}) and X^{∗}(T_{sc}) are the lattices of weights and

Y^{∗} = X^{∗}(T_{sc})/X^{∗}(T_{der}),

then

Y^{∗} = Hom(Y_{∗}, Q/Z)

and

C(\overline{Q}_{𝔭}) = Hom(Y^{∗}, \overline{Q}_{𝔭}^{×}) .

Replacing T by \overline{T}, we obtain \overline{Y}^{∗} and \overline{Y}_{∗}, but

\overline{Y}_{∗} ≃ Y_{∗}

and

\overline{Y}^{∗} ≃ Y^{∗} .

The isomorphisms are canonical.

If we apply the local duality of Tate for finite 𝔊(\overline{Q}_{𝔭}/Q_{𝔭}) modules, we have only to check that the cup product of {e_{ρ,σ}} with any element of

H^{0}(𝔊(\overline{Q}_{𝔭}/Q_{𝔭}), Y^{∗})

is trivial. An element of this group is represented by a λ ∈ X^{∗}(T_{sc}) with σλ − λ ∈ X^{∗}(T_{der}) for all σ or by a λ̄ ∈ X^{∗}(\overline{T}_{sc}) with σλ̄ − λ̄ ∈ X^{∗}(\overline{T}_{der}). The cup product is

f_{ρ,σ} = {∏_{τ} c_{ρ,σ,τ}^{⟨λ̄,ρστμ̄_{1}^{∨}⟩}} {∏_{τ} c_{ρ,σ,τ}^{⟨λ,ρστμ_{1}^{∨}⟩}}^{−1} .

To be definite, we take λ̄ to be h(λ) where h is an element of I(\overline{Q}_{𝔭}) taking T to \overline{T} and

h(λ)(hth^{−1}) = λ(t) .

We write

⟨λ, ρστμ1^{∨}⟩ = ⟨λ − ρστλ, ρστμ1^{∨}⟩ + ⟨λ, μ1^{∨}⟩ .

Since λ − ρστλ is a weight of T_{der}, it is equal to the restriction to T_{der} of a weight λ′ of T, and

⟨λ − ρστλ, ρστμ1^{∨}⟩ = ⟨λ′, mμ^{∨}⟩ − ⟨λ′, μ2^{∨}⟩ .

Each c_{ρ,σ,τ} is an m^{th} root of unity and

c_{ρ,σ,τ}^{⟨λ′,mμ^{∨}⟩} = 1 .

We treat ⟨λ̄, ρστμ̄1^{∨}⟩ in the same manner, writing

λ̄ − ρστλ̄ = h(λ) − ρστh(λ) = h(λ′) + h(λ′′)

with

λ′′ = (1 − h^{−1}ρστ(h))ρστ(λ) .

λ′′ is a weight of T_{ad}, and so

⟨h(λ′) + h(λ′′), μ2^{∨}⟩ = ⟨h(λ′), μ2^{∨}⟩ = ⟨λ′, μ2^{∨}⟩ .

We conclude that

f_{ρ,σ} = {∏_{τ} c_{ρ,σ,τ} }^{⟨λ̄,μ̄_{1}^{∨}⟩−⟨λ,μ_{1}^{∨}⟩} .

However,

∏_{τ} c_{ρ,σ,τ} = ρ (∏_{τ} b_{σ,τ}) (∏_{τ} b_{ρ,στ}) (∏_{τ} b_{ρσ,τ})^{−1} (∏_{τ} b_{ρ,σ})^{−1} .

We may replace στ by τ in the second factor. The first three terms then form a boundary and {f_{ρ,σ}} is cohomologous to

b_{ρ,σ}^{−[k_{𝔭}:Q_{𝔭}](⟨λ̄,μ̄_{1}^{∨}⟩−⟨λ,μ_{1}^{∨}⟩)} .

However,

⟨λ̄, μ̄1^{∨}⟩ − ⟨λ, μ1^{∨}⟩ = ⟨λ, h^{−1}(μ̄1^{∨}) − μ1^{∨}⟩ = 𝔪⟨λ, h^{−1}(μ̄^{∨}) − μ^{∨}⟩ ,

and

⟨λ, h^{−1}(μ̄^{∨}) − μ^{∨}⟩

is integral because

h^{−1}(μ̄^{∨}) − μ^{∨}

is a coweight of T_{sc}. Since

b_{ρ,σ}^{m} = a_{ρ,σ}

and

{a_{ρ,σ}^{[k_{𝔭}:Q_{𝔭}]}}

is trivial, the lemma is proved.

Suppose k′_{𝔭} ⊆ k_{𝔭} are two finite Galois extensions of Q_{𝔭} that split T. There is a homomorphism w → w′ from W_{k_{𝔭}/Q_{𝔭}} to W_{k′_{𝔭}/Q_{𝔭}} and we may pull back the cocycle {b_{w′}}, but the result may not be cohomologous to {b_{w}}.

Lemma A.3. There is a cocycle c_{w} with values in the units of k_{𝔭}^{×} such that w → b_{w}b_{w′}^{−1} is cohomologous in T (k_{𝔭}) to c_{w}^{ν^{∨}}.

We begin by recalling the manner in which the homomorphism from W_{k′_{𝔭}/Q_{𝔭}} to W_{k_{𝔭}/Q_{𝔭}} is defined [1]. Let {a_{ρ,σ}} be a fundamental 2-cocycle for k_{𝔭}/Q_{𝔭}. For each ρ′ in 𝔊(k′_{𝔭}/Q_{𝔭}) choose a representative ρ̄ in 𝔊(k_{𝔭}/Q_{𝔭}) A fundamental 2-cocycle for k′_{𝔭}/Q_{𝔭} is then

a′_{ρ′,σ′} = N_{k_{𝔭}/k_{𝔭′}} (a_{ρ̄,σ̄}a_{γ,ρ̄σ̄}^{−1}) ∏_{β∈𝔊(k_{𝔭}/k′_{𝔭})} a_{β,γ}

with

γ = ρ̄σ̄ρσ̄^{−1} .

An expression for a′_{ρ′σ′} which is more useful to us is ([1], p. 188)

∏_{β∈𝔊(k_{𝔭}/k′_{𝔭})} a_{ρβ,σ̄}a_{β,ρ̄}a_{β,ρ̄σ̄}^{−1}

if ρ is any lifting of ρ′ to 𝔊(k_{𝔭}/Q_{𝔭}). We apply the coboundary relation to the first factor to obtain

∏_{β} ρa_{β,σ̄}a_{ρ,βσ̄}a_{ρ,β}^{−1}a_{β,ρ̄}a_{β,ρ̄σ̄}^{−1} .

An element of W_{k_{𝔭}/Q_{𝔭}} may be written as xρσ̄ with x ∈ k_{𝔭}^{×}, ρ ∈ 𝔊(k_{𝔭}/k′_{𝔭}), and σ′ ∈ 𝔊(k′_{𝔭}/Q_{𝔭}). It is mapped to

w′ = {∏_{β∈𝔊(k_{𝔭}/k′_{𝔭})} βx} {∏_{β∈𝔊(k_{𝔭}/k′_{𝔭})} a_{β,ρ}} σ′ .

Then b_{w′} is the product of

{∏_{β} βx^{ν^{∨}}} {∏_{β} a_{β,ρ}^{ν^{∨}}}

and

∏_{α′∈𝔊(k′_{𝔭}/Q_{𝔭})} ∏_{β∈𝔊(k_{𝔭}/k′_{𝔭})} σa_{β,α}^{σαμ^{∨}}a_{σ,βᾱ}^{σαμ^{∨}}a_{σ,β}^{−σαμ^{∨}}a_{β,σ̄}^{σαμ^{∨}}a_{β,σ̄ᾱ}^{−σαμ^{∨}}

Here σ is the lifting ρσ̄ of σ′ and α any lifting of α′ to 𝔊(k_{𝔭}/Q_{𝔭}). If

c = ∏_{α′} ∏_{β} a_{β,ᾱ}^{αμ^{∨}}

the first term yields σ(c) and the last c^{−1}. Since we are only interested in the cohomology class of w → b_{w′}, we may drop these two terms. The second term yields

∏_{τ∈𝔊(k_{𝔭}/Q_{𝔭})} a_{σ,τ}^{στμ^{∨}} .

The third and fourth yield

{∏_{β} a_{α,β}^{−1}a_{α,σ̄}}^{ν^{∨}} .

Collecting the information at our disposal, we see that the lemma is valid with

c_{w} = ∏_{β} xβ(x)^{−1}a_{β,ρ}a_{σ,β}^{−1}a_{β,σ̄} .

It must be verified that c_{w} is a unit, but that is a consequence of the next lemma applied to the trivial torus T = GL(1) and both k_{𝔭} and k′_{𝔭}.

Lemma A.4. If λ is a rational character of T over Q_{𝔭} and v the image of w under the homomorphism W_{k_{𝔭}/Q_{𝔭}} → W_{Q_{𝔭}/Q_{𝔭}} = Q_{𝔭}^{×}, then

|λ(b_{w})| = |v|^{⟨λ,μ^{∨}⟩} .

If w = x × σ, the left side is equal to

|x|^{⟨λ,ν^{∨}⟩} ∏_{τ} |a_{σ,τ} |^{⟨λ,μ^{∨}⟩} = {∏_{τ} |x|^{⟨λ,μ^{∨}⟩}} {∏_{τ} |a_{σ,τ} |^{⟨λ,μ^{∨}⟩}} .

Since

(στ)^{−1}σ = τ^{−1}σ^{−1}(a_{σ,τ}) × τ^{−1}

in W_{k_{𝔭}/Q_{𝔭}}, v is equal to

{∏_{τ} τ(x)} {∏_{τ} τ^{−1}σ^{−1}(a_{σ,τ})} .

The lemma follows.

Let 𝔨 be the completion of the maximal unramified extension Q_{𝔭}^{un} of Q_{𝔭}. We have still to explain in detail how the element b of G(𝔨) introduced in [13] is defined. Let D_{1} be the image of GL(1) in T under ν^{∨}. It is an algebraic subgroup over Q_{𝔭}. Let W_{k_{𝔭}/Q_{𝔭}}^{0} be the inverse image of the units in Q_{𝔭}^{×} under the

homomorphism W_{k_{𝔭}/Q_{𝔭}} → W_{Q_{𝔭}/Q_{𝔭}} = Q_{𝔭}^{×}.

Lemma A.5. If k_{𝔭} is sufficiently large, the cocycle {b_{w}} is cohomologous to a cocycle {b′_{w}} with the following two properties:

(i) the restriction of {b′_{w}} to W_{k_{𝔭}/Q_{𝔭}}^{0} takes values in D_{1}(k_{𝔭});

(ii) the image of b′_{w} in T/D_{1}(k_{𝔭}) lies in T/D_{1}(Q_{𝔭}^{un}).

The second property is a consequence of the first, because the first implies that b′_{w} is invariant under W_{k_{𝔭}/Q_{𝔭}}^{0} modulo D_{1}, and the image of W_{k_{𝔭}/Q_{𝔭}}^{0} in 𝔊(k_{𝔭}/Q_{𝔭}) is the inertial group. To obtain a cocycle with the first property, we apply results from Chap. X, §7, of [18]. We may as well suppose that D_{1} is trivial, and hence that {b_{w}} = {b_{σ}} is a cocycle of 𝔊(k_{𝔭}/Q_{𝔭}).

We have a diagram of fields

k_{𝔭}^{un}
⧸  ⧹
k_{𝔭}  Q_{𝔭}^{un}
⧹  ⧸
Q_{𝔭}

and we may regard {b_{σ}} as a cocycle of 𝔊(k_{𝔭}^{un}/Q_{𝔭}). By the corollary of Prop. 11 of [18], its restriction to 𝔊(k_{𝔭}^{un}/Q_{𝔭}^{un}) is cohomologous to the trivial cocycle, and may therefore be assumed to be trivial, for we are willing to enlarge the field k_{𝔭}. Thus {b_{σ}} is the lifting to 𝔊(k_{𝔭}/Q_{𝔭}) of a cocycle of

𝔊(k_{𝔭} ∩ Q_{𝔭}^{un}/Q_{𝔭})

and is trivial on the inertial group. Consequently, {b_{w}} is trivial on W_{k_{𝔭}/Q_{𝔭}}^{0}.

Our purposes demand a strengthening of the previous lemma.

Lemma A.6. Suppose k_{𝔭} is sufficiently large and l_{𝔭} is the maximal unramified extension of Q_{𝔭} in k_{𝔭}. Then {b_{w}} is cohomologous to a product {b′_{w}b′′_{w}}, where {b′_{w}} is the lifting of a cocycle of W_{l_{𝔭}/Q_{𝔭}} with values in T (l_{𝔭}) and {b′′_{w}} is of the form b′′_{w} = d_{w}^{ν^{∨}} where w → d_{w} is a cocycle of W_{k_{𝔭}/Q_{𝔭}} with values in the units of k_{𝔭}^{×}.

We begin with an extension k_{𝔭} over which T splits, and let l_{𝔭} be the unramified extension of Q_{𝔭} with

[l_{𝔭} : Q_{𝔭}] = [k_{𝔭} : Q_{𝔭}] = n .

We shall prove the lemma not for k_{𝔭} but for k′_{𝔭}, the composition of k_{𝔭} and l_{𝔭}, in which l_{𝔭} is the maximal unramified subfield. Schematically we have:

k′_{𝔭}
⧸  ⧹
l_{𝔭}  k_{𝔭}
⧹  ⧸
k_{𝔭} ∩ l_{𝔭}
Q_{𝔭}

If we appeal to Lemma A.3, we see that it is sufficient to take the cocycle {b_{w}} associated to the Weil group W_{k_{𝔭}/Q_{𝔭}}, and then prove that its lifting to W_{k′_{𝔭}/Q_{𝔭}} can be factored as {b′_{w}b′′_{w}}.

The Galois group 𝔊(l_{𝔭}/Q_{𝔭}) is cyclic of order 𝔫 and is generated by the Frobenius element which we shall, during the present proof, denote by Φ. We take a uniformizing parameter ϖ for Q_{𝔭}, which could be 𝔭 itself, and take the fundamental cocycle c_{ρ,σ} of the extension l_{𝔭}/Q_{𝔭} to be

c_{Φ^{i},Φ^{j}} = \{ \begin{matrix}1, & 0 ≦ i, j < n, i + j < n, \\ ϖ, & 0 ≦ i, j < n, i + j ≧ n . \end{matrix}

We may also simplify matters by supposing that the lattice of coweights of T is the free 𝔊(k_{𝔭}/Q_{𝔭})-module generated by μ^{∨}.

Suppose we can find a chain {c_{σ}} of 𝔊(l_{𝔭}/Q_{𝔭}) with values in T (l_{𝔭}) and boundary {c_{ρ,σ}^{ν^{∨}}}. Then we may define a cocycle {d_{w}} of W_{l_{𝔭}/Q_{𝔭}} with values in T (l_{𝔭}) by

d_{w} = x^{ν^{∨}}c_{σ}, w = x × σ .

Suppose in addition that if

c_{Φ} = ∏_{τ∈𝔊(k_{𝔭}/Q_{𝔭})} c_{Φ}(τ)^{τμ^{∨}}

then

∏_{τ} |c_{Φ}(τ)| = |ϖ| .

I claim that if {b_{w′}} and {d_{w′}} denote the liftings of {b_{w}} and {d_{w}} to W_{k′_{𝔭}/Q_{𝔭}}, then {b_{w′}d_{w′}^{−1}} is cohomologous to a cocycle w′ → c_{w′}^{ν^{∨}} where {c_{w′}} is a cocycle with values in the group of units of k′_{𝔭}. To see this we pull back {a_{σ}} and {c_{σ}} to cochains {a_{σ′}} and {c_{σ′}} of 𝔊(k′_{𝔭}/Q_{𝔭}). Their boundaries are obtained by pulling back {a_{ρ,σ}^{ν^{∨}}} and {c_{ρ,σ}^{ν^{∨}}} to {a_{ρ′,σ′}^{ν^{∨}}} and {c_{ρ′,σ′}^{ν^{∨}}}. By local class-field theory, the two cocycles {a_{ρ′,σ′}} and {c_{ρ′,σ′}} are cohomologous, and

a_{ρ′,σ′} = e_{ρ′}ρ′(e_{σ′})e_{ρ′,σ′}^{−1}c_{ρ′,σ′} .

Thus {a_{σ′}} and {e_{σ′}^{ν^{∨}}c_{σ′}} have the same boundary. Because of our simplifying assumption, T (k′_{𝔭}) has no cohomology in dimension 1, and

a_{σ′} = fe_{σ′}^{ν^{∨}}c_{σ′}σ′(f^{−1}) .

In particular,

w′ → u_{w′} = b_{w′}f^{−1}d_{w′}^{−1}w′(f) = b_{w′}f^{−1}d_{w′}^{−1}σ′(f)

takes values in D_{1}(k′_{𝔭}). With the simplifying assumption that the lattice of coweights is freely generated over 𝔊(k_{𝔭}/Q_{𝔭}) by μ^{∨},

D_{1}(k′_{𝔭}) = {x^{ν^{∨}} | x ∈ k′_{𝔭}} .

To establish this claim, I must show that if λ is a rational character of D_{1}, then

|λ(u_{w′})| = 1

for all w′. It will be enough to show this for a rational character of T defined over Q_{𝔭}. Clearly,

|λ(f^{−1}σ′(f))| = 1

and, by Lemma A.4,

|λ(b_{w′})| = |v|^{⟨λ,μ^{∨}⟩}

if v is the image of w′ in Q_{𝔭}^{×}. We check that

|λ(d_{w′})| = |λ(d_{w})| = |v|^{⟨λ,μ^{∨}⟩}

if w′ maps to w in W_{k_{𝔭}/Q_{𝔭}}.

This is easily seen to be so if w = x × 1 with x ∈ l_{𝔭}^{×}, and so the point is to verify it when w = 1 × Φ. Then v = ϖ and

|λ(d_{w})| = ∏_{𝔊(k_{𝔭}/Q_{𝔭})} |c_{Φ}(τ)|^{⟨λ,τμ^{∨}⟩} .

Since ⟨λ, τμ^{∨}⟩ = ⟨λ, μ^{∨}⟩, the right side equals

{∏ |c_{Φ}(τ)|}^{⟨λ,μ^{∨}⟩}

and is, by assumption, |ϖ|^{⟨λ,μ^{∨}⟩}.

To completely prove the lemma we must establish the existence of the chain {c_{σ}}. We first remark that there is an a in k_{𝔭}^{×} for which

ϖ = Nm_{k′_{𝔭}/k_{𝔭}}a .

Set

c_{Φ} = ∏_{τ∈𝔊(k′_{𝔭}/l_{𝔭})} τ(a)^{τμ^{∨}}

and

c_{Φ^{i}} = c_{Φ}Φ(c_{Φ}) … Φ^{i−1}(c_{Φ}), 0 ≦ i < n .

The chain {c_{σ}} then takes values in T (l_{𝔭}) and will have the boundary {c_{ρ,σ}^{ν^{∨}}} if

∏_{i=0}^{𝔫−1} Φ^{i}(c_{Φ}) = ϖ^{ν^{∨}} .

The product on the left is

∏_{𝔊(k′_{𝔭}/Q_{𝔭})} τ(a)^{τμ^{∨}} .

If we take the product over 𝔊(k′_{𝔭}/k_{𝔭}) and then over 𝔊(k_{𝔭}/Q_{𝔭}), we obtain

∏_{𝔊(k_{𝔭}/Q_{𝔭})} ϖ^{τν^{∨}} = ϖ^{ν^{∨}} .

Finally,

∏_{τ} |c_{Φ}(τ)| = ∏_{𝔊(k′_{𝔭}/l_{𝔭})} |τ(a)| = |a|^{[k′_{𝔭}:l_{𝔭}]} = |a|^{[k′_{𝔭}:k_{𝔭}]} = |ϖ| .

In [13] I took 𝔨 to be Q_{𝔭}^{un}, but I should have taken it to be the closure of Q_{𝔭}^{un}, for I am otherwise unable to prove the next lemma. I shall denote the Frobenius automorphism of 𝔨 by σ.

Lemma A.7. If d ∈ 𝔨 and |d| = 1, then the equation

d = cσ(c^{−1}), c ∈ 𝔨,

can be solved.

Since the map c → cσ(c^{−1}) takes Q_{𝔭} to 1, its image is closed, and so we must only verify that it is dense in the group of units of 𝔨^{×}. If y is a unit, we can always find a unit x such that

y ≡ xσ(x^{−1})(mod 𝔭) .

Moreover, the equation

(1 + ap^{k})σ(1 + ap^{k})^{−1} ≡ (a − σ(a))p^{k}(mod 𝔭^{k})

and the simple fact that

a − σ(a) ≡ b(mod 𝔭)

can be solved with an integer a for any given integer b allows us to approximate any y ≡ 1(mod 𝔭). Since D_{1}(𝔨) ≃ 𝔨^{×}, the lemma may be applied to D_{1}(𝔨) as well.

The element b introduced in [13] can now be defined. It lies in I^{0}(𝔨). It is not uniquely determined, but the set

(4)  {cbσ(c^{−1}) | c ∈ I^{0}(𝔨)}

is. We start from a given T, a given, but sufficiently large, k_{𝔭}, and a given fundamental class {a_{ρ,σ}} for k_{𝔭}/Q_{𝔭} and the associated cocycle {b_{w}}. According to Lemma A.6, {b_{w}} is cohomologous to {b′_{w}b′′_{w}}, where b′_{w} and b′′_{w} have the properties specified there. We have the usual homomorphisms

W_{k_{𝔭}/Q_{𝔭}} → W_{Q_{𝔭}/Q_{𝔭}} → Z .

We choose a w_{0} that maps to 1 ∈ Z and set b = b_{w_{0}}. The previous lemma shows that the collection (4) is independent of the particular w_{0} chosen.

The cocycle {b′_{w}} is not unique; it might be replaced by

b′_{w}xw(x^{−1})u_{w}

with x ∈ T (k_{𝔭}), u_{w} ∈ D_{1}(k_{𝔭}). However, x and u_{w} are not arbitrary. The absolute value |λ(u_{w})| must be 1 for any rational character of D_{1}, and the image of x in T/D_{1} must lie in T/D_{1}(l_{𝔭}). By Hilbert’s Theorem 90, there is a v in T (l_{𝔭}) such that

x ≡ v(mod D_{1}(k_{𝔭})) .

Let x = vz. Then

b′_{w}xw(x^{−1})u_{w} = b′_{w}(vw(v^{−1}))(zw(z^{−1})u_{w}) .

We apply Lemma A.7 to zw_{0}(z^{−1})u_{w_{0}} to conclude that the set (4) remains the same. To change the fundamental class {a_{ρ,σ}} does not affect the class of {b_{w}}, and hence does not affect (4). Finally, Lemmas A.2, A.3, and A.7 show that it is not affected by the choice of T and k_{𝔭}.

T has been so taken that its image in I_{ad}^{0} is anisotropic. By the definition of a Frobenius pair, there is therefore a positive rational number r such that

|λ(γ)| = |ϖ|^{τ⟨λ,ν^{∨}⟩}

for all rational characters of T. The element ϖ is again a uniformizing parameter of Q_{𝔭}, and absolute values are taken in \overline{Q}_{𝔭}.

In addition to the group I^{0} (or H^{0}) over Q, I introduced in [13] a group \overline{J}^{0} (or \overline{G}^{0}) over Q_{𝔭}. Its definition did not involve T, but it is easily seen that it is the connected group generated by T and the one-parameter root groups corresponding to roots α for which ⟨α, ν^{∨}⟩ = 0. Consequently, D_{1} lies not only in the center of I^{0} but also in the center of J^{0} and the image of {a_{σ}}, or what is the same, of {b_{w}} in I_{ad}^{0} or J_{ad}^{0}, yields elements of H^{1}(Q_{𝔭}, I_{ad}^{0}) and of H^{1}(Q_{𝔭}, J_{ad}^{0}) which can be used to twist I^{0} and J^{0}, thereby obtaining new groups I and J. The twisting of I^{0} can in fact be extended to a global twisting, but that will not be discussed yet.

Changing {a_{σ}} or {b_{w}} within its cohomology class has the usual effect on J(Q_{𝔭}) and on I(Q_{𝔭}). If {b_{w}} is replaced by xb_{w}w(x^{−1}), then

J(Q_{𝔭}) → {xgx^{−1} | x ∈ J(Q_{𝔭})} .

Since it is easy to keep track of such changes, I feel free to modify {b_{w}} within its class, and indeed to replace \{b_{w}\} by \{b′_{w}\}, where \{b′_{w}\} satisfies the conditions of Lemma A.6, for \{b′′_{w}\} commutes with J(\overline{Q}_{𝔭}). Thus

J(Q_{𝔭}) = {g ∈ J^{0}(𝔨) | bσ(g)b^{−1} = g} .

I claim that

J(Q_{𝔭}) = {g ∈ G(𝔨) | bσ(g)b^{−1} = g} .

When proving this, I may take b to be defined by the cocycle {b′_{w}} constructed in the proof of Lemma A.6. Choose k_{𝔭}, l_{𝔭}, and k′_{𝔭} as in the proof of that lemma with [k_{𝔭} : Q_{𝔭}] = [l_{𝔭} : Q_{𝔭}] = 𝔫. Then

bσ(b) … σ^{n−1}(b) = c = x^{ν^{∨}}, |x| < 1 .

Iterating the relation

g = bσ(g)b^{−1}

we obtain

g = cσ^{n}(g)c^{−1}

and

g = c^{−1}σ^{−n}(g)c ,

or, more generally,

g = c^{m}σ^{mn}(g)c^{−m}

and

g = c^{−m}σ^{−mn}(g)c^{m}

for every positive integer m. We may choose a sequence m_{i} so that {σ^{m_{i}n}(g)} and {σ^{−m_{i}n}(g)} converge. Then {c^{−m_{i}}gc^{𝔪_{i}}} and {c^{m_{i}}gc^{−m_{i}}} converge. Since G is a matrix group, we see, by passing to a larger field with respect to which T can be diagonalized and taking the form of c into account, that this is possible only if c commutes with g. Since the connected component of the centralizer of any positive power of c in G is J^{0}, the centralizer of c in G is connected and equals J^{0} [23].

Although the groups of this paper are simple enought that the existence of the global twisting of I^{0} demanded by the formalism of [13] is clear, it turns out nonetheless to be useful to say a few words about the construction of the cocycle defining this global twisting.

Recall that we started with a Cartan subgroup T of I^{0} defined over Q such that T_{ad}, the image of T in I_{ad}^{0}, is anisotropic at ∞ and 𝔭. If μ̄^{∨} is the coweight of T_{ad} obtained by composing μ^{∨} with T → T_{ad}, then the twisting at 𝔭 is given by the cocycle

α_{𝔭} = \{\overline{a}_{σ}\}

with

\overline{a}_{σ} = ∏_{τ∈𝔊(k_{𝔭}/Q_{𝔭})} a_{σ,τ}^{στμ^{∨}}, a ∈ 𝔊(k_{𝔭}/Q_{𝔭}) .

We define a twisting cocycle at ∞ in exactly the same fashion

α_{∞} = \{\overline{a}_{σ}\}

with

\overline{a}_{σ} = ∏_{τ∈𝔊(C/R)} a_{σ,τ}^{−στμ̄^{∨}}, σ ∈ 𝔊(C/R) .

We have changed the sign in the exponent, but that has no effect on the resultant cohomology class.

Lemma A.8. If I is the group over R obtained by twisting I^{0} by α_{∞}, then I_{ad}(R) is compact.

Let 𝔊(C/R) = {1, σ}. We may take the fundamental cocycle to be

a_{1,1} = a_{1,σ} = a_{σ,1} = 1, a_{σ,σ} = −1 .

Then

\overline{a}_{1} = 1 \overline{a}_{σ} = (−1)^{μ̄^{∨}} .

Every root of T in I^{0} or in I is imaginary. All we must do is verify that the roots of T in I are compact. Let β be a root of T in I^{0} and choose root vectors X_{β}, X_{−β} with

[X_{β}, X_{−β}] = H_{β},

where

λ(H_{β}) = ⟨λ, β^{∨}⟩ .

Since σ(β) = −β,

σ(X_{β}) = cX_{−β} σ(X_{−β}) = dX_{β} .

It is easily seen that c must be real and that cd = 1. Examining the two forms of SL(2) over R, one sees that c > 0 if and only if β is compact. On the other hand, β is compact if and only if ⟨β, μ^{∨}⟩ = 0. If β is not compact, then ⟨β, μ^{∨}⟩ = +1. When we twist by α_{𝔭}, the new action on X_{β} is

X_{β} → (−1)^{⟨β,μ^{∨}⟩}cX_{−β} .

Thus c is replaced by (−1)^{⟨β,μ^{∨}⟩}c, and compact roots remain compact while noncompact roots become compact.

The global twisting of I is by an element α of H^{1}(Q, T_{ad}) whose image in H^{1}(R, T_{ad}) is α_{∞}, in H^{1}(Q_{𝔭}, T_{ad}) is α_{𝔭}, and whose image in H^{1}(Q_{l}, T_{ad}), l ≠ 𝔭, is trivial. Its existence follows from standard results in Galois cohomology, which we will now describe. We first state the appropriate lemma formally.

Lemma A.9. Let T be a torus over Q and μ^{∨} a coweight of T. Suppose T is anisotropic at ∞ and 𝔭. Then

α_{∞} = {∏_{τ∈𝔊(C/R)} a_{σ,τ}^{−στμ^{∨}}} ,
α_{𝔭} = {∏_{τ∈𝔊(k_{𝔭}/Q_{𝔭})} a_{σ,τ}^{στμ^{∨}}}

represent cohomology classes in H^{1}(R, T) and H^{1}(Q_{𝔭}, T), respectively. There is an element α in H^{1}(Q, T) whose local components are trivial everywhere except at ∞ and 𝔭, where they equal α_{∞} and α_{𝔭}.

Let K be a finite Galois extension of Q that splits T. Then

T (K) = X_{∗}(T) ⊗ K^{×},
T (A_{K}) = X_{∗}(T) ⊗ I_{K} .

If C_{K} = I_{K}/K^{×} is the idèle-class group, set

T_{C} = X_{∗}(T) ⊗ C_{K} .

The exact sequence

1 → T (K) → T (A_{K}) → T_{C} → 1

leads to

H^{1}(𝔊(K/Q), T (K)) → H^{1}(𝔊(K/Q), T (A_{K})) → H^{1}(𝔊(K/Q), T_{C}) .

Let β be the element of the middle group with component α_{∞} at ∞, α_{𝔭} at 𝔭, and 1 elsewhere. All we have to do is verify that its image in the final group is trivial. The middle group is

⊕_{v}H^{1}(𝔊(K_{v}/Q_{v}), T (K_{v}))

and we must verify that the product of the images ᾱ_{∞}, ᾱ_{𝔭} of α_{∞} and α_{𝔭} in H^{1}(𝔊(K/Q), T_{C}) is trivial. The Tate-Nakayama isomorphisms are

H^{i}(𝔊(K_{v}/Q_{v}), X_{∗}(T) ≃ H^{i+2}(𝔊(K_{v}/Q_{v}), T (K_{v}))

and

H^{i}(𝔊(K/Q), X_{∗}(T) ≃ H^{i+2}(𝔊(K/Q), T_{C})

There is a diagram

H^{i}(𝔊(K_{v}/Q_{v}), X_{∗}(T) ≃ H^{i+2}(𝔊(K_{v}/Q_{v}), T (K_{v}))
↓  ↓
H^{i}(𝔊(K/Q_{v}), X_{∗}(T) ≃  H^{i+2}(𝔊(K/Q_{v}), T_{C})

The left vertical arrow is corestriction. The right vertical is the composition of

H^{j}(𝔊_{v}, T (K_{v})) ≃ H^{j}(𝔊, T (K ⊗ Q_{v})) → H^{j}(𝔊, T (A_{K})) → H^{j}(𝔊, T_{c})

with j = i + 2, 𝔊_{v} = 𝔊(K_{v}/Q_{v}), 𝔊 = 𝔊(K/Q). The place v of Q has been extended in some way, no matter which, to K. It can be verified without too much difficulty, although it is more than a mere formality, that the diagram is commutative. One examines the proof of the Tate-Nakayama theorems and recalls at the same time the relation between the local and the global fundamental classes. I forego the details, although I have no reference to furnish the reader.

Suppose in particular that i = −1. The corestriction takes the element of H^{−1}((K_{v}/Q_{v}), X_{∗}(T)) represented by λ with Nm_{K_{v}/Q_{v}}λ = 0 to the element of H^{−1}(𝔊(K/Q), V_{∗}(T)) represented by the same λ. Therefore ᾱ_{∞} corresponds to the element of H^{−1}(𝔊(K/Q), X_{∗}(T)) represented by −μ^{∨} and ᾱ_{𝔭} to the element represented by μ^{∨}, and ᾱ_{∞} · ᾱ_{𝔭} is trivial.

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