Modular Forms and ℓ-adic Representations^{∗}

R. P. Langlands

International Summer School on Modular Functions Antwerp 1972

∗ First appeared in Springer Lecture Notes, vol. 349, 1972 Antwerp 1972 2 1. Introduction. This report is another attempt on the part of its author to come to terms with the circumstance that L-functions can be introduced not only in the context of automorphic forms, with which he has had some experience, but also in the context of diophantine geometry. That this circumstance can be the source of deep problems was, I believe, first perceived by E. Artin. He was, to be sure, concerned with forms on GL(1) and with varieties of dimension 0. This remains the only case in which results of any profundity have been obtained. These have been hard won. Their mathematical germ is the theory of cyclotomic fields; itself easy only in comparison to the general theory.

One assertion of the theory, a simple statement of existence, is that an L-functions for GL(1, A(Q)) associated to a Grössencharakter χ trivial on the connected component of GL(1, R), that is, a Dirichlet L-function, is an Artin L-function associated to a character χ′ of the Galois group of a cyclotomic field. In order that such an assertion really have concrete reciprocity laws, such as the quadratic, as consequences, it must be supplemented by a procedure for obtaining χ from χ′. This is usually given locally, not merely at almost every place, but at every place.

There are similarities, as well as dissimilarities, between the role played by cyclotomic equations in the study of GL(1)/Q and that played by elliptic modular varieties in the study of GL(2)/Q. According to the Eichler–Shimura theory, the non-trivial part of the zeta-function of each of the varieties M_{K} ⊗Q studied in a previous lecture can be expressed as a product of L-functions associated to automorphic forms on GL(2, A(Q)). Both Eichler and Shimura, as well as Igusa and Deligne, have contented themselves with results valid for, in one sense or another, almost all primes. This may represent the most important step. Nonetheless, for reasons explained at Nice and elsewhere, I want a result valid at all primes.

The zeta-functions and the automorphic forms can be brought together because of the close relation of both to certain cohomology groups. These groups are introduced, and evaluated in terms of automorphic forms, in §2. In §3 they are interpreted as groups in the étale theory. Combining the results of §2 and §3, we formulate in §4 the problem with which these lectures are concerned.

To explain this problem, which is only partially solved in the later paragraphs, we formulate it as the solution to an existence question. Suppose π′ = ⊗π′_{v} is a representation of GL(2, A(Q)) occurring in the space of automorphic forms. Let W (C/R) be the Weil group of C over R. W (C/R) is an extension of the Galois group 𝔊(C/R) by C^{×}. π′_{∞} is associated to a two-dimensional complex representation σ_{∞} of W (C/R). Suppose σ_{∞} restricted to C^{×} is equivalent to

z → ( \begin{matrix}z^{m}z̄^{n} & 0 \\ 0 & z^{m′}z̄^{n′} \end{matrix} )

with m, n, m′, and n′ in Z. I would then expect that π′ is associated to a diophantine object of some sort (a “motive”—a word with as yet no satisfactory definition!) which is (again in a sense not yet made precise) of rank two. In lieu of precise definitions, which would be premature, we can look for the objects themselves.

Since we are working over Q, any such object should yield for each p and ℓ different from p a two-dimensional representation σ′_{p} of 𝔊(\overline{Q}_{p}/Q_{p}) over some finite extension of Q_{ℓ}. We can twist both σ′_{p} and π′_{p} by a quasi-character of Q_{p}^{×}, at least if it takes values in Q^{×}. The twistings define L-functions and ε-factors. If the object is in any sense associated to π′ these must be the same for σ′_{p} as for π′_{p}. The implications of this demand are discussed at greater length in §4.

In these lectures we take m′ = n, n′ = m, the most interesting case, and, an unfortunate but natural restriction, m ≠ n; and for each ℓ associate to π′ a two-dimensional ℓ-adic representation of 𝔊(\overline{Q}/Q). To obtain a first, and tentative, form of an existence theorem, we must establish the local relations described above.

As was observed this problem cannot yet be completely solved. That it can be treated when there is any ramification whatsoever is only possible because of the recent results of Deligne on the behavior of the modular varieties at primes where they do not have good reduction. Theorems 7.1 and 7.5 are the best we can do at present. To prove them, one has to turn the basic problem, which concerns two very abstract objects, into an elementary assertion. Since the ℓ-adic representations were introduced to study congruences, they contain a great deal of elementary information. To reveal it, all one has to do is unravel the definitions. This is done in §7. However it is a Grothendieck definition we have to unravel; and that is not so easy for ordinary mathematicians.

The most powerful tool available for the study of the representations of GL(2, A(Q)) on the space of automorhpic forms is the Selberg trace formula. We exploit it in §6. The elementary manipulations needed to compare the results of §6 and §7 and to prove the theorems are carried out in §5.

As I have just hinted, the analysis of the ℓ-adic cohomology required in §7 is far from trivial. To carry it out properly requires a command of an elaborate theory, which, when I began to prepare this report a year ago, was almost completely foreign to me, and which I have certainly not yet mastered. This has no doubt resulted in many obscurities and lacunae, but, I hope, no mistakes.

Antwerp 1972 4 I observe in particular that Proposition 7.12, which plays an important role in the discussion, is an extension of the available forms of the Lefschetz theorem which should have been, but was not, verified.

If this report is of any value, it will be because of all I have learned from friends in Bonn, Bures, and New Haven. Conversations and correspondence with Casselman, Deligne, Harder, Lang, and Rapoport have been decisive in its genesis and execution.

2. de Rham Cohomology. Let A be the adèle ring of Q and let A_{f} be the subring of A formed by the elements with coordinate 0 at ∞. Let Z_{f} be the ring of integral elements in A_{f}. Finally let K be an open compact subgroup of G(A_{f}), where G = GL(2), and let

K_{∞} = \{( \begin{matrix}a & b \\ −b & a \end{matrix} ) ∈ G(R)\} .

The map

g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) → \frac{ai + b}{ci + d}

identifies G(R)/K_{∞} with the complex plane minus the real line.

Since

G(A)/K_{∞}K = G(R)/K_{∞} × G(A_{f})/K

and the second factor is discrete, G(A)/K_{∞}K is also a complex manifold. G(Q) acts on it to the left.

Lemma 2.1. If K is amall enough, G(Q) acts freely.

If γ ∈ G(Q) has a fixed point it is conjugate to an element of K_{∞}K. Thus its eigenvalues λ, μ are complex conjugates and, together with their inverses, algebraic integers. Thus they are roots of unity. If, for example,

K ⊆ {k ∈ G(Z_{f}) | k ≡ 1(mod 5)}

then λ + μ = 2 and γ = 1.

I always take K so small that G(Q) acts freely. Then G(Q)\G(A)/K_{∞}K is a complex manifold M_{K}^{o}(C).

Suppose we are given a rational representation μ of G which is defined over Q and which acts on the vector space L and thus, in particular, a representation of G(Q) on L(Q) or on L(C).

L(C) ×_{G(Q)} G(A)/K_{∞}K

Antwerp 1972 5 is a sheaf of complex vector spaces over M_{K}^{o}(C), locally free of rank equal to the dimension of L. This sheaf will be denoted F_{μ}^{K}(C) or, when this is at all compatible with clarity, F_{μ}(C).

The de Rham cohomology groups

H^{i}(M_{K}^{o}, F_{μ}(C))

as well as the groups with compact support

H_{c}^{i}(M_{K}^{o}, F_{μ}(C))

have been studied by many people. We have to review the results of their efforts. We shall be especially concerned with the image

H_{p}^{i}(M_{K}, F_{μ}(C))

of

H_{c}^{i}(M_{K}^{o}, F_{μ}(C)) → H^{i}(M_{K}^{o}, F_{μ}(C)).

By its very definition the sheaf F_{μ}(C) lifts to the trivial sheaf

L(C) × G(A)/K

over the manifold G(A)/K. Any q-form ω with values in F_{μ}(C) lifts to a form ω_{0} on G(A)/K with values in L(C). if g ∈ G(A) let g_{∞} be its projection on G(R) and define η = η(ω) by

η(g) = μ^{−1}(g_{∞}) ω_{0}(g).

The forms η obtained in this way are characterized by:

(i) If γ ∈ G(Q) and if L_{γ} denotes left translation of a form by γ,

L_{γ}η = η.

(ii) If k ∈ K_{∞} and if R_{k} denotes right translation by k,

R_{k}η = μ(k^{−1})η.

(iii) If X_{1}, ⋯, X_{q} belong to 𝔤, the Lie algebra of G(R), and hence define left-invariant vector fields

on G(A)/K then

η(g) : (X_{1}, ⋯, X_{q}) → 0

if one of X_{1}, ⋯, X_{q} belongs to 𝔨, the Lie algebra of K_{∞}.

The boundary operator is easily expressed in terms of η. dω corresponds to dη where

dη(X_{0}, ⋯, X_{q})

is defined to be the sum of

∑_{i=0}^{q}(−1)^{i}(X_{i} + μ(X_{i})) η(X_{0}, ⋯, X̂_{i}, ⋯, X_{q})

and

∑_{i<j}(−1)^{i+j}η([X_{i}, X_{j}], X_{0}, ⋯, X̂_{i}, ⋯, X̂_{j}, ⋯, X_{q}).

Suppose now for simplicity that μ is irreducible, so that μ restricted to Z, the centre of G, is of the form z → μ(z)I, where μ(z) is a scalar. Let V (μ, K) be the space of infinitely differentiable functions φ on G(Q) \ G(A)/K satisfying

φ(zg) = μ^{−1}(z)φ(g)

for z in Z^{o}(R), the connected component of Z(R). The universal enveloping algebra 𝒜 of 𝔤 and the group K_{∞} both act on V (μ, K) by right translations. Call this action r. η may be regarded as a function on the q^{th} exterior power ∧^{q}(𝔤/𝔨) with values in L(C) ⊗ V (μ, K). Condition (ii) may be written

(1)  η(kX_{1}, ⋯, kX_{q}) = (μ ⊗ r)(k)(η(X_{1}, ⋯, X_{q})).

Of course, X → kX is the adjoint of k. The boundary dη becomes the sum of

(2)  ∑_{i=0}^{q} (1 ⊗ r(X_{i}) + μ(X_{i}) ⊗ 1) η(X_{0}, ⋯, X̂_{i}, ⋯, X_{q})

and

(3)  ∑_{i<j}(−1)^{i+j}η([X_{i}, X_{j}], X_{0}, ⋯, X̂_{i}, ⋯, X̂_{j}, ⋯, X_{q}).

These definitions admit of immediate extensions to any space U on which 𝒜 and K_{∞} act, in a consistent manner, according to a representation s. We introduce namely the space C^{q}(U, μ) of functions on Λ^{q}(𝔤/𝔨) with values in L(C) ⊗ U which satisfy (1), with r replaced by s. Since

s(kX) = s(k)s(X)s(k^{−1})

Antwerp 1972 7 if X ∈ 𝔤 and k ∈ K_{∞}, the boundary dη given by (2) and (3), except that r must again be replaced by s, lies in C^{q+1}(U, μ). Thus we can introduce the cohomology groups H^{q}(U, μ) or, if we want to stress the role of s rather than that of U, H^{q}(s, μ). These groups depend covariantly on U. We identify H^{q}(M_{K}^{o}, F_{μ}(C)) with H^{q}(V (μ, k), μ). If V_{c}(μ, K) is the space of functions in V (μ, K) with compact support modulo Z^{o}(R), we may also identify H_{c}^{q}(M_{K}^{o}, F_{μ}(C)) and H_{c}^{q}(V_{c}(μ, K), μ).

Let V_{sp}(μ, K) be the space of all functions φ in V (μ, K) for which

∫_{N(Q)\N(A)} φ(ng) dn = 0

for all g in G(A) and for which

|μ(detg)|^{1/2}Xφ(g)

is square integrable on Z^{o}(R)G(Q) \ G(A) for all X in the universal enveloping algebra. N is the group of matrices of the form

( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) .

Proposition 2.2. The map

H^{1}(V_{sp}(μ, K), μ) → H^{1}(V (μ, K), μ)

is an injection and its image is H_{p}^{1}(M_{K}, F_{μ}(C)).

In order to prove this proposition we have to introduce a large, almost overwhelming, number of auxiliary spaces. W (μ, K) will be the space of infinitely differentiable functions on B(Q)\G(A)/K which satisfy

φ(zg) = μ^{−1}(z) φ(g), z ∈ Z^{o}(R),

taken modulo functions which vanish on some set of the form

S(M, Ω) = \{nak | n ∈ N(A), a = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) ∈ A(A), |\frac{α}{β}| ≥ M, k ∈ Ω\}.

B is the group of super-triangular matrices and Ω is a compact set such that ΩK_{∞}K = Ω. We demand moreover that G(A) = B(A)Ω. Given M there is an N > M and an infinitely differentiable function ε_{M} such that

ε_{M}(hgk) ≡ ε_{M}(g)

if

h ∈ Z^{o}(R)N(A)B(Q)

Antwerp 1972 8 and k ∈ K and such that ε_{M} is 1 on S(N, Ω) and 0 off S(M, Ω). φ and ε_{M}φ represent the same element of W (μ, K). In order to avoid clumsy expressions, we speak, taking all necessary care, of the elements of W (μ, K) as though they were themselves functions. W_{S}(μ, K) is the subspace of W (μ, K) formed by those functions φ for which |μ(detg)|^{1/2}Xφ(g) is square integrable on Z^{o}(R)B(Q) \ S(M, Ω) for all X in the unversal enveloping algebra. This condition is satisfied for all pairs M, Ω if it is satisfied for one. W_{p}(μ, K) consists of those functions φ in W (μ, K) for which

∫_{N(Q)\N(A)} φ(ng) dn

vanishes, as a function of g, on at least one S(M, Ω).

W_{sp}(μ, K) = W_{s}(μ, K) ∩ W_{p}(μ, K).

Finally we let V_{s}(μ, K) be the set of all φ in V (μ, K) for which |μ(detg)|^{1/2}Xφ(g) is squareintegrable on Z^{o}(R)G(Q) \ G(A) for all X in the unversal enveloping algebra.

Lemma 2.3. The two maps V (μ, K) → W (μ, K) and V_{s}(μ, K) → W_{s}(μ, K) are surjective with kernels V_{c}(μ, K).

The two maps are of course obtained simply be regarding an element of the first space as an element of the second. One knows, from the standard reduction theory, that there exist an M and an Ω such that γ ∈ G(Q), γS(M, Ω) ∩ S(M, Ω) not empty imply γ ∈ B(Q). Given φ in W (μ, K), set ψ = ε_{M}φ and consider

φ_{1}(g) = ∑_{B(Q)\G(Q)} ψ(γg).

φ_{1} lies in V (μ, K) and determines the same element of W (μ, K) as φ. Since the complement of the image of S(M, Ω) in Z^{o}(R)G(Q) \ G(A) is relatively compact, φ_{1} lies in V_{s}(μ, K) if it lies in W_{s}(μ, K). The last assertion of the lemma follows from reduction theory.

The universal enveloping algebra and the group K_{∞} act in a consistent way on the space W (μ, K) and its variants, so that the cohomology groups such as H^{q}(W (μ, K), μ) are defined.

Lemma 2.4. The groups H^{q}(W_{p}(μ, K), μ) and H^{q}(W_{sp}(μ, K), μ) are 0.

G(A) is a finite disjoint union

∪_{i}G(Q)g_{i}G^{o}(R)K.

G^{o}(R) is the connected component of G(R). Let

Γ_{i} = G(Q) ∩ g_{i}G^{o}(R)Kg_{i}^{−1}.

The group G(Q) is a disjoint union

∪_{j}Γ_{i}h_{ij}B(Q).

Let h̄_{ij} and ḡ_{i} be the projections of h_{ij} and g_{i} on G(R) and let

S^{o}(M, Ω) = S(M, Ω) ∩ G^{o}(R).

It is a consequence of the reduction theory that, if M is given and N is, for a given Ω, sufficiently large, the map

∪_{i} ∪_{j} Δ_{ij} \ S^{o}(N, Ω)

into B(Q) \ S(M, Ω)/K given by

Δ_{ij}g ∈ Δ_{ij} \ S^{o}(N, Ω) → h_{ij}^{−1}g_{i}ḡ_{i}^{−1}h̄_{ij}g

is injective with an image which differs from B(Q) \ S(M, Ω)/K by a relatively compact set, modulo Z^{o}(R)B(Q). Here Δ_{ij} is the projection of

h_{ij}^{−1}Γ_{i}h_{ij} ∩ B(Q) ∩ G^{o}(R)G(A_{f})

on G(R). Since the operators X in the universal enveloping algebra are left-invariant we can, when proving the lemma, pull back φ in W (μ, K) or W_{s}(μ, K) to a function on Δ_{ij} \S^{o}(N, Ω) and study it there. Thus W (μ, K) becomes a direct sum

⊕_{i,j}W (i, j).

In the same way W_{s}(μ, K) is isomorphic to

⊕_{i,j}W_{s}(i, j).

We study these spaces individually. For simplicity we suppress the i and j from the notations. In particular Δ_{ij} is now Δ, a discrete subgroup of B(R) ∩ G^{o}(R) such that Δ ∩ N(R) \ N(R) is compact and Δ ∩ N(R) \ Δ is finite. If A is the group of diagonal matrices and A^{o}(R) is the connected component of A(R),

G^{o}(R) = N(R)A^{o}(R)K_{∞}.

Antwerp 1972 10 W is the space of infinitely differentiable functions φ on Δ \ G^{o}(R) satisfying

φ(zg) = μ^{−1}(z)φ(g), z ∈ Z^{o}(R),

taken modulo those which vanish on some set of the form

S(M) = \{nak | n ∈ N(R), a = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) ∈ A^{o}(R), \frac{α}{β} > M, k ∈ K_{∞}\}.

W_{s} is defined in a similar way.

Let Δ ∩ N(R) be generated by

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

and if φ belongs to W (μ) let

φ(n, a, k) = φ(nak) = ∑_{z∈Z}exp (\frac{2πizx}{N})φ_{z}(a, k),

where

n = ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) .

φ_{z}(a, k) is an infinitely differentiable function of a and k. We need to know what conditions to place on the φ_{z} in order that φ belong to W_{s}.

The matrices

\begin{aligned}U & = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) \\ H & = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) \end{aligned}

span 𝔤 modulo 𝔨. Any X in the universal enveloping algebra may be written as a finite sum

∑_{m,ℓ} Adk^{−1}(U^{m})Adk^{−1}(H^{ℓ})Y_{m,ℓ}(k)

where each Y_{m,ℓ}(k) is some finite linear combination of elements of the universal enveloping of 𝔨 with coefficients which are infinitely differentable functions of k with bounded derivatives. Applying

Adk^{−1}(U^{m})Adk^{−1}(H^{ℓ})

to φ we obtain

(\frac{α}{β})^{m} ∑_{z} (\frac{2πiz}{N} )^{m} exp (\frac{2πizx}{N})H^{i}φ_{z}(a, k).

Antwerp 1972 11 H belongs to the Lie algebra of A(R) and H^{ℓ}φ_{z} is to be interpreted accordingly. Integrating the square of the absolute value of this expression times |μ(detg)| over S(M) modulo (Δ∩N(R))Z^{o}(R) we obtain

∑_{z} (\frac{2πz}{N} )^{2m} ∫_{\frac{α}{β} ≥M} ∫_{Z^{o}(R)\K_{∞}} (\frac{α}{β})^{2m−1}|H^{ℓ}φ_{z}(a, k)|^{2} |μ(detak)| da dk.

To be more precise the first integral is taken over the subset of Z^{o}(R) \ A^{o}(R) represented by matrices

( \begin{matrix}α & 0 \\ 0 & β \end{matrix} )

satisfying the indicated condition. If φ lies in W_{s} this expression must be finite for all ℓ and m. If Δ contains a matrix

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

then

(4)  exp (\frac{−2πiyz}{N} )φ_{z}(a, −k) = φ_{z}(a, k)

for all z.

If the collection {ψ_{z}(a, k) | z ≠ 0} satisfies (4) and if there is a φ in W_{s} such that for each ℓ ≥ 0 there is a finite set of integers ℓ_{j} ≥ 0 and constants C_{ℓ} and r_{ℓ} such that

|H^{ℓ}ψ_{z}(a, k)| ≤ C_{ℓ}|\frac{α}{β}|^{r_{ℓ}} ∑ |H^{ℓ_{j}}φ_{z}(a, k)|

for all z ≠ 0 and |\frac{α}{β} | ≥ 1 then

∑_{z≠o} exp (\frac{2πizx}{N})ψ_{z}(a, k)

is also in W_{s}.

We can now begin the proof of Lemma 2.4. It is of course enough to prove that the groups H^{q}(W_{p}, μ) and H^{q}(W_{sp}, μ) are 0. W_{p} consists of those φ in W for which φ_{0}(a, k) is identically 0 and W_{sp} is the intersection of W_{s} and W_{p}.

We represent 𝔮/𝔨 and 𝔟/𝔷 where 𝔟 is the Lie algebra of B(R) and 𝔷 that of Z(R). It has a basis formed by U and H. An element of C^{q}(W, μ) is represented by a function on S(M) × 𝔟 with values in L(C), which for convenience we provide with an inner product. Because of (1) it is determined by its values on N(R)A^{o}(R) ∩ S(M). Expand the function in a Fourier series

η(na, X_{1}, ⋯, X_{q}) = ∑_{z} exp (\frac{2πizx}{N})η_{z}(a, X_{1}, ⋯, X_{q}).

Antwerp 1972 12 Then

(U + μ(U))η(na, X_{1}, ⋯, X_{q})

is equal to

(5)  ∑ (\frac{α}{β} \frac{2πiz}{N} + μ(U)) exp (\frac{2πizx}{N})η_{z}(a, X_{1}, ⋯, X_{q})

and

(H + μ(H))η(na, X_{1}, ⋯, X_{q})

is equal to

(6)  ∑ exp (\frac{2πizx}{N})(H + μ(H))η_{z}(a, X_{1}, ⋯, X_{q}).

The operator

A_{z} = \frac{α}{β} \frac{2πiz}{N} + μ(U)

on L(ℂ) is invertible if z ≠ 0 because μ(U) is nilpotent. Moreover for any ℓ ≥ 0 and any ε > 0 there is a constant C_{ℓ} such that

∥H^{ℓ}A_{z}^{−1}∥ ≤ \frac{C_{ℓ}}{|z|} |\frac{β}{α}|

for |\frac{α}{β} | ≥ ε. If

∑_{z} exp (\frac{2πizx}{N})φ_{z}(a, k)

lies in W_{s} the form determined by

ν(na, X_{1}, ⋯, X_{q}) = \{ ∑_{z≠0} exp (\frac{2πizx}{N})A_{z}^{−1}φ_{z}(a, 1)\}λ(X_{1}, ⋯, X_{q}),

where λ is a linear form on 𝔤/𝔨, lies in C^{q}(W_{sp}, μ).

We show now that H^{q}(W_{p}, μ) and H^{q}(W_{sp}, μ) are 0. This will prove Lemma 2.4. It is clear from (5) that any cycle in C^{0}(W_{p}, μ), and a fortiori in C^{0}(W_{sp}, μ), is 0. If η is a cycle in C^{1}(W_{p}, μ) set

ν(na) = ∑_{z≠0} exp (\frac{2πizx}{N})A_{z}^{−1}η_{z}(a, U).

If η belongs to C^{1}(W_{sp}, μ) then the element of C^{0}(W_{p}, μ) corresponding to ν lies in C^{0}(W_{sp}, μ). Replacing η by η − dν if necessary, we may suppose η(U) = 0. Since

[U, H] = −2U

we see, upon computing dη(U, H), that η(H) is also 0. Finally any form in C^{2}(W_{p}, μ) or C^{2}(W_{sp}, μ) is clearly the boundary of a form ν which vanishes on U.

Antwerp 1972 13 Corollary 2.5. The image of H^{q}(V_{sp}(μ, K), μ) in H^{q}(V_{s}(μ, K), μ) is contained in the image of H^{q}(V_{c}(μ, K), μ).

Given a cycle η in C^{q}(V_{sp}(μ, K), μ) choose, by the previous two lemmas, a form ν in C^{q−1}(V_{s}(μ, K), μ) such that η − dν is 0 in C^{q}(W_{s}(μ, K), μ). By Lemma 2.3, η − dν lies in C^{q}(V_{c}(μ, K), μ).

Proposition 2.2 itself will follow from the two lemmas below.

Lemma 2.6. The map

H^{1}(V_{sp}(μ, K), μ) → H^{1}(V_{s}(μ, K), μ)

is surjective.

Lemma 2.7. The maps

H^{q}(V_{sp}(μ, K), μ) → H^{q}(V (μ, K), μ)

are injective.

Let W_{e}(μ, K) be the image in W (μ, K) of the infinitely differentiable functions on the set N(A)B(Q) \ G(A)/K. Set

W′(μ, K) = W_{e}(μ, K) + W_{sp}(μ, K)

and let V′(μ, K) be the inverse image in V (μ, K) of W′(μ, K). Since

V′(μ, K) \ V (μ, K) ≃ W_{sp}(μ, K) \ W_{p}(μ, K),

the maps

H^{q}(V′(μ, K), μ) → H^{q}(V (μ, K), μ)

are isomorphisms. We shall prove Lemma 2.7 with V (μ, K) replaced by V′(μ, K).

There are some standard facts in representation theory that need to be recalled. If s is a representation of G(R) on a Hilbert space and X belongs to the Lie algebra of G(R), a vector u is said to belong to the domain of s(X) if

s(X)u = lim_{t→0} \frac{s(exp tX)u − u}{t}

exists in U. In general u is said to be in the domain of s(X_{n} ⋯ X_{1}) if u is in the domain of s(X_{1}), s(X_{1})u is in the domain of s(X_{2}), and so on. u is infinitely differentiable if it is in the domain of every s(X_{1} ⋯ X_{n}). The space of such vectors will be denoted U^{∞}. If s is unitary and u and v belong to the domain of s(X), then

(s(X)u, v) = −(u, s(X)v).

If U′ ⊆ U^{∞} is a dense G(R)-invariant subspace of U and if u ∈ U is such that

v ∈ U^{∞} → (X_{1} ⋯ X_{n}v, u)

extends to a continuous function on U for all X_{1}, ⋯, X_{n} then u ∈ U^{∞}.

Let L_{s}(μ, K) be the space of all measurable functions φ on G(Q) \ G(A)/K such that

φ(zg) = μ^{−1}(z)φ(g), z ∈ Z^{o}(R),

and such that

|μ(detg)|^{1/2}φ(g)

is square integrable on Z^{o}(R)G(Q) \ G(A). The representation r of G(R) on L_{s}(μ, K) by right translation is not unitary but r_{1} is, where r_{1}(h) takes φ to φ′ with

φ′(g) = |μ(deth)|^{1/2}φ(gh).

Since r and r_{1} have the same infinitely differentiable vectors, it follows from Sobolev’s lemma that

L_{s}^{∞}(μ, K) = V_{s}(μ, K).

If L_{sp}(μ, K) is the subspace of L_{s}(μ, K) consisting of those φ for which

∫_{N(Q)\N(A)} φ(ng) dn = 0

for almost all g, then L_{sp}(μ, K) is closed and invariant and

L_{sp}^{∞}(μ, K) = V_{sp}(μ, K).

Let L_{se}(μ, K) be the orthogonal complement of L_{sp}(μ, K) in L_{s}(μ, K). Then

V_{s}(μ, K) = V_{se}(μ, K) ⊕ V_{sp}(μ, K)

if

V_{se}(μ, K) = L_{se}^{∞}(μ, K).

If φ belongs to V′(μ, K) write

φ = φ_{1} + φ_{2}

where φ_{1} maps to W_{sp}(μ, K) and φ_{2} has support in some G(Q)S(M, Ω) and in S(M, Ω) satisfies

φ_{2}(ng) = φ_{2}(g), n ∈ N(A).

Of course φ_{1} and φ_{2} are both to be functions on G(Q) \ G(A)/K. The possibility of such a construction is assured by Lemma 2.3 If ψ belongs to L_{sp}(μ, K) then, as long as M is, for a given Ω, sufficiently large

(ψ, φ_{1}) = ∫_{Z^{o}(R)G(Q)\G(A)} |μ(detg)|ψ(g)\overline{φ_{1}(g)} dg

is determined by φ alone. There exists a unique φ_{p} in the space L_{sp}(μ, K) such that

(ψ, φ_{1}) = (ψ, φ_{p}).

If X_{1}, ⋯, X_{n} belongs to 𝔮 and if

φ′ = r(X_{1}) ⋯ r(X_{n})φ

then φ′_{p} is equal to r(X_{1}) ⋯ r(X_{n})φ_{p} in the sense of distributions; so φ_{p} lies in V_{sp}(μ, K). If

V_{e}(μ, K) = {φ ∈ V′(μ, K) | φ_{p} = 0},

then

V′(μ, K) = V_{e}(μ, K) ⊕ V_{sp}(μ, K).

Lemma 2.7 follows immediately.

To prove Lemma 2.6 we show that

H^{1}(V_{se}(μ, K), μ) = 0.

This requires a little more preparation. If s is a representations of G on a Hilbert space U, we set

H^{q}(s, μ) = H^{q}(U, μ) = H^{q}(U^{∞}, μ).

Lemma 2.8. (a) Suppose the representation

g → |μ(detg)|^{1/2}s(g)

is unitary and let ω be the Casimir operator of g. If

s(ω) − μ(ω)

has a bounded inverse, then

H^{q}(s, μ) = 0

for all q.

(b) Let λ^{q} be the representation of K_{∞} on the q^{th} exterior power of 𝔤/𝔨 and let μ̃ be the contra-

gredient of μ. If

Hom_{K_{∞}}(λ^{q} ⊗ μ̃, s) = 0

then

H^{q}(s, μ) = 0.

The second part of the lemma is an immediately consequence of the definition, because

C^{q}(U^{∞}, μ) = Hom_{K_{∞}}(Λ^{q}𝔤/𝔨, L(C ⊗ U^{∞}))

and the right-hand side is isomorphic to

Hom_{K_{∞}}(Λ^{q}𝔤/𝔨 ⊗ Hom_{C}(L(C), C), U^{∞}).

To prove the first part, we recall a formula of Kuga, or at least the simple case of it we need. As usual,

𝔤 = 𝔨 ⊕ 𝔪

where 𝔨 and 𝔪 are orthogonal with respect to a non-degenerate G(R)-invariant form whose restriction to the derived algebra 𝔤′ is the Killing form. Introduce an inner product on L(C) with respect to which μ(X), X ∈ 𝔪, is hermitian while μ(X), X ∈ 𝔨 ∩ 𝔤′ is skew-hermitian. If X ∈ 𝔤′ the operator s(X) is skew-hermitian. The inner products on L(C) and U yield one on L(C) ⊗ U^{∞}. Define one on C^{q}(U^{∞}, μ) by

(η, ν) = ∑_{i} (η(Y_{i}), ν(Y_{i}))

where (Y_{i}) is an orthonormal basis, with respect to the Killing form, of Λ^{q}𝔤/𝔨 ≃ Λ^{q}𝔪. There is an operator δ of degree −1 on the complex C^{q}(U^{∞}, μ) such that

(dη, ν) = (η, δν).

Set

Δ = dδ + δd.

If ω is the Casimir operator then

−Δ = s(ω) ⊗ 1 − 1 ⊗ μ(ω) = s(ω) − μ(ω).

Since μ is assumed irreducible, μ(ω) is a scalar.

There is perhaps no harm in verifying this formula for the group under consideration. Let Y_{1} and Y_{2} be an orthonormal basis of 𝔪. Then Y_{1} ∧ Y_{2} is an orthonormal basis of Λ^{2}𝔪 and [Y_{1}, Y_{2}] lies in 𝔨. If η is a 0-form

((dη), ν) = ({s(Y_{1}) + μ(Y_{1})}η, ν(Y_{1})) + ({s(Y_{2}) + μ(Y_{2})}η, ν(Y_{2})).

So

δν = −{(s(Y_{1}) − μ(Y_{1}))ν(Y_{1}) + (s(Y_{2}) − μ(Y_{2}))ν(Y_{2})}.

Here s(Y_{i}) − μ(Y_{i}) is an abbreviation for

s(Y_{1}) ⊗ 1 − 1 ⊗ μ(Y_{i}).

If η is a 1-form,

(dη, ν) = ({s(Y_{1}) + μ(Y_{1})}η(Y_{2}) − {s(Y_{2}) + μ(Y_{2})}η(Y_{1}), ν(Y_{1}, Y_{2}))

so that δν takes

\begin{aligned}Y_{1} & → \{s(Y_{2}) − μ(Y_{2})\}ν(Y_{1}, Y_{2}) \\ Y_{2} & → −\{s(Y_{1}) − μ(Y_{1})\}ν(Y_{1}, Y_{2}). \end{aligned}

Let Z be an element of 𝔨 ∩ 𝔤′ on which the Killing form takes the value −1. Then

ω = Y_{1}^{2} + Y_{2}^{2} − Z^{2}.

If η is a 0-form then

s(Z)η + μ(Z)η = 0

and

{s(ω) − μ(ω)}η = ∑_{i=1}^{2}{s(Y_{i}^{2}) − μ(Y_{1}^{2})}η.

The formulas above show that the right-hand side is

−δdη = −(dδ + δd)η.

Since the value of the Killing form on

( \begin{matrix}x & y \\ z & −x \end{matrix} )

is 4(x^{2} + yz) we may take

Y_{1} = ( \begin{matrix}\frac{1}{2} & 0 \\ 0 & −\frac{1}{2} \end{matrix} ) , Y_{2} = ( \begin{matrix}0 & \frac{1}{2} \\ \frac{1}{2} & 0 \end{matrix} ) , Z = ( \begin{matrix}0 & \frac{1}{2} \\ −\frac{1}{2} & 0 \end{matrix} ) ,

so that

[Y_{1}, Y_{2}] = Z [Z, Y_{1}] = −Y_{2} [Z, Y_{2}] = Y_{1}.

If η is a 1-form, condition (1) implies that

\begin{aligned}\{s(Z) + μ(Z)\}η(Y_{1}) & = −η(Y_{2}) \\ \{s(Z) + μ(Z)\}η(Y_{2}) & = η(Y_{1}). \end{aligned}

Now δdη takes Y_{1} to

{s(Y_{2}) − μ(Y_{2})}({s(Y_{1}) + μ(Y_{1})}η(Y_{2}) − {s(Y_{2}) + μ(Y_{2})}η(Y_{1}))

and dδη takes Y_{1} to

−{s(Y_{1}) + μ(Y_{1})}({s(Y_{1}) − μ(Y_{1})}η(Y_{1}) + {s(Y_{2}) − μ(Y_{2})}η(Y_{2})).

Adding, we obtain

− ∑_{i=1}^{2}{s(Y_{i}^{2}) − μ(Y_{i}^{2})}η(Y_{1})

plus

{s([Y_{2}, Y_{1}]) − μ([Y_{2}, Y_{1}])}η(Y_{2}) = {s(Z) − μ(Z)}{s(Z) + μ(Z)}η(Y_{1})

which is simply

−s(ω)η(Y_{1}) + μ(ω)η(Y_{1}).

Antwerp 1972 19 The complete verification of the formula proceeds along the same lines.

Returning to the lemma, we observe that (s(ω) − μ(ω))^{−1} commutes with s(g), g ∈ G(R), so that it takes U^{∞} to U^{∞}. If η belongs to C^{q}(U^{∞}, μ) and dη = 0, set

ν = (s(ω) − μ(ω))^{−1}η.

Since dΔ = dΔ, we have

dδdν = Δdν = 0.

Thus

(δdν, δdν) = (dδ dν, dν) = 0

and both δdν and dν are 0. Consequently η = dδν bounds.

In order to apply Lemma 2.8 we must recall the structure of the space L_{se}(μ, K). Suppose χ is a quasi-character of A(Q) \ A(A) such that

χ(z) = μ^{−1}(z), z ∈ Z^{o}(R).

Suppose in fact that

a = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) → |αβ|^{1/2}χ(a)

is a character. Consider the space B(χ, K) of all measurable functions φ on G(A)/K, satisfying

φ(nag) = |\frac{α}{β}|^{1/2}χ(a)φ(g)

for n in N(A) and a in A(A), for which

∫_{B(A)\G(A)} |μ(detg)| |φ(g)|^{2}ρ^{−1}(g) dḡ.

Here ρ is a positive function on G(A) such that

ρ(nagu) = |\frac{α}{β}|ρ(g)

if n ∈ N(A, a ∈ A(𝔸), and u ∈ K_{∞}K. Moreover dḡ is the measure associated to ρ as in Theorem 1.1 of [2.2]. The representation π(χ) of G(R) on B(χ, K) by right translations is unitary. If

χ : ( \begin{matrix}α & 0 \\ 0 & α^{−1} \end{matrix} ) → α^{is}

Antwerp 1972 20 for α > 0 in R then

π(ω, χ) = −\frac{s^{2} + 1}{4}.

Of course μ(ω) is positive or 0; so π(ω, χ) − μ(ω) is a real number less than or equal to −\frac{1}{4}.

If φ is an infinitely differentiable function on N(A)B(A) \ G(A)/K with compact support modulo N(A)B(Q)Z^{o}(R) satisfying φ(zg) = μ^{−1}(z)φ(g) for z ∈ Z(R), then

φ(g, χ) = ∫_{A(Q)Z^{o}(R)\A(A)} |\frac{α}{β}|^{− \frac{1}{2}} χ^{−1}(a)φ(ag) da

is a function on B(χ, K). If 𝔄 is the set of quasi-characters of the form indicated, 𝔄 carries a measure, the translation of the Haar measure, and

∫_{𝔄} ∥φ(g, χ)∥^{2} dχ < ∞.

There is thus a unique direct integral

(7)  ∫_{𝔄} B(χ, K) dχ

containing these functions. If

s(g) = ∫_{𝔄} π(g, χ) dχ

then s(ω) − μ(ω) is clearly invertible.

From the theory of Eisenstein series one knows that L_{se}(μ, K) is a direct sum of G(R)-invariant subspaces

L_{se}(μ, K) = L_{se}^{0}(μ; K) ⊕ L_{se}^{1}(μ, K)

the second of which is G(R)-isomorphic to a subspace of (7). The first is a direct sum, over the quasicharacters of the idèle-class group such that ν(a) = μ^{−1}(a) if a > 0 lies in R and ν(detk) = 1 if k ∈ K, of one-dimensional spaces on which G(R) acts according to g → ν(detg). H^{1}(L_{se}^{1}(μ, K), μ) = 0 because of Lemma 2.8(a) and H^{1}(L_{se}^{0}(μ, K), μ) = 0 because of Lemma 2.8(b). Lemma 2.6 now follows.

Lemma 2.9. If the representation s of G(R) on U is such that

g → |μ(detg)|^{1/2}s(g)

is unitary and if

s(ω) − μ(ω) = 0

then

H^{q}(U, μ) ≃ Hom_{K_{∞}}(∧^{q}𝔤/𝔨 ⊗ Hom_{C}(L(C), C), U^{∞}).
The right side is isomorphic to C^{q}(U^{∞}, μ). If η belongs to C^{q}(U^{∞}, μ) then
0 = (Δη, η) = (dη, dη) + (δη, δη).

In particular dη = 0, so the boundary operator is trivial.

For a given μ there are exactly three irreducible admissible representations of G(ℝ), satisfying π(z) = μ^{−1}(z) for z ∈ Z^{o}(R), for which π(ω) = μ(ω). There are (i) μ̃, the contragredient of μ, (ii) g → sgn(detg)μ̃(g), (iii) an infinite-dimensional representation π = π(μ) such that g → |μ(deg g)|^{1/2}π(g) is unitary. If

trace μ ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) = \frac{α^{n}β^{m} − α^{m}β^{n}}{α − β} , n > m,

then g → |detg|^{−1/2}π(g) is, in the notation of Chapter 12 of Jacquet–Langlands, the representation π(σ), if σ is the two-dimensional representation of the Weil group W_{ℂ/ℝ} induced from the representation z → z^{−n}z̄^{−m} of ℂ^{∗}. It follows from the previous lemma and the known structure of π(μ) that H^{1}(π(μ), μ) is two dimensional.

We define L_{s}(μ) and L_{sp}(μ) in the same way as we define L_{s}(μ, K) and L_{sp}(μ, K) except that we drop the condition of K-invariance. G(𝔸) acts on L_{sp}(μ) by right translations. This representation r is a direct sum of irreducible representations

(8)  ⊕_{π}V^{π}.

Each π is a tensor product

π = π_{∞} ⊗ π_{f} = π_{∞} ⊗ (⊗_{p}π_{p}).

We write accordingly

V^{π} = V_{∞}^{π} ⊗ V_{f}^{π}.

If V_{f}^{π}(K) is the space of K-invariant vectors in V_{f}^{π}, then V_{f}^{π}(K) is finite dimensional and

L_{sp}(μ, K) = ⊕_{π}V_{∞}^{π} ⊗ V_{f}^{π}(K).

Each π_{∞} is infinite dimensional and the set {π_{∞}(ω) | π ∈ A, V_{f}^{π}(K) ≠ 0} is discrete. A is the set of π occurring in the sum (8). If A′ = {π ∈ A | π_{∞} ≠ π(μ)} and A′′ = {π ∈ A | π_{∞} = π(μ)}, set

\begin{aligned}U^{1} & = ⊕_{π∈A′}V_{∞}^{π} ⊗ V_{f}^{π}(K) \\ U^{2} & = ⊕_{π∈A′′}V_{∞}^{π} ⊗ V_{f}^{π}(K). \end{aligned}

Then

H^{q}(V_{sp}(μ, K)) ≃ H^{q}(U^{1}, μ) ⊕ H^{1}(U^{2}, μ).

By Lemma 2.8 the first of these groups is 0. Because the set of π in A′′ for which V_{f}^{π}(K) ≠ 0 is finite the second is

⊕_{π_{∞}} ≃_{π(μ)} H^{q}(π_{∞}, μ) ⊗ V_{f}^{π}(K).

Theorem 2.10. The group H_{p}^{1}(M_{K}, F_{μ}(C)) is isomorphic to the direct sum over those π occurring in the representation of G(A) on L_{sp}(μ) for which π_{∞} = π(μ) of

H^{1}(π_{∞}, μ) ⊗ V_{f}^{π}(K).

Moreover for these π, the group H^{1}(π_{∞}, μ) is two dimensional.

Suppose g ∈ G(A_{f}) K′ is another open compact subgroup of G(A_{f}), and g^{−1}K′g ⊆ K. The map h → hg of G(A) to itself factors to a map

R(g) : M_{K′}^{o}(C) → M_{K}^{o}(C).

We map the inverse image R^{∗}(g)F_{μ}^{K}(C) isomorphically to F_{μ}^{K′}(C) by sending a point h × (v × hg) in a fibre of the first to v × h in a fibre of the second. Thus g defines maps, all denoted by R(g),

\begin{aligned}H^{q}(M_{K}^{o}, F_{μ}(C)) & → H^{q}(M_{K′}^{o}, F_{μ}(C)) \\ H_{c}^{q}(M_{K}^{o}, F_{μ}(C)) & → H_{c}^{q}(M_{K′}^{o}, F_{μ}(C)) \end{aligned}

and, most importantly for us,

H_{p}^{q}(M_{K}, F_{μ}(C)) → H_{p}^{q}(M_{K′}, F_{μ}(C)).

The corresponding maps

\begin{aligned}H^{q}(V (μ, K), μ) & → H^{q}(V (μ, K′), μ) \\ H^{q}(V_{c}(μ, K), μ) & → H^{q}(V_{c}(μ, K′), μ) \\ H^{1}(V_{sp}(μ, K), μ) & → H^{1}(V_{sp}(μ, K′), μ) \end{aligned}

are simply those yielded by right translation by g, which takes, for example, V (μ, K) to V (μ, K′).

The map R(g) : M_{K′}(C) → M_{K}(C) is a local homeomorphism and the inverse image of every point is finite. In fact the inverse image of a point represented by h is represented by A(h) = {h′ | h′g = hk, k ∈ B} if B is a set of coset representatives for K/g^{−1}K′g. The fibre of the direct image R_{∗}(g)F_{μ}^{K′}(C) at the point corresponding to h is

⊕_{h′∈A(h)}L(C) × h′.

We map it to the fibre of F_{μ}^{K}(C) at h by sending

⊕v(h′) × h′ → ( ∑ v(h′)) × h.

This yields a map

(9)  R_{∗}(g)F_{μ}^{K′}(C) → F_{μ}^{K}(C)

and hence mappings R(g) on the cohomology groups. In particular it yields

R(g) : H_{p}^{1}(M_{K′}, F_{μ}(C)) → H_{p}^{1}(M_{K}, F_{μ}(C)).

This corresponds to the map

H^{1}(V_{sp}(μ, K′), μ) → H^{1}(V_{sp}(μ, K), μ)

determined by the linear transformation V_{sp}(μ, K′) → V_{sp}(μ, K) which sends φ′ to φ with

φ(h) = ∑_{K/g^{−1}K′g} φ(hkg^{−1}).

I remark that starting from the isomorphism F_{μ}^{K′}(C) → R^{∗}(g)F_{μ}^{K}(C) we obtain by functoriality

(10)  R_{∗}(g)F_{μ}^{K′}(C) → R_{∗}(g)R^{∗}(g)F_{μ}^{K}(C).

The fibre of R_{∗}(g)R^{∗}(g)F_{μ}^{K}(C) at x in M_{K}(C) is the direct sum

⊗_{x′→x}F_{μ}^{K}(C)_{x},

if F_{μ}^{K}(C)_{x} is the fibre of F_{μ}^{K}(C) at x. The map ⊕v(x′) → ∑ v(x′) on fibres yields a map, the trace,

(11)  R_{∗}(g)R^{∗}(g)F_{μ}^{K}(ℂ) → F_{μ}^{K}(ℂ).

The composition of (10) and (11) gives us (9).

Given g we take K′ = K ∩ gKg^{−1} and let

T (g) : H_{p}^{1}(M_{K}, F_{μ}(ℂ)) → H_{p}^{1}(M_{K}, F_{μ}(ℂ))

be R(1)R(g). In terms of H^{1}(V_{sp}(μ, K), K) it is determined by the linear transformation T (g) : φ → φ′ with

φ′(h) = ∑_{K/K′} φ(hkg)

of V_{sp}(μ, K).

If f_{g} is the characteristic function of KgK divided by meas(K) then T (g) is simply the restriction to V_{sp}(μ, K) of

r(f) = ∫_{G(𝔸_{f})} f_{g}(h)r(h) dh

which acts on L_{sp}(μ). Thus the algebra generated over C by the operators T (g) consists of the restrictions to V_{sp}(μ, K) of the operators r(f), where f ∈ ℋ_{C}(K), the algebra of compactly supported functions on G(A_{f}) biinvariant under K. The corresponding action of ℋ_{C}(K) on

H^{1}(V_{sp}(μ, K), μ) ≃ ⊕H^{1}(π_{∞}, μ) ⊗ V_{f}^{π}(K)

is given by the actions π_{f}(f) of f ∈ ℋ_{C}(K) on the V_{f}^{π}(K).

The representation of ℋ_{C}(K) on V_{f}^{π}(K) is irreducible and in fact {π_{f}(f) | f ∈ ℋ_{ℂ}(K)} is the set of all linear transformations of V_{f}^{π}(K). By Proposition 11.1.1 of Jacquet–Langlands the representations π of G(A) occurring in the decomposition of r are mutually inequivalent. The set of such π we called A. Therefore the various representations π_{f} of ℋ_{C}(K) are also equivalent. We deduce immediately:

Antwerp 1972 25 Proposition 2.11. The algebra of linear transformations of H_{p}^{1}(M_{K}, F_{μ}(C)) commuting with the T (g), g ∈ G(A_{f}) is a direct sum over those π in A for which π_{∞} ≃ π(μ) and V_{f}^{π}(K) ≠ 0 of 2 × 2 matrix algebras.

3. ℓ-adic Cohomology. The next step is to define new sheaves, whose cohomology is closely related to that of F_{μ}(C) but which will have a meaning in the étale topology. We may of course introduce

F_{μ}^{K}(Q) = F_{μ}(Q) = L(A) ×_{G(A)} G(A)/K_{∞}K.

For convenience we introduce

L(A_{f}) = L(Q) ⊗_{Q} A_{f}

on which G(A_{f}) acts via the representation μ. We choose an open compact subgroup of L(A_{f}) stable under K and denote it by L(Z_{f}). If L(Z) = L(Q) ∩ L(Z_{f}) then L(Z_{f}) = L(Z) ⊗_{Z} Z_{f}. If g = g_{∞}g_{f} lies in G(A), set gL(Z_{f}) = μ(g)L(Z_{f}) = μ(g_{f})L(Z_{f}) and set

gL(Z) = L(Q) ∩ gL(Z_{f}).

G(Q) acts on

∪_{g∈G(A)} gL(Z) × g = ∪_{g∈G(A_{f})/K} gL(Z) × G(R)gK

to the left and K_{∞}K acts on it to the right. Dividing out by K_{∞}K we obtain

∪_{g∈G(A_{f})/K} gL(Z) × G(R)gK/K_{∞}K.

G(Q) acts freely on this space and if we divide by the action of G(Q) we obtain a sheaf F_{μ}(Z) of Z-modules over M_{K}^{o}(C). This sheaf is locally free. If we tensor F_{μ}(Z) with Q we obtain F_{μ}(Q). If ℓ is a prime and n ≥ 0 we may tensor with Z/ℓ^{n}Z to obtain the sheaf F (Z/ℓ^{n}Z).

The groups H^{q}(M_{K}^{o}, F_{μ}(Z)) and H_{c}^{q}(M_{K}^{o}, F_{μ}(Z)) are finitely generated. We have canonical isomorphisms

H^{q}(M_{K}^{o}, F_{μ}(Q)) ≃ H^{q}(M_{K}^{o}, F_{μ}(Z)) ⊗_{Z} Q

and

H^{q}(M_{K}^{o}, F_{μ}(C)) ≃ H^{q}(M_{K}^{o}, F_{μ}(Q)) ⊗_{Q} C,

as well as exact sequences

0 → H^{q}(M_{K}^{o}, F_{μ}(Z)) ⊗ Z/ℓ^{n}Z → H^{q}(M_{K}^{o}, F_{μ}(Z/ℓ^{n}Z)) → H^{q+1}(M_{K}^{o}, F_{μ}(Z)) ∗ Z/ℓ^{n}Z → 0.

If A is an abelian group A ∗ Z/ℓ^{n}Z is the set of elements of order dividing ℓ^{n} in A. The map a → ℓ^{n−m}a sends A ∗ Z/ℓ^{n}Z to A ∗ Z/ℓ^{m}Z if n ≥ m and the diagram

0 ⟶ H^{q}(M_{K}^{o}, F_{μ}(Z)) ⊗ Z/ℓ^{n}Z ⟶ H^{q}(M_{K}^{o}, F_{μ}(Z/ℓ^{n}Z)) ⟶ H^{q+1}(M_{K}^{o}, F_{μ}(Z)) ∗ Z/ℓ^{n}Z ⟶ 0
↓ ↓ ↓
0 ⟶ H^{q}(M_{K}^{o}, F_{μ}Z)) ⊗ Z/ℓ^{m}Z ⟶ H^{q}(M_{K}^{o}, F_{μ}(Z/ℓ^{m}Z)) ⟶ H^{q+1}(M_{K}^{o}, F_{μ}(Z)) ∗ Z/ℓ^{m}Z ⟶ 0

in which the first two vertical arrows are yielded by the projection

Z/ℓ^{n}Z ⟶ Z/ℓ^{m}Z

is commutative. Thus

lim_{⟵_{n}} H^{q}(M_{K}^{o}, F_{μ}(Z/ℓ^{n}Z)) ≃ H^{q}(M_{K}^{o}, F_{μ}(Z)) ⊗_{Z} Z_{ℓ}

if Z_{ℓ} is the ring of ℓ-adic integers. A fortiori

lim_{⟵_{n}} H^{q}(M_{K}^{o}, F_{μ}(Z/ℓ^{n}Z)) ⊗_{Z_{ℓ}} Q_{ℓ} ≃ H^{q}(M_{K}^{o}, F_{μ}(Q)) ⊗_{Q} Q_{ℓ}.

The groups on the right may be identified with H^{q}(M_{K}^{o}, F_{μ}(Z_{ℓ})) and with H^{q}(M_{K}^{o}, F_{μ}(Q_{ℓ})) respectively.

Let A be any of the rings C, Q, Z, Z_{ℓ}, Q_{ℓ}, or Z/ℓ^{n}Z. If g ∈ G(A_{f}) and we let gF_{μ}(A) be defined in the same way as F_{μ}(A) except that L(Z) is replaced by gL(Z), then the analogues of all the above assertions remain true. Of course

gF_{μ}(A) = F_{μ}(A)

if A is C, Q, or Q_{ℓ}. If g^{−1}K′g ⊆ K, we have, just as in the previous paragraphs, maps

\begin{aligned}R(g) & : H^{q}(M_{K}^{o}, F_{μ}(A)) ⟶ H^{q}(M_{K′}, gF_{μ}(A)) \\ R(g) & : H^{q}(M_{K′}^{o}, gF_{μ}(A)) ⟶ H^{q}(M_{K}^{o}, F_{μ}(A)). \end{aligned}

In particular if we take K′ = gKg^{−1} ∩ K and choose the scalar matrix a(g) so that a(g)g stablizes L(Z_{f}), we may define

(1)  T (g) : H^{q}(M_{K}^{o}, F_{μ}(A)) ⟶ H^{q}(M_{K}^{o}, a^{−1}(g)F_{μ}(A))

as the composition

H^{q}(M_{K}^{o}, F_{μ}(A))⟶^{R(g)}H^{q}(M_{K′}^{o}, gF_{μ}(A))⟶H^{q}(M_{K′}^{p}, a^{−1}(g)F_{μ}(A))⟶^{R(g)}H^{q}(M_{K}^{o}, a^{−1}(g)F_{μ}(a))

in which the middle arrow is obtained from the embedding gF_{μ}(A) → a^{−1}(g)F_{μ}(A). These maps are all compatible with the indentifications made above. Exactly analogous assertions are valid for cohomology with compact support. Observe also that in the limit (1) yields

H^{q}(M_{K}^{o}, F_{μ}(Q_{ℓ})) ⟶ H^{q}(M_{K}^{o}, a^{−1}(g)F_{μ}(Q_{ℓ})) ≃ H^{q}(M_{K}^{o}, F_{μ}(Q_{ℓ})).

In the following we shall be interested in H^{1}(M_{K}^{o}, F_{μ}(Q_{ℓ})) and H_{c}^{1}(M_{K}^{o}, F_{μ}(Q_{ℓ})) which we regard as the tensor product over Z_{ℓ} of Q_{ℓ} with

lim_{←} H^{1}(M_{K}^{o}, F_{μ}(Z/ℓ^{n}Z))

and

lim_{←} H_{c}^{1}(M_{K}^{o}, F_{μ}(Z/ℓ^{n}Z))

respectively. These last two groups we shall later identify with groups given by the étale cohomology. We first remark a consequence of Propositions 2.11. Note that the image H_{p}^{1}(M_{K}^{o}, F_{μ}(Q_{ℓ})) of H_{c}^{1}(M_{K}^{o}, F_{μ}(Q_{ℓ})) in H^{1}(M_{K}^{o}, F_{μ}(Q_{ℓ})) may be identified with

H_{p}^{1}(M_{K}, F_{μ}(Q)) ⊗ Q_{ℓ}.

Let ℋ\overline{Q}(K) be the subalgebra of ℋ_{C}(K) formed by the linear combinations of the functions f_{g} with coefficients from \overline{Q}. For this we must have an imbedding of \overline{Q} in C. We fix once and for all such an imbedding as well as an imbedding of \overline{Q} in \overline{Q}_{ℓ}. Set

ℋ_{\overline{Q}_{ℓ}}(K) = ℋ\overline{Q}(K) ⊗\overline{Q} \overline{Q}_{ℓ}.

The first algebra acts on H_{p}^{1}(M_{K}, F_{μ}(\overline{Q})); the second acts on H_{p}^{1}(M_{K}, F_{μ}(\overline{Q}_{ℓ})). Tensoring the first action with C, we obtain the action of ℋ_{C}(K) on H_{p}^{1}(M_{K}, F_{μ}(C)). Let ρC^{π} be the representation of ℋ_{C}(K) on V_{f}^{π}(K). Proposition 2.11 yields the following assertion immediately.

Antwerp 1972 28 Proposition 3.1. For each π in A for which V_{f}^{π}(K) ≠ 0 there is a representation ρ\overline{Q}^{π} of ℋ\overline{Q}(K) which extended to ℋ_{C}(K) yields ρC^{π}. The space H_{p}^{1}(M_{K}, F_{μ}(\overline{Q})) is a direct sum over \{π ∈ A | π_{∞} ≃ π(μ) and V_{f}^{π}(K) ≠ 0\} of tensor products U\overline{Q}^{π} ⊗ V\overline{Q}^{π}(K). U\overline{Q}^{π} is two-dimensional. Moreover each of these subspaces is invariant under the action of ℋ\overline{Q}(K) and this algebra acts on U\overline{Q}^{π} ⊗ V\overline{Q}^{π}(K) according to 1 ⊗ ρ\overline{Q}^{π}. The same assertions are valid for ρ\overline{Q}_{ℓ}^{π}, the linear extension of ρ\overline{Q}^{π} to ℋ\overline{Q}(K), if U\overline{Q}_{ℓ}^{π} = U\overline{Q}^{π} ⊗ \overline{Q}_{ℓ} and V\overline{Q}_{ℓ}^{π}(K) = V\overline{Q}^{π}(K) ⊗ \overline{Q}_{ℓ}.

The manifold M_{K}^{o}(C) is the set of complex points on a nonsingular algebraic curve defined over Q which I denote M_{K}^{o} ⊗ Q. The next step is to interpret the sheaves F_{μ}(Z/ℓ^{n}Z) as sheaves in the étale topology of M_{K}^{o} ⊗ \overline{Q}.

Choose K_{0} normal in K so that K_{0} acts trivially on

L(Z_{f})/ℓ^{n}L(Z_{f}) ≃ L(Z/ℓ^{n}Z).

I map F_{μ}(Z/ℓ^{n}Z) to

L(Z/ℓ^{n}Z) ×_{K/K_{0}} M_{K_{0}}(C)

by means of hv ×h → v ×v if v lies in L(Z/ℓ^{n}Z) and h in G(A) projects to x in M_{K}(C). The action of K/K_{0} is, by the way, given by v × x → k^{−1}v × xk. The map just introduced is well-defined because γhv × γhk with γ in G(Q) and k in Kis sent to k^{−1}v × xk. It is a local homeomorphism and a bijection. In fact if hv × h and h_{1}v_{1} × h_{1} have the same image h_{1} = γhk with γ in G(Q) and k in K, then v_{1} × xk is equivalent to v × x modulo the action of K/K_{0}; so v_{1} = k^{−1}v and

h_{1}v_{1} × h_{1} = γhv × γhk

defines the same element of F_{μ}(Z/ℓ^{n}Z) as hv × h.

The product

L(Z/ℓ^{n}Z) ×_{K/K_{0}} (M_{K_{0}}^{o} ⊗ Q)

is defined as a group object in the category of schemes étale over M_{K}^{o} ⊗ Q. Tensoring with \overline{Q} we obtain a sheaf for the étale topology of M_{K}^{o} ⊗ \overline{Q}. Because we have an imbedding of \overline{Q} in C, the étale cohomology groups of this sheaf, with or without compact support, may be identified with those of F_{μ}(Z/ℓ^{n}Z). We may, and shall, also regard H_{p}^{1}(M_{K}, F_{μ}(Q_{ℓ})) as the tensor product of Q_{ℓ} with the image of

lim_{←} H_{c}^{1}(M_{K}^{o}, F_{μ}(Z/ℓ^{n}Z)) ⟶ lim_{←} H^{1}(M_{K}^{o}, F_{μ}(Z/ℓ^{n}Z)),

Antwerp 1972 29 where of course the two groups are to be taken in the étale cohomology.

Both maps R(g), and hence T (g), may be defined in the étale cohomology, once we have established that R^{∗}(g)F_{μ}^{K}(Z/ℓ^{m}Z) and gF_{μ}^{K′}(Z/ℓ^{m}Z) are isomorphic. Recall that there is a map R(g) : M_{K′}^{o} ⊗ Q → M_{K}^{o} ⊗ \overline{Q}, which when applied to the C-valued points yields the map R(g) already introduced. Tensoring with \overline{Q}, we obtain R(g) : M_{K′}^{o} ⊗ \overline{Q} → M_{K}^{o} ⊗ \overline{Q}.

R^{∗}(g) applied to F_{μ}^{K}(Z/ℓ^{n}Z), regarded now as a sheaf in the étale topology and, in fact, at first as a sheaf over M_{K}^{o} ⊗ Q, gives

(2)  (M_{K′} ⊗ Q) ×_{M_{K}⊗Q}(L(Z/ℓ^{n}Z) ×_{K/K_{0}}(M_{K_{0}}^{o} ⊗ Q)).

The sheaf gF_{μ}^{K′}(Z/ℓ^{n}Z) is

(3)  gL(Z/ℓ^{n}Z) ×_{K′/K′_{0}}(M_{K′_{0}}^{o} ⊗ Q).

We may suppose g^{−1}K′_{0}g ⊆ K_{0}. We may map

(4)  gL(Z/ℓ^{n}Z) × (M_{K′_{0}}^{o} ⊗ Q)

to (2) by taking a cartesian product of maps to the two factors. The map to M_{K′}^{o} ⊗ Q is obtained by composing R(e) : M_{K′_{0}}^{p} ⊗ Q → M_{K′}^{o} ⊗ Q with projection on the second factor of (4). The map g^{−1} × R(g) from (4) to L(Z/ℓ^{n}Z) × (M_{K_{0}}^{o} ⊗ Q) followed by the projection from this space to its quotient by K/K_{0} is the map to the second factor of (2). The map from (4) to (2) is easily seen to factor through (3). The simplest way to see that the resultant map from (3) to (2) is an isomorphism is to look at its effect on the C-valued points and then to invoke the comparison theorem. In terms of our representation of the C-valued points as coset spaces of G(A), the map in question is: gv × h → h × v × hg. It is easily seen to be an isomorphism and, indeed, when the redefinition of the sheaves F_{μ}(Z/ℓ^{n}Z) is taken into account, to be the isomorphism used, explicitly or implicitly, in the original discussion. The isomorphism over \overline{Q} is obtained by base change.

We have abbreviated an expression like

(M_{K}^{o} ⊗ Q) ×_{Spec Q}Spec \overline{Q}

to M_{K}^{o} ⊗ \overline{Q}. From the proper form one sees that 𝔊(\overline{Q}/Q) acts on M_{K} ⊗ \overline{Q} through its action on the second factor. To be specific s ∈ 𝔊(\overline{Q}/Q) acts on Spec \overline{Q} to the left and the corresponding action on \overline{Q} is x → s^{−1}(x). We shall denote the action of s on M_{K}^{o} ⊗ \overline{Q}, for any K, by L(s).

Antwerp 1972 30

It is clear that the maps L(s) and R(g) commute. Hence we may also let L(s) act on F_{μ}(Z/ℓ^{n}Z)—now pulled back to the scheme M_{K} ⊗ \overline{Q}. Since

F_{μ}(Z/ℓ^{n}Z) ⟶^{L(s)} F_{μ}(Z/ℓ^{n}Z)
↓ ↓
M_{K} ⊗ \overline{Q} ⟶^{L(s)} M_{K} ⊗ \overline{Q}

is commutative and L(s) is invertible the upper right corner may be regarded, by means of the diagonal map to the lower left corner, as L^{∗}(s)F_{μ}(Z/ℓ^{n}Z) and the inverse of the upper horizontal arrow is a map

L^{∗}(s)F_{μ}(Z/ℓ^{n}Z) ⟶ F_{μ}(Z/ℓ^{n}Z).

These maps yield a representation s → ρ(s) of 𝔊(\overline{Q}/Q) on the groups H^{q}(M_{K}^{o}, F_{μ}(Z/ℓ^{n}Z)) and, if one takes into account the completion of M_{K}^{o}⊗Qintroduced in [3.1], on H_{c}^{q}(M_{K}^{o}, F_{μ}(Z/ℓ^{n}Z)). Hence, because of the obvious consistency, the group 𝔊(\overline{Q}/Q) also acts on H^{q}(M_{K}^{o}, F_{μ}(Q_{ℓ})), H_{c}^{q}(M_{K}^{o}, F_{μ}(Q_{ℓ})), and on H_{p}^{q}(M_{K}, F_{μ}(Q_{ℓ})). ρ(s) acts to the right while T (g) acts to the left. The two actions, because of their construction, commute.

We take H_{p}^{1}(M_{K}, F_{μ}(Q_{ℓ})), tensor with \overline{Q}_{ℓ} to obtain H_{p}^{1}(M_{K}, F_{μ}(\overline{Q}_{ℓ})), and apply Proposition 3.1 to see that ρ, a representation over \overline{Q}_{ℓ}, is a direct sum

⊗_{π}σ(π) ⊗ 1

where σ(π) is a two-dimensional representation on U\overline{Q}_{ℓ}^{π} and 1 acts on V\overline{Q}_{ℓ}^{π}(K).

4. The basic problem. We may, and it is in fact convenient to do so, form the direct limit

lim_{⟶_{K}} H_{p}^{1}(M_{K}, F_{μ}(\overline{Q}_{ℓ})).

Proposition 3.1 and standard facts about tensor products show that it is a direct sum

(1)  ⊕_{\{π∈A | π_{∞}≃π(μ)\}}U\overline{Q}_{ℓ}^{π} ⊗ V\overline{Q}_{ℓ}^{π}

where

V\overline{Q}_{ℓ}^{π} = lim_{⟶_{K}} V\overline{Q}_{ℓ}^{π}(K).

The operators R(g) yield a representation of G(A) on V\overline{Q}_{ℓ}^{π} and V\overline{Q}_{ℓ}^{π}(K) may be regarded as the space of K-invariant vectors in V\overline{Q}_{ℓ}^{π}. The space of K-invariant vectors in (1) may be identified with Antwerp 1972 31 H_{p}^{1}(M_{K}, F_{μ}(\overline{Q}_{ℓ})). This justifies our failure, in Proposition 3.1, to include the dependence of U\overline{Q}_{ℓ}^{π} on K in the notation. This shows also that the representation σ(π) depends only on π and not on K and, since K may be chosen arbitrarily small, that it is defined for all π for which π_{∞} ≃ π(μ) for some μ.

Given such a π let

π′(g) = |detg|^{−1/2}π(g), g ∈ G(Q).

As we observed in the second paragraph, π′ is any representation of G(A) occurring in the space of cusp forms such that π′_{∞} is a π(σ) where σ is a two-dimensional representation of W_{C/R} whose restriction to C^{×} is of the form

z → ( \begin{matrix}z^{−n}z̄^{−m} & 0 \\ 0 & z^{−m}z̄^{−n} \end{matrix} )

with m ≠ n.

By restriction σ = σ(π) yields for every prime p and in particular for every p ≠ ℓ, a condition we shall always impose, a representation σ_{p} = σ_{p}(π) of the decomposition group 𝔊(\overline{Q}_{p}/Q_{p}). Of course only the class of σ_{p} is uniquely determined.

Let 𝒲(\overline{Q}_{p}/Q_{p}) be the subgroup of 𝔊(\overline{Q}_{p}/Q_{p}) formed by those s which project to an integral power of the Frobenius in 𝔊(F̄_{p}/F_{p}) if F_{p} = GF (p). Recall also that we have fixed an imbedding of \overline{Q} in \overline{Q}_{ℓ}. Suppose σ_{p} is a continuous two-dimensional representation of 𝔊(\overline{Q}_{p}/Q_{p}) over some finite extension of Q_{ℓ} such that

trace σ_{p}(s) ∈ \overline{Q}

for all s in 𝒲(\overline{Q}_{p}/Q_{p}).

I want to describe how to associate, in so far as it is possible at present, to such a σ_{p} an irreducible admissible representation π(σ_{p}) of GL(2, Q_{p}). There are two cases to consider separately. Either there exists a finite Galois extension F of Q_{p} such that σ_{p} factors through the Galois group, 𝔊(F^{un}/Q_{p}), of the maximal unramified extension of F over Q_{p} or it does not. In the second case I say, for lack of a better terminology, that σ_{p} is special.

Suppose first of all that σ_{p} is not special. 𝔊(F^{un}/F) is a normal subgroup of 𝔊(F^{un}/Q_{p}) and is injected into the abelian quotient group 𝔊(F̄_{p}/F_{p}). It is therefore central. If s ∈ 𝔊(F^{un}/Q_{p}) let σ_{p}(s) = σ_{1}σ_{2} where σ_{1} is semi-simple and σ_{2} is unipotent, the two matrices being supposed to commute. Some power s^{n} of s is central. Since σ_{p}(s^{n}) = σ_{1}^{n}σ_{2}^{n}, σ_{2}^{n} commutes with all σ_{p}(t). So does σ_{2}. Then s → σ_{1} is a representation σ′_{p} of 𝔊(F^{un}/Q_{p}) by semi-simple matrices. σ′_{p} Antwerp 1972 32 restricted to the finitely generated group 𝒲(F^{un}/Q_{p}) is a representation by matrices with entries from a finitely generated subfield k of \overline{Q}_{p}. Imbedding k in C we obtain a complex representation of 𝒲(F^{un}/Q_{p}) which when combined with the standard homomorphism of the Weil group W_{F/Q_{p}} onto 𝒲(F^{un}/Q_{p}) yields a representation σ′′_{p} of W_{F/Q_{p}} by complex semi-simple matrices. The irreducible admissible representation π(σ′′_{p}) of GL(2, Q_{p}), in so far as it is known to exist, has been described in Chapter 12 of Jacquet–Langlands. We set π(σ_{p}) = π(σ′′_{p}). It should of course be observed that the class of σ′′_{p} does not depend on the imbedding of k in C, because it has been assumed that trace σ(s) lies in \overline{Q} for s in 𝒲(F^{un}/Q_{p}).

For convenience I recall how π(σ_{p}) is defined for the representations of most interest to us. Suppose σ′′_{p} is the direct sum of two one-dimensional representations λ and ν, which may be regarded as quasi-characters of Q_{p}^{×}. Let ρ(λ, ν) be the representations of G(Q_{p}) by right translations in the space of locally constant functions φ on G(Q_{p}) satisfying

φ (( \begin{matrix}α & x \\ 0 & β \end{matrix} ) g) = λ(α)ν(β)|\frac{α}{β}|^{1/2}φ(g).

If λν^{−1} is not of the form α → |α|^{±1} then ρ(λ, ν) is irreducible and is in fact π(σ′′_{p}). Otherwise it has a composition series with two terms, one of which π(σ′′_{p}) is finite dimensional. The other σ(λ, ν) is a so-called special representation.

To see the significance of this terminology, suppose now that σ_{p} itself is special. Let Q_{p}^{t} be the union of all finite tamely ramified extensions of Q_{p}. 𝔊(\overline{Q}_{p}/Q_{p}^{t}) is a pro p-group. Thus any ℓ-adic representation is trivial on a subgroup of finite index. There is a finite Galois extension F of Q_{p} such that σ_{p} is trivial on 𝔊(\overline{Q}_{p}/F^{t}). σ_{p} is thus a representation of 𝔊(F^{t}/Q_{p}). Moreover, 𝔊(F^{t}/F^{un}) is a normal subgroup isomorphic to ∏_{q≠p} Z_{q}. Enlarging F is necessary, we suppose that the restriction of σ_{p} to 𝔊(F^{t}/F^{un}) factors through the projection on Z_{ℓ}. The action of 𝔊(F^{un}/F) on 𝔊(F^{t}/F^{un}) is such that the n^{th} power of the Frobenius, relative to F, sends s to s^{q^{n}} if q is the number of elements in the finite field corresponding to F. Thus if λ is an eigenvalue of σ_{p}(s) so is λ^{q^{n}}. Since σ_{p}(s) has only a finite number of eigenvalues they are all roots of unity. Enlarging F is necessary we may suppose that σ_{p}(s) is unipotent if s projects to 1 in Z_{ℓ} and hence for all s. Because σ_{p} is special, σ_{p}(s) cannot be 1 for that s projecting to 1. Let it have the form

( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) .

This may of course be effected by an appropriate choice of basis. Since 𝔊(F^{t}/F^{un}) is normal in 𝔊(F^{t}/Q_{p}), σ_{p} is a representation of the form

s → ( \begin{matrix}μ(s) & ∗ \\ 0 & ν(s) \end{matrix} )

where μ and ν are quasi-characters. Restricted to 𝒲(F^{t}/Q_{p}) they yield as before two quasicharacters, which we again call μ and ν, of Q_{p}^{×}. Because of our assumption on the traces these take values in \overline{Q}. Taking the action of 𝔊(F^{t}/Q_{p}) on 𝔊(F^{t}/F^{un}) into account we see that μν^{−1} takes α to |α|^{−1}. We take π(σ_{p}) to be the special representation σ(μ, ν).

We consider again a π occurring in the space of cusp forms for which π_{∞} ≃ π(μ) and let π′(g) = |detg|^{−1/2}π(g). π′ may be written as a tensor product over the valuations of Q, π′ = ⊗_{v}π′_{v}.

Conjecture: If σ_{p} = σ_{p}(π) then trace σ_{p}(s) ∈ \overline{Q} for all s in 𝒲(\overline{Q}_{p}/Q_{p}) and π′_{p} is π(σ_{p}).

In order to lend some credibility to this conjecture, I shall prove in these lectures that it is valid if π′_{p} = π(τ_{p}) where τ_{p} is the direct sum of two one-dimensional representations of the Weil group or if π′_{p} = σ(λ, ν) is a special representation.

5. Some auxiliary considerations. p is now fixed. Let A_{f}^{p} be the subring of A_{f} formed by those elements whose component at p is 0. G(A_{f}) is a direct product G(A_{f}^{p})G(Q_{p}). There is thus a decomposition V_{f}^{π} = V π ; p̂ ⊗ V_{p}^{π}. G(A_{f}^{p}) acts on the first and G(Q_{p}) on the second factor. Set K^{p} = K ∩ G(A_{f}^{p}) and K_{p} = K ∩ G(Q_{p}). We consider only those K for which K = K^{p}K_{p}. They form a cofinal family. V_{f}^{π}(K) is then a tensor product

V π ; p̂(K^{p}) ⊗ V_{p}^{π}(K_{p}).

The dimension of the second factor is m(π_{p}, K_{p}), the multiplicity with which the trivial representation of K_{p} occurs in π_{p}.

If g ∈ G(A_{f}^{p}), we may consider T (g) to be acting on V π ; p̂(K_{p}). The action extends to a representation of ℋ_{C}(K^{p}), the algebra of compactly supported K^{p}-biinvariant functions on G(A_{f}^{p}), which we denote πp̂.

Let A^{1}(μ) be the set of all π in A such that π_{∞} ≃ π(μ) and π′_{p} ≃ π(τ_{p}), where τ_{p}, a representation of the Weil group of Q_{p} by semi-simple matrices, is a direct sum of two one-dimensional representations. I recall that π′_{p} : g_{p} → |detg_{p}|^{− \frac{1}{2}} π_{p}(g_{p}). We consider τ^{1}, a double representation of ℋ_{C}(K^{p}) and the Weil group of Q_{p}, which is defined as

⊕_{π∈A^{1}(μ)}m(π_{p}, K_{p}){πp̂ ⊗ τ_{p}}.

Observe that the sum is effectively finite and that τ_{p} is uniquely determined by π_{p}.

This is not the only double representation we want to consider. Let A^{2}(μ) be the set of π in A for which π_{∞} ≃ π(μ) while π′_{p} is a special representation σ(λ_{p}, ν_{p}). The second double representation to be considered is

τ^{2} = ⊕_{π∈A^{2}(μ)}m(π_{p}, K_{p}){πp̂ ⊗ (λ_{p} ⊕ ν_{p})}.

In this paragraph we find formulae for trace (τ^{1}(f_{g}, s)) and trace (τ^{2}(f_{g}, s)) if s is an element of the Weil group mapping to a non-unit in Q_{p}^{×}. Actually the representations of the Weil group considered in this paragraph factor through Q_{p}^{×}. However to treat them as representations of Q_{p}^{×} would be slightly misleading.

The formulae we obtain are quite pretty. To describe them exactly requires some preparation and, in particular, the introduction of some sets on which G(A_{f}^{p}) acts. They may appear strange at first. I hope their significance becomes clear in the course of this and the following paragraph.

Let F be an imaginary quadratic extension of Q which splits at p. Choosing a basis of F, a vector space over Q, we may regard G as the group of invertible linear transformations of F. The centralizer of F, which acts on itself by multiplication, is an algebraic subgroup H_{f} of G. There is a v in G(Q_{p}) such that, if A is the group of diagonal matrices,

vA(Q_{p})v^{−1} = H(Q_{p}).

We fix such a v = v(F) and set

V (Q_{p}) = vN(Q_{p})v^{−1}.

We also introduce the set

M(p, F) = H_{F} (Q)V (Q_{p}) \ G(A_{f})/K_{p}

on which G(A_{f}^{p}) acts to the right. Dividing by K^{p} we obtain

M_{K}(p, F) = H_{F} (Q)V (Q_{p}) \ G(A_{f})/K

Antwerp 1972 35 which we turn into a toplogical space by providing it with the discrete topology. K^{p} acts on L(Q_{ℓ}) by means of the projection into G(Q_{ℓ}) composed with the action of G(Q_{ℓ}) on L(Q_{ℓ}) defined by μ. We let it act on

L(Q_{ℓ}) × M(p, F)

by k : (u, y) → (k^{−1}u, yk). Then

L(Q_{ℓ}) ×_{K^{p}} M(p, F)

is a sheaf over M_{K}(p, F). I denote it by F_{μ}(Q_{ℓ}), a notation already used in another context. But as it was then used for a sheaf over another space there should be no confusion. The overlapping notation is of course deliberate.

If g ∈ G(A_{f}^{p}) and g^{−1}K′g ⊆ K, with K′_{p} = K_{p}, we may introduce

R(g) : M_{K′}(p, F) → M_{K}(p, F)

as well as the maps

R^{∗}(g)F_{μ}(Q_{ℓ}) → F_{μ}(Q_{ℓ})

between two sheaves over M_{K′}(p, F) and

R_{∗}(g)F_{μ}(Q_{ℓ}) → F_{μ}(Q_{ℓ})

between two sheaves over M_{K}(p, F). To be explicit, the first sends the point represented by (u × hg) × h to the point represented by gu × h. The second sends

⊕_{k∈g^{−1}K′g\K} u × h′

where h′ is defined by h′gk = h, to

( ∑ k^{−1}g^{−1}u) × h.

If s belongs to the Weil group of Q_{p} and s maps to b in Q_{p}^{×}, then left translation of G(A_{f}) by

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

factors to yield maps L(s) from M_{K}(p, F) to itself or from M(p, F) to itself. There is also a map

L^{∗}(s)F_{μ}(Q_{ℓ}) → F_{μ}(Q_{ℓ}).

Both ends of the arrow are sheaves on M_{K}(p, F). The map is obtained by factoring

(u × v ( \begin{matrix}1 & 0 \\ 0 & b \end{matrix} ) v^{−1}h) × h → u × h.

Now suppose K′ = K ∩ gKg^{−1}. We form the map
φ(g, s) = R(e) × R(g)L(s) = R(e) × L(s)R(g) : M_{K′}(p, F) → M_{K}(p, F) × M_{K}(p, F).

A point x′ which is the inverse image of (x, x) on the diagonal will be called a fixed point of φ(g, s). The maps of sheaves introduced earlier yield in the present circumstances a linear transformation first from the fibre F_{μ}(Q_{ℓ})_{x} of F_{μ}(Q_{ℓ}) at x, which is also the fibre of R^{∗}(g)L^{∗}(s)F_{μ}(Q_{ℓ}) at x′, to the fibre F_{μ}(Q_{ℓ})_{x′} and then from this space, a subspace of the fibre of R_{∗}(e)F_{μ}(Q_{ℓ}) at x, to F_{μ}(Q_{ℓ})_{x}. This composition we denote

φ_{x′}(g, s) : F_{μ}(Q_{ℓ})_{x} → F_{μ}(Q_{ℓ})_{x}.

Lemma 5.1. If s maps to a non-unit b in Q_{p}^{×} then the set of fixed points of φ(g, s) is finite. It is in fact empty for all but a finite number of isomorphism classes of imaginary quadratic extensions which split at p.

We have chosen a basis of F. We may therefore identify A_{f}(F) = F ⊗ A_{f}, the ring of finite adèles of F, with A_{f} ⊕ A_{f}. For our purposes a module is just an open compact subgroup M(Z_{f}) of A_{f}(F). It must be of the form

∏_{p} M(Z_{p})

with M(Z_{p}) ⊆ F ⊗ Q_{p}. A module is an order if it is a subring of A_{f}(F) containing 1. G(A_{F}) acts transitively on the modules. Two modules will be said to lie in the same genus if one can be transformed to the other by an element of H(A_{f}) and to the same class if one can be transformed to the other by an element of H(Q). As on p. 251 of Jacquet–Langlands every genus contains a unique order. Since the stablizer of a module is open and H(Q) \ H(A_{f}) is compact every genus contains only a fininte number of classes.

Let n be a positive integer. It follows readily from Lemma 7.3.1 of Jacquet–Langlands that if γ ∈ H(Q) = F^{×} and γ is not a scalar matrix, that is not in Q^{×}, then the set of orders containing nγ is finite. Thus the number of genera Γ such that nγ : M(Z_{f}) → M(Z_{f}) for M(Z_{f}) in Γ is finite.

Returning to the lemma we represent a fixed point x′ by h in G(A_{f}). Set

g(s) = v ( \begin{matrix}1 & 0 \\ 0 & b \end{matrix} ) v^{−1} ∈ G(Q_{p}).

There is a k in K, a u in V (Q_{p}), and a γ in H(Q) such that

(1)  g(s)hgk = uγh.

Let

h = hp̂h_{p}

with hp̂ in G(A_{f}^{p}) and h_{p} in G(Q_{p}). Decompose k in the same manner. The equation (1) yields

g(s)h_{p}k_{p} = uγh_{p}

or

(2)  k_{p} = h_{p}^{−1}g^{−1}(s)uγh_{p}

and

hp̂gkp̂ = γhp̂

or

(3)  gkp̂ = h−1 ; p̂γhp̂.

Considering eigenvalues, we infer from (2) and (3) that γ must lie in a compact subset of H(A_{f}) and hence in a finite subset of H(Q).

Choose a module M(Z_{f}) fixed by K. The map h → hM(Z_{f}) maps H(Q)\G(A_{f})/K to the set of classes of modules. The inverse image of each class is finite. We choose a set of representtives for the fixed points. We have to show that this set is finite. Since H(Q_{p})V (Q_{p}) \ G(Q_{p})/K_{p} is finite we may assume h_{p} lies in a fixed finite set. Choose n in Z such that ng takes M(Z_{f}) into itself. Suppose also that for all γ which can enter into the equation (1), and for the fixed finite set of h_{p}, nγ takes h_{p}M(Z_{p}) into itself. From (3) we deduce that nγ takes hM(Z_{f}) into itself. Since there are only a finite number of γ under consideration and, because of (2) and our assumption on s, none of them lie in Q^{×}, the set of classes in which hM(Z_{f}) can lie is finite. It follows that the set of representatives is itself finite.

The second assertion of the lemma follows from the observation that (2) and (3) force the trace and determinant of γ to lie in a compact subset of A_{f} and thus in a finite subset of Q.

Set

A(F) = ∑ trace φ_{x′}(g, s).

Antwerp 1972 38 The sum is taken over the fixed points of φ(g, s). It will be important to us but we should write it in a more useful form.

It was agreed at the beginning to take K so small that no element of G(Q) is conjugate to an element of K_{∞}K. If K_{p} is given we can choose K^{p} so small that for no element γ of an imaginary quadratic field can γ be integral at p and such that the equations

trace k_{q} = trace γ
detk_{q} = Nmγ

are solvable for all q ≠ p with k = (k_{q}) in K^{p}. This we assume henceforth. Since K plays only an auxiliary role, we can afford to make such an assumption.

Given h in G(A_{f}) which represents a fixed point, suppose that in addition to equation (1) an equation

g(s)hgk̄ = ūγ̄h

of the same type also holds. Then

k̄−1 ; p̂kp̂ = h−1 ; p̂γ̄^{−1}γhp̂

and

k̄_{p}^{−1}k_{p} = h_{p}^{−1}γ̄^{−1}ū^{−1}uγh_{p}.

The eigenvalues of γ̄^{−1}ū^{−1}uV are those of γ̄^{−1}γ. Our assumption implies therefore that γ = γ̄. Thus γ is uniquely determined by h, and kp̂ is in its turn uniquely determined by γ and h.

B is the group of supertriangular matrices. Let G(Q_{p}) be the disjoint union

∪_{i}B(Q_{p})g_{i}K_{p}.

If h_{i} = vg_{i}, with v = v(F), then G(Q_{p}) is also the disjoint union

∪_{i}V (Q_{p})H(Q_{p})h_{i}K_{p}.

Let U_{i} be the projection of

g_{i}K_{p}g_{i}^{−1} ∩ B(Q_{p})

on A(Q_{p}). If h_{p} ∈ V (Q_{p})H(Q_{p})h_{i}K_{p} the equation (2) can be satisfied with some u if and only if g^{−1}(s)γ lies in vU_{i}v^{−1}.

We observe next that at a fixed point representated by h the trace of φ_{x′}(g, s) is

trace μ(γ)

if γ satisfies (1). To see this observe that the corresponding map of F_{μ}(Q_{ℓ})_{x} to F_{μ}(Q_{ℓ})_{x′} is the composition

((u × g(s)hg) × hg) × h → (u × hg) × h → gu × h.

Now passing from F_{μ}(Q_{ℓ})_{x′} tp F_{μ}(Q_{ℓ})_{x} we send

(gu × h) → kgu × hk^{−1}

because of equation (1). Thus

trace φ_{x′}(g, s) = trace μ(k_{ℓ}g_{ℓ}) = trace μ(g_{ℓ}k_{ℓ}) = trace μ(γ)

because of equation (3). k_{ℓ} and g_{ℓ} are the components of k and g at ℓ.

Lemma 5.2. Let f_{g} be the characteristic function K^{p}gK^{p}, a subset of G(A_{f}^{p}), divided by the measure of K^{p} and let χ_{i} be the characteristic function of U_{i}. Then A(F) is the measure of H(Q) \ H(A_{f}) times the sum over γ in H(Q) of

trace μ(γ) \{∑_{i} \frac{χ_{i}(v^{−1}g^{−1}(s)γv)}{meas(vU_{i}v^{−1})}\} ∫_{H(A_{f}^{p})\G(A_{f}^{p})} f_{g}(h^{−1}γh) dh.

Our discussion to this point shows that A(F) is equal to

(4)  ∑_{γ} ∑_{i} ∑_{h} χ_{i}(v^{−1}g^{−1}(s)γv) trace μ(γ).

The outer sum is over γ in H(Q). The inner sum is over a set of coset representatives for H(Q)V (Q_{p}) \ G(A_{f})/K′ for which

h_{p} ∈ vB(Q_{p})g_{i}K_{p}

and

h−1 ; p̂γhp̂ ∈ gK^{p}.

Let {k_{α}} be a set of coset representatives for K/K′, each of which is taken to lie in K^{p}. I claim that it is possible to replace the sum over coset representatives of H(Q)V (Q_{p})\G(A_{f})/K′ by a sum over hk_{α}, α varying and h varying over a collection of coset representatives of H(Q)V (Q_{p}) \ G(A_{f})/K. Suppose in fact that

(5)  hk_{α} = γuhk_{β}k′

with γ in H(Q), u in V (Q_{p}), and k′ in K′. Then γu lies in hKh^{−1}. Since γ and γu have the same eigenvalues at each place, our basic assumption implies that γ = 1. Examining equation (5) away from p and recalling that k_{α} and k_{β} lie in K^{p}, we conclude that α = β, as claimed.

The set K^{p}gK^{p} is the disjoint union

∪k_{α}gK^{p}.

Thus

h−1 ; p̂γhp̂ ∈ K^{p}gK^{p}

if and only if

k_{α}^{−1}h−1 ; p̂γhp̂k_{α} ∈ gK^{p}

for some α, and this α is then unique. A(F) is thus equal to

∑_{γ} ∑_{i} ∑_{h} trace μ(γ)χ_{i}(v^{−1}g^{−1}(s)γv)f_{g}(h−1 ; p̂γhp̂) meas K^{p}.

The inner sum is taken over a set of representatives for the double cosets

H(Q)V (Q_{p})vU_{i}v^{−1} \ G(A_{f}^{p})V (Q_{p})H(Q_{p})/K^{p} = H(Q)vU_{i}v^{−1} \ G(A_{f}^{p})H(Q_{p})/K^{p}.

A(F) may be written then as

∑_{γ} ∑_{i} trace μ(γ)\frac{χ_{i}(v^{−1}g^{−1}(s)γv)}{meas(vU_{i}v^{−1})} ∫_{H(Q)\G(A_{f}^{p})H(Q_{p})} f_{g}(h−1 ; p̂γhp̂) dh.

The step from here to the assertion of the lemma is short.

There are still more spaces like M_{K}(p, F) to be constructed. Let D be the quaternion algebra over Q split everywhere but at p and ∞ and let G′ be the multiplicative group of D. G′ and G are isomorphic as algebraic groups over Q_{q} if q ≠ p. This observation yields isomorphisms G′(Q_{q}) → G(Q_{q}) as well as G′(A_{f}^{p}) → G(A_{f}^{p}). We choose one such isomorphism. Any two would differ by an inner automorphism, so it does not really matter which one we take. Let

V_{i} = {detk | k ∈ U_{i}}

Antwerp 1972 41 and

W_{i} = {g ∈ G′(Q_{p}) | Nmg ∈ V_{i}}.

Then M_{K}^{i}(p, D) is the space

G′(Q) \ G′(A_{f})/K^{p}W_{i}.

I have taken the liberty of using the isomorphism introduced, to imbed K^{p} in G(A_{f}^{p}). We may also introduce

M^{i}(p, D) = G′(Q) \ G′(A_{f})/W_{i}

as well as the sheaf F_{μ}(Q_{ℓ}) on M_{K}^{i}(p, D) defined by

L(Q_{ℓ}) ×_{K^{p}} M^{i}(p, D).

G(A_{f}^{p}) acts on M^{i}(p, D) to the right by means of the isomorphism between G(A_{f}^{p}) and G′(A_{f}^{p}), which we might as well use to identify the two groups. We may also define the maps R(g). If s belongs to the Weil group and maps to b in Q_{p}^{×} let g(s) now be any element of G′(Q_{p}) such that

Nmg(s) = b.

Let L(s) be the map of M_{K}^{i}(p, D) to itself obtained by factoring the map h → g(s)h on G′(A_{f})/W_{i}.

We can again introduce the necessary maps from one sheaf to another, as well as the correspondence φ(g, s) and, if x′ is a fixed point of the correspondence, the linear transformation φ_{x′}(g, s). The introduction of

A(D) = ∑_{i} ∑ trace φ_{x′}(g, s)

where the inner sum is over the fixed points of φ(g, s) on M_{K′}^{i}(p, D) is more than justified by the next lemma.

Lemma 5.3. The set M_{K}^{i}(p, D) is finite.

We may regard Z, the centre of G, also as the centre of G′. G′(Q) is discrete in G(A_{f}) and

G′(Q)Z^{o}(R) \ G′(A) ≃ Z^{o}(R) \ G′(R) × G′(Q) \ G′(A_{f}).

The left side is known to be compact. Since K^{p}W_{i} is open in G(A_{f}) the lemma follows.

Lemma 5.4. Let δ_{i} be the characteristic function of W_{i}. If γ ∈ G′(Q) let G′(γ) be its centralizer in G′ and let {γ} be the conjugacy class of γ in G′(Q). A(D) is equal to the sum over all {γ} of

trace μ(γ) meas (G′(γ, Q) \ G′(γ, A_{f})) meas (G′(γ, Q_{p}) \ G′(Q_{p}))

times

\{∑_{i} \frac{δ_{i}(g^{−1}(s)γ)}{meas W_{i}}\} ∫_{G(γ,A_{f}^{p})\G(A_{f}^{p})} f_{g}(h^{−1}γh) dh.

Suppose h in G′(A_{f}) represents a fixed point on M′_{K}(p, D). There is then an equation

g(s)hgkk_{p} = γh

with γ in G′(Q), k in K^{p}, and k_{p} in W_{i}. This equation, for a given h and γ, can be solved for k and k_{p} if and only if g^{−1}(s)γ ∈ W_{i} and

h−1 ; p̂γhp̂ ∈ gK^{p}.

If h represents the fixed point x′ and

x = L(s)R(g)x′ = R(e)x′

the map from the fibre of L^{∗}(s)R^{∗}(g)F_{μ}(Q_{ℓ}) at x′, which is also F_{μ}(Q_{ℓ})_{x′} to F_{μ}(Q_{ℓ})_{x′} is given by composition

((u × g(s)hg) × hg) × h → (u × hg) × h → gu × h.

Passing from F_{μ}(Q_{ℓ})_{x′}, which is contained in the fibre of R_{∗}(e)F_{μ}(Q_{ℓ}) at x, to F_{μ}(Q_{ℓ})_{x} we send

gu × h → kk_{p}gu × hk_{p}^{−1}k^{−1} = kgu × hk_{p}^{−1}k^{−1}.

Thus

trace φ_{x′}(g, s) = trace μ(k_{ℓ}g_{ℓ}) = trace μ(γ).

Let me observe also that an equation

g(s)hgk̄k̄_{p} = γ̄h

implies that

h^{−1}γ̄^{−1}γh = k̄_{p}^{−1}k̄^{−1}kk_{p}

and hence, by our assumption, that γ = γ̄. A(D) is thus equal to

∑_{γ} ∑_{i} ∑_{h} trace μ(γ).

γ runs over G(Q). h runs over those elements of a set of representatives for the double coset space M_{K′}^{i}(p, D) which satisfy g^{−1}(s)γ ∈ W_{i} and h−1 ; p̂γhp̂ ∈ gK^{p}. As before we replace the set of representatives for M_{K′}^{i}(p, D) by hk_{α}, α varying as before and h varying over a set of representatives for G′(Q) \ G′(A_{f})/K^{p}W_{i}. The sum is then

∑_{γ} ∑_{i} ∑_{h} trace μ(γ)δ_{i}(g^{−1}(s)γ)f_{g}(h−1 ; p̂γhp̂) meas K^{p}

which is equal to

∑_{γ} ∑_{i} trace μ(γ)\frac{δ_{i}(g^{−1}(s)γ)}{meas W_{i}} ∫_{G′(Q)\G′(A_{f})} f_{g}(h−1 ; p̂γhp̂) dh.

This expression is readily transformed into that of the lemma.

A(F) and A(D) will occur in our formula for trace τ^{1}(f_{g}, s). However to express this trace completely, still more supplementary terms must be introduced. N is the group of matrices of the form

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

and A^{+}(Q) is the group of diagonal matrices

( \begin{matrix}α & 0 \\ 0 & β \end{matrix} )

with α and β in Q and αβ > 0. We form the space

C(p) = N(A_{f})A^{+}(Q) \ G(A_{f})/K_{p}

as well as

C_{K} = C_{K}(p) = N(A_{f})A^{+}(Q) \ G(A_{f})/K.

If g ∈ G(A_{f}^{p}), R(g) is as usual the transformation of C yielded by right translation of G(A_{f}) by g. If g^{−1}K′g ⊆ K and K′_{p} = K_{p}, in particular if K′ = K ∩ gKg^{−1}, then R(g) : C_{K′} → C_{K}. If s is in the Weil group and maps to b in Q_{p}^{×}, we again take

g(s) = ( \begin{matrix}1 & 0 \\ 0 & b \end{matrix} )

and let L(s) : C_{K} → C_{K} be left translation by g(s).

We introduce once again

φ(g, s) = R(e) × L(s)R(g) : C_{K′} → C_{K} × C_{K}.

x′ is a fixed point of φ(g, s) if it lies in the inverse image of some point (x, x) on the diagonal. h in G(A_{f}) is the representative of a fixed point if there is a γ in A^{+}(Q), a k in K, and an n in N(A_{f}) such that

(6)  g(s)hgk = nγh.

If a similar equation

g(s)hgk̄ = n̄γ̄h

also obtains, then

k^{−1}k̄ = h^{−1}(γ^{−1}n^{−1}n̄γ̄)h.

Since γ^{−1}n^{−1}n̄γ̄ and γ^{−1}γ̄ have the same eigenvalues, our basic assumption implies that γ = γ̄. Let

γ = ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) .

It is determined for a given g and s solely by x′. Since

hkh^{−1} = g^{−1}(s)nγ

we have |a|_{p} = 1 and |d|_{p} = |b|_{p}. In particular if, as we assume, b is a non-unit, then |a| ≠ |d|. Let a′ and d′ be a and d cleared of any common divisor or sign and let

trace μ ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) = ∑_{{1+j=m \atop i≥ℓ} ; j≥ℓ} a^{i}d^{j}.

If |a| > |d|, set

ψ(γ) = (d′ − a′) trace μ(γ) + a′a^{m−ℓ}d^{ℓ}

and if |a| < |d| set

ψ(γ) = a′a^{m−ℓ}d^{ℓ}.

Take B to be the sum over all fixed points in C_{K′} of ψ(γ).

Antwerp 1972 45 Lemma 5.5. Assume that the image b of s in Q_{p}^{×} is not a unit. Then B is equal to the measure of A^{+}(Q) \ A(A_{f}) times the sum over γ in A^{+}(Q) of the eigenvalue μ(γ) smallest in absolute value multiplied by

|b|_{p}^{− \frac{1}{2}} \{∑_{i} \frac{χ_{i}(g^{−1}(s)γ)}{meas U_{i}}\} min \{|\frac{a}{d}|^{\frac{1}{2}} , |\frac{d}{a}|^{\frac{1}{2}} \}

times

\{∏_{q≠p} |\frac{(a − d)^{2}}{ad} |q^{\frac{1}{2}} ∫_{A(A_{f}^{p})\G(A_{f}^{p})} f_{g}(h^{−1}γh) dh\} .

If h_{p} lies in B(Q_{p})g_{i}K_{p} then (6) has a solution for a given h and γ if and only if g^{−1}(s)γ lies in U_{i} and np̂ in N(A_{f}^{p}) exists such that

(7)  h−1 ; p̂np̂γhp̂ ∈ gK^{p}.

If

V = V^{p}V_{p} = γh(K ∩ h^{−1}γ^{−1}N(A_{f})γh)h^{−1}γ^{−1}

then np̂ is uniquely determined modulo V^{p} and kp̂ is determined by np̂.

The sum B is thus equal to

∑_{γ} ∑_{i} ∑_{h,np̂} χ_{i}(g^{−1}(s)γ)ψ(γ).

The inner sum is taken over the coset representatives of N(A_{f})A^{+}(Q) \ G(A_{f})/K′ and N(A_{f}^{p})/V^{p} which satisfy h_{p} ∈ B(Q_{p})g_{i}K_{p} and relation (7). The eigenvalue λ(γ) of μ ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) of smallest absolute value is a^{m−ℓ}d^{ℓ} if |a| < |d| and d^{m−ℓ}a^{ℓ} if |d| < |a|. Thus

ψ(γ) = λ(γ) min{a′, d′}.

We cannot immediately indulge in the usual device of replacing the coset representatives of C_{K′} by hk_{α}, where h runs over a set of coset representatives of C_{K} and k_{α} over a set of representatives of K/K′. To see what modification is necessary, we suppose hk_{α} and hk_{β} determine the same coset in C_{K′}, so that

hk_{α} = nγhk_{β}k̄

with n in N(A_{f}), γ in G(Q), and k̄ in K′. The equation implies that nγ is conjugate to an element of K and it follows then, from our assumption, that γ = 1. Hence

k_{β}^{−1}k_{α}k̄ = k_{β}^{−1}h^{−1}nhk_{β} ∈ k_{β}^{−1}h^{−1}N(A_{f})hk_{β} ∩ K.

For a given β the number of α satisfying this condition is the index

[k_{β}^{−1}h^{−1}N(A_{f})hk_{β} ∩ K : k_{β}^{−1}h^{−1}N(A_{f})hk_{β} ∩ K′].

Since K_{p} = K′_{p}, K = K^{p}K_{p}, and K′ = (K′ ∩ G(A_{f}^{p}))K_{p}, we may replace h by hp̂, A_{f} by A_{f}^{p}, K by K^{p}, and K′ by K′ ∩ G(A_{f}^{p}). There is also no harm in supposing that k_{β} = 1. Set

W = N(A_{f}^{p}) ∩ hp̂K^{p}h−1 ; p̂.

We want to compute

[W : W ∩ hp̂(K′ ∩ G(A_{f}^{p}))h−1 ; p̂].

The smaller group is

(8)  W ∩ hp̂gK^{p}g^{−1}h−1 ; p̂

because K′ is gKg^{−1} ∩ K. We are also supposing the relation (7) is satisfied. When that is so

hp̂gK^{p}g^{−1}h−1 ; p̂ = np̂γhp̂K^{p}h−1 ; p̂γ^{−1}n−1 ; p̂

and the intersection (8) is

W ∩ np̂γW γ^{−1}n−1 ; p̂ = W ∩ γW γ^{−1}.

The index [W : W ∩ γW γ^{−1}] is clearly ∏_{q≠p} |a′|_{q}^{−1}.

We may therefore replace the sum over coset representatives of C_{K′} by a sum over hk_{α} provided we replace λ(γ) min\{a′, d′\} by λ(γ) min\{1, \frac{d}{a}\} that is, provided we divide by ∏_{q≠p} |a′|_{q}^{−1}. As usual we drop the sum over α and replace condition (7) by

h−1 ; p̂np̂γhp̂ ∈ K^{p}gK^{p}.

It must of course be observed that the group V or V^{p}, which at first sight depends on h and k_{α}, depends in fact only on h.

Taking the above discussion into account, I rewrite the sum B as

∑_{γ} ∑_{i} ∑_{h} ∑_{np̂} χ_{i}(g^{−1}(s)γ) min\{1, \frac{d}{a}\}λ(γ).

Here h = hp̂h_{p} runs over those elements in a set of coset representatives for C_{K} for which h_{p} lies in B(Q_{p})g_{i}K_{p} and np̂ runs over a set of coset representatives of N(A_{f}^{p})/V^{p} for which

h−1 ; p̂np̂γhp̂ ∈ K^{p}gK^{p}.

The sum over np̂ may be written as

(9)  \frac{meas K^{p}}{meas V^{p}} ∫_{N(A_{f}^{p})} f_{g}(h−1 ; p̂np̂γhp̂) dnp̂

multiplied by

(10)  χ_{i}(g^{−1}(s)γ) min \{1, \frac{d}{a}\}λ(γ)|b|_{p}^{−1}.

Observe that

γ^{−1}V^{p}γ = hK^{p}h^{−1} ∩ N(A_{f}^{p}) = U^{p}

depends only on h and not on γ. Changing variables one sees that (9) is equal to

(11)  \frac{meas K^{p}}{meas U^{p}} ∫_{N(A_{f}^{p})} f_{g}(h−1 ; p̂γnp̂hp̂) dnp̂.

We may take our coset representatives for C_{K} to be of the form hg_{i}, where h = hp̂h_{p} and h_{p} lies in A(Q_{p}). For each i, h then varies over a set of coset representatives for

(12)  N(A_{f}^{p})A^{+}(Q) \ G(A_{f}^{p})A(Q_{p})/K^{p}U_{i}.

The expression (10) depends on hg_{i} only through g_{i}. Hence to compute B we sum (9), or rather (11), over coset representatives for (12). Choose a product measure on G(A_{f}^{p})A(Q_{p}). The given measure on N(A_{f}^{p}), combined with the measure on A^{+}(Q) which assigns the measure 1 to each point, yields a measure on N(A_{f}^{p})A^{+}(Q). Taking quotients we obtain a measure on N(A_{f}^{p})A^{+}(Q)\ G(A_{f}^{p})A(Q_{p}). The measure of the double coset in (12) represented by h is

\frac{meas K^{p} · meas U_{i}}{meas U^{p}} .

The sum of (11) over coset representatives for (12) is therefore

(13)  \frac{1}{meas U_{i}} ∫_{N(A_{f}^{p})A^{+}(Q)\G(A_{f}^{p})A(Q_{p})} \{ ∫_{N(A_{f}^{p})} f_{g}(h−1 ; p̂γnhp̂) dn\} dh.

The map

n → n′ = γ^{−1}n^{−1}γn

is a homeomorphism of N(A_{f}^{p}) with itself and

dn′ = \{ ∏_{q≠p} |1 − \frac{d}{a}|_{1}\} dn.

Since

|1 − \frac{d}{a}|_{q} = |\frac{d}{a}|_{q}^{1/2}|\frac{(a − d)^{2}}{ad} |_{q}^{1/2}

and

|\frac{d}{a}|_{p}\{ ∏_{q≠p} |\frac{d}{a}|_{q}\} = |\frac{a}{d}|

while

|\frac{a}{d}|_{p} = |b|_{p}^{−1}

a change of variables in (13) yields the product of

|b|_{p}^{−1/2}|\frac{a}{d}|^{1/2} ∏_{q≠p} |\frac{(a − d)^{2}}{ad} |_{q}^{1/2}

and

\frac{1}{meas U_{i}} ∫_{N(A_{f}^{p})A^{+}(Q)\G(A_{f}^{p})A(Q_{p})} \{ ∫_{N(A_{f}^{p})} f_{g}(h−1 ; p̂n^{−1}γnhp̂) dn\} dh.

The double integral is

∫_{A^{+}(Q)\G(A_{f}^{p})A(Q_{p})} f_{g}(h_{p}^{−1}γhp̂) dh

or

meas (A^{+}(Q) \ A(A_{f})) ∫_{A(A_{f}^{p})\G(A_{f}^{p})} f_{g}(h^{−1}γh) dh.

We obtain the lemma simply by reassembling the above data.

There are further spaces, simpler than the previous ones, to be introduced. Let Ḡ = GL(1)

and let

K̄ = K̄^{p}K̄_{p} = {detk | k ∈ K}.

We introduce the space

M̄_{K} = Ḡ^{+}(Q) \ G(A_{f})/K̄

Antwerp 1972 49 where Ḡ^{+}(Q) is the set of positive elements in Ḡ(Q). If as before V_{i} is the set of detk, k ∈ U_{i}, we set

M̄_{K}^{i} = Ḡ^{+}(Q) \ G(A_{f})/K̄^{p}V_{i}.

Each of these spaces is finite.

ḡ is detg and ḡ(s) is just b, the image of s in Q_{p}^{×} = ḡ(Q_{p}). The maps R(g) : M̄_{K}^{i} → M̄_{K}^{i} are defined by right multiplication by ḡ and L(s) : M̄_{K}^{i} → M̄_{K}^{i} by left multiplication by ḡ(s). We do not introduce any sheaves unless μ is one-dimensional. Assume for now that this is so. μ is then of the form g → ν(detg). If M̄^{i} = Ḡ^{+}(Q) \ Ḡ(A_{f})/V_{i} we set

F_{ν}(Q_{ℓ}) = L(Q_{ℓ}) ×K̄^{p} M̄^{i}

where k : v × y → ν(k_{ℓ}^{−1})v × yk. Once this is done we can introduce

φ(g, s) : ∪M̄_{K}^{i} → ∪M̄_{K}^{i} × M̄_{K}^{i}

as well as the various maps between sheaves. We let Ā be the product of

[K : K′] | (|b|_{p}^{−1} + 1)

with the sum over the fixed points x′ of φ(g, s) on ∪M̄_{K}^{i} of trace φ_{x′}(g, s).

Lemma 5.6. If μ is one-dimensional, Ā is equal to

(|b|_{p}^{−1} + 1) ∑_{i} { ∑ χ(b) ∫_{G(A_{f}^{p})} f_{g}(h)χ(deth) dh}.

The inner sum is over all quasi-characters χ of Ḡ(Q) \ Ḡ(A) which are trivial on K̄^{p}V_{i} and such that χ(z)ν(z) = 1 of z ∈ Ḡ(R) is positive.

If h in Ḡ(A_{f}) represents a fixed point x′

bhḡk̄ = δh

with k̄ in K̄^{p}V_{i} and δ in Ḡ^{+}(Q). In general the map from the fibre at bhḡ to that at h may be represented by

v × bhḡ → ḡv × h = ν(ḡ_{ℓ})v × h.

When bhḡk̄ = δh the point represented by the expression on the right is also represented by

ν(k̄_{ℓ}ḡ_{ℓ})v × bhḡ.

Thus

trace y_{x′}(g, s) = trace ν̄(k̄_{ℓ}ḡ_{ℓ}) = trace ν(δ).

δ is uniquely determined by b and ḡ. Thus

Ā = ∑_{i} Ā_{i}

where Ā_{i} is 0 if there are no fixed points on M̄_{K}^{i}, that is, if bḡ does not lie in Ḡ^{+}(Q)K̄^{p}V_{i}, but where Ā_{i} is otherwise [K : K′](|b|_{p}^{−1} + 1) times

[Ḡ(A_{f}) : Ḡ^{+}(Q)K̄^{p}V_{i}]ν(δ).
Let ε_{i} be the characteristic function of Ḡ^{+}(Q)K̄^{p}V_{i}. Then
∑ χ(b) ∫_{G(A_{f}^{p})} f_{g}(h)χ(deth) dh

is equal to

[Ḡ(A_{f}) : Ḡ^{+}(Q)K̄^{p}V_{i}] ∫_{G(A_{f}^{p})} |ν(b deth)|^{−1}f_{g}(h)ε_{i}(b deth) dh.

Recall that f_{g} is the characteristic function of K^{p}gK^{p} divided by the measure of K^{p}. If h = k_{1}gk_{2} then

ε_{i}(b deth) = ε_{i}(bḡ).

Thus the integrand vanishes identically if Ā_{i} = 0. Otherwise the integral is equal to

\frac{meas(K^{p}gK^{p})}{meas K^{p}} = [K : K′]

multiplied by

|ν(b det h)|^{−1} = |ν(δ)|.

This equality is a consequence of the product formula applied to ν(δ). The lemma is an immediate consequence of these calculations.

The following proposition will be proved in the next paragraph.

Proposition 5.7. If K_{p} is given and K^{p} is then chosen sufficiently small and if s in the Weil group maps to an element of absolute value less that 1 in Q_{p}^{×}, then the trace of τ^{1}(f_{g}, s) is equal to

−{ ∑_{F} A(F)} − A(D) − B + Ā

where Ā is taken to be 0 if μ is not one-dimensional. The sum is over a set of representatives for the isomorphism classes of imaginary quadratic fields.

We shall also find a formula for trace τ^{2}(f_{g}, s). Define Ā_{0} just as Ā was defined except that the union of M̄_{K}^{i} is replaced by M̄_{K}. Let

W = {g′ ∈ G′(Q_{p}) | Nmg′ ∈ K̄_{p}}

and set

M_{K}(p, D) = G′(Q) \ G′(A_{f})/K^{p}W.

Define A_{o}(D) in the same way as A(D) except that M_{K}(p, D) replaces the union of M_{K}^{i}(p, D).

Proposition 5.8 With the assumptions of the previous proposition the trace of τ^{2}(f_{g}, s) is equal to

(|b|_{p}^{−1} + 1)(A(D) − A_{o}(D)) − (Ā − Ā_{o}).

6. The trace formula. This formula, which will be used to prove Propositions 5.7 and 5.8 has been described, in a form pretty close to that needed, in [4.1] and [6.3]. The preprint [6.1] is also a very good reference. I write it out again here, or at least the part of it we require, with the appropriate changes and comments.

It will save a lot of trouble if we fix some conventions for measures. Let F be a global field, which to avoid irrelevant explanations I take of characteristic zero, and let ψ be a non-trivial character of A(F), trivial on F. For every place v, ψ_{v} defines a non-trivial character ψ_{v} for F_{v}. Let dx_{v} be the measure on F_{v} self-dual with respect to ψ_{v}. If Ω is an n-dimensional analytic manifold over F_{v} and ω = φ(x^{1}, ⋯, x^{n}) dx^{1} ∧⋯∧dx^{n} is an n-form we set |ω|_{v} = |φ(x^{1}, ⋯, x^{n})| dx_{v}^{1} ⋯ dx_{v}^{n}. In particular, if G is a connected algebraic group over F and ω is a right-invariant form of highest degree also defined over F, then |ω|_{v} is a Haar measure on G(F_{v}). If λ is the representation of 𝔊(F̄/F) on the lattice of rational characters of G and L(s, λ_{v}) is defined as, for example, in [6.4], then

dg_{v} = L(1, λ_{v})|ω|_{v}

Antwerp 1972 52 is called the Tamagawa measure on G(F_{v}) determined by ω. It will be convenient to write L(s, G/F_{v}) for L(s, λ_{v}). The product measure ∏_{v} dg_{v} will be called the unnormalized Tamagawa measure on G(A). If r is the multiplicity of the trivial representation in λ, the quotient of the unnormalized Tamagawa measure by lim_{s→1}(s − 1)^{r}L(s, λ) is called the Tamagawa measure. L(s, λ) is defined by a product over all places. We shall use Tamagawa measures locally and unnormalized Tamagawa measures globally. On discrete groups we take the measure which assigns the value 1 to each point. On quotient spaces we take, unless otherwise stated, quotient measures, at least when this is possible, and on the Pontrjagin dual D(M) of a locally compact group M we take the measure dual to that in M.

If Φ is a function on G(A) which satisfies

Φ(zg) = μ(z)Φ(g)

for z in Z^{o}(R), has compact support modulo Z^{o}(R), is bi-invariant with respect to some open subgroup of G(A_{f}), and infinitely differentiable as a function of its coordinate at infinity, then

r(Φ) = ∫_{Z^{o}(R)\G(A)} Φ(h)r(h) dh

operates on L_{sp}(μ) and is of trace class. The trace formula gives its trace.

The reader will notice that in [4.1] and [6.3] Z(A) played the role here assumed by Z^{o}(R). This has an effect on the formula. Moreover the trace formula as described in [4.1] gives the trace not of the representation r on L_{p}(μ) but of the representation on the sum of L_{p}(μ) and L_{se}^{o}(μ). Thus we have to add to it the negative of the trace on L_{se}^{o}(μ). This is

(1)  − ∑_{χ} ∫_{Z^{o}(R)\G(A)} χ(deth)Φ(h) dh.

The sum is over all quasi-charaacters of Ḡ(Q) \ Ḡ(A) such that χ(detz)μ(z) is 1 for z in Z^{o}(R).

We now rewrite, in so far as we need them, the terms (i) to (viii) of the trace formula given on p. 516–517 of Jacquet–Langlands. The term (i) is

(2)  meas (Z^{o}(R)G(Q) \ G(A)) ∑_{γ∈Z(Q)} Φ(γ).

The term (ii) becomes

(2′)  ∑_{{γ}} meas (Z^{o}(R)G(γ, Q) \ G(γ, A)) ∫_{G(γ,A)\G(A)} Φ(h^{−1}γh) dh.

Antwerp 1972 53 The sum is over conjugacy classes in G(Q) whose eigenvalues do not lie in Q. The term (iii) does not occur for a field of characteristic 0. The terms (vi) and (vii), as well as the first part of (v), will vanish for the Φ to which we shall apply the trace formula, so there is no need to write them out explicitly.

Each term mentioned so far, including the ones not written out explicitly, has been invariant, in the sense that it does not change if Φ is replaced by Φ′ with Φ′(h) = Φ(x^{−1}hx). This is not true of (iv), the last part of (v), and (viii). Their sum can however be written as a sum of invariant terms. Let me describe the form taken by these terms when Φ is of the form

Φ(h) = ∏_{v} Φ_{v}(h_{v}).

If a ∈ A(Q_{v}) is a diagonal matrix set

F^{A}(a, Φ_{v}) = Δ(a) ∫_{A(Q_{v})\G(Q_{v})} Φ_{v}(h^{−1}ah) dh

with

Δ(a) = |\frac{(α − β)^{2}}{αβ} |v^{\frac{1}{2}}

if α and β are the eigenvalues of a. I observe that if the representation ρ(λ_{v}, η_{v}), where λ_{v} and η_{v} are two quasi-characters of Q_{v}^{×}, is defined as the previous paragraph, then

trace ρ(Φ_{v}; λ_{v}, η_{v}) = ∫_{A(Q_{v})} F^{A}(a, Φ_{v})λ_{v}(α)η_{v}(β) da

if v is finite. If Q_{v} is R, the right side must be replaced by an integral over Z^{o}(R) \ A(R) and the formula is only valid when λ_{v}η_{v} equals μ^{−1} on Z^{o}(R).

The distribution Φ_{v} → F^{A}(a, Φ_{v}) is invariant. There is for each v and each γ in A(Q) another invariant distribution Φ_{v} → D(γ, Φ_{v}), rather complicated to write out explicitly, such that the remaining parts of the trace formula may be combined into

(3)  meas (Z^{o}(R)Z(Q) \ Z(A)) ∑_{γ∈A(Q)} ∑_{v} {D(γ, Φ_{v}) ∏_{w≠v} F^{A}(γ, Φ_{w})}.

All but finitely many of the terms in this double sum are 0.

Given g in G(A_{f}^{p}) and an s in the Weil group which maps to an element of absolute value less than 1 in Q_{p}^{×}, we shall choose Φ^{1} and Φ^{2} so that

(4)  trace r(Φ^{1}) = − trace τ^{1}(f_{g}, s)

and

(5)  trace r(Φ^{2}) = − trace τ^{2}(f_{g}, s).

Write h in G(A) as h_{∞}h_{p}h_{f}^{p} with h_{∞} in G(R), h_{p} in G(Q_{p}), and h_{f}^{p} in G(A_{f}^{p}). The functions Φ^{1} and Φ^{2} will have the form

(6)  Φ^{1}(h) = Φ_{∞}(h_{∞})Φ_{p}^{1}(h_{p}, s)f_{g}(h_{f}^{p})

and

(7)  Φ^{2}(h) = Φ_{∞}(h_{∞})Φ_{p}^{2}(h_{p}, s)f_{g}(h_{f}^{p}).

The function Φ_{∞} is of course to be infinitely differentiable, with compact support modulo Z^{o}(R), and to satisfy

Φ_{∞}(zg) = μ(z)Φ_{∞}(g)

for z in Z^{o}(R). If π_{∞} is an irreducible admissible representation of G(R) which agrees with μ^{−1} in Z^{o}(R) then π_{∞}(Φ_{∞}) is defined. We demand that the trace of π_{∞}(Φ_{∞}) be zero unless π_{∞} ≃ π(μ), when it is to be −1, unless π_{∞} ≃ μ̃ or sgn(detg)μ̃, when it is to be 1. μ̃ is as before the representation contragredient to μ. The existence of Φ_{∞}, at least for μ trivial on Z, is proved by Duflo–Labesse as an application of a Paley–Wiener theorem for the projective group. The existence in general is proved in a similar manner.

The conditions imposed on Φ_{∞} entail several properties that will be important to us. First of all

F^{A}(γ, Φ_{∞}) = 0

if γ ∈ A(R). This equation is what forces the terms of the trace formula that we have not written out explicitly to vanish. It means moreover that (3) is equal to

(8)  meas (Z^{o}(R)Z(Q) \ Z(A)) ∑_{γ∈A(Q)} D(γ, Φ_{∞}) ∏_{w≠∞} F^{A}(γ, Φ_{w}).

Let G′(R) be the multiplicative group of a quaternion algebra over R. As in §15 of Jacquet– Langlands, the measure on G(R) determines one on G′(R). The centre of G′(R), which to simplify the notation we identify with Z(R), is also provided with a measure. If z ∈ Z(R)

Φ_{∞}(z) = \frac{−1}{meas (Z^{o}(R) \ G′(R))} trace μ(z).

If γ in G(R) has two distinct real eigenvalues

∫_{G(γ,R)\G(R)} Φ_{∞}(h^{−1}γh) dh = 0

but if γ has complex eigenvalues

∫_{G(γ,R)\G(R)} Φ_{∞}(g^{−1}γg) dg = \frac{1}{meas (Z^{o}(R) \ G(γ, R))} trace μ(γ).

All these facts are consequences of the general theory of harmonic analysis on reductive Lie groups. We also need to evaluate D(γ, Φ_{∞}). For μ trivial on Z this has been done in Duflo–Labesse [6.3]; otherwise one has either to carry out their proof for general μ or to appeal to a forthcoming paper (tentatively entitled “The Fourier transform of some tempered distributions”) by J. Arthur, who discusses the problem for all groups of real rank 1. In any case, D(γ, Φ_{∞}) is zero if γ has eigenvalues α and β of opposite sign. Otherwise let

ε|\frac{β}{α}|d(|\frac{α}{β}|)

where ε is some constant, be the measure on Z(R) \ A(R). Then D(γ, Φ_{∞}) is

\frac{1}{4ε} min \{|\frac{α}{β}|^{\frac{1}{2}} , |\frac{β}{α}|\}

times the eigenvalue of μ(γ) of the smallest absolute value.

Φ_{p}^{1}(h, s) and Φ_{p}^{2}(h, s) will be locally constant functions on G(Q_{p}) with compact support. They will be chosen to satisfy the following conditions, in which π_{p} denotes an irreducible admissible representation of G(Q_{p}) and b is the image of s in Q_{p}^{×}.

(i) If π_{p} is infinite dimensional and π = π(τ_{p}), where τ_{p}, a complex representation of the Weil group, is the direct sum of two one-dimensional representations, then

trace π_{p}(Φ_{p}^{1}(s)) = |b|_{p}^{− \frac{1}{2}} m(π_{p}, K_{p}) trace τ_{p}(s)
trace π_{p}(Φ_{p}^{2}(s)) = 0.

(ii) If π_{p} = σ(λ_{p}, η_{p}) is a special representation then

trace π_{p}(Φ_{p}^{1}(s)) = 0
trace π_{p}(Φ_{p}^{2}(s)) = |b|_{p}^{− \frac{1}{2}} m(π_{p}, K_{p})(λ_{p}(b) + η_{p}(b)).

(iii) If π_{p} = π_{p}(τ_{p}) where τ_{p} is the direct sum of λ_{p} and η_{p}, two quasi-characters of Q_{p}^{×} with λ_{p}^{−1}η_{p}(x) = |x|_{p}, so that π_{p} is one-dimensional and if π′_{p} = σ(λ_{p}, η_{p}), then

trace π_{p}(Φ_{p}^{1}(s)) = |b|_{p}^{− \frac{1}{2}} \{m(π_{p}, K_{p}) + m(π′_{p}, K_{p})\} trace τ_{p}(s)
trace π_{p}(Φ_{p}^{2}(s)) = −|b|_{p}^{− \frac{1}{2}} m(π′_{p}, K_{p})(λ_{p}(b) + η_{p}(b)).

(iv) If π_{p} is absolutely cuspidal

trace π_{p}(Φ_{p}^{1}(s)) = trace π_{p}(Φ_{p}^{2}(s)) = 0.

Φ_{p}^{1}(s) and Φ_{p}^{2}(s) are by no means unique. To establish their existence, as well as some of their additional properties, requires some preparation. I recall that if γ in G(Q_{p}) has eigenvalues γ_{1}, γ_{2}, which need not lie in Q_{p}, then

Δ(γ) = |\frac{(γ_{1} − γ_{2})^{2}}{γ_{1}γ_{2}} |p^{\frac{1}{2}} .

If T is a Cartan subgroup of G, over Q_{p}, and f a locally constant function on G(Q_{p}) with compact support, we set as usual

F^{T} (γ, f) = Δ(γ) ∫_{T (Q_{p})\G(Q_{p})} f(g^{−1}γg) dg

for γ regular in T (Q_{p}).

The collection of functions γ → F^{T} (γ, f) on the regular elements in the Cartan subgroups is not arbitrary. To characterize them we apply a result of Shalika; but first an observation. A regular element in G is an element whose centralizer is of dimension 2. Thus h is regular if and only if it is not a scalar matrix. We introduce the map φ from the variety of Ĝ of regular elements in G to the affine plane X given by

φ : h → (trace h, deth).

It is possible to introduce the notion of a regular two-form on Ĝ relative to X. First of all a two-form on a Zaariski-open subset Y of Ĝ will be called relatively trivial if Y can be covered by Zariski-open sets Y_{α}, mapping X_{α} in X, such that on Y_{α} the form is a sum of β dλ dη where β and λ are regular on Y_{α} and η is the pullback to Y_{α} of a regular function on X_{α}. To give a two-form on Ĝ regular relative to X, one gives a Zariski-open covering {Y_{α}} of Ĝ, and on each Y_{α} a regular two-form ω_{α}, defined only modulo relatively trivial two-forms, such that ω_{α} − ω_{β} is trivial on Y_{α} ∩ Y_{β}.

The fibres of φ are smooth and two-dimensional. The restriction of a regular relative two-form ω to a fibre yields a well-defined two-form on the fibre. If γ is regular, the map

h → h^{−1}γh

defines an isomorphism of G(γ) \ G with the fibre over (trace γ, detγ). G(γ) is the centralizer of γ. Let ω_{γ} be the pullback to G(γ) \ G of the restriction of ω to this fibre.

Suppose on the other hand that λ is a non-zero invariant form on degree 4 on G and η a nonzero invariant form of degree 2 on G(γ). It is possible to find a regular two-form ν on G(γ)\G, with pullback ν′ to G, and a two-form η′ on G regular relative to G(γ) \ G and right-invariant under G such that the restriction to η′ to G(γ) is η and such that λ = η′ν′. ν is uniquely determined by λ and η and we denote it \frac{λ}{η}.

Lemma 6.1. Suppose non-zero invariant forms λ and η, of degrees 4 and 2 respectively, are given on G and A. There is then a unique two-form ω on Ĝ regular relative to X which is invariant under the adjoint action of G and satisfies

(x_{o} − y_{o})\frac{λ}{η} = ω_{γ}

if

γ = ( \begin{matrix}x_{o} & 0 \\ 0 & y_{o} \end{matrix} )

is a regular element of A. The restriction of ω to any fibre is non-zero.

It is enough to verify this for

η = \frac{dX}{X} \frac{dY}{Y}

and

λ = \frac{dα dβ dγ dδ}{(αδ − βγ)^{2}}

when

h = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) .

I define ω as follows. If β ≠ 0

ω = −\frac{dα dβ}{β} .

If γ ≠ 0

ω = \frac{dα dγ}{γ}

Antwerp 1972 58 and if α − δ ≠ 0

ω = \frac{1}{α − δ} dγ dβ.

Since one of β, γ and α − δ is non-zero at any point of G, ω is defined. The compatability is a consequence of the relative triviality of

dα + dδ

and

α dδ + δ dα − β dγ − γ dβ.

The entire lemma can now be verified by direct, although somewhat lengthy, calculations. I omit them.

Suppose λ and η, and hence ω, have been chosen. If γ is the regular let η_{γ} be the two-form on G(γ) defined by

\frac{λ}{η_{γ}} ω_{γ}.

If the centralizer of γ is a Cartan subgroup and x and y are the rational characters given by some diagonalization, and if η and λ are chosen as in the proof of the lemma, then

η_{γ} = \frac{1}{x(γ) − y(γ)} \frac{dx}{x} \frac{dy}{y} .

If

γ = ( \begin{matrix}z_{o} & x_{o} \\ 0 & z_{o} \end{matrix} )

then

η_{γ} = \frac{1}{x_{o}} \frac{dz dx}{z^{2}}.

Suppose then λ and η have been fixed, as above or in some other way. If for each Cartan subgroup T defined over Q_{p} we have introduced on T a non-zero invariant two-form η_{T}, also defined over Q_{p}, then for any locally constant function f with compact support we may consider the functions F^{T} (γ, f). Let c(γ) in Q_{p} be the constant defined by

c(γ)η_{γ} = η_{T}

and let

E^{T} (γ, f) = \frac{|c(γ)|_{p}}{Δ(γ)|detγ|p^{\frac{1}{2}}} \frac{L(1, T/Q_{p})}{L(1, G/Q_{p})}F^{T} (γ, f).

Antwerp 1972 59 As before λ determines a measure not only on G(Q_{p}) but also on G′(Q_{p}), if G′ is the multiplicative group of a quaternion algebra which is not split at p.

We shall call a family {a^{T} }, where a^{T} is a locally constant function on T̂(Q_{p}) = Ĝ(Q_{p})∩T (Q_{p}) a Shalika family if the following conditions are satisfied.

(i) If T′ = h^{−1}T h with h in G(Q_{p}) and γ ∈ T (Q_{p}), then

a^{T′}(h^{−1}γh) = A^{T} (γ).

(ii) The support of a^{T} is relatively compact on T (Q_{p}). (iii) There is a locally constant function ξ on Z(Q_{p}) with compact support such that if z ∈ Z(Q_{p})

a^{A}(γ) = ξ(z)

for γ in the intersection of some neighborhood of z in A(Q_{p}) with Â(Q_{p}). (iv) There is another locally constant function ζ on Z(Q_{p}) with compact support such that if T is

a Cartan subgroup associated to a quadratic extension of Q_{p} then for each z in Z(Q_{p})

a^{T} (γ) = ξ(z) = \frac{|c(γ)|_{p}}{|detγ|p^{\frac{1}{2}}} \frac{L(1, T/Q_{p})}{L(1, G/Q_{p})} meas (T (Q_{p}) \ G′(Q_{p}))ζ(z)

on the intersection of some neighborhood of z in T (Q_{p}) with T̂(Q_{p}). As in §15 of Jacquet– Langlands, we have felt free to regard T also as a Cartan subgroup of G′.

Lemma 6.2. There exists a locally constant function f on G(Q_{p}) with compact support such that

a^{T} ≡ E^{T} (f)

for all T if and only if {a^{T} } is a Shalika family. ξ(z) is then the integral of f over the orbits of ( \begin{matrix}z & 1 \\ 0 & z \end{matrix} ) with respect to the measure defined by ω and

ζ(z) = f(z).

I observe that

\frac{|c(γ)|_{p}}{Δ(γ)|detγ|p^{\frac{1}{2}}}

is a constant which depends only on T and not on γ. Moreover E^{T} (γ, f) depends on ω but not on η_{T}.

Consider a family {E^{T} (f)}. The necessity of (i) is clear. Condition (iii), as well as condition (ii) together with the local constancy for T = A, is a consequence of the familiar relation

(9)  Δ(γ) ∫_{N(Q_{p})} ∫_{G(Z_{p})} f(k^{−1}n^{−1}γnk) dk dn = |\frac{α}{β}|^{\frac{1}{2}} ∫_{N(Q_{p})} ∫_{G(Z_{p})} f(k^{−1}γnk) dn dk

for

γ = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} )

and for dn = dx when

n = ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} )

because the right-hand side is equal to

(10)  ∫∫ f (k^{−1} ( \begin{matrix}α^{−1} & 0 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}z & 1 \\ 0 & z \end{matrix} ) ( \begin{matrix}α & 0 \\ 0 & 1 \end{matrix} ) k) \frac{dα}{|α|^{2}} dk

when γ is close to z. We should of course recall that

(11)  dh = |\frac{β}{α}| dx \frac{dα}{|α|} \frac{dβ}{|β|} dk

is, for

h = ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) k

a Haar measure on G(Q_{p}). To be more precise, the double integral on the left side of (9) is really an integral over A(Q_{p}) \ G(Q_{p}) with respect to the quotient of the measure of (11) by \frac{dα}{|α|} \frac{dβ}{|β|}. The integral in (10) is however an integral over G(γ_{o}, Q_{p}) \ G(Q_{p}), where

γ_{o} = ( \begin{matrix}z & 1 \\ 0 & z \end{matrix} ) ,

with respect to the quotient of the measure in (11) by the measure

\frac{du dv}{|u|^{2}}

on

G(γ_{o}) = \{( \begin{matrix}u & v \\ 0 & u \end{matrix} )\} .

Recalling the definition of the local Tamagawa measures as well as the explicit form for η_{γ_{o}}, we obtain (ii).

The condition (ii) and the local constancy of E^{T} (f) for a Cartan subgroup corresponding to a quadratic extension of Q_{p} follows readily from, for example, the proof of Lemma 7.3.2 of Antwerp 1972 61 Jacquet–Langlands. Their Lemma 7.3.1, together with a little calculation, implies (iv) when f is the characteristic function of G(Z_{p}). Its validity in general is then a consequence of a theorem of Shalika [6.6].

When {a^{T} } = {F^{T} (f)} we write ξ(z, f) and ζ(z, f) to stress the dependence of ξ and ζ on f. It is clear that ζ(z, f) can be specified arbitrarily. Consider the family of functions on Z(Q_{p}) formed by the ξ(z, f) corresponding to those f for which ζ(z, f) vanishes identically. This family is linear and translation invariant. It contains moreover a positive function with support in an arbitrarily small neighborhood of 1. To see this, take f to be a function which is positive at ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) and vanishes at

h = ( \begin{matrix}a & b \\ c & d \end{matrix} )

if |a + d − 2| ≥ ε, or |ad − bc − 1| ≥ ε or |b| + |c| ≤ ε where ε is a small positive number. It follows easily that the family contains every locally constant function with compact support.

To complete the proof of the lemma, it has only to be verified that f exists when {a^{T} } is a Shalika family for which both ξ and ζ vanish. This is easy because there are only a finite number of conjugacy classes of Cartan subgroups and the map

T̂(Q_{p}) × T (Q_{p}) \ G(Q_{p}) → G(Q_{p})

which sends t×h to h^{−1}th is a local homeomorphism and in fact either a double or simple covering of an open subset of G(Q_{p}).

We now take Φ_{p}^{1}(s), when s maps to a non-unit b, to be any locally constant function with compact support for which

F^{A}(γ, Φ_{p}^{1}(s)) = |b|_{p}^{− \frac{1}{2}} ∑_{i} \frac{χ_{i}(g^{−1}(s)γ) + χ_{i}(g^{−1}(s)γ̃)}{meas U_{i}}

if

g(s) = ( \begin{matrix}1 & 0 \\ 0 & b \end{matrix} )

and

γ̃ = ( \begin{matrix}β & 0 \\ 0 & α \end{matrix} )

for

γ = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} )

Antwerp 1972 62 and for which

F^{T} (γ, Φ_{p}^{1}(s)) = Δ(γ)\{ ∑_{i} \frac{χ_{V_{i}} (b^{−1}detγ)}{meas W_{i}}\} meas (T (Q_{p}) \ G′(Q_{p}))

if T is a Cartan subgroup associated to a quadratic extension of Q_{p}. That Φ_{p}^{1}(s) exists is assured by the previous lemma. ξ(z, Φ_{p}^{1}(s)) is zero and

Φ_{p}^{1}(z, s) = − ∑_{i} \frac{χ_{V_{i}} (b^{−1}detz)}{meas W_{i}}.

This relation will be needed for the trace formula.

Φ_{p}^{2}(s) is specified by demanding that

F^{A}(γ, Φ_{p}^{2}(s)) = 0

while F^{T} (γ, Φ_{p}^{2}(s)) equals

Δ(γ)(|b|_{p}^{−1} + 1)\{ − ∑_{i} \frac{χ_{V_{i}}(b^{−1}detγ)}{meas W_{i}} + \frac{χ_{V} (b^{−1}detγ)}{meas W}\} meas (T (Q_{p}) \ G′(Q_{p}))

if T is not split. For symmetry we denote the group introduced as K̄_{p} by V. We remark explicitly that

Φ_{p}^{2}(z, s) = −(|b|^{−1} + 1)\{ − ∑_{i} \frac{χ_{V_{i}}(b^{−1}detz)}{meas W_{i}} + \frac{χ_{V} (b^{−1}detz)}{meas W}\}.

In order to verify that these functions satisfy the required conditions we recall that if π is an irreducible admissible representation of G(Q_{p}) then its character is a function χ_{π} and for every locally constant function f on G(Q_{p}) with compact support the trace of π(f) is given by

\frac{1}{2} ∑ ∫_{T (Q_{p})} F^{T} (γ, f)χ_{π}(γ)Δ(γ) dγ.

The sum is taken over a set of representatives for the conjugacy classes of Cartan subgroups of G(Q_{p}). This fact is also valid if π admits a finite composition series whose terms are irreducible and admissible.

In particular for one of the representations ρ = ρ(λ, ν) introduced in §4 we have as a consequence of Proposition 7.6 of Jacquet–Langlands

trace ρ(f) = \frac{1}{2} ∫_{A(Q_{p})} F^{A}(γ, f)\{λ(α)ν(β) + ν(α)λ(β)\} dγ

if α and β are the eigenvalues of γ. It follows immediately that trace ρ(Φ_{p}^{2}(s)) = 0 and that trace ρ(Φ_{p}^{1}(s)) is |b|_{p}^{− \frac{1}{2}} (λ(b) + ν(b)) times the number of i for which γ → λ(α)ν(β) is trivial on U_{i}. This latter number is also equal to the number of i for which ν(α)λ(β) is trivial on U_{i}. The conditions on trace π(Φ_{p}^{1}(s)) and trace π(Φ_{p}^{2}(s)) for π infinite-dimensional and of the form π(τ_{p}), τ_{p} = λ ⊕ ν, follow from:

Antwerp 1972 63 Lemma 6.3. The multiplicity with which the trivial representation of K_{p} occurs in ρ(λ, ν) is equal to the number of i for which the quasi-character η : γ → λ(α)ν(β) of A(Q_{p}) is trivial on U_{i}.

Extend η to B(Q_{p}) by making it trivial on N(Q_{p}). Recall that

G(Q_{p}) = ∪B(Q_{p})g_{i}K_{p}.

Thus ρ(λ, ν) restricted to K_{p} is the direct sum of the representations of K_{p} induced from the representations η_{i} : k → η(g_{i}kg_{i}^{−1}) of K_{p} ∩ g_{i}^{−1}B(Q_{p})g_{i}. The relevant multiplicity is just the number of i for which η_{i} is trivial and η_{i} is trivial if and only if η is trivial on U_{i}.

If α is the quasi-character x → |x|_{p} then, for any quasi-character χ of Q_{p}^{×}, ρ(α^{\frac{1}{2}} χ, α^{− \frac{1}{2}} χ) has a composition series of length 2 in which the special representation σ(α^{\frac{1}{2}} χ, α^{− \frac{1}{2}} χ) and the one-dimensional representation π(α^{\frac{1}{2}} χ, α^{− \frac{1}{2}} χ) : h → χ(deth) appear.

If π = π(α^{\frac{1}{2}} χ, α^{− \frac{1}{2}} χ) then

\frac{1}{2} ∫_{A(Q_{p})} F^{A}(γ, Φ_{p}^{1}(s))χ_{π}(s)Δ(γ) dγ

is, since Δ(γ) is equal to |b|_{p}^{− \frac{1}{2}} on the support of F^{A}(γ, Φ_{p}^{1}(s)), equal to

|b|_{p}^{−1}{ ∑_{i} δ_{i}(χ)}χ(b).

Here δ_{i}(χ) is 1 or 0 according as χ is or is not trivial on V_{i}. The previous lemma, applied to the pair λ = α^{\frac{1}{2}} χ, ν = α^{− \frac{1}{2}} χ, shows that ∑_{i} δ_{i}(χ) is the multiplicity with which the trivial representation of K_{p} occurs in ρ(α^{\frac{1}{2}} χ, α^{− \frac{1}{2}} χ).

The character of σ(α^{\frac{1}{2}} χ, α^{− \frac{1}{2}} χ) is the difference of the characters of ρ(α^{\frac{1}{2}} χ, α^{− \frac{1}{2}} χ) and π(α^{\frac{1}{2}} χ, α^{− \frac{1}{2}} χ). Thus

(12)  \frac{1}{2} ∫_{A(Q_{p})} F^{A}(γ, Φ_{p}^{1}(s))χ_{π}(s)Δ(γ) dγ

is equal to ∑_{i} δ_{i}(χ)χ(b) if π is this special representation. The following lemma implies that (12) vanishes when π is absolutely cuspidal.

Antwerp 1972 64 Lemma 6.4. If π is an absolutely cuspidal representation the support of its character does not contain any matrix whose eigenvalues lie in Q_{p} and have different absolute values.

For p ≠ 2, this has been known for some time (cf. [6.5] and [6.7]). In general it is a simple consequence of a recent theorem of Casselman [6.2]. I omit the proof.

If Φ_{p}^{1}(s) is replaced by Φ_{p}^{2}(s) then (12) vanishes for every π. The condition (iii) is a consequence of condition (ii), which we shall now verify together with condition (iv). We have to show that

(13)  \frac{1}{2} ∑_{T}′ ∫_{T (Q_{p})} F^{T} (γ, f)χ_{π}(γ)Δ(γ) dγ

is equal to − ∑_{i} δ_{i}(χ)χ(b) if π = σ(α^{\frac{1}{2}} χ, α^{− \frac{1}{2}} χ) and f = Φ_{p}^{1}(s) and that it is equal to 0 if f is Φ_{p}^{1}(s) or Φ_{p}^{2}(s) and π is absolutely cuspidal. If δ(χ) is 1 or 0 acording as χ is or is not trivial on V, then the multiplicity of the trivial representation of K_{p} in σ(α^{\frac{1}{2}} χ, α^{− \frac{1}{2}} χ) is clearly

(14)  ∑_{i} δ_{i}(χ) − δ(χ).

We have to show that when π is this special representation and f is Φ_{p}^{2}(s) the expression (13) is equal to (|b|_{p}^{−1} +1)χ(b) times (14). I observe that the prime indicates that we omit the split Cartan subgroup from the summation.

If π is special or absolutely cuspidal and if π′ is, as in Theorem 15.1 of Jacquet–Langlands, the corresponding representation of G′(Q_{p}), then the expression (13) is equal to

(15)  −\frac{1}{2} ∑_{T}′ ∫_{T (Q_{p})} F^{T} (γ, f)χ_{π′}(γ)Δ(γ) dγ.

We may regard this as a sum over conjugacy classes of Cartan subgroups of G′(Q_{p}). If f′ is a function on G′(Q_{p}) it is possible to define F^{T} (γ, f′) in the same way as we defined F^{T} (γ, f). If Ψ_{p}^{1}(s) is the sum over i of the characteristic functions of the sets {h ∈ G′(Q_{p}) | Nmh ∈ bV_{i}} divided by their measures, then

F^{T} (γ, Φ_{p}^{1}(s)) = F^{T} (γ, Ψ_{p}^{1}(s)).

Take Ψ_{p}^{2}(s) to be (|b|_{p}^{−1} + 1) times the difference of the characteristic function of the set {h ∈ G′(Q_{p}) | Nmh ∈ bV } divided by its measure and the function Ψ_{p}^{1}(s). Then (15), for f = Φ_{p}^{k}(s), is equal to

− ∫_{G′(Q_{p})} Ψ_{p}^{k}(h, s)χ_{π′}(h) dh.

Antwerp 1972 65 The required relations follow immediately.

It is clear that the functions Φ^{1} and Φ^{2} defined by (6) and (7) satisfy (4) and (5). To prove Propositions 5.7 and 5.8 we apply the trace formula to calculate the left sides of (4) and (5).

Let χ be a quasi-character appearing in the sum (1).

∫_{Z^{o}(R)\G(R)} χ_{∞}(deth)Φ_{∞}(h) dh = trace π(Φ_{∞})

if π : h → χ_{∞}(deth). Our conditions on Φ_{∞} are such that this is 0 if μ is not one-dimensional and 1 if it is. For similar reasons

∫_{G(Q_{p})} χ_{p}(deth)Φ_{p}^{1}(h, s) dh = (|b|_{p}^{−1} + 1){ ∑_{i} δ_{i}(χ_{p})}χ_{p}(b)

and

∫_{G(Q_{p})} χ_{p}(deth)Φ_{p}^{2}(h, s) dh

is equal to

(|b|_{p}^{−1} + 1){δ(χ_{p}) − ∑_{i} δ_{i}(χ_{p})}χ_{p}(b).

Since

∫_{Z^{o}(R)\G(A)} χ(deth)Φ^{k}(h) dh

is equal to

∫_{Z^{o}(R)\G(R)} χ_{∞}(deth)Φ_{∞}(h) dh ∫_{G(Q_{p})} χ_{p}(deth)Φ_{p}^{k}(s, h) dh ∫_{G(A_{f}^{p})} χ(deth)f_{g}(h) dh,

it follows easily from Lemma 5.6 that the contribution of (1) to the trace formula for Φ^{1} is −Ā. Its contribution to the trace formula for Φ^{2}is Ā − Ā_{o}.

If the eigenvalues of γ do not lie in Q

∫_{G(γ,A)\G(A)} Φ^{k}(h^{−1}γh) dh

is

∫_{G(γ,R)\G(R)} Φ_{∞}(h^{−1}γh) dh ∫_{G(γ,Q_{p})\G(Q_{p})} Φ_{p}^{k}(s, h^{−1}γh) dh ∫_{G(γ,A_{f}^{p})\G(A_{f}^{p})} f_{g}(h^{−1}γh) dh.

The first integral is 0 if γ has real eigenvalues, otherwise it is

\frac{trace μ(γ)}{meas (Z^{o}(R) \ G(γ, R))}.

Antwerp 1972 66 Consider the second for those matrices γ with eigenvalues on Q_{p}. It is 0 for k = 2. If k = 1 we choose a set of representations for the conjugacy classes of γ which split in Q_{p} but not in R as follows. For each of the imaginary quadratic fields occurring in Proposition 5.7, we consider the corresponding group H. For each F we choose from each pair {γ, γ̃} in H(Q) corresponding to an element of F not in Q and its conjugate we choose one element γ. This yields the required set of representatives. For such a γ the centralizer G(γ) is H and the second integral equals

\frac{∑_{i} χ_{i}(v^{−1}g^{−1}(s)γ_{v}) + χ_{i}(v^{−1}g^{−1}(s)γ̃_{v})}{meas vU_{i}v^{−1}}

if v = v(F) is defined as before. Observe that

χ_{i}(v^{−1}g^{−1}(s)γ_{v}) = 0

if γ is a scalar matrix—we are assuming b is not a unit.

Since

meas (Z^{o}(R)H(Q) \ H(A)) = meas (H(Q) \ H(A_{f})) meas (Z^{o}(R) \ H(R))

the contribution of the terms in (2′) corresponding to a γ split in Q_{p} but not in R to the trace formula for Φ^{1} is ∑_{F} A(F). They contribute nothing to the trace formula for Φ^{2}.

If γ is not split in Q_{p} the second integral equals

\{\frac{∑_{i} χ_{V_{i}}(b^{−1}detγ)}{meas W_{i}}\} meas (G′(γ, Q_{p}) \ G′(Q_{p}))

if k = 1 and

(|b|_{p}^{−1} + 1) \{\frac{χ_{v}(b^{−1}detγ)}{meas W} − ∑_{i} \frac{χ_{V_{i}}(b^{−1}detγ)}{meas W_{i}}\} meas (G′(γ, Q_{p}) \ G′(Q_{p}))

if k = 2. The contributions of the terms in (2′) corresponding to a γ split neither in Q_{p} nor in R to the trace formula for Φ^{1} is, since these conjugacy classes may be identified with the conjugacy classes of non-scalars in G′(Q), just that part of the sum in Lemma 5.4 corresponding to such classes.

Consider the contribution of (2) to the trace formula for Φ^{1}. If γ belongs to Z(Q)

Φ^{1}(γ) \{\frac{− trace μ(γ)}{meas (Z^{o}(R) \ G′(R))}\} \{\frac{− ∑_{i} χ_{V_{i}}(b^{−1}detγ)}{meas W_{i}}\} f_{g}(γ).

If we also regard γ as an element of G′(Q) then G′(γ) = G′. Since

meas (Z^{o}(R)G(Q) \ G(A)) = meas (Z^{o}(R)G′(Q) \ G′(A))

and

meas (Z^{o}(R)G′(Q) \ G′(A)) = meas (G′(Q) \ G′(A_{f})) meas (Z^{o}(R) \ G′(R)),

the contribution of (2) to the trace formula is just the remaining part of the sum in Lemma 5.4.

Using similar formulae and an obvious variant of Lemma 5.4 we show that the contribution of (2) and that part of (2′) corresponding to γ which split neither in R nor in Q_{p} to the trace of Φ^{2} is

(|b|_{p}^{−1} + 1)(A(D) − A_{o}(D)).

To complete the proofs of Propositions 5.7 and 5.8 all we have to do is show that (8) contributes nothing to the trace formula for Φ^{2} and that its contribution for Φ^{1} is B. Since we are no longer dealing with a function given by a product we must replace

∏_{w≠∞} F^{A}(γ, Φ_{w})

by

F^{A}(γ, Φ_{p}^{k}(s))\{ ∏_{q≠p} |\frac{(a − d)^{2}}{ad} |q^{\frac{1}{2}} \} ∫_{A(A_{f}^{p})\G(A_{f}^{p})} f_{g}(h^{−1}γh) dh.

The eigenvalues of γ are a and d. The first term of this product vanishes if k = 2. If k = 1 it is equal to

|b|_{p}^{− \frac{1}{2}} ∑_{i} \frac{χ_{i}(g^{−1}(s)γ) + χ_{i}(g^{−1}(s)γ̃)}{meas U_{i}}.

After we sum over γ we may replace

χ_{i}(g^{−1}(s)γ) + χ_{i}(g^{−1}(s)γ̃)

by

χ_{i}(g^{−1}(s)γ)

and multiply by 2. Since D(γ, ϕ_{∞}) is

\frac{1}{4ε} min \{|\frac{α}{β}|^{\frac{1}{2}} , |\frac{β}{α}|^{\frac{1}{2}} \}

times the eigenvalue of μ(γ) of smallest absolute value, we may apply Lemma 5.5 to conclude that the required contribution is B, provided we show that

(16)  \frac{2 meas (Z^{o}(R)Z(Q) \ Z(A))}{4ε} = meas (A^{+}(Q) \ A(A_{f})).

This assertion is independent of the choice of forms on Z and A giving the Tamagawa measures. To avoid fuss, I cheat a little and take them to be \frac{dz}{z} and \frac{da}{a} \frac{db}{b}. Then ε = 1,

meas (Z^{o}(R)Z(Q) \ Z(A)) = 1

and

meas (A^{+}(Q) \ A(A_{f})) = \frac{1}{2}.

We had as a matter of fact already cheated, because the unnormalized Tamagawa measure of the set of ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) in (Z \ A)(A) for which |\frac{a}{b}| = 1 taken modulo (Z \ A)(Q) should have stood in front of (3) and (8). We have in effect made a tautologous use of the fact that this measure is 1. It would have been better not to do so.

7. The Lefschetz formula. We begin by recalling those cases of the conjecture that we are trying to prove. Suppose π is an irreducible representation of G(A) occurring in L_{sp}(μ) for which π_{∞} ≃ π(μ). We define π′ as in §4.

Theorem 7.1. If π′_{p} ≃ π(τ_{p}), where τ_{p} is the direct sum of two one-dimensional complex representations of the Weil group, then trace σ_{p}(s) ∈ \overline{Q} for all s in 𝔚(Q_{p}/Q_{p}) and π′_{p} ≃ π(σ_{p}).

In order to reduce the theorem to an assertion we are in a position to prove we digress.

Lemma 7.2. Suppose λ is a continuous finite-dimensional representation of the Galois group 𝔊(\overline{Q}_{p}/Q_{p}) over a finite extension of Q_{ℓ}. If trace λ(s) ∈ \overline{Q} whenever s ∈ 𝔚(Q_{p}/Q_{p}) and the image of s in Q_{p}^{×} has an absolute value which is sufficiently small then it lies in \overline{Q} for all s in 𝔚(\overline{Q}_{p}/Q_{p}).

If the absolute value of the image of s in Q_{p}^{×} is small, so are the absolute values of the images of s^{n}, n ≥ 1. Thus trace λ(s^{n}) ∈ \overline{Q} for n ≥ 1. It follows that all eigenvalues of λ(s) lie in \overline{Q}. In proving the lemma we may therefore suppose that λ is absolutely irreducible. If the absolute value of the image of s is different from 1, then the absolute value of the image of some power s^{n} of s, Antwerp 1972 69 with n positive or negative, is very small so that the eigenvalues of λ(s^{n}) and hence those of λ(s) lie in \overline{Q}.

Supposing λ absolutely irreducible, we may apply the argument of §4 to show that λ factors, through 𝔊(F^{un}/Q_{p}) where F is a finite Galois extension of Q_{p}. If s ∈ 𝔚(F^{un}/F) then λ(s) is a scalar. Moreover if s ≠ 1 the image of s has absolute value different from 1. In any case the eigenvalues of λ(s) lie in \overline{Q}. Since some power of any element of 𝔚(F^{un}/Q_{p}) lies in 𝔚(F^{un}/F) this is true in general.

Lemma 7.3. Suppose λ is a continuous finite-dimensional representation of 𝔊(\overline{Q}_{p}/Q_{p}) over a finite extension of Q_{ℓ} such that trace λ(s) ∈ \overline{Q} for all s ∈ 𝔚(\overline{Q}_{p}/Q_{p}). Suppose ν is a continuous finitedimensional complex representation of 𝔚(\overline{Q}_{p}/Q_{p}) and suppose that

trace λ(s) = trace ν(s)

for those s in 𝔚(\overline{Q}_{p}/Q_{p}) whose images in Q_{p}^{×} have an absolute value which is sufficiently small. Then the equality is valid for all s in 𝔚(\overline{Q}_{p}/Q_{p}).

Since we are only dealing with traces we may suppose that both λ and ν are direct sums of absolutely irreducible representations. Choosing the finite Galois extension F of Q_{p} sufficiently large, we may suppose that both the restriction of λ to 𝔚(\overline{Q}_{p}/Q_{p}) and ν factor through 𝔚(F^{un}/Q_{p}) and that

trace λ(s) = trace ν(s)

if s ∈ 𝔚(F^{un}/F) maps to an element of Q_{p}^{×} of absolute value less than 1. Choose an s_{o} of this type which generates 𝔚(F^{un}/F).

If p is a polynomial with coefficients in \overline{Q} and s maps to an element of sufficiently small absolute value, then

trace p(λ(s_{o}))λ(s) = trace p(ν(s_{o}))ν(s).

Let λ_{j}, 1 ≤ j ≤ r, be the restrictions of λ to the various eigenspaces of λ(s_{o}) and let the polynomials p_{j} be such that p_{j}(λ(s_{o})) are the corresponding projections. We suppose, as we may, that they have constant term 0. Then

trace λ_{j}(s) = trace p_{j}(λ(s_{o}))λ(s)

for those s which map to an element in Q_{p}^{×} of small absolute value. Moreover

trace p_{j}^{m}(λ(s_{o}))λ^{n}(s_{o}) = trace p_{j}^{m}(ν(s_{o}))ν^{n}(s_{o}), m, n ≥ 0.

Thus p_{j}(ν(s_{o})) is also a projection and corresponds to the same eigenvalue as p_{j}(λ(s_{o})). Since

trace ν_{j}(s) = trace p_{j}(ν(s_{o}))ν(s)

if s maps to an element of small absolute value we may consider the λ_{j} separately. In other words we may suppose that λ(s_{o}) and ν(s_{o}) are the same scalar ε.

Given s in 𝔚(F^{un}/Q_{p}) choose n sufficiently large that

trace λ(ss_{o}^{n}) = trace ν(ss_{o}^{n}).

Then

ε^{n} trace λ(s) = ε^{n} trace ν(s).

The lemma follows.

Suppose K = K^{p}K_{p} is given. Both ℋ\overline{Q}(K^{p}), the algebra of \overline{Q}-linear combinations of the functions f_{g}, g ∈ G(A_{f}^{p}), and 𝔊(\overline{Q}_{p}/Q_{p})

H_{p}^{1}(M_{K}, F_{μ}(\overline{Q}_{ℓ})) = H_{p}^{1}(M_{K}, F_{μ}(Q_{ℓ})) ⊗_{Q_{ℓ}} \overline{Q}_{ℓ}.

This double representation we call ρ.

Lemma 7.4. Suppose there are two subspaces X ⊆ Y of H_{p}^{1}(M_{K}, F_{μ}(\overline{Q}_{ℓ})) invariant under ρ such that if ρ′ is the double representation on the quotient

(1)  trace ρ′(f_{g}, s) = trace τ^{1}(f_{g}, s)

whenever s maps to an element in Q_{p}^{×} of absolute value less than 1. Suppose moreover that ρ′ factors through some 𝔊(F^{un}/Q_{p}), where F is a finite Galois extension of Q_{p}. Then Theorem 6.1 is valid if π_{f} contains the trivial representation of K.

Proposition 5.7 implies that the right side of (1) is rational. Thus the left side is also. Choose a π satisfying the conditions of Theorem 7.1. By a theorem of Casselman [7.1] and Miyake [7.3], π is uniquely determined by \{π_{q} | q ≠ p\}. Thus we can find α_{1}, ⋯, α_{r} in \overline{Q} and g_{1}, ⋯, g_{r} in G(A_{f}^{p}) such that

∑_{i} α_{i}ρ(f_{g_{i}})

is, in the notation of Proposition 3.1, the projection on U\overline{Q}^{π} ⊗ V\overline{Q}^{π}(K). Then

∑_{i} α_{i} trace ρ(f_{g_{i}}, s) = trace ( ∑_{i} α_{i}ρ(f_{g_{i}}, s)) = dim V\overline{Q}^{π}(K) trace σ_{p}(s)

if σ = σ(π). If σ′_{p} is the representation of 𝔊(\overline{Q}_{p}/Q_{p}) or of 𝔚(\overline{Q}_{p}/Q_{p}) on

U\overline{Q}^{π} ⊗ V\overline{Q}^{π}(K) ∩ X \ U\overline{Q}^{π} ⊗ V\overline{Q}^{π}(K) ∩ Y

then

∑_{i} α_{i} trace ρ′(f_{g_{i}}, s) = trace σ′_{p}(s).

Moreover if

deg σ′_{p} = dim V\overline{Q}^{π}(K) · deg σ_{p} = 2 dim V\overline{Q}^{π}(K)

then σ′_{p} is equivalent to dim V\overline{Q}^{π}(K) · σ_{p}.

Suppose π^{1} lies in A(μ) and π1 ; p̂ is defined as at the beginning of §5. If V_{f}^{π_{1}}(K) ≠ 0 then

∑ α_{i}π1 ; p̂(f_{g_{i}})

is 0 if π^{1} ≠ π and is the identity if π^{1} = π. By the very definition of the double representation τ^{1}

∑_{i} α_{i} trace τ^{1}(f_{g_{i}}, s) = dim V\overline{Q}^{π}(K) trace τ_{p}(s).

Lemmas 7.2 and 7.3 imply that trace σ′_{p}(s) lies in \overline{Q} for all s in 𝔚(\overline{Q}_{p}/Q_{p}) and is equal to dim V\overline{Q}^{π}(K) trace τ_{p}(s). In particular the degrees of σ′_{p} and dim V\overline{Q}^{π}(K) · τ_{p} are the same. Since τ_{p} is of degree 2 we conclude that σ′_{p} = dim V\overline{Q}^{π}(K)σ_{p} and that trace σ_{p}(s) = trace τ_{p}(s) for s ∈ 𝔚(\overline{Q}_{p}/Q). However we have assumed that no subrepresentation of σ′_{p} is special. In particular σ_{p} is not special. Referring to the definitions, we conclude that π′ = π(σ_{p}).

There is only one other case in which we can do anything about the conjecture at present.

Theorem 7.5. If π′_{p} is a special representation σ(λ_{p}, ν_{p}) then trace σ_{p}(s) ∈ \overline{Q} for s ∈ 𝔚(\overline{Q}_{p}/Q_{p}) and π′_{p} = π(σ_{p}).

To verify this theorem we shall use the following lemma.

Lemma 7.6. Suppose X ⊆ Y are two subspaces of H_{p}^{1}(M_{k}, F_{μ}(\overline{Q}_{ℓ})) invariant under the double representation ρ and let ρ′ be the double representation on the quotient. Suppose

trace ρ′(f_{g}, s) = trace τ^{2}(f_{g}, s)

whenever s in 𝔚(\overline{Q}_{p}/Q_{p}) maps to b in Q_{p}^{×} with |b|_{p} < 1. Suppose moreover that ρ′ as a representation of 𝔊(\overline{Q}_{p}/Q_{p}) is a direct sum of two-dimensional special representations. Then π′_{p} = π(σ_{p}) for every π in A^{2}(μ) for which V_{f}^{π}(K) ≠ 0.

If π′_{p} = σ(λ_{p}, ν_{p}) the conclusion of the lemma amounts to the assertion that σ_{p} is special, and that trace σ_{p}(s) ∈ \overline{Q} and is equal to λ_{p}(b) + ν_{p}(b) if s ∈ 𝔚(\overline{Q}_{p}/Q_{p}).

The structure of two-dimensional special representations is such that for some sufficiently large finite Galois extension F of Q_{p} the space of vectors invariant under the restriction of ρ′ to 𝔊(\overline{Q}_{p}/F^{un}) has dimension \frac{1}{2} deg ρ′. For no F is the dimension of this space larger. It follows that if ρ′ is in any way the direct sum of two-dimensional representations then each of them is special.

Choose α_{1}, ⋯, α_{r} in \overline{Q} and g_{1}, ⋯, g_{r} in G(A_{f}^{p}) so that if π^{1} lies in A(μ) and V_{f}^{π^{1}}(K) ≠ 0 then

∑ α_{i}π1 ; p̂(f_{g_{i}})

is 0 if π^{1} ≠ π and is the identity if π^{1} = π. The proof of Lemma 6.4 may be imitated to show that

trace σ_{p}(s) = λ_{p}(b) + ν_{p}(b)

for s ∈ 𝔚(\overline{Q}_{p}/Q_{p}). It is of course implicit in this equality that both sides lie in \overline{Q}. The proof also shows that the direct sum over those π in A^{2}(μ) for which V_{f}^{π}(K) ≠ 0 of dim V\overline{Q}^{π}(K)σ_{p} is a subrepresentation of ρ′ regarded as a representation of 𝔊(\overline{Q}_{p}/Q_{p}) alone. However it follows from Lemma 7.3 that deg ρ′ = deg τ^{2}. Consequently this direct sum is not merely a subrepresentation but in fact all of ρ′. In particular each σ_{p} is special.

Since we are only interested in the action of 𝔊(\overline{Q}_{p}/Q_{p}) we may replace M_{K}^{o} ⊗Q by M_{K}^{o} ⊗Q_{p} = (M_{K}^{o} ⊗ Q) ⊗Q Q_{p} and M_{K}^{o} ⊗ \overline{Q} by M_{K}^{o} ⊗ \overline{Q}_{p} = (M_{K}^{o} ⊗ Q_{p}) ⊗_{Q_{p}} \overline{Q}_{p}. Moreover for our purposes it is sufficient to take K of the form K^{p}K_{p} where

K^{p} ⊆ {h ∈ G(Z_{p}^{f}) | h ≡ 1 (mod M)}
K_{p} = {h ∈ G(Z_{p}) | h ≡ 1 (mod p^{m})}.

Antwerp 1972 73 However we must be able to take M, which is prime to p, and m arbitrarily large. It will be useful to assume that K^{p} is so small that if K_{1} = K^{p}G(Z_{p}) then each component of M_{K_{1}} ⊗ \overline{Q} has genus greater than zero.

We choose then m arbitrarily and fix it in the discussion to follow. Let R be the ring of integers in Q_{p}(ζ) where ζ is a p^{m}th root of unity. 𝔊(\overline{Q}_{p}/Q_{p}) acts on R and hence on Spec R. I recall that the action of s on Spec R is dual to that of s^{−1} on R. Thus the group acts on the left. In [3.1] it was seen that there is a scheme M_{K}^{o} \overline{⊗}R over Spec R and an action of 𝔊(\overline{Q}_{p}/Q_{p}) on this scheme as well as an isomorphism

(M_{K}^{o} \overline{⊗}R) ⊗_{R} \overline{Q}_{p} ≃ M_{K}^{o} ⊗ \overline{Q}_{p}

compatible with the actions of the Galois group on both sides. The action of the Galois group on the left is given by its action on the two factors. Another scheme M_{K} \overline{⊗}R proper and flat over Spec R was also introduced. The Galois group acted on M_{K} ⊗ R and there was an open immersion M_{K}^{o} \overline{⊗}R → M_{K} \overline{⊗}R compatible with the actions. Moreover the complement of M_{K}^{o} \overline{⊗}R in M_{K} \overline{⊗}R with its induced subscheme structure is also flat over Spec R. Finally the map M_{K} \overline{⊗}R → Spec R is smooth except at a finite number of points in the special fibre, all of which lie in M_{K}^{o} \overline{⊗}R.

There is a family of sheaves F_{μ}(Z/ℓ^{m}Z) on M_{K}^{o} \overline{⊗}R whose pullbacks to M_{K}^{o} ⊗ \overline{Q}_{p} yield the family {F_{μ}(Z/ℓ^{n}Z)}, defined by pulling back the corresponding family on M_{K}^{o} ⊗Q. Let F_{μ!}(Z/ℓ^{n}Z) be the extension of F_{μ}(Z/ℓ^{n}Z) by zero to M_{K} \overline{⊗}R or M_{K} ⊗ \overline{Q}_{p}. We were interested in the ℓ-adic cohomology of the family \{F_{μ}(Z/ℓ^{n}Z)\} on M_{K}^{o} ⊗ \overline{Q} and we denoted it by H^{i}(M_{K}^{o}, F_{μ}(Q_{ℓ})). For reasons which will soon become apparent, we denote it now by H^{i}(M_{K}^{o} ⊗ \overline{Q}, F_{μ}(Q_{ℓ})); but replace it immediately by the isomorphic group H^{i}(M_{K}^{o} ⊗ \overline{Q}_{p}, F_{μ}(Q_{ℓ})). The group formerly denoted H_{c}^{i}(M_{K}^{o}, F_{μ}(Q_{ℓ})) is H_{c}^{i}(M_{K}^{o} ⊗ \overline{Q}, F_{μ}(Q)), which is isomorphic to H_{c}^{i}(M_{K}^{o} ⊗ \overline{Q}_{p}, F_{μ}(Q_{ℓ})). I recall that this is by definition H^{i}(M_{K} ⊗ \overline{Q}_{p}, F_{μ!}(Q_{ℓ})).

Let φ be the immersion of M_{K}^{o} \overline{⊗}R in M_{K} \overline{⊗}R or of M_{K}^{o} ⊗ \overline{Q}_{p} in M_{K} ⊗ \overline{Q}_{p}. The isomorphism

φ^{∗}F_{μ!}(Z/ℓ^{n}Z) ≃ F_{μ}(Z/ℓ^{n}Z)

yields by adjunction

F_{μ!}(Z/ℓ^{n}Z) → φ_{∗}F_{μ}(Z/ℓ^{n}Z).

More generally, we have in the derived category

(4)  F_{μ!}(Z/ℓ^{n}Z) → R_{φ_{∗}}F_{μ}(Z/ℓ^{n}Z).

This yields a map from

H_{c}^{i}(M_{K}^{o} ⊗ \overline{Q}_{p}, F_{μ}(Z/ℓ^{n}Z)) ≃ H^{i}(M_{K} ⊗ \overline{Q}_{p}, F_{μ!}(Z/ℓ^{n}Z))

to

H^{i}(M_{K} ⊗ \overline{Q}_{p}, R_{φ_{∗}}F_{μ}(Z/ℓ^{n}Z)) ≃ H^{i}(M_{K}^{o} ⊗ \overline{Q}_{p}, F_{μ}(Z/ℓ^{n}Z)).

These are of course the maps used implicitly throughout the report; so in the limit the image of the left side in the right is the group formerly denoted H_{p}^{i}(M_{K}, F_{μ}(Q_{ℓ})), but which will now be denoted H_{p}^{i}(M_{K} ⊗ \overline{Q}_{p}, F_{μ}(Q_{ℓ})).

Let G(Z/ℓ^{m}Z) be the mapping cone of (4). This is a complex

⋯ 0 → F_{μ!}(Z/ℓ^{n}Z) → G^{0}(Z/ℓ^{n}Z) → G^{1}(Z/ℓ^{n}Z) → ⋯

in which F_{μ!}(Z/ℓ^{m}Z) is the term of degree −1. We have a long exact sequence

⋯ → H^{i}(M_{K} ⊗\overline{Q}_{p}, F_{μ!}(Z/ℓ^{n}Z)) → H^{i}(M_{K}^{o} ⊗\overline{Q}_{p}, F_{μ}(Z/ℓ^{n}Z)) → H^{i}(M_{K} ⊗\overline{Q}_{p}, G(Z/ℓ^{n}Z)) → · · · .

The complex G(Z/ℓ^{n}Z) is exact on M_{K}^{o} ⊗ \overline{Q}_{p}. If M_{K}^{∞} ⊗ \overline{Q}_{p} is the complement of M_{K}^{o} ⊗ \overline{Q}_{p} in M_{K} ⊗ \overline{Q}_{p} with its reduced subscheme structure, let i be the imbedding M_{K}^{∞} ⊗ \overline{Q}_{p} → M_{K} ⊗ \overline{Q}_{p}. In the derived category the complex G(Z/ℓ^{n}Z) is isomorphic to

⋯ → 0 → i_{∗}i^{∗}G^{o}(Z/ℓ^{n}Z) → · · · .

We need to know the cohomology of this complex. This is the same as that of the complex

(5)  0 → i^{∗}G^{o}(Z/ℓ^{n}Z) → · · · .

Suppose M̃_{K} ⊗\overline{Q}_{p} is the sum of the strict henselisations of M_{K} ⊗\overline{Q}_{p} at the points of M_{K}^{∞} ⊗\overline{Q}_{p}. We know [3.1] that M̃_{K} ⊗ \overline{Q}_{p} is a direct sum over

C_{K} = N(A_{f})A^{+}(Q) \ G(A_{f})/K

of schemes M_{K} ⊗ \overline{Q}_{p}|h each of which is the spectrum of a strict henselian ring which is also a discrete valuation ring, with residue field \overline{Q}_{p}. h is to be thought of as a representative of the given double coset in G(A_{f}) or as the double coset itself, according to the context. x(K, h) will denote a uniformizing parameter for this ring. M̃_{K}^{o} ⊗ \overline{Q}_{p}, with summands M_{K}^{o} ⊗ \overline{Q}_{p}|h, will be the fibre product of M̃_{K} ⊗ \overline{Q}_{p} with M_{K}^{o} ⊗ \overline{Q}_{p}. It is the spectrum of a field A(K, h, \overline{Q}_{p}).

Antwerp 1972 75

Suppose K_{o} = K_{o}^{p}K_{p} with K_{o}^{p} ⊆ K^{p}. The fibre over M_{K} ⊗ \overline{Q}_{p}|h in M̃_{K_{o}}^{o} ⊗ \overline{Q}_{p} is

∐_{{ho∈C_{Ko} \atop ho→h}} M_{K}^{o} ⊗ \overline{Q}_{p}|h_{o}.

Let r = r(h) be the order of the cyclic group

(6)  h_{o}^{−1}N(A_{f})h_{o} ∩ K/h_{o}^{−1}N(A_{f})h_{o} ∩ K_{o}.

The map M_{K_{o}} ⊗ \overline{Q}_{p}|h_{o} → M_{K}^{o} ⊗ \overline{Q}_{p}|h takes x(K, h) to a unit times x(K_{o}, h_{o})^{r}. If K_{o} is normal in K then K/K_{o} acts in a simple way. An element k of K takes M_{K_{o}} ⊗ \overline{Q}_{p}|h_{o} to M_{K_{o}}^{o} ⊗ \overline{Q}_{p}|h′_{o} if h_{o}k and h′_{o} represent the same element of C_{K_{o}}. With this action the group (6) becomes the Galois group of A(K_{o}, h_{o}, \overline{Q}_{p}) over A(K, h, \overline{Q}_{p}).

To compute the cohomology groups of (5) we make use of the fact (SGA 4. VIII.5.2) that they are the cohomology groups of the pullback of F_{μ}(Z/ℓ^{n}Z) to M̃_{K}^{o} ⊗\overline{Q}_{p} viz., the direct sum over C_{K} of the cohomology groups of the pullbacks to M_{K}^{o} ⊗ \overline{Q}_{p}|h. These may be computed as Galois cohomology and, indeed, since the order of the Galois module defined by F_{μ}(Z/ℓ^{n}Z) is prime to p as cohomology groups for

𝔊^{p}(K, h, \overline{Q}_{p}) = 𝔊(A^{p}(K, h, \overline{Q}_{p})/A(K, h, \overline{Q}_{p}))

if A^{p}(K, h, \overline{Q}_{p}) is the direct limit of all finite Galois extensions of A(K, h, \overline{Q}_{p}) of degree prime to p. We may identify Spec (A^{p}(K, h, \overline{Q}_{p})) with

lim_{\overleftarrow{Ko}} M_{K_{o}}^{p} ⊗ \overline{Q}_{p}|h

and hence 𝔊^{p}(K, h, \overline{Q}_{p}) with h^{−1}N(A_{f}^{p})h ∩ K. There is of course a K_{o} such that the pullback of F_{μ}(Z/ℓ^{n}Z) is

L(Z/ℓ^{n}Z) ×_{K/K_{o}} M_{K_{o}}^{o} ⊗ \overline{Q}_{p}|h.

Every map of the inverse limit to this scheme factors through a map of M_{K_{o}}^{o} ⊗ \overline{Q}_{p}|h of the form v × identity with v ∈ L(Z/ℓ^{n}Z). The Galois module associated to F_{μ}(Z/ℓ^{n}Z) is thus L(Z/ℓ^{n}Z) with h^{−1}N(A_{f}^{p})h ∩ K acting in the usual way.

Since 𝔊^{p}(K, h, \overline{Q}_{p}) is isomorphic to Z_{f}^{p}, the cohomology groups are easily calculated. In degree 0 we obtain the invariants L_{h}^{o}(Z/ℓ^{n}Z) of L(Z/ℓ^{n}Z) with respect to the actions of h^{−1}N(A_{f}^{p})h∩ K. In degree 1 we obtain L_{o}^{h}(Z/ℓ^{n}Z), the quotient of L(Z/ℓ^{n}Z) by the subgroup generated by elements of the form (k − 1)v, k ∈ h^{−1}N(A_{f}^{p})h ∩ K. In higher degrees the groups vanish.

If we apply the properties of the mapping cone, then take a limit and tensor with Q_{ℓ}, we obtain the following lemma.

Antwerp 1972 76 Lemma 7.7. There is an exact sequence

0 → H^{o}(M_{K} ⊗ \overline{Q}_{p}, F_{μ!}(Q_{ℓ})) → H^{o}(M_{K}^{o} ⊗ \overline{Q}_{p}, F_{μ}(Q_{ℓ})) →

→ ⊕_{C_{K}} L_{h}^{o}(Q_{ℓ}) → H^{1}(M_{K} ⊗ \overline{Q}_{p}, F_{μ!}(Q_{ℓ})) → H^{1}(M_{K}^{o} ⊗ \overline{Q}_{p}, F_{μ}(Q_{ℓ})) →
→ ⊕_{C_{K}} L_{o}^{h}(Q_{ℓ}) → H^{2}(M_{K} ⊗ \overline{Q}_{p}, F_{μ!}(Q_{ℓ})) → H^{2}(M_{K}^{o} ⊗ \overline{Q}_{p}, F_{μ}(Q_{ℓ})) → 0

in which L_{h}^{o}(Q_{ℓ}) is the set of invariants of h^{−1}(N(Q_{ℓ}))h in L(Q_{ℓ}) and L_{o}^{h}(Q_{ℓ}) is the quotient of L(Q_{ℓ}) by the sum of the ranges of k − 1, k ∈ h^{−1}N(Q_{ℓ})h.

The terms of higher degree have been left out of this exact sequence because they vanish anyway. The complex G^{•}(Z/ℓ^{n}Z) may also be taken as a complex of sheaves over M_{K} \overline{⊗}R as may, if i now denotes the imbedding M_{K}^{∞}\overline{⊗}R → M_{K} \overline{⊗}R, i_{∗}i^{∗}G^{•}(Z/ℓ^{n}Z). We can also regard it, by taking an inverse image, as a sheaf on the special fibre (M_{K} \overline{⊗}R) ⊗ \overline{F}_{p}, which we denote M_{K} \overline{⊗} \overline{F}_{p} or on (M_{K} \overline{⊗}R) ⊗ R^{un} = M_{K} \overline{⊗}R^{un}. R^{un} is the maximal unramified extension of R.

The diagram

(7)  M_{K} \overline{⊗} \overline{F}_{p} → M_{K} \overline{⊗}R^{un} ← M_{K} ⊗ \overline{Q}_{p}

yields

H^{i}(M_{K} \overline{⊗} \overline{F}_{p}, i_{∗}i^{∗}G^{•}(Z/ℓ^{n}Z))
↑
H^{i}(M_{K} \overline{⊗}R^{un}, i_{∗}i^{∗}G^{•}(Z/ℓ^{n}Z))
↓
H^{i}(M_{K} ⊗ \overline{Q}_{p}, i_{∗}i^{∗}G^{•} (Z/ℓ^{n}Z)).

One establishes, using standard techniques, that both arrows are isomorphisms. The groups on the left may be calculated in the same way as those on the right. We employ, mutatis mutandis, the same notations for the objects required in the course of calculations.

We can now seriously begin to set the stage for the application of Lemma 7.4. The arrow on the left of (7) determines, for all sheaves we are considering, isomorphisms of cohomology groups—before or after limits are taken. We have therefore a commutative diagram

(8)

⊕_{C_{K}} L_{h}^{o}(Q) ⟶ H^{1}(M_{K} \overline{⊗} \overline{F}_{p}, F_{μ!}(Q_{ℓ})) ⟶ H_{p}^{1}(M_{K} \overline{⊗} \overline{F}_{p}, F_{μ}(Q_{ℓ})) ⟶ 0
↓  ↓  ↓
⊕_{C_{K}} L_{h}^{o}(Q_{ℓ}) ⟶ H^{1}(M_{K} ⊗ \overline{Q}_{p}, F_{μ!}(Q_{ℓ})) ⟶ H_{p}^{1}(M_{K} ⊗ \overline{Q}_{p}, F_{μ}(Q_{ℓ})) ⟶ 0

in which the rows are exact and the left hand arrow is the identity. To show that the arrow on the right is injective, we have just to establish this for the middle arrow. We employ the results [7.2] and the works therein cited.

Antwerp 1972 77

There is a finite extension of the quotient field of R^{un} such that M_{K} ⊗ \overline{Q}_{p} has a stable model over its ring of integers S. Set (M_{K} \overline{⊗}R^{un}) ⊗_{R^{un}} S = M_{K} \overline{⊗}S. M_{K} \overline{⊗}S → Spec S is smooth except perhaps at those points in the special fibre lying over a point of M̄_{K} \overline{⊗}R at which M̄_{K} \overline{⊗}R → Spec R is not smooth. In particular the arithmetical surface M_{K} \overline{⊗}S is regular except at a finite number of points in the special fibre. Let M′_{K} \overline{⊗}S be a regular scheme obtained by a desingularization of M_{K} \overline{⊗}S over this finite set. Because of our assumption on K^{p}, no irreducible component of M_{K} \overline{⊗}F_{p} is rational. We may therefore suppose that M′_{K} \overline{⊗}S is a minimal model. Fixing a commutative diagram

S ↙ ↘ \overline{F}_{p}  \overline{Q}_{p}

↖ ↗ R

we obtain another

(9)  \begin{matrix}M′_{K} \overline{⊗} \overline{F}_{p} & ⟶^{x} & M′_{K} \overline{⊗}S & ⟵^{y} & M′_{K} \overline{⊗}\overline{Q}_{p} \\ ↓f & & ↓g & & ↓h \\ M_{K} \overline{⊗} \overline{F}_{p} & ⟶^{u} & M_{K} \overline{⊗}R^{un} & ⟵^{v} & M_{K} \overline{⊗}\overline{Q}_{p} \end{matrix}

in which the right hand arrow is an isomorphism.

There is a spectral sequence

H^{i}(M_{K} \overline{⊗} \overline{F}_{p}, u^{∗}R^{j}v_{∗}F_{μ!}(Z/ℓ^{n}Z)) ⟹ H^{i+j}(M_{K} ⊗ \overline{Q}_{p}, F_{μ!}(Z/ℓ^{n}Z)).

As a consequence there is an exact sequence

(10)

0 → H^{1}(M_{K} \overline{⊗} \overline{F}_{p}, u^{∗}R^{o}v_{∗}F_{μ!}(Z/ℓ^{n}Z)) → H^{1}(M_{K} ⊗ \overline{Q}_{p}, F_{μ!}(Z/ℓ^{n}Z)) →
→ H^{o}(M_{K} \overline{⊗} \overline{F}_{p}, u^{∗}R^{1}v_{∗}F_{μ!}(Z/ℓ^{n}Z)) → H^{2}(M_{K} \overline{⊗} \overline{F}_{p}, u^{∗}R^{o}v_{∗}F_{μ!}(Z/ℓ^{n}Z)).

To establish the asserted injectivity in the diagram (8) we need only show that

u^{∗}v_{∗}F_{μ!}(Z/ℓ^{n}Z) ≃ F_{μ!}(Z/ℓ^{n}Z).

Let F′_{μ!}(Z/ℓ^{n}Z) be the pullback of F_{μ!}(Z/ℓ^{n}Z) by means of any of the vertical arrows in (9). Since h is an isomorphism

u^{∗}v_{∗}F_{μ!}(Z/ℓ^{n}Z) ≃ u^{∗}v_{∗}h_{∗}F_{μ!}(Z/ℓ^{n}Z) ≃ u^{∗}g_{∗}y_{∗}F′_{μ!}(Z/ℓ^{n}Z).

Antwerp 1972 78 Since g is proper, the base change theorem for proper morphisms shows that the group on the right is isomorphic to

f_{∗}x^{∗}y_{∗}F′_{μ!}(Z/ℓ^{n}Z).

The proof of Lemma 1.12 of [7.2] shows that on M′_{K} \overline{⊗}S the special fibre M′_{K} \overline{⊗} \overline{F}_{p} is a divisor with normal crossings. Just as in SGA 7.1.3 one sees that

x^{∗}y_{∗}F′_{μ!}(Z/ℓ^{n}Z) ≃ F′_{μ!}(Z/ℓ^{n}Z).

It remains to be shown that

f_{∗}F′_{μ!}(Z/ℓ^{n}Z) ≃ F_{μ!}(Z/ℓ^{n}Z).

The left side is f_{∗}f^{∗}F_{μ!}(Z/ℓ^{n}Z). Since M_{K} \overline{⊗}R^{un} is normal and g is birational, this final isomorphism is a consequence of Zariski’s main theorem.

Let M̂_{K} be the desingularization M_{K} \overline{⊗}\overline{F}_{p}. It can be obtained by tensoring \overline{F}_{p} with a desingularization of M̂_{K} \overline{⊗}F_{p} of M_{K} \overline{⊗}F_{p}. Let

q : M̂_{K} \overline{⊗} \overline{F}_{p} → M_{K} \overline{⊗} \overline{F}_{p}

be the map giving the desingularization and let F̂_{μ!}(Z/ℓ^{n}Z) be q^{∗}F_{μ!}(Z/ℓ^{n}Z). From the Leray spectral sequence we obtain

0 → H^{1}(M_{K} \overline{⊗} \overline{F}_{p}, R^{o}q_{∗}F̂_{μ!}(Z/ℓ^{n}Z)) → H^{1}(M̂_{K} \overline{⊗} \overline{F}_{p}, F̂_{μ!}(Z/ℓ^{n}Z)) → H^{o}(M_{K} \overline{⊗} \overline{F}_{p}, R^{1}q_{∗}F̂_{μ!}(Z/ℓ^{n}Z)).

Since R^{1}q_{∗} = 0 we have

H^{1}(M̂_{K} \overline{⊗} \overline{F}_{p}, F̂_{μ!}(Z/ℓ^{n}Z)) ≃ H^{1}(M_{K} \overline{⊗} \overline{F}_{p}, q_{∗}F̂_{μ!}(Z/ℓ^{n}Z)).

But we have an exact sequence

0 → F_{μ!}(Z/ℓ^{n}Z) → q_{∗}F̂_{μ!}(Z/ℓ^{n}Z) → E(Z/ℓ^{n}Z) → 0

in which E(Z/ℓ^{n}Z) has support at the set of singular points. Thus we have an exact sequence

H^{0}(M_{K} \overline{⊗} \overline{F}_{p}, E(Z/ℓ^{n}Z)) → H^{1}(M_{K} \overline{⊗} \overline{F}_{p}, F_{μ!}(Z/ℓ^{n}Z) → H^{1}(M_{K} \overline{⊗} \overline{F}_{p}, q_{∗}F̂_{μ!}(Z/ℓ^{n}Z)) → 0

and hence a surjection

(11)  H^{1}(M_{K} \overline{⊗} \overline{F}_{p}, F_{μ!}(Z/ℓ^{n}Z)) → H^{1}(M̂_{K} \overline{⊗} \overline{F}_{p}, F̂_{μ!}(Z/ℓ^{n}Z)).

This yields a surjection

H_{p}^{1}(M_{K} \overline{⊗} \overline{F}_{p}, F_{μ}(Q_{ℓ})) → H_{p}^{1}(M̂_{K} \overline{⊗} \overline{F}_{p}, F̂_{μ}(Q_{ℓ})).

The notation on the right is to be interpreted in the obvious way.

L(s) and R(g) continue to act on M̂_{K}^{o} \overline{⊗}\overline{F}_{p}, the inverse image of M_{K}^{o} \overline{⊗} \overline{F}_{p} in M̂_{K} \overline{⊗} \overline{F}_{p}. They also act on M_{K} \overline{⊗}\overline{F}_{p} and M̂_{K} \overline{⊗} \overline{F}_{p}. Moreover one defines a map

gF̂_{μ}^{K′}(Z/ℓ^{n}Z) → R^{∗}(g)F̂_{μ}^{K}(Z/ℓ^{n}Z)

as in the third paragraph and verifies, by examining its effect on fibres, that it is an isomorphism. Thus T (g) acts on H_{p}^{1}(M̂_{K} \overline{⊗} \overline{F}_{p}, F̂_{μ}(Q_{ℓ})). So does L(s). Since all the necessary compatabilities are satisfied, Theorem 7.1 will be a consequence of the next lemma.

Lemma 7.8. Let ρ′ be the double representation of ℋ_{Q}(K^{p}) and 𝔊(\overline{Q}_{p}/\overline{Q}_{p}) on H_{p}^{1}(M̂_{K} \overline{⊗}\overline{F}_{p}, F̂_{μ}(Q_{ℓ})). If g ∈ G(A_{f}^{p}) and s in 𝔚(\overline{Q}/Q_{p}) maps to b in Q_{p}^{×} with |b|_{p} < 1 then the trace of ρ′(f_{g}, s) is

−{ ∑_{F} A(F)} − A(D) − B + Ā.

We begin with an observation about the operator L(s) on cohomology. We have set

(M_{K} \overline{⊗}R) ⊗_{R} F_{p} = M_{K} \overline{⊗}F_{p}

and we may write

M_{K} \overline{⊗} \overline{F}_{p} = (M_{K} \overline{⊗}R) ⊗_{R} \overline{F}_{p} = (M_{K} \overline{⊗}F_{p}) ⊗_{F_{p}} \overline{F}_{p}.

If we use the representation in the middle the action L(s) of s in 𝔊(\overline{Q}_{p}/Q_{p}) on M_{K} \overline{⊗} \overline{F}_{p} is defined by its action on the factors M_{K} \overline{⊗}R and \overline{F}_{p}. Howover the action on M_{K} \overline{⊗}R, together with the trivial action on Spec F_{p}, yields an action on M_{K} \overline{⊗}F_{p}. This can be combined with the trivial action on F_{p} to yield, in terms of the representation on the right an action L_{1}(s) on M_{K} \overline{⊗} \overline{F}_{p}. The trivial action on M_{K} \overline{⊗}F_{p} combined with the action on Spec \overline{F}_{p} yields an action L′_{2}(s) on M_{K} \overline{⊗}\overline{F}_{p}. L(s) is the product of L_{1}(s) and L′_{s}(s). Similar remarks apply to the actionon M̂_{K}^{o} \overline{⊗}\overline{F}_{p} or on M̂_{K} \overline{⊗} \overline{F}_{p} on which we now concentrate our attention.

Antwerp 1972 80

To make L_{1}(s) act on the cohomology we observe that the broken arrow in the diagram below yields a sheaf over M̂_{K}^{o} \overline{⊗}\overline{F}_{p} which may be taken as L_{1}^{∗}(s)F_{μ}(Z/ℓ^{n}Z). The inverse of the upper horizontal arrow yields

F_{μ}(Z/ℓ^{n}Z) ⟶^{L_{1}(s)} F_{μ}(Z/ℓ^{n}Z)
↓  ↓
M̂_{K}^{o} ⊗ \overline{F}_{p} ⟶ M̂_{K}^{o} \overline{⊗} \overline{F}_{p}

an isomorphism between L_{1}^{∗}(s)F_{μ}(Z/ℓ^{n}Z) and F_{μ}(Z/ℓ^{n}Z).

Suppose s → b ∈ Q_{p}^{×} and |b|_{p} < 1 so that s maps to a positive power, the mth, of the Frobenius. Because of our definitions L′_{2}(s^{−1}), the inverse of L′_{2}(s), acts as as the mth power of the Frobenius endomorphism of the second factor Spec \overline{F}_{p}. Therefore, in so far as its effect on the cohomology is concerned, L′_{2}(s) may be replaced by L_{2}(s), the mth power of the Frobenius endomorphism of the first factor M̂_{K}^{o} \overline{⊗}F_{p}, the geometric Frobenius (SGA 5 XV). The same remarks apply to M̂_{K} \overline{⊗} \overline{F}_{p} and the sheaves F̂_{μ!}(Z/ℓ^{n}Z).

The actions on cohomology groups are defined by the correspondence: φ_{1} × φ_{2} = R(e) × R(g)L_{1}(s)L_{2}(s) which may be regarded as mapping M̂_{K′}^{o} \overline{⊗} \overline{F}_{p} to M̂_{K}^{o} \overline{⊗} \overline{F}_{p} × M̂_{K}^{o} \overline{⊗} \overline{F}_{p} or as a mapping M̂_{K′} \overline{⊗} \overline{F}_{p} to M̂_{K} \overline{⊗} \overline{F}_{p} × M̂_{K} \overline{⊗} \overline{F}_{p}. In addition one needs the map

φ_{2}^{∗}F̂_{μ}^{K}(Z/ℓ^{n}Z) → φ_{1}^{∗}a^{−1}(g)F̂_{μ}^{K}(Z/ℓ^{n}Z)

which is obtained as a composition R^{∗}(g)L_{1}^{∗}(s)L_{2}^{∗}(s)F̂_{μ}^{K}(Z/ℓ^{n}Z) → R^{∗}(g)F̂_{μ}^{K}(Z/ℓ^{n}Z) → gF̂_{μ}^{K′}(Z/ℓ^{n}Z) →

→ a^{−1}(g)F̂_{μ}^{K}(Z/ℓ^{n}/Z) → R^{∗}(e)a^{−1}(g)F̂_{μ}^{L}(Z/ℓ^{n}Z)

as well as the trace map (SGA 4 XVII)

R_{∗}(e)R^{∗}(e)(a^{−1}(g)F̂_{μ}^{K}(Z/ℓ^{n}Z)) → a^{−1}(g)F̂_{μ}^{K}(Z/ℓ^{n}Z).

If we apply the analogue of Lemma 7.7 for the space M̂_{K} ⊗\overline{F}_{p} we see that the trace of ρ′(f_{g}, s) is

trace ρ_{1}(f_{g}, s) − trace ρ_{2}(f_{g}, s) + trace ρ_{3}(f_{g}, s) − trace ρ_{4}(f_{g}, s).

Here ρ_{1} and ρ_{4} are the representations on H^{i}(M̂_{K} \overline{⊗} \overline{F}_{p}, F̂_{μ!}(Q_{ℓ})) with i = 1 and 0 respectively. The representation on H^{o}(M̂_{K}^{o} \overline{⊗}\overline{F}_{p}, F̂_{μ}(Q_{ℓ})) is ρ_{3} and ρ_{2} is the representation on the tensor product of Q_{ℓ} with

lim_{⟵} H^{o}(M_{K} \overline{⊗} \overline{F}_{p}, i_{∗}i^{∗}G^{•}(Z/ℓ^{n}Z))

Antwerp 1972 81 which is a group we have explicitly calculated. Notice also that we are exploiting the circumstance that M̂_{K} \overline{⊗} \overline{F}_{p} is isomorphic to M_{K} \overline{⊗} \overline{F}_{p} in a neighborhood of M_{K}^{∞}\overline{⊗} \overline{F}_{p}. Let ρ_{5} be the representation on H^{2}(M̂_{K} \overline{⊗} \overline{F}_{p}, F̂_{μ!}(Q_{ℓ})). The trace of ρ′(f_{g}, s) is the sum of

trace ρ_{1}(f_{g}, s) − trace ρ_{4}(f_{g}, s) − trace ρ_{5}(f_{g}, s)

and

trace ρ_{3}(f_{g}, s) + trace ρ_{5}(f_{g}, s)

and

− trace ρ_{2}(f_{g}, s).

Lemma 7.9. Under the assumptions of Lemma 7.8, the sum of trace ρ_{3}(f_{g}, s) and trace ρ_{5}(f_{g}, s) is equal to Ā.

Let T (Z/ℓ^{n}Z) be the sheaf of ℓ^{n}th roots of unity. As usual we introduce the dual complexes

DF̂_{μ!}^{K}(Z/ℓ^{n}Z) = R = Hom(F̂_{μ!}^{K}(Z/ℓ^{n}Z), T (Z/ℓ^{n}Z)).

If φ denotes, for any K, the imbedding M̂_{K}^{o} \overline{⊗} \overline{F}_{p} → M̂_{K} \overline{⊗} \overline{F}_{p} and μ̃ the representation contragredient to μ, then

DF̂_{μ!}^{K}(Z/ℓ^{n}Z) ≃ R = φ_{∗}(F̂ K ; μ̃(Z/ℓ^{n}Z) ⊗ T (Z/ℓ^{n}Z)).

The map

(12)  φ_{2}^{∗}F̂_{μ!}^{K}(Z/ℓ^{n}Z) → φ_{1}^{∗}a^{−1}(g)F̂_{μ!}^{K}(Z/ℓ^{n}Z)

yields

D(φ_{1}^{∗}gF̂_{μ!}^{K}(Z/ℓ^{n}Z)) → D(φ_{2}^{∗}F̂_{μ!}^{K}(Z/ℓ^{n}Z))

or

(13)  R = φ_{∗}φ_{1}^{∗}gF̂ K ; μ̃(Z/ℓ^{n}Z) → R = φ_{∗}φ_{2}^{∗}F̂ K ; μ̃(Z/ℓ^{n}Z)

or (SGA 5.I.1.12)

(14)  φ_{1}^{∗}Da^{−1}(g)F̂_{μ!}^{K}(Z/ℓ^{n}Z) → R =^{!}φ_{2}DF̂_{μ!}^{K}(Z/ℓ^{n}Z).

There is a map of sheaves over M̂_{K}^{o} \overline{⊗} \overline{F}_{p}

(15)  φ_{1}^{∗}(a^{−1}(g)F̂ K ; μ̃(Z/ℓ^{n}Z) ⊗ T (Z/ℓ^{n}Z)) → φ_{2}^{∗}(F̂ K ; μ̃(Z/ℓ^{n}Z) ⊗ (Z/ℓ^{n}Z)).

which can be constructed in the following manner. We start with the isomorphism (SGA 5.XV)

F̂ K ; μ̃(Z/ℓ^{n}Z) → L_{2}^{∗}(s)F̂ K ; μ̃(Z/ℓ^{n}Z).

Then we lift by means of L_{1}^{∗}(s) and compose with the map F̃ K ; μ̃(Z/ℓ^{n}Z) → L_{1}^{∗}(s)F̂ K ; μ̃(Z/ℓ^{n}Z) defined by the upper horizontal arrow in the following diagram

↓  ↙  ↓_{M̂_{K}^{o} \overline{⊗} \overline{F}_{p}  ⟶^{L_{1}(s)} M̂_{K}^{o} \overline{⊗} \overline{F}_{p} .}^{F̂ K ; μ̃(Z/ℓ^{n}Z) ⟶^{L_{1}(s)} F̂ K ; μ̃(Z/ℓ^{n}Z)}

Finally we lift by R^{∗}(g) and compose with the resultant of

R^{∗}(e)a^{−1}(g)F̂ K ; μ̃(Z/ℓ^{n}Z) ⟶^{∼} a^{−1}(g)F̂ K′ ; μ̃(Z/ℓ^{n}Z) → gF̂ K′ ; μ̃(Z/ℓ^{n}Z) ⟶^{∼} R^{∗}(g)F̂ K ; μ̃(Z/ℓ^{n}Z)

to obtain

φ_{1}^{∗}(gF̂ K ; μ̃(Z/ℓ^{n}Z)) → φ_{2}^{∗}(F̄ K ; μ̃(Z/ℓ^{n}Z)).

To complete the construction of (15) we observe that the processes of taking an inverse image and tensoring with T (Z/ℓ^{n}Z) commute. The map (13) is obtained from (15) by applying R = φ_{∗}.

There is a lack of symmetry between the formulae (12) and (14) that should be commented upon and corrected. The map R(g) : M̂_{K′} \overline{⊗} \overline{F}_{p} → M̂_{K} \overline{⊗} \overline{F}_{p} is unramified over M̂_{K}^{o} \overline{⊗} F_{p} so that as long as we were working with sheaves over the latter space or their extensions by zero, we did not need to distinguish beteen R =^{!}R(g) and R^{∗}(g). In (14) we do. To restore the symmetry we write (12) as

(16)  φ_{2}^{∗}F̂_{μ!}^{K}(Z/ℓ^{n}Z) → R =^{!}φ_{1}a^{−1}(g)F̂_{μ!}^{K}(Z/ℓ^{n}Z).

The preceding discussion will be of use again later, but its immediate purpose is the proof of Lemma 7.9. The map (16) defines (SGA 5.III)

Φ_{i} : H^{i}(M̂_{K} \overline{⊗} \overline{F}_{p}, F̂_{μ!}^{K}(Z/ℓ^{n}Z)) → H^{i}(M̂_{K} \overline{⊗} \overline{F}_{p}, a^{−1}(g)F̂_{μ!}(Z/ℓ^{n}Z))

and (14) defines

Ψ_{i} : H^{i}(M̂_{K} \overline{⊗} \overline{F}_{p}, Da^{−1}(g)F̂_{μ!}^{K}(Z/ℓ^{n}Z)) → H^{i}(M̂_{K}^{o} \overline{⊗} \overline{F}_{p}, DF̂_{μ}^{K}(Z/ℓ^{n}Z))

which may also be written as

Ψ_{i} : H^{i}(M̂_{K}^{o} \overline{⊗} \overline{F}_{p}, a^{−1}(g)F̂ K ; μ̃(Z/ℓ^{n}Z)) → H^{i}(M̂_{K}^{o} \overline{⊗} \overline{F}_{p}, F̂ K ; μ̃(Z/ℓ^{n}Z)).

When we take the limit and tensor with Q_{ℓ}, Φ_{2} becomes ρ_{5}(f_{g}, s). By the duality theorem

H^{o}(M̂_{K} \overline{⊗} \overline{F}_{p}, Da^{−1}(g)F̂_{μ!}^{K}(Z/ℓ^{n}Z)) ≃ Hom(H^{2}(M̂_{K} \overline{⊗} \overline{F}_{p}, a^{−1}(g)F̂_{μ!}^{K}(Z/ℓ^{n}Z)), Z/ℓ^{n}Z)

and Φ_{2} and Ψ_{0} are adjoint. It will be enough to see what happens to Ψ_{0} upon passage to the limit.

The group

H^{o}(M̂_{K}^{o} \overline{⊗} \overline{F}_{p}, a^{−1}(g)F̂ K ; μ̃(Z/ℓ^{n}Z))

is the group of sections

M̂_{K}^{o} \overline{⊗} \overline{F}_{p} → a^{−1}L(Z/ℓ^{n}Z) ×_{K/K_{o}}(M̂_{K_{o}}^{o} \overline{⊗} \overline{F}_{p}).

When restricted to a connected component Y of M̂_{K}^{o} \overline{⊗} \overline{F}_{p}, such a section must be of the form

Y ⟶^{∼} a^{−1}(g)v ×_{K_{1}/K_{0}}X

where X is a connected component of M̂_{K_{o}}^{o} \overline{⊗} \overline{F}_{p} whose stabilizer K_{1}/K_{o} and where a^{−1}(g)v is fixed by every element of K_{1}/K_{o}.

It is shown in [3.1] that the set of connected components of M_{K_{o}}^{o} \overline{⊗} \overline{F}_{p} is the union over the double cosets of B(Q_{p}) \ G(Q_{p})/K_{p}, which as usual we index by i, of

M̄_{K_{o}}^{i} = Ḡ^{+}(Q) \ Ḡ(A_{f})/K̄_{o}^{p}V_{i}

where each V_{i} is {α ∈ Q_{p}^{×} | α ≡ 1 (mod p^{m})}. The action of k ∈ K/K_{o} ≃ K^{p}/K_{o}^{p} is left multiplication by detk. In particular the stablizer of any connected component contains all cosets represented by k with detk = 1. If deg μ > 1 there can thus be cohomology in degree 0 at a finite stage but none in the limit so the trace of ρ_{5}(f_{g}, s) is zero. If k ∈ K stabilizes any component then detk ∈ Ḡ^{+}(Q)detK_{o} and hence lies in detK_{o}. Consequently if deg μ = 1 the action of k on a^{−1}(g)L(Z/ℓ^{n}Z) is equal to that of some k_{o} and is therefore trivial. Thus if deg μ = 1

H^{o}(M̂_{K}^{o} \overline{⊗} \overline{F}_{p}, F̂μ̃(Z/ℓ^{n}Z))

is isomorphic to the direct sum over ∪_{i}M̄_{K}^{i} = M̄_{K} of the fibres of the sheaf

(17)  a^{−1}(g)F K ; ν̃(Z/ℓ^{n}Z) = ∪_{i}a^{−1}(g)L̃(Z/ℓ^{n}Z) ×_{K̄^{p}}M̄^{i}

defined in the same was as F_{ν}(Q_{ℓ}), in the prelude to Lemma 5.6, except that Z/ℓ^{n}Z replaces Q_{ℓ} and that ν̃, which is defined by ν̃(detg) = μ̃(g), replaces ν. In other words it is the group of sections of this sheaf over M̄_{K}^{∧}.

The operator Ψ_{o} is determined by (15). Its effect on a section of a^{−1}(g)F̂ L ; μ̃(Z/ℓ^{n}Z) has to be determined in steps. The first step is to pull back a section to R^{∗}(e)a^{−1}(g)F̂ K ; μ̃(Z/ℓ^{n}Z) and then to use the isomorphism of this sheaf with a^{−1}(g)F̂ K′ ; μ̃(Z/ℓ^{n}Z). This corresponds to pulling back a section of a^{−1}(g)F K ; ν̃(Z/ℓ^{n}Z) to R^{∗}(e)a^{−1}(g)F K ; ν̃(Z/ℓ^{n}Z) and then using the following Cartesian diagram, where R(h) : M̄_{K′}^{∧} → M̄_{K}^{∧} is multiplication by deth, with h = e,

(18)  \begin{matrix}ha^{−1}(g)F K′ ; ν̃(Z/ℓ^{n}Z) & ⟶ & a^{−1}(g)F K ; ν̃(Z/ℓ^{n}Z) \\ ↓ & & ↓ \\ M̄_{K′}^{∧} & ⟶^{R(h)} & M̄_{K}^{∧} \end{matrix}

where the upper arrow is defined by the action of R(h) on the second factors and h^{−1} on the first in

ha^{−1}(g)F K′ ; ν̃(Z/ℓ^{n}Z) = ∪_{i}ha^{−1}(g)L̃(Z/ℓ^{n}Z)×_{K′}M̄^{i}

to obtain a section of a^{−1}(g)F K′ ; ν̃(Z/ℓ^{n}Z) over M̄_{K′}^{1}.

The next step is an application of the embedding a^{−1}(g)F̂ K′ ; μ̃(Z/ℓ^{n}Z) → gF̂ K′ ; μ̃(Z/ℓ^{n}Z) which in terms of sections over M̄_{K′}^{∧} is an application of the imbedding a^{−1}(g)F K′ ; ν̃(Z/ℓ^{n}Z) → gF K′ ; ν̃(Z/ℓ^{n}Z). In case it has been puzzling the reader, I observe that Fμ̃(Z/ℓ^{n}Z) is defined by means of the lattice dual to L(Z_{f}). Since a(g)g fixes L(Z_{f}), (a(g)g)^{−1} fixes the dual lattice and the imbedding is defined. The next step is to use the isomorphism

gF̂ K′ ; μ̃(Z/ℓ^{n}Z) ⟶^{∼} R^{∗}(g)F̂ K ; μ̃(Z/ℓ^{n}Z)

to obtain a section of the sheaf on the right. This corresponds in terms of sections over M̄_{K′}^{∧} to using the isomorphism

gF K′ ; ν̃(Z/ℓ^{n}Z) ⟶^{∼} R^{∗}(g)F K ; ν̃(Z/ℓ^{n}Z)

given by (18).

Antwerp 1972 85

The map defined by L_{1}(s) is no problem. If b = p^{m}x with x a unit, let L_{1}(s) be the map on M̄_{K}^{∧} and on the sheaves F K ; ν̃(Z/ℓ^{n}Z) defined by left multiplication by x. The broken arrow in the diagram

(19)  ↓  ↙  ↓_{M̄_{K}^{∧}  ⟶^{L_{1}(s)}  M̄_{K}^{∧}}^{F K ; ν̃(Z/ℓ^{n}Z) ⟶^{L_{1}(s)} F K ; ν̃(Z/ℓ^{n}Z)}

turns the upper right corner into L_{1}^{∗}(s)F K ; ν̃(Z/ℓ^{n}Z). The map defined by L_{1}(s) is the upper horizontal arrow.

The action of L_{2}(s) on F̂ K ; μ̃(Z/ℓ^{n}Z) is given (SGA 4.XV.1) by the broken arrow in the commutative diagram.

(20)

↙^{L_{2}(s)}
F̂ K ; μ̃(Z/ℓ^{n}Z) ⟵ L_{2}^{∗}(s)F̂ K ; μ̃(Z/ℓ^{n}Z) ←−− F̂ K ; μ̃(Z/ℓ^{n}Z)
↓  ↓  ↙
M̂_{K}^{o} \overline{⊗} \overline{F}_{p}  ⟵^{L_{2}(s)}  M̂_{K}^{o} \overline{⊗} \overline{F}_{p}

We must pull back the action of L_{2}(s) by L_{1}^{∗}(s), compose with the action of L_{1}(s), pull back the result by R^{∗}(g), and then take the trace with respect to R(g)L_{1}(s)L_{2}(s). Taking the trace with respect to R(g) erases the pullback by R^{∗}(g) and replaces it by multiplication by

\frac{[K : K′]}{[K̄ : K̄′]}

where, for example, K̄ = {detk | k ∈ K}. This number is the degree of R(g) on a connected component. Of course we must then apply the trace at the level of sheaves over M̄_{K′}^{∧}, and M̄_{K}^{∧}. The trace with respect to L_{1}(s) of the pullback of the action of L_{2}(s) by L_{1}^{∗}(s) gives this action back again. The trace with respect to L_{1}(s) applied to the action of L_{1}(s) replaces a section of the sheaf defined by the left vertical arrow in (19) by the section of the sheaf defined by the right vertical arrow obtained by applying L_{1}(s).

Antwerp 1972 86

Taking the trace with respect to L_{2}(s) of the actions of L_{2}(s) has the effect of replacing a section of the sheaf defined by the skew arrow on the right in (20) by the unique section of the sheaf defined by the arrow on the left which makes

F̂ K ; μ̃(Z/ℓ^{n}Z) ⟵^{L_{2}(s)} F̂ K ; μ̃(Z/ℓ^{n}Z)
↑  ↗
M̂_{K}^{o} \overline{⊗} \overline{F}_{p}  ⟵^{L_{2}(s)}  M̂_{K}^{o} \overline{⊗} \overline{F}_{p}

commutative.

If we let L_{2}(s) acting on M̄_{K}^{∧} be multiplication by p^{m} ∈ Ḡ(Q_{p}) then, when we represent a section of F̂ K ; μ̃(Z/ℓ^{n}Z) by a section of F K ; ν̃(Z/ℓ^{n}Z), the effect of L_{2}(s) is to take the section defined by the arrow on the right in the diagram below and replace it by the one defined by the arrow on the left which makes the diagram commutative

F K ; ν̃(Z/ℓ^{n}Z) ⟵^{L_{2}(s)} F K ; ν̃(Z/ℓ^{n}Z)
↓  ↓
M̄_{K}^{∧}  ⟵^{L_{2}(s)}  M̄_{K}^{∧}

If we observe that under the map M̄_{K′}^{∧} → M̄_{K}^{∧} the inverse image of every point contains [K̄ : K̄′] points, we may summarize the preceding discussion as follows. We have shown that

H^{o}(M̂_{K}^{o} \overline{⊗} \overline{F}_{p}, a^{−1}(g)F̂ K ; μ̃(Z/ℓ^{n}Z)) ≃ H^{o}(M̂_{K}^{∧}, Fν̃(Z/ℓ^{n}Z))

and that the map

H^{o}(M̂_{K}^{o} \overline{⊗} \overline{F}_{p}, a^{−1}(g)F̂ K ; μ̃(Z/ℓ^{n}Z)) → H^{o}(M̂_{K}^{o} \overline{⊗} \overline{F}_{p}, F̂ K ; μ̃(Z/ℓ^{n}Z))

defined by the map

(21)  φ_{1}^{∗}(a^{−1}(g)F̂ K ; μ̃(Z/ℓ^{n}Z)) → φ_{2}^{∗}(F̂ K ; μ̃(Z/ℓ^{n}Z)),

itself defined as (15) was, except that T (Z/ℓ^{n}Z) need not be taken into account, corresponds to [K : K′] times the map

H^{o}(M̄_{K}^{∧}, a^{−1}(g)F K ; ν̃(Z/ℓ^{n}Z)) → H^{o}(M̄_{K}^{∧}, F K ; ν̃(Z/ℓ^{n}Z))

Antwerp 1972 87 which assigns to the section defined by the arrow in the right of the diagram below that defined by the arrow on the left

Fν̃(Z/ℓ^{n}Z) ⟵ a^{−1}(g)Fν̃(Z/ℓ^{n}Z)
↑  ↑
M̄_{K}^{∧}  ⟵  M̄_{K}^{∧}

The horizontal arrows are defined by the maps on M̄_{K}^{∧} and M̄^{∧} given by multiplication by b detg.

The elements of

H^{o}(M̂_{K}^{o} \overline{⊗} \overline{F}_{p}, F̂ K ; μ̃(Z/ℓ^{n}Z) ⊗ T (Z/ℓ^{n}Z))

are obtained from those of

H^{o}(M̂_{K}^{o} \overline{⊗} \overline{F}_{p}, F̂ K ; μ̃(Z/ℓ^{n}Z))

by tensoring with a fixed non-zero section of T (Z/ℓ^{n}Z). The two groups are in particular isomorphic. The action defined by (15) differs from that defined by (21) only in its effect on this section: There is none until the last stage, when we apply L_{2}(s). This introduces an additional factor p^{m} = |b|_{p}^{−1}. Taking the limit and tensoring with Q_{ℓ}, we see readily that

trace ρ_{f}(f_{g}, s) = \frac{|b|_{p}^{−1}}{1 + |b|_{p}^{−1}} Ā.

A similar but easier discussion, since there is no need to introduce duality, shows that

trace ρ_{3}(f_{g}, s) = \frac{1}{1 + |b|_{p}^{−1}} Ā

and proves Lemma 7.9.

The number B discussed in Lemma 5.5 was defined as a sum of certain expressions ψ(γ). With the same notations as there, set

ψ_{1}(γ) = (d′ − a′) trace μ(γ)

if |a| > |d| and let it be 0 otherwise. Set

ψ_{2}(γ) = a′a^{m−ℓ}d^{ℓ}.

Then ψ(γ) = ψ_{1}(γ) + ψ_{2}(γ) and B is B_{1} + B_{2}.

Antwerp 1972 88 Lemma 7.10. Under the assumptions of Lemma 7.8, the trace of ρ_{2}(f_{g}, s) is B_{2}.

ρ_{2}(f_{g}, s) may be taken to be the double representation on the tensor product with Q_{ℓ} of the projective limit of the groups

H^{o}(M̃_{K}^{o} \overline{⊗} \overline{F}_{p}, F_{μ}(Z/ℓ^{n}Z))

which are equal to

⊕_{C_{K}} H^{o}(𝔊^{p}(K, h, \overline{F}_{p}), L(Z/ℓ^{n}Z)).

To be more precise, the sum is over a set of representatives in G(A_{f}) of the double cosets in C_{K}.

If K_{1} = K_{1}^{p}K_{p} and g^{−1}K_{1}g ⊆ K then R(g) : M̃_{K_{1}}^{o} \overline{⊗} \overline{F}_{p} → M̃_{K}^{o} \overline{⊗}\overline{F}_{p}. It takes M_{K_{1}}^{o} \overline{⊗} \overline{F}_{p}|h to M_{K}^{o} \overline{⊗} \overline{F}_{p}|hg. Our conventions are such that it yields a well-defined map from

𝔊^{p}(K_{1}, h, \overline{F}_{p}) ≃ h^{−1}N(A_{f}^{p})h ∩ K_{1}

to

𝔊^{p}(K, hg, \overline{F}_{p}) ≃ g^{−1}h^{−1}N(A_{f}^{p})hg ∩ K.

Namely, it sends the element of the Galois group represented by the matrix n to that represented by g^{−1}ng.

The action of ρ_{2}(f_{g}, s) is defined by

(22)  φ_{2}^{∗}F_{μ}^{K}(Z/ℓ^{n}Z) → φ_{1}^{∗}a^{−1}(g)F_{μ}^{K}(Z/ℓ^{n}Z)

which has been defined as the composite of a sequence of operations. Without repeating their definition, we shall describe their effect on the Galois cohomology in degree 0. Of course the operations over M̃_{K′}^{o} \overline{⊗} \overline{F}_{p} have to be preceded by a lifting by the inverse image functor and followed by the trace map.

If K_{o} = K_{o}^{p}K_{p} is normal in K and K′_{o} = K_{o} ∩ gK_{o}g^{−1} and h_{1} = g(s)hg we have

M_{K′_{o}}^{o} \overline{⊗} \overline{F}_{p}|h ⟶^{R(g)L_{1}(s)L_{2}(s)} M_{K_{o}}^{o} \overline{⊗} \overline{F}_{p}|h_{1}.

These maps define an imbedding

𝔊^{p}(K′, h, \overline{F}_{p}) → 𝔊(K, h_{1}, \overline{F}_{p})

which in terms of matrices takes n in h^{−1}N(A_{f}^{p})h∩K′ to g^{−1}ng in h_{1}^{−1}N(A_{f}^{p})h_{1}∩K. The lifting by R^{∗}(g)L_{1}^{∗}(s)L_{2}^{∗}(s) is, in terms of Galois cohomology, the restriction map defined by this imbedding.

Antwerp 1972 89 The effect of (22) is to map L(Z/ℓ^{n}Z) to a^{−1}(g)L(Z/ℓ^{n}Z), on which h^{−1}N(A_{f}^{p})h ∩ K′ acts in the usual way, by v → gv. The last step is the trace with respect to R(e) and it is just corestriction with respect to the imbedding 𝔊^{p}(K′, h, \overline{F}_{p}) → 𝔊^{p}(K, h, \overline{F}_{p}).

Suppose h_{1} and h represent the same element of C_{K}, that is, h represents a fixed point of φ(g, s) in C_{K′}, so that g(s)hgk = nγh with k ∈ K, n ∈ N(A_{f}), γ ∈ A^{+}(Q). To pass from cohomology with respect to 𝔊^{p}(K, h, \overline{F}_{p}) to that with respect to 𝔊^{p}(K, h_{1}, \overline{F}_{p}), we have to use the map from L(Z/ℓ^{n}Z) → L(Z/ℓ^{n}Z) given by k. Recall that it was shown during the proof of Lemma 5.5 that when h represents a fixed point the index

(23)  [h^{−1}N(A_{f}^{p})h ∩ K : h^{−1}N(A_{f}^{p})h ∩ K′]

is equal to a′.

To compute the trace of ρ_{2}(f_{g}, s) we have to take the sum over a set of representatives of the fixed points of φ(g, s) on C_{K′} of the trace on

(24)  lim_{←} H^{o}(𝔊^{p}(K, h, \overline{F}_{p}), L(Z/ℓ^{n}Z)) ⊗ Q_{ℓ} = L_{h}^{p}(Q_{ℓ}).

Under the present circumstances, corestriction in degree zero becomes in the limit simply multiplication by the index (23). Thus the trace of ρ_{2}(g, s) on (24) is a′ times the trace gk on L_{h}^{o}(Q_{ℓ}), the invariants in L(Q_{ℓ}) of h^{−1}N(Q_{ℓ})h. This is a′ times the trace of nγ on the invariants of N(Q_{ℓ}), which is in turn a′a^{m−ℓ}d^{ℓ} = ψ_{2}(γ).

To complete the proof of Lemma 6.8 we have only to establish one more fact.

Lemma 7.11. Under the conditions of Lemma 7.8, the alternating sum of the traces of the operators on H^{i}(M̂_{K} \overline{⊗} \overline{F}_{p}, F̂_{μ!}(Q_{ℓ})) defined by g and s is

∑_{F} A(F) + A(D) + B_{1}.

As explained in the introduction we use a variant of the Lefschetz fixed point formula [7.4] to prove this. Suppose X and Y are complete, non-singular, but not necessarily connected, curves over an algebraically closed field and φ = φ_{1} × φ_{2} is a morphism Y → X × X. Let F (Q_{ℓ}) = {F (Z/ℓ^{n}Z)} be a constructible étale sheaf of Q_{ℓ} vector spaces over X. If a is a non-zero ℓ-adic integer aF (Q_{ℓ}) is by definition the sheaf defined by {aF (Z/ℓ^{n}Z)}. The map aF (Q_{ℓ}) → F (Q_{ℓ})

Antwerp 1972 90 given by aF (Z/ℓ^{n}Z) ↪ F (Z/ℓ^{n}Z) yields an isomorphism of cohomology groups. If b is another non-zero ℓ-adic integer define \frac{1}{b}F (Q_{ℓ}) by \{\frac{1}{b}F (Z/ℓ^{n}Z)\} where by definition

\frac{1}{b}F (Z/ℓ^{n}Z) = F (Z/ℓ^{n}Z).

Map F (Q_{ℓ}) to \frac{1}{b}F (Q_{ℓ}) by b : F (Z/ℓ^{n}Z) → F (Z/ℓ^{n}Z). This map also yields an isomorphism of cohomology groups. In general if α = \frac{a}{b} is a non-zero ℓ-adic number we may combine the two operations to obtain αF (Q_{ℓ}) and well-defined isomorphisms between the cohomology groups of this sheaf and those of F (Q_{ℓ}). We also have isomorphisms between the fibres F (Q_{ℓ})_{x} ≃ αF (Q_{ℓ})_{x} where, for example,

F (Q_{ℓ})_{x} = lim_{←} F (Z/ℓ^{n}Z)_{x} ⊗ Q_{ℓ}.

Fix α ∈ Q_{ℓ}^{∗}. Suppose we have a consistent collection of maps

Φ : φ_{2}^{∗}F (Z/ℓ^{n}Z) → R^{!}φ_{1}αF (Z/ℓ^{n}Z).

As was observed earlier this yields maps Φ^{i} : H^{i}(X, F (Q_{ℓ})) → H^{i}(X, αF (Q_{ℓ})) ≃ H^{i}(X, F (Q_{ℓ})).

A fixed point of φ is a closed point y which maps to a point (x, x) in the diagonal. Let X_{x} and Y_{y} be the strict localizations of X and Y at x and y respectively. We have a commutative diagram

Y_{y} ⟶ X_{x} × X_{x}
↓ψ  ↓ψ
Y ⟶ X × X

ψ^{∗}(Φ) defines a map Φ_{y} from H^{o}(X_{x}, ψ^{∗}F (Q_{ℓ})) to itself and, by definition, H^{o}(X_{x}, ψ^{∗}F (Q_{ℓ})) is F (Q_{ℓ})_{x}.

Φ also defines

DΦ : φ_{1}^{∗} \frac{1}{α}DF (Z/ℓ^{n}Z) → R_{φ_{2}}^{!}DF (Z/ℓ^{n}Z)

and homomorphisms

DΦ_{y} : \frac{1}{α}DF (Z/ℓ^{n}Z)_{x} → DF (Z/ℓ^{n}Z)_{x}.

This is a homomorphism not from one Z/ℓ^{n}Z module to another but from one complex of modules to another. We may still take a direct limit to obtain DF (Q_{ℓ})_{x} ≃ \frac{1}{α}DF (Q_{ℓ})_{x} as well as

DΦ_{y} : DF (Q_{ℓ})_{x} ⟶ DF (Q_{ℓ})_{x}.

The trace of DΦ_{y} is the alternating sum of the traces on the cohomology groups.

We do not prove the following proposition.

Antwerp 1972 91 Proposition 7.12. Suppose that at every fixed point, y → (x, x), φ = φ_{1} × φ_{2} has the form φ_{1}^{∗}(t_{x}) = ut_{y}^{a}, φ_{2}^{∗}(t_{x}) = vt_{y}^{d}, with a ≠ d, where t_{x} and t_{y} are uniformizing parameters at x and y and u and v are units in the local ring of Y at y. Then the alternating sum of the traces of the Φ^{i} is equal to the sum over those fixed points y with d > a of trace Φ_{y} plus the sum over the fixed points with a > d of trace DΦ_{y}.

The correspondence which figures in Lemma 7.11 certainly satisifes the conditions of this proposition. At a fixed point in M̂_{K}^{o} \overline{⊗} \overline{F}_{p}, a is 1 and d is p^{m} with, by assumption, m ≥ 1. Since M̂_{K} \overline{⊗} \overline{F}_{p} and M_{K} \overline{⊗} \overline{F}_{p} are isomorphic away from the singular points of M_{K} \overline{⊗} \overline{F}_{p}, all of which lie in M_{K}^{o} \overline{⊗} \overline{F}_{p}, it follows from [3.1] that at a fixed point in M̂_{K}^{∞}\overline{⊗} \overline{F}_{p} the integer a is prime to p and d is divisible by p^{m}. To prove Lemma 7.11, we examine the contributions of the fixed points in M_{K′}(p, F), ∪_{i}M_{K′}^{i}(p, D), and C_{K′} separately.

We start with a point x in M_{K}(p, F) represented by h in G(A_{f}). Let M̂_{K} \overline{⊗} \overline{F}_{p}|h be the strict localization of M̂_{K} \overline{⊗} \overline{F}_{p}, or, what is the same, of M_{K} \overline{⊗} \overline{F}_{p} at this point. If K_{o} = K_{o}^{p}K_{p} with K_{o}^{p} ⊆ K^{p} the map M̂_{K_{o}}^{o} \overline{⊗} \overline{F}_{p} → M̂_{K}^{o} \overline{⊗} \overline{F} is étale. Consequently if h_{o} and h represent the same point in M_{K}(p, F) the map

M̂_{K_{o}} \overline{⊗} \overline{F}_{p}|h_{o} → M_{K} \overline{⊗} \overline{F}_{p}|h

is an isomorphism. If K_{o} is normal in K and sufficiently small

F (Z/ℓ^{n}Z)_{x} = (L(Z/ℓ^{n}Z) ×_{K/K_{o}} M̂_{K_{o}}^{o} \overline{⊗} \overline{F}_{p}) ×_{M_{K}^{o} \overline{⊗} \overline{F}_{p}} M̂_{K}^{o} \overline{⊗} \overline{F}_{p}|h

is canonically isomorphic to

L(Z/ℓ^{n}Z) × M̂_{K_{o}}^{o} \overline{⊗} \overline{F}_{p}|h ≃ L(Z/ℓ^{n}Z) × M̂_{K}^{o} \overline{⊗} \overline{F}_{p}|h.

The isomorphism here is also canonical and we may identify F (Z/ℓ^{n}Z)_{x} with L(Z/ℓ^{n}Z) and F (Q_{ℓ})_{x} with L(Q_{ℓ}). Since this identification depends upon h we write F (Z_{ℓ})_{x} as L(Q_{ℓ}) × h.

It is shown in [3.1] that the correspondence R(e) × R(g)L_{1}(s)L_{2}(s) acts on M_{K}(p, F) as the correspondence φ(g, s) used in §5 to define A(F). Moreover if (h_{1}, h_{2}) with nγh_{2} = g(s)h_{1}gk, represents a point in the image, the map L(Q_{ℓ}) × h_{2} → L(Q_{ℓ}) × h_{1} on fibres is v × h_{2} → gkv × h_{1} and this is the map used there. It follows that the contribution of the fixed points in M_{K}(p, F) to the alternating sum of the traces of the Φ^{i} is A(F).

One sees in the same way that the contribution of the fixed points in ∪_{i}M_{K}^{i}(p, D) = M̂_{K}(p, D) to the alternating sum of the traces is A(D).

Antwerp 1972 92

It is also shown in [3.1] that R(e) × R(g)L_{1}(s)L_{2}(s) acts on the points of C_{K} as the correspondence φ(g, s) used to define the number B. If h represents a fixed point, there is an equation g(s)hgk = nγh with

γ = ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} )

a and d have the same sign. Let a′ and d′ be relatively prime positive integers with a′ : d′ = a : d. If h represents x′ and x′ → (x, x) let t_{x} and t_{x′} be uniformizing parameters at x and x′. Then φ_{1}^{∗}(t_{x′}) = ut_{x}^{a′}, φ_{2}^{∗}(t_{x′}) = vt_{x}^{d′} where u and v are units in the local ring at x′. If a′ < d′ there is no contribution to the alternating sum of the traces because the fibre of F (Q_{ℓ}) is 0 at a point of C_{K}.

To compute the trace of the map

DF̂_{μ!}^{K}(Q_{ℓ})_{x} → DF̂_{μ!}^{K}(Q_{ℓ})_{x}

defined by (14), we use the description of (14) in terms of (15). If M_{K} \overline{⊗} \overline{F}_{p}|h is the strict localization of M_{K} \overline{⊗} \overline{F}_{p} at the point represented by h and

M_{K}^{o} \overline{⊗} \overline{F}_{p}|h = M_{K}^{o} \overline{⊗} \overline{F}_{p} ×_{M_{K} \overline{⊗} \overline{F}_{p}} M_{K} \overline{⊗} \overline{F}_{p}|h,

it is a question of determining the effect of (15) on the cohomology groups

H^{i}(M_{K}^{o} \overline{⊗} \overline{F}_{p}|h, a^{−1}(g)Fμ̃(Z/ℓ^{n}Z))

at a fixed point. This involves considerations in the Galois cohomology discussed already in the proof of Lemma 7.10; so we can be brief. Only degrees 0 and 1 need be considered.

In degree 0 the difference between the present situation and that of Lemma 7.10 is that μ is replaced by μ̃ and (22) by (15). The steps involved are restriction by 𝔊^{p}(K′, h, \overline{F}_{p}) → 𝔊^{p}(K, h, \overline{F}_{p}), which is defined in terms of matrices by the imbedding

h^{−1}N(A_{f}^{p})h ∩ K′ ↪ h^{−1}N(A_{f}^{p})h ∩ K.

This is followed by

a^{−1}(g)L̃(Z/ℓ^{n}Z) ⊗ T (Z/ℓ^{n}Z) ↪ gL̃(Z/ℓ^{n}Z) ⊗ T (Z/ℓ^{n}Z)

and then by

k^{−1}g^{−1} : gL̃(Z/ℓ^{n}Z) ⊗ T (Z/ℓ^{n}Z) → L̃(Z/ℓ^{n}Z) × T (Z/ℓ^{n}Z).

Antwerp 1972 93 At this stage an element of 𝔊^{p}(K′, h, \overline{F}_{p}) represented by n acts on the right as k^{−1}g^{−1}ngk. The last step is corestriction with respect to the imbedding 𝔊^{p}(K′, h, \overline{F}_{p}) → 𝔊^{p}(K′h, \overline{F}_{p}) defined by n → k^{−1}g^{−1}ngk tensored with multiplication by p^{m} on T (Z/ℓ^{n}Z).

An easy calculation shows that the index

[h^{−1}N(A_{f})h ∩ K : k^{−1}g^{−1}(h^{−1}N(A_{f})h ∩ K′)gk]

is equal to ∏_{q≠p} |d′|_{q}^{−1}. The trace in degree 0 is thus d′ times the trace of μ̃(γ^{−1}) on the invariants of N(Q_{ℓ}) and this product is, in the notation of Lemma 5.5, d′a^{ℓ}d^{m−ℓ}. The restriction in degree 1 amounts to multiplication by

[h^{−1}N(A_{f}^{p})h ∩ K : h^{−1}N(A_{f}^{p})h ∩ K′] = a′.

Corestriction, when we interpret the limit of the first cohomology groups as L_{o}^{h}(Q_{ℓ}), has no effect. Thus the trace in degree 1 is a′a^{m−ℓ}d^{ℓ}. Since

ψ_{1}(γ) = d′d^{m−ℓ}a^{ℓ}a − a′a^{m−ℓ}d^{ℓ}

we are done with Lemma 7.11.

Theorem 7.5 remains to be proven. We have an injection

0 → H_{p}^{1}(M_{K} \overline{⊗} \overline{F}_{p}, F_{μ}(Q_{ℓ})) → H_{p}^{1}(M_{K} ⊗ \overline{Q}_{p}, F_{μ}(Q_{ℓ}))

and a surjection

H_{p}^{1}(M_{K} \overline{⊗} \overline{F}_{p}, F_{μ}(Q_{ℓ})) → H_{p}^{1}(M̂_{K} \overline{⊗} \overline{F}_{p}, F̂_{μ}(Q_{ℓ})) → 0.

On all spaces involved we have a double representation of ℋ_{Q}(K^{p}) and 𝔊(\overline{Q}_{p}/Q_{p}). Let ρ^{o} be the double representation on the kernel of the arrow in the second diagram.

Lemma 7.13. If s → b in Q_{p}^{×} and |b|_{p} < 1 the trace of ρ^{o}(f_{g}, s) is equal to

(A(D) − A_{o}(D)) − (|b|_{p}^{−1} + 1)^{−1}(Ā − Ā_{o}).

There is an exact sequence

0 → H^{o}(M_{K}^{o} \overline{⊗} \overline{F}_{p}, F_{μ}(Q_{ℓ})) → H^{o}(M̂_{K}^{o} \overline{⊗} \overline{F}_{p}, F̂_{μ}(Q_{ℓ})) →
→ H^{o}(M_{K}^{o} \overline{⊗} \overline{F}_{p}, E(Q_{ℓ})) → H^{1}(M_{K}^{o} \overline{⊗} \overline{F}_{p}, F_{μ}(Q_{ℓ})) → H^{1}(M̂_{K}^{o} \overline{⊗} \overline{F}_{p}, F̂_{μ}(Q_{ℓ}))

Antwerp 1972 94 ρ^{o} is also the representation on the kernel of the arrow in the second line. The trace in which we are interested is therefore the alternating sum of the traces on the groups in the first row. The representation on the group in the middle is ρ_{3}; and we have already seen, during the proof of Lemma 7.9, that its trace is (|b|_{p}^{−1} + 1)^{−1}Ā. One proves in the same way that the trace on the first group is (|b|_{p}^{−1} + 1)^{−1}Ā_{o}. It is only necessary to take into consideration the different structure of the set of connected components of M_{K_{o}} \overline{⊗} \overline{F}_{p}, for K_{o} = K_{o}^{p}K_{p} ⊆ K.

The sheaves E(Z/ℓ^{n}Z) are supported on the finite set M_{K}(p, D) of closed points in M_{K} \overline{⊗} \overline{F}_{p} [3.1]. We also use M_{K}(p, D) to denote this set with its reduced subscheme structure and let i denote the corresponding immersion. We do the same for M̂_{K}(p, D) in M̂_{K} \overline{⊗} \overline{F}_{p}. M̂_{K}(p, D) is the fibre of M_{K}(p, D) in M̂_{K} \overline{⊗} F_{p}. Let q be the map: M̂_{K}(p, D) → M_{K}(p, D). There is an exact sequence

0 → i_{∗}i^{∗}F_{μ}(Z/ℓ^{n}Z) → i_{∗}q_{∗}i^{∗}F̂_{μ}(Z/ℓ^{n}Z) → E(Z/ℓ^{n}Z) → 0

from which it is easily seen that the trace on H^{o}(M_{K}^{o} \overline{⊗} \overline{F}_{p}, E(Q)) is A(D) − A_{o}(D).

The representation ρ^{o} acts in effect on an invariant subspace of H_{p}^{1}(M_{K} ⊗ \overline{Q}_{p}, F_{μ}(Q_{ℓ})). Call this subspace X^{o}. Suppose that, over a finite extension E_{ℓ} of Q_{ℓ}, we can find another subspace Y invariant at least under the restriction of ρ to 𝔊(Q_{p}/F^{un}), where F is a finite extension of Q_{p}, so that on Y this restriction is a direct sum of two-dimensional indecomposable representations. Let X be the sum of the one-dimensional subspaces of Y invariant under 𝔊(Q_{p}/F). Then 2 dim X = dim Y. Suppose also that X^{o} ⊆ X.

By assumption we can find a decomposition Y = ⊕Y_{j} of Y as a direct sum of two-dimensional subspaces, on each of which the representation of 𝔊(Q_{p}/F^{un}) is indecomposable. If X_{j} = X ∩ Y_{j} then X = ⊕_{j}X_{j}. We may also suppose that

X^{o} ⊗ F = ⊕_{X_{j}⊆X^{o}⊗F} X_{j}.

On the other hand, by an obvious refinement of Proposition 3.1, if E_{ℓ} is suffficiently large

H_{p}^{1}(M_{K} ⊗ \overline{Q}_{p}, F_{μ}(E_{ℓ})) = ⊕_{π} ⊕_{k} U_{E_{ℓ}}^{π} ⊗ v_{k}^{π}

where {v_{k}^{π}} is a basis of V_{E_{ℓ}}^{π}(K). Since X^{o} ⊗ F is invariant under the double representation, it is the direct sum of its intersections with the spaces U_{E_{ℓ}}^{π} ⊗ V_{E_{ℓ}}^{π}(K). Moreover if, for a given π, it intersects U_{E_{ℓ}}^{π} ⊗ V_{E_{ℓ}}^{π}(K) in a non-zero subspace, then its intersection with each U_{E_{ℓ}}^{π} ⊗ v_{k}^{π} is non-zero.

If the projection of Y^{j} to U_{E_{ℓ}}^{π} ⊗ v_{k}^{π} is non-zero on X^{j} it must be surjective. It follows that whenever

(25)  (X^{o} ⊗ F) ∩ (U_{E_{ℓ}}^{π} ⊗ V_{E_{ℓ}}^{π}(K)) ≠ 0

the representation σ_{p}(π) is special. Moreover the dimension of this intersection is one-half the dimension of U_{E_{ℓ}}^{π} ⊗V_{E_{ℓ}}^{π}(K). Let Y^{o} be the sum over those π satisfying (25) of U_{E_{ℓ}}^{π} ⊗V_{E_{ℓ}}^{π}(K). Since ρ acts to the right, it follows easily from Lemma 7.13 and the structure of special representations that the trace of the restriction of ρ(f_{g}, s) to Y^{o} is

(|b|_{p}^{−1} + 1)(A(D) − A_{o}(D) − (Ā − Ā_{o})).

Theorem 7.5 therefore follows from Proposition 5.8 and Lemma 7.9 once we have proved the existence of such a subspace Y.

Let S be the finite extension of R^{un} figuring in the diagram (9). By [7.2], we may suppose that

M′_{K} \overline{⊗}S = (M′_{K} \overline{⊗}S_{o}) ⊗_{S_{o}} S

where S_{o} ⊆ S is a finite extension of R and M′_{K} \overline{⊗}S_{o} is a desingularization of M_{K} \overline{⊗}R. The field F is to contain S_{o}. Let M̂′_{K} \overline{⊗} \overline{F}_{p} be the desingularization of M′_{K} \overline{⊗} F_{p}. We have a commutative diagram

M̂′_{K} \overline{⊗} \overline{F}_{p} ⟶ M′_{K} \overline{⊗} \overline{F}_{p}
↓  ↓
M̂_{K} \overline{⊗} \overline{F}_{p} ⟶ M_{K} \overline{⊗} \overline{F}_{p}

Let, for example, F̂′_{μ!}(Z/ℓ^{n}Z) be the inverse image on M̂′_{K} \overline{⊗} \overline{F}_{p} of the sheaf F_{μ!}(Z/ℓ^{n}Z). There is a commutative diagram

(26)

H_{p}^{1}(M̂′_{K} \overline{⊗} \overline{F}_{p}, F̂′_{μ}(Q_{ℓ})) ⟵ H_{p}^{1}(M′_{K} \overline{⊗} \overline{F}_{p}, F′_{μ}(Q_{ℓ}))
↑  ↑
H_{p}^{1}(M̂_{K} \overline{⊗} \overline{F}_{p}, F̂_{μ}(Q_{ℓ})) ⟵ H_{p}^{1}(M_{K} \overline{⊗} \overline{F}_{p}, F_{μ}(Q_{ℓ}))

in which the horizontal arrows are surjections and the right vertical arrow is an injection. Let X_{o} be the kernel of the lower horizontal arrow and X that of the upper.

There is another commutative diagram which may be attached to the diagram (26)

(27)

H_{p}^{1}(M′_{K} \overline{⊗} \overline{F}_{p}, F′_{μ}(Q_{ℓ})) ⟶ H_{p}^{1}(M′_{K} ⊗ \overline{Q}_{p}, F′_{μ}(Q_{ℓ}))
↑  ↑
H_{p}^{1}(M_{K} \overline{⊗} \overline{F}_{p}, F_{μ}(Q_{ℓ})) ⟶ H_{p}^{1}(M_{K} ⊗ \overline{Q}_{p}, F_{μ}(Q_{ℓ}))

The horizontal arrows are injective and the vertical arrow on the right is an isomorphism. We may therefore regard X_{o} and X, with X_{o} ⊆ X, as subspaces of H_{p}^{1}(M′_{K} ⊗ \overline{Q}_{p}, F′_{μ}(Q_{ℓ})).

The group 𝔊(\overline{Q}_{p}/F) acts, in a compatible way, on all the spaces appearing in (26) and (27). With the following lemma the proof of Theorem 7.5 is complete.

Lemma 7.14. There is a subspace Y of H_{p}^{1}(M′_{K} ⊗ \overline{Q}_{p}, F′_{μ}(Q_{ℓ})) with X ⊆ Y and with 2 dim X = dim Y such that over some finite extension of Q_{ℓ} the representation of 𝔊(\overline{Q}_{p}/F^{un}) on Y is the direct sum of two-dimensional indecomposable representations. Moreover X is the sum of all onedimensional subspaces of Y invariant under 𝔊(\overline{Q}_{p}/F).

The proof that follows is simply a bowdlerization of a conversation with Deligne. It relies of course on the Picard–Lefschetz theory of SGA 7. If

q′ : M̂_{K}^{o′} \overline{⊗} \overline{F}_{p} → M_{K}^{o′} \overline{⊗} \overline{F}_{p}

let E′(Q_{ℓ}) be defined by

0 → F′_{μ}(Q_{ℓ}) → q′_{∗}F̂′_{μ}(Q_{ℓ}) → E′(Q_{ℓ}) → 0.

There is an exact sequence

(28)  \begin{matrix}0 → H^{o}(M_{K}^{o′} \overline{⊗} \overline{F}_{p}, F′_{μ}(Q_{ℓ})) → H^{o}(M̂_{K}^{o′} \overline{⊗} \overline{F}_{p}, F̂′_{μ}(Q_{ℓ})) → \\ → H^{o}(M_{K}^{o′} \overline{⊗} \overline{F}_{p}, E′(Q_{ℓ})) → X → 0. \end{matrix}

Moreover

H^{o}(M_{K}^{o′} \overline{⊗} \overline{F}_{p}, E′(Q_{ℓ})) = ⊕_{x}E′(Q_{ℓ})_{x}

and

(29)  E′(Q_{ℓ})_{x} ≃ F′(Q_{ℓ})_{x} = F′(Q_{ℓ})_{x} ⊗_{Z_{ℓ}} Z_{ℓ}.

The sum is over all double points of M′_{K} \overline{⊗} \overline{F}_{p} and the isomorphism (29) is determined up to a sign which depends upon the choice of an order for the pair of branches passing through x.

Let q′ also be the map M̂′_{K} \overline{⊗} \overline{F}_{p} → M′_{K} \overline{⊗} \overline{F}_{p} and let X_{!} be the kernel of

H^{1}(M′_{K} \overline{⊗} \overline{F}_{p}, F′_{μ!}(Q_{ℓ})) → H^{1}(M̂′_{K} \overline{⊗} \overline{F}_{p}, F̂′_{μ!}(Q_{ℓ})).

After extending by zero, we may regard E′(Q_{ℓ}) as a sheaf on M′_{K} \overline{⊗} \overline{F}_{p}. There is an analogue of (28). Moreover

H^{o}(M′_{K} \overline{⊗} \overline{F}_{p}, E′(Q_{ℓ})) = H^{o}(M_{K}^{o′} \overline{⊗} \overline{F}_{p}, E′(Q_{ℓ})),

so that we have a commutative diagram

(30)  ↘  ↓_{X}^{⊕F′(Q_{ℓ})_{x} \begin{matrix}⊗ \\ Z & ℓ \end{matrix} Z_{ℓ}  ⟶ X_{!}}

in which all arrows are surjective.

A spectral sequence similar to the one which led to (10) yields H^{1}(M′_{K} ⊗ \overline{Q}_{p}, F′_{μ!}(Q_{ℓ})) → H^{0}(M′_{K} \overline{⊗} \overline{F}_{p}, u^{∗}R^{1}v_{∗}F′_{μ!}(Q_{ℓ})) →

→ H^{2}(M′_{K} \overline{⊗} \overline{F}_{p}, F′_{μ!}(Q_{ℓ})) → H^{2}(M′_{K} ⊗ \overline{Q}_{p}, F′_{μ!}(Q)) → 0.

The meaning of the maps u and v is more or less the same as in (10). The sheaf u^{∗}R^{1}v_{∗}F′_{μ!}(Q_{ℓ}) has support in the set of double points and, since F′_{μ}(Q_{ℓ}) is locally constant in M_{K}^{o′} \overline{⊗}S, its fibre at x is, once an order has been chosen for the branches at this point, isomorphic to

F′_{μ}(Q_{ℓ})_{x} ⊗_{Z_{ℓ}} T_{ℓ}^{−1}

T_{ℓ} is lim_{←} T (Z/ℓ^{n}Z) and T (Z/ℓ^{n}Z) is the group of the ℓ^{n}th roots of unity. 𝔊(\overline{Q}_{p}/F), and in fact 𝔊(F^{un}/F), act on T_{ℓ} to the right. The action is the direct limit of ζ → τ^{−1}(ζ), ζ ∈ T (Z/ℓ^{n}Z). T_{ℓ}^{−1} is the contragredient module.

If F′ is the limit of the finite Galois extensions of F^{un} whose order is a power of ℓ the action of 𝔊(Q_{p}/F^{un}) on H^{1}(M′_{K} ⊗ \overline{Q}_{p}, F′_{μ!}(Q_{ℓ})) can be factored through 𝔊(F′/F^{un}). With respect to the action of 𝔊(F^{un}/F) to the right given by σ → τ^{−1}στ, 𝔊(F′/F^{un}) is canonically isomorphic to T_{ℓ}. There is therefore a pairing

𝔊(F′/F^{un}) × T_{ℓ}^{−1} → Z_{ℓ},

and hence for each σ ∈ 𝔊(F′/F^{un}) a linear map

ε(σ) : T_{ℓ}^{−1} → Z_{ℓ}.

σ − 1 annihilates H^{1}(M′_{K} \overline{⊗} \overline{F}_{p}, F′_{μ!}(Q_{ℓ})) and the diagram

⊕F′_{μ}(Q_{ℓ})_{x} ⊗ T_{ℓ}^{−1}  ⟶^{⊕1⊗ε(σ)} F′_{μ}(Q_{ℓ})_{x} ⊗ Z_{ℓ} → X
↑  ↓
H^{1}(M′_{K} ⊗ \overline{Q}_{p}, F′_{μ!}(Q_{ℓ})) ⟶^{σ−1} H^{1}(M′_{K} ⊗ \overline{Q}_{p}, F′_{μ!}(Q_{ℓ}))

is commutative. All we need do to verify the lemma is to show that if σ ≠ 1 so that ε(σ) ≠ 0 then the composite of the top arrows is surjective on the kernel of

(31)  ⊕_{x}F_{μ}(Q_{ℓ})_{x} ⊗ T_{ℓ}^{−1} → H^{2}(M′_{K} \overline{⊗} \overline{F}_{p}, F′_{μ!}(Q)).

For this purpose the action of 𝔊(F^{un}/F) is irrelevant so we may as well identify T_{ℓ}^{−1} with Z_{ℓ} and ε(σ) with a constant, which by an appropriate choice of σ we may take to be 1.

The remainder of the discussion will be easier to follow if we make use of a graph of Z whose edges are formed by the set U of double points of M′_{K} \overline{⊗} \overline{F}_{p} and whose vertices are formed by the set V of irreducible components of M′_{K} \overline{⊗} \overline{F}_{p} or, what is the same, of M̂′_{K} \overline{⊗} \overline{F}_{p}. A vertex lies on an edge if the corresponding double point lies in the component represented by the vertex.

If μ is one-dimensional let W be empty, otherwise let it be the set of vertices, or irreducible components, which do not contract to a point in M_{K} \overline{⊗} \overline{F}_{p}. All the vertices and edges lying in the same connected component of Z − W must map to the same irreducible component of M_{K} \overline{⊗} \overline{F}_{p} and if μ is not one-dimensional to a single point. We saw during the proof of Lemma 7.9 that if μ is one-dimensional then F_{μ}(Q_{ℓ}) is constant on each connected component of M_{K} \overline{⊗} \overline{F}_{p}.

These observations allow us to identify

⊕_{x}F (Q_{ℓ})_{x} ⊗ Z_{ℓ}

with

(32)  L(Q_{ℓ}) ⊗ (⊕_{x}Q) = ⊕_{x}L(Q_{ℓ}).

The identification is unique up to a composition with ⊕_{x}A_{x} where A_{x} is an isomorphism of L(Q_{ℓ}) and x → A_{x} is constant on the edges lying in the same connected component of Z − W.

It follows from the proof of Lemma 7.9 that if deg μ > 1 then

H^{o}(N, F′_{μ}(Q_{ℓ})) = 0

for each component N of M̂^{o′} \overline{⊗} \overline{F}_{p} lying in W. Thus

H^{o}(M̂^{o′} \overline{⊗} \overline{F}_{p}, F′_{μ}(Q_{ℓ})) ≃ ⊕_{N∈V −W} L(Q_{ℓ}).

The map of this space into (32) is given by

⊕v_{N} → ⊕_{x}(v_{N_{1}(x)} − v_{N_{2}(x)})

where N_{1}(x) and N_{2}(x) are the components containing the two branches passing through x. Recall that these branches were ordered. It can happen that N_{1}(x) and N_{2}(x) are the same. In any case the image is of the form L(Q_{ℓ}) ⊗ P with P ⊆ ⊕_{x}Q.

The group

H^{2}(M′_{K} \overline{⊗} \overline{F}_{p}, F′_{μ!}(Q_{ℓ}))

is isomorphic to

⊕_{N∈V −W} L(Q_{ℓ}) ⊗ T_{ℓ}^{−1}.

According to our argument, we may ignore the T_{ℓ}^{−1}. The map (31), after identification of its source with (32), is

⊕_{x}v_{x} → ⊕_{N}( ∑_{N=N_{i}(x)}(−1)^{i}v_{x}).

Thus the kernel is of the form L(Q_{ℓ}) ⊗ Q, where Q ⊆ ⊕_{x}Q is the orthogonal complement of P with respect to the standard inner product on ⊕_{x}Q.

With this the lemma is proved and the report is concluded.

References

1.1. P. Deligne, Formes modulaires et représentations ℓ-adiques, Sem. Bourbaki, exp. 355, 1969. 1.2. M. Eichler, Quaternare quadratische Formen und die Riemannsche Vermutung für die Kon-

gruenzzetafunktion, Arch. Math. 5 (1954). 1.3. J. Igusa, Kroneckerian model of fields of elliptic modular functions, Amer. J. Math. 81

(1959). 1.4. Y. Ihara, Hecke polynomials as congruence ζ-functions in elliptic modular case, Ann. of

Math. 85 (1967). 1.5. G. Shimura, Correspondences modulaires et les fonctions ζ de courbes algébriques, J. Math.

Soc. Japan 10 (1958).

These are the principal references for the main topic of this report, which is, in spirit, closely related to [1.4] and owes a great deal to it. Of the articles, [1.1] is most recent and most comprehensive. It contains much not referred to here. Another recent article is that of Piateckii– Shapiro, which appears in the present volume.

2.1. R.P. Langlands, On the Functional Equations satisfied by Eisenstein series, mimeographed

notes. 2.2. G. Mackey, Induced representations of locally compact groups I, Ann. of Math. 55 (1952). 2.3. S. Murakami, Cohomologies of vector-valued forms on compact, locally symmetric Riemann

manifolds, Proc. of Symp. in Pure Math., Providence (1966).

The material in §2 is the specialization to a very simple case of rather general theories, for which [2.1] and [2.3] are possible references.

3.1. P. Deligne and M. Rapoport, Schémas modulaires des courbes elliptiques, this volume. 3.2 A. Grothendieck et al., Séminaire de geometrie algébrique, 4, 5, 7.

[3.2] is of course the general reference for étale topology.

4.1. H. Jacquet and R.P. Langlands, Automorphic forms on GL(2), Springer Lecture Notes,

v. 114. 6.1. J. Arthur, The Selberg trace formula for groups of F -rank one, to appear. 6.2. W. Casselman, The restriction of a representation of GL_{2}(k) to GL_{2}(o), to appear. 6.3. M. Duflo and J.-P. Labesse, Sur la formule des traces de Selberg, Ann. Éc. Norm. Sup.,

v. 4 (1971). 6.4. R.P. Langlands, On Artin’s L-functions, Rice University Studies, v. 56 (1970). 6.5. J. Shalika, A theorem on semi-simple p-adic groups, Ann. of Math., v. 95 (1972). 6.6. , Representations of the two-by-two unimodular group over local fields, Seminar on

representations of Lie groups, Institute for Advanced Study (1965).

Antwerp 1972 101 6.7. A Silberger, P GL_{2} over the p-adics: its representations, spherical functions, and Fourier

analysis, Springer Lecture Notes, v. 166.

7.1. W. Casselman, On some results of Atkin and Lehner, to appear.

7.2. P. Deligne and D. Mumford, The irreducibility of the space of curves of a given genus, Publ. Math. I.H.E.S., N^{o} 35.

7.3. T. Miyake, On automorphic forms on GL_{2} and Hecke operators, Ann. of Math., v. 94 (1971).

7.4. J.-L Verdier, The Lefschetz fixed-point formula in étale cohomology, Driebergen conference on Local Fields, Springer, 1967.