Problems in the Theory of Automorprhic Forms^{†}

To Salomon Bochner In Gratitude

1. There has recently been much interest, if not a tremendous amount of progress, in the arithmetic theory of automorphic forms. In this lecture I would like to present the views not of a number theorist but of a student of group representations on those of its problems that he finds most fascinating. To be more precise, I want to formulate a series of questions which the reader may, if he likes, take as conjectures. I prefer to regard them as working hypotheses. They have already led to some interesting facts. Although they have stood up for a fair length of time to the most careful scrutiny I could give, I am still not entirely easy about them. Indeed even at the beginning in the course of the definitions, which I want to make in complete generality, I am forced, for lack of time and technical competence, to make various assumptions.

I should perhaps apologize for such a speculative lecture. However, there are some interesting facts scattered amongst the questions. Moreover, the unsolved problems in group representations arising from the theory of automorphic forms are much less technical than the solved ones, and their significance can perhaps be more easily appreciated by the outsider.

Suppose G is a connected reductive algebraic group defined over a global field F. F is then an algebraic number field or a function field in one variable over a finite field. Let 𝔸(F) be the adèle ring of F. G_{𝔸(F)} is a locally compact topological group with G_{F} as a discrete subgroup. The group G_{𝔸(F)} acts on the functions on G_{F} \ G_{𝔸(F)}. In particular, it acts on L^{2}(G_{F} \ G_{𝔸(F)}). It should be possible, although I have not done so and it is not important at this stage, to attach a precise meaning ot the assertion that a given irreducible representation π of G_{𝔸(F)} occurs in L^{2}(G_{F} \ G_{𝔸(F)}). If G is abelian it would mean that π is a character of G_{F} \ G_{𝔸(F)}. if G is not abelian it would be true for at least those representations which act on an irreducible invariant subspace of L^{2}(G_{F}) \ G_{𝔸(F)}.

If G is GL(1) then to each such π one, following Hecke, associates an L-function. If G is GL(2) then Hecke has also introduced, without explicitly mentioning group representations, some L-functions. The problems I want to discuss center about the possibility of defining L-functions for

† Appears in Lectures in Modern Analysis and Applications III, C. T. Taam, ed., Springer Lecture Notes in Mathematics 170, 1970, pp. 18–61 all such π and proving that they have the analytic properties we have grown used to expecting of such functions. I shall also comment on the possible relations of these new functions to the Artin L-functions and the L-functions attached to algebraic varieties.

Given G, I am going to introduce a complex analytic group Ĝ_{F}. To each complex analytic representation σ of Ĝ_{F} and each π I want to attach an L-function L(s, σ, π). Let me say a few words about the general way in which I want to form the function. G_{𝔸(F)} is a restricted direct product ∐_{𝔭} G_{F_{𝔭}}. The product is taken over the primes, finite and infinite, of F. It is reasonable to expect, although to my knowledge it has not yet been proved in general, that π can be represented as ∏_{𝔭} ⊗π_{𝔭} where π_{𝔭} is a unitary representation of G_{F_{𝔭}}.

I would like to have first associated to any algebraic group G defined over F_{𝔭} a complex analytic group Ĝ_{F_{𝔭}} and to any complex analytic representation σ_{𝔭} of Ĝ_{F_{𝔭}} and any unitary representation π_{𝔭} of G_{F_{𝔭}} a local L-function L(s, σ_{𝔭}, π_{𝔭}) which, when 𝔭 is non-archimedean, would be of the form

∏_{i=1}^{n} \frac{1}{1 − α_{i}|ϖ_{𝔭}|^{s}}

where n is the degree of σ_{𝔭}. Some of the α_{i} may be zero. For 𝔭 infinite it would be, basically, a product of Γ-functions. L(s, σ_{𝔭}, π_{𝔭}) would depend only on the equivalence classes of σ_{𝔭} and π_{𝔭}. I would also like to have defined for every non-trivial additive character ψ_{F_{𝔭}} of F_{𝔭} a factor ε(s, σ_{𝔭}, π_{𝔭}, ψ_{F_{𝔭}}) which, as a function of s, has the form ae^{bs}.

There would be a complex analytic homomorphism of Ĝ_{F_{𝔭}} into Ĝ_{F} determined up to an inner automorphism of Ĝ_{F}. Thus σ determines for each 𝔭 a representation of σ_{𝔭} of Ĝ_{F_{𝔭}}. I want to define

L(s, σ, π) = ∏_{𝔭} L(s, σ_{𝔭}, π_{𝔭}) .  (A)

Of course it has to be shown that the product converges in a half-plane. We shall see how to do this. Then we will want to prove that the function can be analytically continued to a function meromorphic in the whole complex plane. Let ψ_{F} be a non-trivial character of F \ 𝔸(F) and let ψ_{F_{𝔭}} be the restriction of ψ_{F} to F_{𝔭}. We will want ε(s, σ_{𝔭}, π_{𝔭}, ψ_{F_{𝔭}}) to be 1 for all but finitely many 𝔭. We will also want

ε(s, σ, π) = ∏_{𝔭} ε(s, σ_{𝔭}, π_{𝔭}, ψ_{F_{𝔭}})

to be independent of ψ_{F}. The functional equation should be

L(s, σ, π) = ε(s, σ, π)L(1 − s, σ̃, π)

if σ̃ is the representation contragredient to σ.

We are asking for too much too soon. What we should try to do is to define the L(s, σ_{𝔭}, π_{𝔭}) and the ε(s, σ_{𝔭}, π_{𝔭}, ψ_{F_{𝔭}}) when there is no ramification, verify that there is ramification at only a finite number of primes, and show that if the product in (A) is taken only over the unramified primes it converges for Re s sufficiently large. As we learn how to prove the functional equation we shall be able to make the definitions at the unramified primes. By the way, we introduce the additive characters, whose appearance must appear rather mysterious, only because we can indeed prove some things and know better than to leave them out.

What does unramified mean in our context? First of all for 𝔭 to be unramified G will have to be quasi-split over F_{𝔭} and split over an unramified extension. In that case there is, as we shall see, a canonical conjugacy class of maximal compact subgroups of G_{F_{𝔭}}. For 𝔭 to be unramified, the restriction of π_{𝔭} to any one of these groups will have to contain the identity representation. There is also a condition to be imposed on ψ_{F_{𝔭}}. Although it is not very important I would like to mention it explicitly. If 𝔭 is non-archimedean the largest ideal of F_{𝔭} on which ψ_{F_{𝔭}} is trivial will have to be O_{F_{𝔭}}, the ring of integers in F_{𝔭}. If F_{𝔭} is ℝ then ψ_{F_{𝔭}}(x) will have to be e^{2πix} and if F_{𝔭} is ℂ then ψ_{F_{𝔭}}(z) will have to be e^{4πi Re z}. We want ε(s, σ_{𝔭}, π_{𝔭}, ψ_{F_{𝔭}}) to be 1 if 𝔭 is unramified.

2. Ĝ_{F} can be identified for a connected reductive group over any field F. Take first a quasi-split group G over F which splits over the Galois extension K. Choose a Borel subgroup B of G which is defined over F and let T be a maximal torus of B which is also defined over F. Let L be the group of rational characters of T. Write G as G^{0}G^{1} where G^{0} is abelian and G^{1} is semi-simple. Then G^{0} ∩ G^{1} is finite. If T^{0} = G^{0} and T^{1} = T ∩ G^{1} then T = T^{0}T^{1}. Let L_{+}^{0} be the group of rational characters of T^{0} and let L_{−}^{0} be the elements of L_{+}^{0} which are 1 on T^{0} ∩ T^{1}. Let L_{−}^{+} be the group generated by the roots of T^{1}. If R is any field let E_{R}^{1} = L_{−}^{1} ⊗_{ℤ} R. The Weyl group Ω acts on L_{−}^{1} and therefore on E_{R}^{1}. Let ( · , · ) be a non-degenerate bilinear form on E_{ℂ}^{1} which is invariant under Ω. Suppose also that its restriction to E_{ℝ}^{1} is positive definite. Let

L_{+}^{1} = \{λ ∈ E_{ℂ}^{1} | 2\frac{(λ, α)}{(α, α)} ∈ ℤ for all roots α\} .

Set L_{−} = L_{−}^{0} ⊕ L_{−}^{1} and L_{+} = L_{+}^{0} ⊕ L_{+}^{1}. We may regard L as a sublattice of L_{+}. It will contain L_{−}.

Let α_{1}, …, α_{ℓ} be the simple roots of T^{1} with respect to B and let

(A_{ij}) = 2\frac{(α_{i}, α_{j})}{(α_{i}, α_{i})}

be the Cartan matrix. If σ belongs to 𝔊(K/F) and λ belongs to L then σλ, where σλ(t) = σ(λ(σ^{−1}t)), also belongs to L. Thus 𝔊(K/F) acts on L. It also acts on L_{−} and L_{+} and the actions on these three lattices are consistent. Moreover the roots α_{1}, …, α_{ℓ} are permuted amongst themselves and the Cartan matrix is left invariant.

If R is any field containing ℚ let E_{R} = L ⊗_{ℤ} R and let Ê_{R} = Hom_{R}(E_{R}, R). The lattices

L̂_{+} = Hom(L_{−}, ℤ) = Hom(L_{−}^{0}, ℤ) ⊕ Hom(L_{−}^{1}, ℤ) = L_{+}^{0} ⊕ L̂_{+}^{1}
L̂ = Hom(L, ℤ)
L̂_{−} = Hom(L_{+}, ℤ) = Hom(L_{+}^{0}, ℤ) ⊕ Hom(L_{+}^{1}, ℤ) = L̂_{−}^{0} ⊕ L̂_{−}^{1}

may be regarded as subgroups of Ê_{ℂ}. IF E_{R}^{0} = L_{−}^{0} ⊗_{ℤ} R then E_{R} = E_{R}^{0} ⊗ E_{R}^{1}. With the obvious definitions of Ê_{R}^{0} and Ê_{R}^{1} we have Ê_{R} = Ê_{R}^{0} ⊕ Ê_{R}^{1}. Let ( · , · ) also denote the form on Ê_{ℂ}^{1} adjoint to the given form on E_{ℂ}^{1}. To be precise if λ and μ belong to E_{ℂ}^{1}, if λ̂ and μ̂ belong to Ê_{ℂ}^{1}, and if ⟨η, λ̂⟩ = (η, λ) and ⟨η, μ̂⟩ = (η, μ) for all η in E_{ℂ}^{1} then (λ, μ) = (λ̂, μ̂).

If α is a root define its coroot α̂ in Ê_{ℂ}^{1} by the condition:

⟨λ, α̂⟩ = 2 \frac{(λ, α)}{(α, α)}

for all λ in E_{ℂ}^{1}. The coroots generate L̂_{−}^{1}. Moreover

(α̂, β̂) = 4 \frac{(α, β)}{(α, α)(β, β)}

and

2 \frac{(α̂, β̂)}{(α̂, α̂)} = 2 \frac{(α, β)}{(β, β)} .

Thus the matrix

(Â_{ij}) = (2 \frac{(α̂_{i}, α̂_{j})}{(α̂_{i}, α̂_{i})})

is the transpose of (A_{ij}). The linear transformation Ŝ_{i} of Ê_{ℂ}^{1} defined by

Ŝ_{i}(α̂_{j}) = α̂_{j} − Â_{ij}α̂_{i} = α̂_{j} − A_{ji}α̂_{i}

is contragredient to the linear transformation S_{i} of Ê_{ℂ}^{1} defined by

S_{i}(α_{j}) = α_{j} − A_{ij}α_{i} .

Thus the group Ω̂ generated by {Ŝ_{i}|1 ≤ i ≤ ℓ} is canonically isomorphic to the finite group Ω and, by a well-known theorem (cf. Chapter VII of [7]) (Â_{ij}) is the Cartan matrix of a simply-connected complex group Ĝ_{+}^{1}. Let B̂_{+}^{1} be a Borel subgroup of Ĝ_{+}^{1} and let T̂_{+}^{1} be a Cartan subgroup in B̂_{+}^{1}. We identify the simple roots of T̂_{+}^{1} with respect to B̂_{+}^{1} with α̂_{1}, …, α̂_{ℓ} and the free vector space over ℂ with basis {α̂_{1}, …, α̂_{ℓ}} with Ê_{ℂ}^{1}. We may also identify Ω and Ω̂. The roots of T̂_{+}^{1} are the vectors ωα̂_{i}, ω ∈ Ω, 1 ≤ i ≤ ℓ. If ωα_{i} = α then ωα̂_{i} = α̂ because

⟨λ, ωα̂_{i}⟩ = ⟨ω^{−1}λ, α̂_{i}⟩ = 2 \frac{(ω^{−1}λ, α_{i})}{(α_{i}, α_{i})} = 2 \frac{(λ, ωα_{i})}{(ωα_{i}, ωα_{i})} = 2 \frac{(λ, α)}{(α, α)}

Thus the roots of T̂_{+}^{1} are just the coroots. If λ belongs to Ê_{ℂ}^{1} then

2 \frac{(λ, α̂)}{(α̂, α̂)} = ⟨α, λ⟩

so that

L̂_{+}^{1} = \{λ ∈ Ê_{ℂ}^{1} | 2 \frac{(λ, α̂)}{(α̂, α̂)} ∈ ℤ for all coroots α̂\}

and is therefore just the set of weights of T̂_{+}^{1}.

Let

Ĝ_{+}^{0} = Hom_{ℤ}(L̂_{+}^{0}, ℂ^{∗}) .

Ĝ_{+}^{0} is a reductive complex Lie group. Set Ĝ_{+} = Ĝ_{+}^{0} × Ĝ_{+}^{1}. If T̂_{+}^{0} = Ĝ_{+}^{0} and T̂_{+} = T̂_{+}^{0} × T̂_{+}^{1} then L̂_{+} is the set of complex analytic characters of T̂_{+}. If

Ẑ = {t ∈ T̂_{+} | λ(t) = 1 for all λ in L̂}

then Ẑ is a normal subgroup of Ĝ_{+} and Ĝ = Ĝ_{+}/Ẑ is also a complex Lie group. 𝔊(K/F) acts in a natural fashion on L̂_{−}, L̂, and L̂_{+}. The action leaves the set {α̂_{1}, … α̂_{ℓ}} invariant. 𝔊(K/F) acts naturally on Ĝ_{+}^{0}. I want to define an action on Ĝ_{+}^{1} and therefore an action on Ĝ_{+}. Choose H_{1}, …, H_{ℓ} in the Lie algebra of T̂_{+}^{1} so that

λ(H_{i}) = ⟨α_{i}, λ⟩

for all λ in L̂_{+}^{1}. Choose root vectors X_{1}, …, X_{ℓ} belonging to the coroots α̂_{1}, …, α̂_{ℓ} and root vectors Y_{1}, …, Y_{ℓ} belonging to their negatives. Suppose [X_{i}, Y_{i}] = H_{i}. If σ belongs to 𝔊(K/F) let σ(α̂_{i}) = α̂_{σ(i)}. There is (cf. Chapter VII of [7]) a unique isomorphism σ of the Lie algebra of Ĝ_{+}^{1} such that

σ(H_{i}) = H_{σ(i)}, σ(X_{i}) = X_{σ(i)}, σ(Y_{i}) = Y_{σ(i)} .

These isomorphisms clearly determine an action of 𝔊(K/F) on the Lie algebra and therefore one on Ĝ_{+}^{1} itself. Since 𝔊(K/F) leaves L invariant its action on Ĝ_{+} can be transferred to Ĝ. If B̂ is the image of B̂_{+} = T̂_{+}^{0} × B̂_{+}^{1} and T̂ the image of T̂_{+} in Ĝ the action leaves B̂ and T̂ invariant. I want to define Ĝ_{F} to be the semi-direct product Ĝ × 𝔊(K/F).

However Ĝ_{F} as defined depends upon the choice of B, T and X_{1} …, X_{ℓ} and Ĝ_{F} comes provided with a Borel subgroup B̂ of its connected component, a Cartan subgroup T̂ of B̂, and a one-to-one correspondence between the simple roots of T with respect to B and those of T̂ with respect to B̂. Suppose G′ is another quasi-split group over F which is isomorphic to G over K by means of an isomorphism φ such that φ^{−1}σ(φ) is inner for all σ in 𝔊(K/F), B′ is a Borel subgroup of G′ defined over F, and T′ is a Cartan subgroup of B′ also defined over F. There is an inner automorphism ψ of G which is defined over K so that φψ takes B to B′ and T to T′. Then φψ determines an isomorphism of L̂ and L̂′ and a one-to-one correspondence between {α_{1}, …, α_{ℓ}} and {α′_{1}, …, α′_{ℓ}} both of which depend only on φ and, as is easily verified, commute with the action of 𝔊(K/F). There is then a natural isomorphism of Ĝ_{+}^{0} with (Ĝ_{+}^{0})′ associated to φ. Moreover there is a unique isomorphism of Ĝ_{+}^{1} with (Ĝ_{+}^{1})′ whose action on the Lie algebra takes H_{i} to H′_{i}, X_{i} to X′_{i}, and Y_{i} to Y′_{i}. The two together define an isomorphism of Ĝ_{+} with Ĝ′_{+}. If we assume that α_{i} corresponds to α′_{i}, 1 ≤ i ≤ ℓ this isomorphism takes Ẑ to Ẑ′ and determines an isomorphism of Ĝ with Ĝ′ which commutes with 𝔊(K/F). This in turn determines an isomorphism φ̂ of Ĝ′_{F} with Ĝ_{F}. In particular taking G′ = G and φ to be the identity we see that Ĝ_{F} is determined up to a canonical isomorphism.

Suppose G is any reductive group over F, K is a Galois extension of F, G′ and G′′ are quasisplit groups over F which split over K, and φ: G′ → G, ψ: G′′ → G are isomorphisms defined over K such that φ^{−1}σ(φ) and ψ^{−1}σ(ψ) are inner for all σ in 𝔊(K/F). Then (ψ^{−1}φ)^{−1}σ(ψ^{−1}φ) is also inner so that there is a canonical isomorphism of Ĝ′_{F} and Ĝ′′_{F}. We are thus free to set Ĝ_{F} = Ĝ′_{F}. Ĝ_{F} depends on K but there is no need to stress this. However we shall sometimes write Ĝ_{K/F} instead of Ĝ_{F}.

3. Although it is a rather simple case, it may be worthwhile to carry out the previous construction when G is GL(n) and K = F. We take T to be the diagonal and B to be the upper triangular matrices. G^{0} is the group of non-zero scalar matrices and G^{1} is SL(n). If λ belongs to L and

λ: ( \begin{matrix}t_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & t_{n} \end{matrix} ) ⟶ t_{1}^{m_{1}} … t_{n}^{m_{n}}

with m_{1}, …, m_{n} in ℤ we write λ = (m_{1}, …, m_{n}). Thus L is identified with ℤ^{n}. We may identify E_{ℝ} with ℝ^{n} and E_{ℂ} with ℂ^{n}. If λ belongs to L_{+}^{0} and

λ: tI → t^{m}

with m in ℤ we write λ = (\frac{m}{n}, …, \frac{m}{n}). Then L_{−}^{0} which is a subgroup of both L and L_{+}^{0} consists of the elements (m, …, m) with m in ℤ. The rank ℓ is n − 1 and

α_{1} = (1, −1, 0, …, 0)
α_{2} = (0, 1, −1, 0, …, 0)
⋮
α_{ℓ} = (0, …, 0, 1, −1)

Thus

L_{−}^{1} = {(m_{1}, …, m_{n}) ∈ L | ∑_{i=1}^{n} m_{i} = 0} .

E_{ℂ}^{1} is the set of all (z_{1}, …, z_{n}) in E_{ℂ} for which

∑_{i=1}^{n} z_{i} = 0 .

The bilinear form on E_{ℂ}^{1} may be taken as the restriction of the form

(z, w) = ∑_{i=1}^{n} z_{i}w_{i}

on E_{ℂ}. Thus

L_{+}^{1} = {(m_{1}, …, m_{n}) | ∑_{i=1}^{n} m_{i} = 0 and m_{i} − m_{j} ∈ ℤ} .

We may use the given bilinear form to identify Ê_{ℂ} with E_{ℂ}. Then the operation “ ” leaves all lattices and all roots fixed. Thus Ĝ_{+}^{0} = Hom(L_{+}^{0}, ℂ). Any non-singular complex scalar matrix tI defines an element of Ĝ_{+}^{0}, namely, the homomorphism

(\frac{m}{n}, …, \frac{m}{n}) ⟶ t^{m} .

We identify Ĝ_{+}^{0} with the group of scalar matrices. Ĝ_{+}^{1} is SL(n, ℂ). There is a natural map of Ĝ_{+}^{0} × Ĝ_{+}^{1} onto GL(n, ℂ). It sends tI × A to tA. The kernel is easily seen to be Ẑ so that Ĝ_{F} is GL(n, ℂ).

4. To define the local L-functions, to prove that almost all primes are unramified, and to prove that the product of the local L-functions over the unramified primes converges for Re s sufficiently large we need some facts from the reduction theory for groups over local fields (cf. [1]). Much progress has been made in that theory, but it is still incomplete. Unfortunately, the particular facts we need do not seem to be in the literature. Very little is lost at this stage if we just assume them. For the groups about which something definite can be said, they are easily verified.

Suppose K is an unramified extension of the non-archimedean local field F and G is a quasisplit group over F which splits over K. Let B be a Borel subgroup of G and T a Cartan subgroup of B, both of which are defined over F. Let v be the valuation on K. It is a homomorphism from K^{∗}, the multiplicative group of K, onto ℤ whose kernel is the group of units. If t belongs to T_{F}, let v(t) in L̂ be defined by ⟨λ, v(t)⟩ = v(λ(t)) for all λ in L. If σ belongs to 𝔊(K/F), then

⟨λ, σv(t)⟩ = ⟨σ^{−1}λ, v(t)⟩ = v(σ^{−1}(λ(σt))) = v(λ(t))

because σt = t and v(σ^{−1}a) = v(a) for all a in K^{∗}. Thus v is a homomorphism of T_{F} into M̂, the groups of invariants of 𝔊(K/F) in L̂. It is in fact easily seen that it takes T_{F} onto M̂.

We assume the following lemma.

Lemma 1. There is a Chevalley lattice in the Lie algebra of G whose stabilizer U_{K} is invariant under 𝔊(K/F). U_{K} is its own normalizer. Moreover, G_{K} = B_{K}U_{K}, H^{1}(𝔊(K/F), U_{K}) = 1, and H^{1}(𝔊(K/F), B_{K} ∩ U_{K}) = 1. If we choose two such Chevalley lattices with stabilizers U_{K} and U′_{K}, respectively, then U′_{K} is conjugate to U_{K} in G_{K}.

If g belongs to G_{K} and σ belongs to 𝔊(K/F), let g^{σ} = σ^{−1}(g). If g belongs to G_{F}, we may write it as = bu with b in B_{K} and u in U_{K}. Then g^{σ} = b^{σ}u^{σ} and u^{σ}u^{−1} = b^{−σ}b. By the lemma, there is a v in B_{K} ∩ U_{K} such that u^{σ}u^{−1} = b^{−σ}b = v^{σ}v^{−1}. Then b′ = bv belongs to B_{F}, u′ = v^{−1}u belongs to U_{F} = G_{F} ∩ U_{K}, and g = b′u′. Thus, G_{F} = B_{F} U_{F}.

If gU_{K}g^{−1} = U′_{K} for some g in G_{K}, then g^{σ}U_{K}g^{−σ} = U′_{K} so that g^{−σ}g belongs to U_{K}, which is its own normalizer. By the lemma, there is u in U_{K} such that g^{−σ}g = u^{σ}u^{−1}. Then g_{1} = gu lies in G_{F} and g_{1}U_{K}g_{1}^{−1} = U′_{K}. Thus, U_{F} and U′_{F} are conjugate in G_{F}.

Let C_{c}(G_{F} , U_{F}) be the set of all compactly supported functions for G_{F} such that f(gu) = f(ug) = f(g) for all u in U_{F} and all g in G_{F}. C_{c}(G_{F} , U_{F}) is an algebra under convolution. It is called the Hecke algebra. If N is the unipotent radical of B let dn be a Haar measure on N_{F} and let \frac{d(bnb^{−1})}{dn} = δ(b) if b belongs to B_{F}. If λ belongs to M̂, choose t in T_{F} such that v(t) = λ. If f belongs to C_{c}(G_{F} , U_{F}), set

f̂(λ) = δ^{1/2}(t){∫_{N_{F} ∩U_{F}} dn}^{−1} ∫_{N_{F}} f(tn)dn .

The group 𝔊(K/F) acts on Ω. Let Ω^{0} be the group of invariant elements. Ω^{0} acts on M̂. Let A(M̂) be the group algebra of M̂ over ℂ, and let Λ^{0}(M̂) be the invariants of Ω^{0} in Λ(M̂). We also assume the following lemma (cf. [12]).

Lemma 2. The map f → f̂ is an isomorphism of C_{c}(G_{F} , U_{F}) and Λ^{0}(M).

Suppose B is replaced by B_{1} and T by T_{1}. Observe that T ≃ B/N and T_{1} ≃ B_{1}/N_{1}. If u in G_{F} takes B to B_{1}, it takes N to N_{1} and defines a map from T to T_{1}. This map does not depend on u. It determines 𝔊(K/F) invariant maps from L_{1} to L and from L̂ to L̂_{1} and thus maps from M̂ to M̂_{1} and from Λ^{0}(M̂) to Λ^{0}(M̂_{1}). Suppose f̂ goes to f̂_{1} and λ̂ goes to λ̂_{1}. If we choose, as we may, u in U_{F}, then

f̂_{1}(λ̂_{1}) = f̂(λ̂) = δ^{1/2}(t){∫_{N_{F} ∩U_{F}} dn}^{−1} ∫_{N_{F}} f(tn)dn .

Let N_{F} ∩ U_{F} = V. Denote the corresponding group associated to N_{1} by V_{1}. Then u V u^{−1} = V_{1}. Choose d(unu^{−1}) = dn_{1}. Since f(ugu^{−1}) = f(g), the expression on the right equals

δ^{1/2}(utu^{−1}){∫_{V_{1}} dn_{1}}^{−1} ∫_{N_{F}} f(utu^{−1}unu^{−1})dn .

If utu^{−1} projects on t_{1} in T_{1}, then δ(utu^{−1}) = δ(t_{1}) and v(t_{1}) = λ̂_{1}. Moreover,

∫ f(utu^{−1}unu^{−1})dn = ∫ f(t_{1}n_{1})dn_{1}

and the diagram

C_{c}(G_{F} , U_{F})
↙  ↘
Λ^{0}(M) ⟶ Λ^{0}(M_{1})

is commutative.

If gU_{F} g^{−1} = U′_{F}, the map f → f′ with f′(h) = f(g^{−1}hg) is an isomorphism of C_{c}(G_{F} , U_{F}) with C_{c}(G_{F} , U′_{F}). It does not depend on g. We can take g in B_{F}. Then

f̂′(λ) = δ^{1/2}(t){∫_{N_{F} ∩U′_{F}} dn}^{−1} ∫_{N_{F}} f(g^{−1}tng)dn .

Since g^{−1}tng = t(t^{−1}g^{−1}tg)g^{−1}ng, the second integral is equal to

∫_{N_{F}} f(tg^{−1}ng)dn .

Since

\frac{d(g^{−1}ng)}{dn} = \{∫_{N_{F} ∩U′_{F}} dn\}^{−1} ∫_{N_{F} ∩U_{F}} dn

we conclude that f̂′(λ̂) = f̂(λ̂) and that the diagram

C_{c}(G_{F} , U_{F}) ⟶ C_{c}(G_{F} , U′_{F})
↘  ↙
Λ^{0}(M̂)

is commutative.

I shall not explicitly mention the commutativity of these diagrams again. However, they are important because they imply that the definitions to follow have the invariance properties which are required if they are to have any sense.

If π is an irreducible unitary representation of G_{F} on H whose restriction to U_{F} contains the identity representation, then

H_{0} = {x ∈ H | π(u)x = x for all u in U_{F} }

is a one-dimensional subspace. If f belongs to C_{c}(G_{F} , U_{F}), then

π(f) = ∫_{G} f(g)π(g)dg

maps H_{0} into itself. The representation of C_{c}(G_{F} , U_{F}) on H_{0} determines a homomorphism χ of C_{c}(G_{F} , U_{F}) or of Λ^{0}(M̂) into the ring of complex numbers. π is determined by χ. To define the local L-functions, we study such homomorphisms. First of all, observe that, if χ is associated to a unitary representation, then

|χ(f)| ≤ ∫_{G_{F}} |f(g)|dg .

Since Λ(M̂) is a finitely generated module over Λ^{0}(M̂), any homomorphism of Λ^{0}(M̂) into ℂ may be extended to a homomorphism of Λ(M̂) into ℂ which will necessarily be of the form

∑ f̂(λ)λ ⟶ ∑ f̂(λ)λ(t)  (B)

for some t in T̂. Conversely, given t the formula (B) determines a homomorphism χ_{t} of Λ^{0}(M̂) into ℂ. We shall show that χ_{t_{1}} = χ_{t_{2}} if and only if t_{1} × σ_{F} and t_{2} × σ_{F}, where σ_{F} is the Frobenius subsitution, are conjugate in Ĝ_{F}. If t belongs to Ĝ and σ belongs to 𝔊(K/F), we shall abbreviate t × σ to tσ. It is known [4] that every semi-simple element of Ĝ_{F} whose projection on 𝔊(K/F) is σ_{F} is conjugate to some tσ_{F} with t in T̂. Thus, there is a one-to-one correspondence between homomorphisms of the Hecke algebra into ℂ and semi-simple conjugacy classes in Ĝ_{F} whose projection on 𝔊(K/F) is σ_{F}.

If ρ is a complex analytic representation of Ĝ_{F} and χ_{t} is the homomorphism of Λ^{0}(M̂) into ℂ associated to π, we define the local L-function to be

L(s, ρ, π) = \frac{1}{det(I − ρ(tσ_{F})|π_{F} |^{2})}

if π_{F} generates the maximal ideal of O_{F}.

T̂ may be identified with Hom_{ℤ}(L̂, ℂ^{∗}). The exact sequence

0 ⟶ ℤ ⟶^{φ} ℂ ⟶^{ψ} ℂ^{∗} ⟶ 0

with φ(z) = \frac{2πi}{log |π_{F} |}z and ψ(z) = |π_{F} |^{−z} leads to the exact sequence

0 ⟶ L = Hom_{ℤ}(L̂, ℤ) ⟶^{φ} E_{ℂ} = Hom_{ℤ}(L̂, ℂ) ⟶^{ψ} T̂ ⟶ 0 .

Let V_{ℂ} be the invariants of 𝔊(K/F) in E_{ℂ} and let W_{ℂ} be the range of σ_{F} −1. Then E_{ℂ} = V_{ℂ} ⊕W_{ℂ}. If w belongs to W_{ℂ} and λ belongs to M̂, then ⟨w, λ⟩ = 0 and, replacing t by tψ(w) does not change χ_{t}. If w = σ_{F} v − v and ψ(v) = s, then

tψ(w)σ_{F} = ts^{−1}σ_{F} (s)σ_{F} = s^{−1}(tσ_{F})s

is conjugate to tσ_{F}. Thus, we have to show that if t_{1} = ψ(v_{1}) and t_{2} = ψ(v_{2}) with v_{1} and v_{2} in _{ℂ}, then t_{1}σ_{F} and t_{2}σ_{F} are conjugate if and only if χ_{t_{1}} = χ_{t_{2}}.

Some preliminary remarks are necessary. We also have a decomposition Ê_{ℂ} = V̂_{ℂ} ⊕ Ŵ_{ℂ} and M̂ = L̂ ∩ V̂_{ℂ}. Let Q̂ be the elements of V̂_{ℂ} obtained by projecting the positive coroots on V̂_{ℂ}. If S is an orbit of 𝔊(K/F) in the set of positive coroots, every element in S has the same projection on V̂_{ℂ}. Since ∑_{α̂∈S} α̂ belongs to V̂_{ℂ}, the projection must be

\frac{1}{n(S)} ∑_{α̂∈S} α̂

if n(S) is the number of elements in S. Let S_{1}, …, S_{m} be the orbits of 𝔊(K/F) in {α̂_{1}, …, α̂_{ℓ}} and set

β̂_{i} = \frac{1}{n(S_{i})} ∑_{α̂∈S_{i}} α̂ .

Every element of Q̂ is a linear combination of β̂_{1}, …, β̂_{m} with non-negative coefficients. Notice that if ω belongs to Ω^{0} and ω acts trivially on M̂, then ω leaves each β_{i} fixed and therefore takes positive roots to positive roots. Thus, it is 1. If we extend the inner product in any way from Ê_{ℝ}^{1} to Ê_{ℝ} and set

Ĉ = {x ∈ V̂_{ℝ} | (β̂_{i}, x) ≥ 0, 1 ≤ i ≤ m}

and

D̂ = {x ∈ Ê_{ℝ} | (α̂_{i}, x) ≥ 0, 1 ≤ i ≤ ℓ} ,

then Ĉ = D̂ ∩ V̂_{ℝ}. Consequently, no two elements of Ĉ belong to the same orbit of Ω^{0}.

Let 𝔤̂_{i} be the subalgebra of the Lie algebra of Ĝ generated by the root vectors belonging to the coroots in S_{i} and their negatives. 𝔤̂_{i} is fixed by 𝔊(K/F). Let Ĝ_{i} be the corresponding analytic group and let T̂_{i} = T̂ ∩ Ĝ_{i}. Let μ_{i} be the unique element of the Weyl group of T̂_{i} which takes every positive root to a negative root. If σ belongs to 𝔊(K/F), then σ(μ_{i}) has the same property, so that σ(μ_{i}) = μ_{i}. Let w be any element in the normalizer of T̂ whose image in Ω̂ is μ_{i}. Then wσ_{F} (w^{−1}) lies in T̂. Its image in T̂/ψ(W_{ℂ}) is independent of w. I claim that this image is 1. To see this write 𝔤̂_{i} as a direct sum ∑_{k=1}^{n} 𝔤̂_{ik} of simple algebras. If [K: F] = n the stabilizer of 𝔤̂_{i1} is \{σ_{F}^{jn_{i}} | 0 ≤ j ≤ \frac{n}{n_{i}} \}. We may suppose that

𝔤̂_{ik} = σ_{F}^{k−1} \binom{}{𝔤_{i1}} .

If Ĝ_{ik} is the analytic subgroup of Ĝ with Lie aglebra 𝔤̂_{ik}, choose w_{1} in the normalizer of T̂ ∩ Ĝ_{i1} so that w_{1} takes the positive roots of 𝔤̂_{i1} to the negative roots. We may choose w to be ∏_{k=0}^{n_{i}−1} σ_{F}^{k} (w_{1}). Then

\begin{aligned}wσ_{F} (w^{−1}) & = (w_{1}σ_{F} (w_{1}^{−1}))(σ_{F} (w_{1})σ_{F}^{2} (w_{1}^{−1})) … (σ_{F}^{n_{i}−1}(w_{1})σ_{F}^{n_{i}}(w_{1}^{−1})) \\ & = w_{1}σ_{F}^{n_{i}}(w_{1}^{−1}) . \end{aligned}

The Dynkin diagram of 𝔤̂_{i1} is connected and the stabilizer of 𝔤̂_{i1} in 𝔊(K/F) acts transitively on it. This means that it is of type A_{1} or A_{2}.

In the first case the diagram reduces to a point and the action of the stabilizer must be trivial, so that w_{1} = σ_{F}^{n_{i}}(w_{1}). In the second case SL(3, ℂ) is the simply-connected covering group of G_{i1}; we may choose the covering map to be such that T̂ ∩ Ĝ_{i1} is the image of the diagonal matrices and σ_{F}^{n_{i}} corresponds to the automorphism

A → ( \begin{matrix}0 & 0 & 1 \\ 0 & −1 & 0 \\ 1 & 0 & 0 \end{matrix} ) ^{t}A^{−1} ( \begin{matrix}0 & 0 & 1 \\ 0 & −1 & 0 \\ 1 & 0 & 0 \end{matrix} )

of SL(3, ℂ). We may take w_{1} to be the image of

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

Then σ_{F}^{n_{i}}(w_{1}) = w_{1}.

μ_{i} acts on V̂ as the reflection in the hyperlane perpendicular to β_{i}. Thus μ_{1}, …, μ_{m} generate Ω^{0}. If ω belongs to Ω^{0}, choose w in the normalizer of T̂ whose image in Ω is ω. The image of wσ_{F} (w^{−1}) in T̂/ψ(W_{ℂ}) depends only on ω. Call it δ_{ω}. Then

δ_{ω_{1}ω_{2}} = w_{1}w_{2}σ_{F} (w_{2}^{−1}w_{1}^{−1}) = w_{1}(w_{2}σ_{F} (w_{2}^{−1}))w_{1}^{−1}(w_{1}σ_{F} (w_{1}^{−1})) = ω_{1}(δ_{ω_{1}})δ_{ω_{1}} .

Since δ_{ω} is 1 on a set of generators, this relation shows that it is identically 1.

Returning to the original problem, we show first that if χ_{t_{1}} = χ_{t_{2}} there is an ω in Ω^{0} such that ω(w_{1}) = t_{2}. Then, if w lies in the normalizer of T̂ in Ĝ and its image in Ω is ω, we will have w(t_{1}σ_{F})w^{−1} = t_{2}wσ_{F} (w^{−1})σ_{F}. Since wσ_{F} (w^{−1}) lies in ψ(W_{ℂ}), the element on the right is conjugate to t_{2}σ_{F}.

If t belongs to T̂, let χ_{t} also denote the homomorphism

∑ f̂(λ)λ → ∑ f̂(λ)λ(t)

of Λ(M̂) into ℂ. If there were no ω such that ω(t_{1}) = t_{2}, there would be an f̂ in Λ(M̂) such that

χ_{t_{2}}(f̂) ≠ χ_{ω(t_{1})}(f̂)

for all ω in Ω^{0}. Let

∏(X − ω(f̂)) = ∑_{k=0}^{n} f̂_{k}X^{k} .

Each f̂_{k} belongs to Λ^{0}(M̂). Applying χ_{t_{1}} and χ_{t_{2}}, we find that

∏_{ω}(X − χ_{ω(t_{1})}(f̂)) = ∑_{k=0}^{n} χ_{t_{1}}(f̂_{k})X^{k} = ∑_{k=0}^{n} χ_{t_{2}}(f̂_{k})X^{k} = ∏_{ω}(X − χ_{ω(t_{2})}(f̂)) .

The polynomial on the right has χ_{t_{2}}(f̂) as a root, but that on the left does not. This is a contradiction.

If t_{1}σ_{F} and t_{2}σ_{F} are conjugate, then for every representation ρ of Ĝ_{F}

trace ρ(t_{1}σ_{F}) = trace ρ(t_{2}σ_{F}) .

Let ρ act on X and if λ belongs to M̂, let t_{λ} be the trace of ρ(σ_{F}) on

X_{λ} = {x ∈ X | ρ(t)x = λ(t)x for all t in T̂} .

If t belongs to ψ(W_{ℂ}), then λ(t) = 1. If ω belongs to Ω^{0} and w in the normalizer of T has image ω in Ω, the X_{ωλ} = ρ(w)X_{λ}. Then t_{ωλ} is the trace of w^{−1}σ_{F} w = w^{−1}σ_{F} (w)σ_{F} on X_{λ}. Since λ(w^{−1}σ_{F} (w)) = 1, we have t_{ωλ} = t_{λ} and

trace ρ(tσ_{F}) = ∑_{λ∈Ĉ} t_{λ} ( ∑_{μ∈S(λ)} μ(t))

if S(λ) is the orbit of λ. If

f̂_{ρ} = ∑_{λ∈Ĉ} t_{λ} ∑_{μ∈S(λ)} μ

then f̂_{ρ} belongs to Λ^{0}(M̂) and

trace ρ(tσ_{F}) = χ_{t}(f̂_{ρ}) .

All we need to show is that the elements f̂_{ρ} generate Λ^{0}(M̂) as a vector space. This is an easy induction argument because every λ in Ĉ is the highest weight of a representation of Ĝ_{F} whose restriction to Ĝ is irreducible.

5. If t belongs to T̂, there is a unique function ϕ_{t} on G_{F} which satisfies ϕ_{t}(ug) = ϕ_{t}(gu) = ϕ_{t}(g) or all u in U_{F} and all g in _{F}, such that

χ_{t}(f) = ∫_{G_{F}} ϕ_{t}(g)f(g)dg

for all f in C_{c}(G_{F} , U_{F}). A formula for ϕ_{t}, valid under very general assumptions, has been found by I. G. MacDonald. However, because of the present state of reduction theory, his assumptions do not cover the cases in which we are interested. I am going to assume that the obvious generalization of his theorem is valid. In stating it we may as well suppose that t belongs ot ψ(V_{ℂ}).

Let N̂ be the unipotent radical of B̂, let 𝔫̂ be its Lie algebra, and let τ be the representation of T̂ × 𝔊(K/F) on 𝔫̂. If t belongs to ψ(V_{ℂ}) , consider the function θ_{t} on M̂ defined by

θ_{t}(λ) = c|π_{F} |^{−⟨ρ,λ⟩}σ_{ω∈Ω^{0}} \frac{det(I − |π_{F} |τ^{−1}(ω(t)σ_{F}))}{det(I − τ^{−1}(ω(t)σ_{F}))}λ^{−1}(ω(t)) .

If n(B̂) is the number of positive roots projection onto β̂ in Q̂,

c = ∏_{β∈Q} \{\frac{1 − |π_{F} |^{n(β̂)⟨ρ,β̂⟩}}{1 − |π_{F} |^{n(β)(⟨ρ,β⟩+1)}} \} .

As it stands, θ_{t}(λ) makes sense only when none of the eigenvalues of τ(ω(t)σ)F are 1 for any ω in Ω^{0}. However, using the results of Kostant [8], we can write it in a form which makes sense for all t. Let ρ̂ be one-half the sum of the positive coroots. ρ̂ belongs to V̂. If λ belongs to M̂ and λ + ρ̂ is non-singular, that is (λ + ρ̂, β̂) ≠ 0 for all β̂ in Q̂, let ω in Ω^{0} take λ + ρ̂ to Ĉ and let χ_{λ} be sgn ω times the character of the representation of Ĝ_{F} with highest weight ω(λ + ρ̂) − ρ̂. If λ + ρ̂ is singular, let χ_{λ} ≡ 0. If

det(I − |π_{F} |τ^{−1}(tσ_{F})) = ∑_{μ∈M̂} b_{μ}μ(t)

then

θ_{t}(λ) = c|π_{F} |^{−⟨ρ,λ⟩} ∑_{μ∈M̂} b_{μ}χ_{μ−λ}((tσ_{F})) .

Clearly b_{μ}is 0 unless

μ = − ∑_{α̂∈S} α̂

where S is a subset of the set of positive coroots invariant under 𝔊(K/F). If U is the collection of such μ, then {ρ̂ + μ|μ ∈ M} is invariant under Ω^{0}. Suppose ρ̂ + μ is non-singular and belongs to Ĉ. Since ⟨α_{i}, ρ̂⟩ = 1 and ⟨α_{i}, μ⟩ is integral, for 1 ≤ i ≤ ℓ, μ itself must belong to Ĉ. This can only happen if μ is 0. Thus if b_{μ} ≠ 0 either ρ̂ + μ is singular or ρ̂ + μ belongs to the orbit of ρ̂ and χ_{μ}(g) ≡ ±1 on Ĝ_{F}. As a consequence θ_{t}(0) is independent of t. Choose t_{0} such that β̂_{i}(t_{0}) = |π|^{−⟨ρ,β̂_{i}⟩} for 1 ≤ i ≤ m. The eigenvalues of τ(ω(t_{0})σ_{F}) are the numbers ζ|π_{F} |^{−⟨ρ,ω^{−1}β̂⟩} where β̂ belongs to Q and ζ is an n(β̂)^{th} root of unity. If ω ≠ 1 there is a β̂_{i} such that ω^{−1}β̂ = −β̂_{i} for some β in Q. Then ⟨ρ, ω^{−1}β̂⟩ = − ⟨ρ, β̂_{i}⟩ = −1 and τ(ω(t_{0})σ_{F}) has |π_{F} | is an eigenvalue. Thus

θ_{t_{0}}(0) = c \frac{det(I − |π_{f}|τ^{−1}(t_{0}σ_{F}))}{det(I − τ^{−1}(t_{0}σ_{F}))} = 1 .

We are going to assume that if t belongs to ψ(V_{ℂ}), a belongs to T_{F}, and λ = v(a), then

ϕ_{t}(a) = θ_{t}(λ) .

If

|χ_{t}(f)| ≤ ∫_{G_{F}} |f(g)|dg

for all f in C_{c}(G_{F} , U_{F}) then ϕ_{t} is bounded. I want to show that if ϕ_{t} is bounded, λ belongs to L̂, λ̄ in D̂ belongs to the orbit of λ under Ω, and t lies in ψ(V_{ℂ}), then

|λ(t)| ≤ |π_{F} |^{−⟨ρ,λ̄⟩} .

Let t = ψ(v). Then v is not determined by t but Re v is and

|λ(t)| = |π_{F} |^{−Re⟨v,λ⟩} .

We will show that if ϕ_{t} is bounded then Re ⟨v, λ⟩ ≤ ⟨ρ, λ̄⟩ for all λ in Ê_{ℝ}. If ω belongs to Ω^{0} and Re ωv lies in Ĉ then Re ⟨ωv, ωλ⟩ = Re ⟨v, λ⟩. With no loss of generality, we may suppose that v lies in C, the analogue of Ĉ. Then, as is well-known,

Re ⟨v, λ⟩ ≤ Re ⟨v, λ̄⟩

and we may as well assume that λ = λ̄. We want to show that Re ⟨v, λ⟩ ≤ ⟨ρ, λ⟩ for all λ in D̂. Since ρ and v both belong to V_{ℂ}, it is sufficient to verify it for λ in Ĉ. Let Ĉ^{0} be the interior of Ĉ. The set of λ in Ĉ for which the assertion is true is closed, convex, and positively homogeneous. Therefore, if it contains M̂ ∩ Ĉ^{0}, it is Ĉ.

Let S be the set of simple coroots α̂ for which Re ⟨v, α̂⟩ = 0. Let Σ_{0} be the positive coroots which are linear combinations of the elements of S and let Σ_{+} be the other positive coroots. If 𝔫̂_{0} is the span of the root vectors associated to the coroots in Σ_{0} and 𝔫̂_{+} is the span of the root vectors associated to the coroots in Σ_{+}, then τ breaks up into the direct sum of a representation τ_{0} on 𝔫̂_{0} and a representation τ_{+} on 𝔫̂_{+}. Let Ĥ be the analytic subgroup of Ĝ_{F} whose Lie algebra is generated by the root vectors associated to the coroots of Σ_{0} and their negatives and let Θ^{0} be the subgroup of Ω^{0} consisting of those elements with representatives in Ĥ. If ω belongs to Ω^{0} and Re ωv = Re v, then ω belongs to Θ^{0}. If Re ωv ≠ Re v, then Re ⟨ωv, λ⟩ < Re ⟨v, λ⟩ for λ in M̂ ∩ Ĉ^{0}.

Write λ = λ_{1} + λ_{2} where λ_{1} is a linear combination of the coroots in S and λ_{2} is orthogonal to these roots. If s = ψ(u) with u in V_{ℂ}, consider

θ′_{s}(λ) = c|π_{F} |^{⟨u−ρ,λ_{2}⟩} \frac{det(I − |π_{F} |τ_{+}^{−1}(sσ_{F}))}{det(I − τ_{+}^{−1}(sσ_{F}))} \{∑_{Θ^{0}} \frac{det(I − |π_{F} |τ_{0}^{−1}(sσ_{F}))}{det(I − τ_{0}^{−1}(sσ_{F}))} |π_{F} |^{⟨ωu−ρ,λ_{1}⟩}\} .

The function θ′_{s} is not necessarily defined for all s. However, the preceding discussion, applied to Ĥ rather than Ĝ, shows that it is defined at t and that θ′_{t}(0) ≠ 0. A simple application of l’Hospital’s rule shows that, as a function of λ, θ′_{t} is the product of |π_{F} |^{⟨v−ρ,λ⟩} and a linear combination of products of polynomials and purely imaginary exponentials in λ_{1}. Thus, it does not vanish identically in any open cone.

Set θ′′_{t} = θ_{t} −θ′_{t}. θ′′_{t} is a linear combination of products of polynomials in λ and an exponential |π_{F} |^{⟨ωv−ρ,λ⟩} with Re ωv ≠ Re v. Thus, if λ belongs to the interior of Ĉ,

lim_{n→−∞} |π_{F} |^{⟨ρ−v,nλ⟩}θ′′_{t}(nλ) = 0

and

lim_{n→−∞} |π_{F} |^{⟨ρ−v,nλ⟩}θ_{t}(nλ) = lim_{n→−∞} |π_{F} |^{⟨ρ−v,nλ⟩}θ′_{t}(nλ) .

If ⟨ρ, λ⟩ is less than Re ⟨v, λ⟩ for some λ in Ĉ, then ⟨ρ, λ⟩ is less than Re ⟨v, λ⟩ for a λ in Ĉ for which θ′_{t}(nλ) does not vanish identically as a function of n. Since ϕ_{t} is bounded,

lim_{n→−∞} |π_{F} |^{⟨ρ−v,nλ⟩}θ′_{t}(nλ) = 0 .

But |π_{F} |^{⟨ρ−v,nλ⟩}θ′_{t}(nλ) is a function of the form

∑_{k=0}^{q} φ_{k}(n)n^{k}

where φ_{k}(n) is a linear combination of purely imaginary exponentials e^{ixn}. It is easy to see that it cannot approach 0 as n approaches −∞.

6. Suppose G is a group defined over the global field F. There is a quasi-split group G′ over F and an isomorphism φ: G → G′ defined over a Galois extension K of F such that, for every σ in 𝔊(K/F), a_{σ} = φ^{σ}φ^{−1} is an inner automorphism of G′. We assume that there is a lattice 𝔤_{O_{F}} over O_{F} in the Lie algebra of G′ such that O_{K}𝔤_{0_{F}} is a Chevalley lattice.

If 𝔭 is a finite prime of K and 𝔓 is a prime of K dividing 𝔭, the group G over F_{𝔭} is obtained from G′ by twisting by the restriction \overline{a} of the cocycle \{a_{σ}\} to 𝔊(K_{𝔓}/F_{𝔭}). Let \overline{G}′ be the adjoint group of G′. If Ū′_{K_{𝔓}} is the stabilizer of the lattice O_{K_{𝔓}}𝔊_{0_{F}} then, for almost all 𝔭, \overline{a} takes values in Ū′_{K_{𝔓}}. If K_{𝔓}/F_{𝔭} is also unramified, then G is quasi-split over F_{𝔭} because H^{1}(𝔊(K_{𝔓}/F_{𝔭}), Ū′_{K_{𝔭}}) = {1}. Let S be the set of those 𝔭, unramified in K, for which \overline{a} takes values in Ū′_{K_{𝔓}}. Let G act on a vector space X over F and let X_{O_{F}} be a lattice in X_{F}. Let U_{F_{𝔭}} be the stabilizer of O_{F_{𝔭}}X_{0_{F}} in G_{F_{𝔭}} and let U′_{F_{𝔭}} be the stabilizer of O_{F_{𝔭}}𝔤_{O_{F}} in G′_{F_{𝔭}}. Then φ(U_{F_{𝔭}}) = U′_{F_{𝔭}} for almost all 𝔭. If 𝔭 is also in S, choose u in Ū′_{F_{𝔭}} so that φ^{σ}φ^{−1} = Ad u^{σ}u^{−1} for all σ in 𝔊(K_{𝔓}/F_{𝔭}). Then φ^{−1}Ad u is defined over F and φ^{−1}Ad u(U′_{F_{𝔭}}) = U_{F_{𝔭}}. Consequently, U_{F_{𝔭}} is one of the compact subgroups of the fourth paragraph.

To show that almost all 𝔭 are unramified, all we need do is observe that if π occurs in L^{2}(G_{F} \ G_{ℤ(F)}), whatever the precise meaning of this is to be, and π = ∏_{𝔭} ⊗π_{𝔭}, then for almost all 𝔭, the restriction of π_{𝔭} to U_{F_{𝔭}} contains the trivial representation.

If 𝔭 is unramified let the homomorphism of C_{c}(G_{F_{𝔭}}, U_{F_{𝔭}}) associated to π_{𝔭} be χ_{t_{𝔭}}. To show that the product of the local L-function converges in a half plane it would be enough to show that there is a positive constant a such that for all unramified 𝔭 every eigenvalue of ρ(t_{𝔭}σ_{F_{𝔭}}) is bounded by |π_{𝔭}|^{−a}. We may suppose that σ_{F_{𝔭}}(t_{𝔭}) = t_{𝔭}. if n = [K : F], then (t_{𝔭}σ_{F_{𝔭}})^{n} = t_{𝔭}^{n} so that we need only show that the eigenvalues of ρ(t_{𝔭}) are bounded by |π_{𝔭}|^{−a}. This we did in the previous paragraph.

7. Once the definitions are made we can begin to pose questions. My hope is that these questions have affirmative answers. The first question is the one initially posed.

Question 1. Is it possible to define the local L-functions L(s, ρ, π) and the local factors ε(s, ρ, π, ψ_{F}) at the ramified primes so that if F is a global field π = ∏ ⊗π_{𝔭}, and

L(s, ρ, π) = ∏_{𝔭} L(s, ρ_{𝔭}, π_{𝔭})

then L(s, ρ, π) is meromorphic in the entire complex plane with only a finite number of poles and satisfies the functional equation

L(s, ρ, π) = ε(s, ρ, π)L(1 − s, ρ̃, π)

and

ε(s, ρ, π) = ∏_{𝔭} ε (s, ρ_{𝔭}, π_{𝔭}, ψ_{F_{𝔭}}) .

The theory of Eisenstein series can be used [9] to give some novel instances in which this question has, in part, an affirmative answer. However, that theory does not suggest any method of attacking the general problem. If G = GL(n) then Ĝ_{F} = GL(n, ℂ). The work of Godement and earlier writers allows one to hope that the methods of Hecke and Tate can, once the representation theory of the general linear group over a local field is understood, be used to answer the first question when G = GL(n) and ρ is the standard representation of GL(n, ℂ). The idea which led Artin to the general reciprocity law suggests that we try to answer it in general by answering a further series of questions. For the sake of precision, but not clarity, I write them down in an order opposite to that in which they suggest themselves. If G is defined over the local field F let Ω(G_{F}) be the set of equivalence classes of irreducible unitary representations of G_{F}.

Question 2. Suppose G and G′ are defined over the local field F, G is quasi-split and G′ is obtained from G by an inner twisting. Then Ĝ_{F} = Ĝ′_{F}. Is there a correspondence F whose domain is Ω(G′_{F}) and whose range is contained in Ω(G_{F}) such that if π = R(π′) then L(s, ρ, π) = L(s, ρ, π′) for every representation ρ of Ĝ_{F} ?

Notice that R is not required to be a function. I do not know whether or not to expect that

ε(s, ρ, π, ψ_{F}) = ε(s, ρ, π′, ψ_{F}) .

One should, but I have not yet done so, look carefully at this question when F is the field of real numbers. For this one will of course need the work of Harish-Chandra.

Supposing that the second question has an affirmative answer, one can formulate a global version.

Question 3.^{∗} Suppose that G and G′ are defined over the global field F, G is quasi-split, and G′ is obtained from G by an inner twisting. Suppose π′ = ∏_{𝔭} ⊗π′_{𝔭} occurs in L^{2}(G′_{F} \ G′_{𝔸(F)}). Choose for each 𝔭 a representation π_{𝔭} of G_{F_{𝔭}} such that π_{𝔭} = R(π′_{𝔭}). Does π = ∏_{𝔭} ⊗π_{𝔭} occur in L^{2}(G_{F} \G_{𝔸(F)})?

Affirmative evidence is contained in papers of Eichler [3] and Shimizu [16] when G = GL(2) and G′ is the group of invertible elements in a quaternion algebra. Jacquet [16], whose work is not yet complete, is obtaining very general results for these groups.

∗ The question, in this crude form, does not always have an affirmative answer (cf. [6]). The proper question is certainly more subtle but not basically different.

Question 4. Suppose G and G′ are two quasi-split groups over the local field F. Let G split over K and let G′ split over K′ with K ⊆ K′. Let ψ be the natural map 𝔊(K′/F) → 𝔊(K/F). Suppse φ is a complex analytic homomorphism from Ĝ′_{K′/F} to Ĝ_{K/F} which makes

Ĝ′_{K′/F} ⟶ 𝔊(K′/F)
↓^{φ}  ↓^{ψ}
Ĝ_{K/F} ⟶ 𝔊(K/F)

commutative. Is there a correspondence R_{φ} with domain Ω(G′_{F}) whose range is contained in Ω(G_{F}) such that if π = R_{φ}π′, then, for every representation ρ of Ĝ_{F} and every non-trivial additive character ψ_{F}, L(s, ρ, π) = L(s, ρ ∘ φ, π′) and ε(s, ρ, π, ψ_{F}) = ε(s, ρ ∘ φ, π′, ψ_{F}) ?

R_{φ} should of course be functorial and, in an unramified situation, if π′ is associated to the conjugacy class t′ × σ′_{F}, then π should be associated to φ(t′ × σ′_{F}). I have not yet had a chance to look carefully at this question when F is the field of real numbers.

The question has a global form.

Question 5. Suppose G and G′ are two quasi-split groups over the global field F. Let G split over K and let G′ split over K′ with K ⊆ K′. Suppose φ is a complex analytic homomorphism from Ĝ′_{K′/F} to Ĝ_{K/F} which makes

Ĝ′_{K′/F} ⟶ 𝔊(K′/F)
↓^{φ}  ↓
Ĝ_{K/F} ⟶ 𝔊(K/F)

commutative. If 𝔓′ is a prime of K′, let 𝔓 = 𝔓′ ∩ K and let 𝔭 = 𝔓′ ∩ F. φ determines a homomorphism φ_{𝔭}: Ĝ′_{K′_{𝔓′} /F_{𝔭}} → Ĝ_{K_{𝔓}/F_{𝔭}} which makes

Ĝ′_{K′_{𝔓′} /F_{𝔭}} ⟶ 𝔊(K′_{𝔓′}/F_{𝔭})
↓  ↓
Ĝ_{K_{𝔓}/F_{𝔭}} ⟶ 𝔊(K_{𝔓}/F_{𝔭})

commutative. If π′ = ∏ ⊗π′_{𝔭} occurs in L^{2}(G′_{F} \ F′_{𝔸(F)}) choose for each 𝔭 a π_{𝔭} = R_{φ_{𝔭}}(π′_{𝔭}). If

π = ∏_{𝔭} ⊗π_{𝔭} does π occur in L^{2}(G_{F} \ G_{𝔸(F)}) ?

An affirmative answer to the third and fifth questions would allow us to solve the first question by examining automorphic forms on the general linear groups.

It is probably worthwhile to point out the difficulty of the fifth question by giving some examples. Take G′ = {1} G = GL(1), K′ any Galois extension of F and K = F. The assertion that, in this case, the last two questions have affirmative answers is the Artin reciprocity law.

Suppose G is quasi-split and G′ = T. We may identify Ĝ′_{F} with T̂ × 𝔊(K/F) which is contained in Ĝ_{F}. Thus we take K′ = K. Let φ be the imbedding. In this case π′ is a character of G′_{F} \ G′_{𝔸(F)}. The fourth question is, with certain reservations, answered affirmatively by the theory of induced representations. The fifth question is, with similar reservations, answered by the theory of Eisenstein series. The reservations are not important. I only want to point out that the theory of Eisenstein series is a prerequisite to the solution of these problems. with G as before, take G′′ = {1} and K′′ = K so that Ĝ′′_{F} = 𝔊(K/F). Let ψ take σ in 𝔊(K/F) to σ to Ĝ_{F}. There is only one choice of π′′. The associated space of automorphic forms on G_{F} \ F_{𝔸(F)} should be the space of automorphic forms associated to the trivial character of G′_{F} \ G′_{𝔸(F)}. For this character all the reservations apply. I point out that the space associated to π′′ is not the obvious one. It is not the space of constant functions. To prove its existence will require the theory of Eisenstein series.

Take G = GL(2) and let G′ be the multiplicative group of a separable quadratic extension K′ of F. Take K = F. Then Ĝ′_{F} is a semi-direct product (ℂ^{∗} × ℂ^{∗}) × 𝔊(K′/F). If σ is the non-trivial element of 𝔊(K′/F), then σ((t_{1}, t_{2})) = (t_{2}, t_{1}). Let φ be defined by

\begin{matrix}φ: (t_{1}, t_{2}) & ⟶ ( \begin{matrix}t_{1} & 0 \\ 0 & t_{2} \end{matrix} ) \\ φ: σ & ⟶ ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \end{matrix}

The existence of R_{φ} in the local case is a known fact (see for example [6]) in the theory of representations of GL(2, F). An affirmative answer to the fifth question can be given by means of the Hecke theory [6] and by other means [15].

Let E be a separable extension of F and let G be the group over F obtained from GL(2) over D by restriction of scalars. Let G′ be GL(2) over F and let K′ = K be any Galois extension containing E. Let X be the homogeneous space 𝔊(K/E) \ 𝔊(K/F). Then Ĝ_{F} is the semi-direct product of ∏_{x∈X} GL(2, ℂ) and 𝔊(K/F). If σ belongs to 𝔊(K/F), then

σ( ∏_{x∈X} A_{x})σ^{−1} = ∏_{x∈X} B_{x}

with B_{x} = A_{xσ}. Define φ by

φ(A × σ)( ∏_{x∈X} A) × σ .

Although not much is known about the fifth question in this case, the paper [2] of Doi and Naganuma is encouraging.

Suppose G and K are given. Let G′ = {1} and let K′ be any Galois extension of F containing K. If F is a local field, the fourth question asks that, to every homomorphism φ of 𝔊(K′/F) into Ĝ_{F} which makes

𝔊(K′/F) ⟶^{φ}  Ĝ_{F}
↘  ↓
𝔊(K/F)

commutative, there be associated at least one irreducible unitary representation of G_{F}. If F is global, the fifth question asks that to φ there be associated a presentation of G_{𝔸(F)} occurring in L^{2}(G_{F} \ G_{𝔸(F)}).

The L-functions we have introduced have been so defined that they include the Artin Lfunctions. However, Weil [17] has generalized the notion of an Artin L-function. The preceding observations suggest a relation between the generalized Artin L-function and the L-functions of this paper. Weil’s definition requires the introduction of some locally compact groups – the Weil groups. If F is a local field, let C_{F} be the multiplicative group of F. If F is a global field, let C_{F} be the idèle class group. If K is a Galois extension of F, the Weil group W_{K/F} is an extension

1 → C_{K} → W_{K/F} → 𝔊(K/F) → 1

of 𝔊(K/F) by C_{K}. There is a canonical homomorphism τ_{K/F} of W_{K/F} onto C_{F}. If F is a global field, 𝔓 a prime of K, and 𝔭 = F ∩ 𝔓, there is a homomorphism α_{𝔭}: W_{K_{𝔓}/F_{𝔭}} → W_{K/F}. α_{𝔭} is determined up to an inner automorphism. If σ is a representation of W_{K/F}, the class of σ_{𝔭} = σ ∘α_{𝔭} is independent of α_{𝔭}. By a representation σ of W_{K/F} we understand a finite dimensional complex representation such that σ(w) is semi-simple for all w in W_{K/F}.

If F is a local field and ψ_{F} a non-trivial additive character of F, then for any representation σ of W_{K/F} we can define (cf. [11]) a local L-function L(s, σ) and a factor ε(s, σ, ψ_{F}). If F is a global field and σ is a representation of W_{K/F}, the associated L-function is

L(s, σ) = ∏_{𝔭} L(s, σ_{𝔭}) .

The product is taken over all primes, including the archimedean ones. If ψ_{F} is a non-trivial character of F \ 𝔸(F), then ε(s, σ_{𝔭}, ψ_{F_{𝔭}}) is 1 for almost all 𝔭,

ε(s, σ) = ∏_{𝔭} ε(s, σ_{𝔭}, ψ_{F_{𝔭}})

is independent of ψ_{F}, and

L(s, σ) = ε(s, σ)L(1 − s, σ̃)

if σ̃ is contragredient to σ.

Question 6. Suppose G is quasi-split over the local field F and splits over the Galois extension K. Let Û_{F} be a maximal compact subgroup of Ĝ_{F}. Let K′ be a Galois extension of F which contains K and let φ be a homomorphism of W_{K′/F} onto Û_{F} which makes

W_{K′/F} ⟶ 𝔊(K′/F)
↓^{φ} ↓
Û_{F}  ⟶ 𝔊(K/F)

commutative. Is there an irreducible unitary representation π(φ) of G_{F} such that, for every representation σ of Ĝ_{F}, L(s, σ, π(φ)) = L(s, σ ∘ φ) and ε(s, σ, π(φ), ψ_{F}) = ε(s, σ ∘ φ, ψ_{F}) ?

Changing φ by an inner automorphism Û_{F} will not change π(φ), or at least not its equivalence class. If F is non-archimedean and K′/F is unramified, the composition of v, the valuation on F, and τ_{K′/F} defines a homomorphism ω of W_{K/F} onto ℤ. If u = t × σ_{F} belongs to Û_{F}, we could define φ by

φ(w) = u^{ω(w)} .

Then π(φ) would be the representation associated to the homomorphism χ_{t} of the Hecke algebra into ℂ.

We can also ask the question globally.

Question 7. Suppose G is quasi-split over the global field F and splits over K. Let K′ be a Galois extension of F containing K and let φ be a homomorphism of W_{K′/F} into Û_{F} which makes

W_{K′/F} ⟶ 𝔊(K′/F)
↓^{φ} ↓
Û_{F}  ⟶ 𝔊(K/F)

commutative. If 𝔓′ is a prime of K′ and 𝔭 = 𝔓′ ∩ F, then φ_{𝔭} = φ ∘ α_{𝔭} takes W_{K′_{𝔓′} /F_{𝔭}} into Û_{F_{𝔭}}. If π(φ) = ∏_{𝔭} ⊗π(φ_{𝔭}), does π(φ) occur in L^{2}(G_{F} \ G_{𝔸(F)}) ?

Both questions have affirmative answers if G is abelian [10] and the correspondence φ → π(φ) is surjective. In this case our L-functions are all generalized Artin L-functions. If G = GL(2) and K = F, it appears that the Hecke theory can be used to give an affirmative answer to both questions if it is assumed that certain of the generalized Artin L-functions have the expected analytic properties. If all goes well, the details will appear in [6].

I would like very much to end this series of questions with some reasonably precise questions about the relation of the L-functions of this paper to those associated to non-singular algebraic varieties. Unfortunately, I am not competent to do so. Since it may be of interest, I would like to ask one question about the L-functions associated to elliptic curves. If C is defined over a local field F of characteristic zero, I am going to associate to it a representation π(C/F) of GL(2, F). If C is defined over a global field F which is also characteristic zero, then for each prime 𝔭, π(C/F_{𝔭}) is defined. Does π = ∏_{𝔭} ⊗π(C/F_{𝔭}) occur in L^{2}(GL(2, F) \ GL(2, 𝔸(F)) ? If so, L(s, σ, π), with σ the standard representation of GL(2, ℂ), whose analytic properties are known [6] will be one of the L-functions associated to the elliptic curve. There are examples on which the question can be tested. I hope to comment on them in [6].

To define π(C/F), I use the result of Serre [14]. Suppose that F is non-archimedean and the jinvariant of C is integral. Take any prime ℓ different from the characteristic of the residue field and consider the ℓ-adic representation. There is a finite Galois extension K of F such that, if A is the maximal unramified extension of K, the ℓ-adic representation can be regarded as a representation of 𝔊(A/F). There is a homomorphism of W_{K/F} into 𝔊(A/F). The ℓ-adic representation of 𝔊(A/F) determines a representation φ of W_{K/F} in GL(2, R), where R is a finitely generated subfield of the ℓ-adic field ℚ_{ℓ}. Let σ be an isomorphism of R with a subfield fo ℂ. Then

ψ: w → |τ_{K/F} (w)|^{1/2}φ^{σ}(w)

is a representation of W_{K/F} in a maximal compact subgroup of GL(2, ℂ). Let π(C/F) be the representation π(ψ) of Question 6. If C has good reduction, the class of ψ is independent of ℓ and σ. I do not know if this is so in general. It does not matter, because we do not demand that π(C/F) be uniquely determined by C.

If the j-invariant is not integral, the ℓ-adic representation can be put in the form

σ → ( \begin{matrix}χ_{1}(σ) & ∗ \\ 0 & χ_{2}(σ) \end{matrix} )

where χ_{1} and χ_{2} are two representations of the Galois group of the algebraic closure of F in the multiplicative group of ℚ_{ℓ}. If A is the maximal abelian extension of F, then χ_{1} and χ_{2} may be regarded as representations of 𝔊(A/F). There is a canonical map of F^{∗}, the multiplicative group of F, into 𝔊(A/F). χ_{1} and χ_{2} thus define characters μ_{1} and μ_{2} of F^{∗}. μ_{1} and μ_{2} take values in ℚ^{∗} and μ_{1}μ_{2}(x) = μ_{1}μ_{2}^{−1}(x) = |x|^{−1}. In, for example, [6], there is associated to the pair of generalized characters x → |x|^{1/2}μ_{1}(x) and x → |x|^{1/2}μ_{2}(x) a unitary representation of GL(2, F), a so-called special representation. This we take as π(C/F).

If F is ℂ, take π(C/F) to be the representation of GL(2, ℂ) associated to the map

s → ( \begin{matrix}\frac{z}{|z|} & 0 \\ 0 & \frac{z̄}{|z|} \end{matrix} )

of ℂ^{∗} = W_{ℂ/ℂ)} into GL(2, ℂ) by Question 6. ℂ^{∗} is of index two in W_{ℂ/ℝ}. The representation of W_{ℂ/ℝ} induced from the character z → \frac{1}{|z|} of ℂ^{∗} has degree 2. If F = ℝ, let π(C/F) be the representation of GL(2, ℝ) associated to the induced representation by Question 6.

8. I would like to finish up with some comments on the relation of the L-functions of this paper to Ramanujan’s conjecture and its generalizations. Suppose π = ∏ ⊗π_{𝔭} occurs in the space of cusp forms. The most general form of Ramanujan’s conjecture would be that for all 𝔭 the character of π_{𝔭} is a tempered distribution [5]. However, neither the notion of a character nor that of a tempered distribution has been defined for non-archimedean fields. A weaker question is whether or not at all unramified non-archimedean primes the conjugacy class in Ĝ_{F} associated to π_{𝔭} meets Û_{F} (cf. [13]). If this is so, it should be reflected in the behavior of the L-functions.

Suppose, to remove all ramification, that G is a Chevalley group and that K = F = ℚ. Suppose also that each π_{𝔭} is unramified. If p is non-archimedean, there is associated to π_{p} a conjugacy class {t_{p}} in G_{ℚ}. We may take t_{p} in T̂. The conjecture is that, for all λ in L̂,

|λ(t_{p})| = 1 .

Since there is no ramification at ∞, one can, as in [9], associate to π_{∞} a semi-simple conjugacy class {X_{∞}} in the Lie algebra of Ĝ_{ℚ}. We may take X_{∞} in the Lie algebra of T̂. The conjecture at ∞ is that, for λ in L̂,

Re λ(X_{∞}) = 0 .

If σ is a complex analytic representation of Ĝ_{ℚ}, let m(λ) be the multiplicity with which λ occurs in σ. Then

L(s, σ, π) = ∏_{λ} \{π \frac{−s(s + λ(∞))}{2}Γ (\frac{2 + λ(X_{∞})}{2}) ∏_{p} \frac{1}{1 − \frac{λ(t_{p})}{p^{s}}} \}^{m(λ)} .

If the conjecture is true, L(s, σ, π) is analytic to the right of Re s = 1 for all σ.

Let F be any non-archimedean local field and G any quasi-split group over F which splits over an unramified extension field. If f belongs to C_{c}(G_{F} , U_{F}), let f^{∗}(g) = f̄(g^{−1}). If f̂ and f̂^{∗} are the images of f and f^{∗} in Λ^{0}(M), then f̂^{∗}(λ) is the complex conjugate of f̂(−λ). If t belongs to T̂, define t^{∗} by the condition that λ(t^{∗}) = \overline{λ(t^{−1})} for all λ in L̂. The complex conjugate of χ_{t}(t^{∗}) is

∑ f̂(−λ)\overline{λ(t)} = ∑ f̂(λ)λ(t^{∗}) = χ_{t^{∗}}(f) .

If χ_{t} is the homomorphism associated to a unitary representation, then χ_{t}(f^{∗}) is the complex conjugate of χ_{t}(t) for all f so that t × σ_{F} is conjugate to t^{∗} × σ_{F} and for any representation ρ of Ĝ_{F}, the complex conjugate of trace ρ(t × σ_{F}) is trace ρ̃(t × σ_{F}) if ρ̃ is the contragredient of ρ. In the case under consideration, when K = F this means that trace ρ(t_{p}) is the complex conjugate of trace ρ̃(t_{p}). A similar argument can be applied at the infinite prime so show that the eigenvalues of ρ(X_{∞}) are the complex conjugates of the eigenvalues of ρ̃(X_{∞}).

Suppose L(s, σ, π) is analytic to the right of Re s = 1 for all σ. Since the Γ-function has no zeros,

∏_{λ} \{∏_{p} \frac{1}{1 − \frac{λ(t_{p})}{p^{s}}} \}^{m(λ)}  (C)

is also. Let σ be ρ ⊗ ρ̃. Then the logarithm of this Dirichlet series is

∑_{p} ∑_{n=1}^{∞} \frac{1}{n} \frac{trace σ^{n}(t_{p})}{p^{ns}} .

Since

trace σ^{n}(t_{p}) = trace ρ^{n}(t_{p})trace ρ̃^{n}(t_{p}) = |trace ρ^{n}(t_{p})|^{2}

the series for the logarithm has positive coefficients. Thus, the original series does too. By Landau’s theorem, it converges absolutely for Re s > 1 and so does the series for its logarithm. In particular,

det (1 − \frac{σ(t_{p})}{p^{s}})

does not vanish for Re s > 1 so that the eigenvalues of σ(t_{p}) are all less than or equal to p in absolute value. If λ is a weight, choose ρ such that mλ occurs in ρ. Then (mλ)(t_{p}) = λ(t_{p})^{m} is an eigenvalue of ρ(t_{p}) and \overline{λ(t_{p})}^{m} is an eigenvalue of ρ̃, so that |λ(t_{p})|^{2m} is an eigenvalue of σ and

|λ(t_{p})| ≤ p^{\frac{1}{2m}}

for all m and all λ. Thus, |λ(t_{p})| ≤ 1 for all λ. Replacing λ by −λ, we see that |λ(t_{p})| = 1 for all λ. Since the function defined by (C) cannot vanish for Re s > 1 when σ = ρ ⊗ ρ̃, the function

∏_{λ} Γ (\frac{2 + λ(X_{∞})}{2})^{m(λ)}

must be analytic for Re s > 1. This implies that

Re λ(X_{∞}) ≥ −1

if m(λ) > 0. The same argument as before leads to the conclusion that Re λ(X_{∞}) = 0 for all λ.

Granted the generalizations of Ramanujan’s conjecture, one can ask about the asymptotic distribution of the conjugacy classes {t_{p}}. I can make no guesses about the answer. In general, it is not possible to compute the eigenvalues of the Hecke operators in an elementary fashion. Thus, Question 7 cannot be expected to lead by itself to elementary reciprocity laws. However, when the groups G_{F_{𝔭}} at the infinite primes are abelian or compact, these eigenvalues should have an elementary meaning. Thus, Question 7, together with some information on the range of the correspondences of Question 3, may eventually lead to elementary, but extremely complicated, reciprocity laws. At the present it is impossible even to speculate.

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