Lectures for the FERMAT CONFERENCE August 1995 S. Gelbart Three Lectures on the Modularity of ρ̄_{E,3} and the Langlands Reciprocity Conjecture

S. Gelbart

Wiles’ work on Fermat’s Last Theorem is based on methods due to FALTINGS, FREY, LANGLANDS, MAZUR, RIBET, SERRE, TAYLOR, and others. My purpose in these Lectures is to explain how the (automorphic representation theoretic methods and) results of LANGLANDS come into the proof, and how these results themselves are proved. An Introduction to each of the Lectures describes more of the topics discussed; but the titles already speak for themselves:

Lecture I: “The Modularity of ρ̄_{E,3}, and Automorphic Representations of Weight One”

Lecture II: “The Langlands Program: Some Results and Methods” Lecture III: “Proof of the Langlands-Tunnell Theorem”

Acknowledgements

I am grateful to V. Berkovich, E. Lapid, K. Ribet, K. Rubin, Z. Rudnick, and especially R. Livne, for helpful comments and suggestions.

The typing of this manuscript is due to the attentive abilities of Miriam Abraham.

Lecture I The Modularity of ρ̄_{E,3}, and Automorphic Representations of Weight One

The following result plays a small but key step in Wiles’ proof of the Shimura-Taniyama-Weil Conjecture:

Proposition 1.4. For an elliptic curve E over ℚ, let ρ̄_{E,p} : G_{ℚ} ⟶ GL_{2}(𝔽_{p}) denote the natural representation of G_{ℚ} = Gal(ℚ̄/ℚ) on the points of E(ℚ̄) of order p. Then if ρ = 3, and ρ̄_{E,3} is irreducible, it must also follow that ρ̄_{E,3} is modular, i.e., there exists a normalized eigen-cuspform

f(z) = ∑_{n=1}^{∞} a_{n}e^{2πinz}

of weight two, and a prime λ of ℚ̄ containing 3, such that

a_{q} ≡ trace(ρ̄_{E,3}(F r_{q}))(mod λ)

of almost all primes q.

Our main purpose in this Lecture is to explain how this result follows from the following special case of Langlands’ Reciprocity Conjecture for Artin L-functions:

Theorem 1.3 (cf. [La] and [Tu]). Suppose that the continuous representation

σ : G_{ℚ} ⟶ GL_{2}(ℂ)

is “odd”, irreducible, and has solvable image in P GL_{2}(ℂ). (Here odd means that if τ denotes complex conjugation in G_{ℚ}, then det(σ(τ)) = −1.) Then there exists a normalized eigen-cuspform

g(z) = ∑_{n=1}^{∞} b_{n}e^{2πinz}

of weight one such that

b_{q} = trace(σ(F r_{q}))

for all but finitely many primes q.

As we shall see, the proof of this result requires working not only over an arbitrary number field, but also with automorphic cuspidal representations (in place of classical cusp forms). Thus the second half of this Lecture will be devoted to recalling the basic representation theory required to reformulate Theorem 1.3 as follows:

Theorem 2.6. For each irreducible representation

σ : G_{ℚ} ⟶ GL_{2}(ℂ)

which is odd and solvable, there is an automorphic “weight one” cuspidal representation of GL_{2}(𝔸_{ℚ}), call it π(σ), with the property that

trace(t_{π_{q}}) = trace(σ(F r_{q}))

for almost every q. (Here t_{π_{q}} denotes the Langlands class in GL_{2}(ℂ) associated to the unramified local component π_{q} of π(σ) = ⊗π_{p}, and “weight one” means that π_{∞} is the principal series representation of GL_{2}(ℝ) induced from the characters 1 and sgn.)

§1. The Modularity of ρ̄_{E,3} 1.1. Galois Representations mod p Let E denote a fixed elliptic curve defined over ℚ. For a chosen prime

p, let E[p] denote the subgroup of E(ℚ̄) consisting of points of order p. Then E[p] ≅ 𝔽_{p}^{2}, regarded as a two-dimensional vector space over 𝔽_{p}. The natural action of the Galois group

G_{ℚ} = Gal(ℚ̄/ℚ)

on E[p] consequently gives rise to a continuous representation

ρ̄_{E,p} : G_{ℚ} ⟶ GL_{2}(𝔽_{p}) ≈ Aut(E[p]),

which is uniquely defined up to its isomorphic class. That ρ̄_{E,p} encodes much of the arithmetic of E is clear for the two following crucial properties of ρ̄_{E,p}:

(a) Write

ω_{p} : G_{ℚ} ⟶ 𝔽_{p}^{x}

for the character giving the action of G_{ℚ} on the p-th roots of unity μ_{p}. Then

(1.1.1)  det(ρ̄_{E,p}) = ω_{p} ;

this results from the existence of a “Weil pairing” E[p] × E[p] ⟶ μ_{p}, compatible with the action of G_{ℚ} and such that Λ_{𝔽_{p}}^{2}(E[p]) ≈ μ_{p} (cf. §V.2 of [Silv]).

(b) If q is any prime number, and Q is a prime of ℚ̄ dividing q, let F r_{q} denote the canonical Frobenius conjugacy class in D_{Q}/I_{Q} (the quotient of the decomposition group at Q by the inertia group at Q). Then

(1.1.2)  trace ρ̄_{E,p}(F r_{q}) ≡ q + 1 − #(E(𝔽_{g})) (mod p)

for almost all primes q, namely those where ρ̄_{E,p} is trivial on (any) I_{Q} (i.e. those q where ρ̄_{E,p} is unramified).

N.B. (i) The invariants trace ρ̄_{E,p}(F r_{q}) (and also det ρ̄_{E,p}(F r_{q})) are well-defined elements of GL_{2}(𝔽) precisely when ρ̄_{E,p} is unramified at q.

(ii) The identity (1.1.2) essentially amounts to the Riemann hypothesis for elliptic curves over finite fields (proved by Hasse); cf. §V.2 of [Silv]).

(iii) Alternatively, the primes q for which (1.1.2) holds can be characterized as those which are different from p and such that E has “good reduction mod q”. Equivalently, let K be the kernel of ρ̄_{E,p}, and Q̄^{K} =^{:} ℚ(E[p]) the corresponding finite Galois extension of ℚ; then

Gal(ℚ(E[p])/ℚ) ≈ Im ρ̄_{E,p} ,

and (1.1.2) holds exactly for those q which are unramified in ℚ(E[p]) (equivalently, those q such that ρ̄_{E,p} is trivial on I_{q}).

1.2. The Modularity of ρ̄_{E,p}

Let S_{k}(Γ_{0}(N), ε) denote the vector space of modular cusp forms f(z) of weight N ≥ 1 and character ε : (ℤ/Nℤ)^{x} ⟶ ℂ^{x}.

Definition. We call ρ̄_{E,p} modular if there exists some (normalized) eigenform

f(z) = ∑_{n=1}^{∞} a_{n}e^{2πinz} ∈ S_{2}(Γ_{0}(N), ε)

(for some N and ε), and a prime λ of ℚ̄ containing p, such that

a_{q} ≡ q + 1 − #(E(𝔽_{q})) (mod λ)

for almost all primes q.

Recall that Wiles’ goal was to prove that E itself is modular, i.e., for some weight two f(z) as above, the identity (as opposed to congruence)

a_{q} = q + 1 − #(E(𝔽_{q}))

holds for almost all q. As discussed elsewhere, what Wiles actually proves is Mazur’s “Modular Lifting Conjecture”: If p is a prime such that

(i) ρ̄_{E,p} is irreducible, and (ii) ρ̄_{E,p} is modular, THEN E ITSELF IS MODULAR. More precisely, Wiles proves that (a) the Modular Lifting Conjecture is true for p = 3 and 5 when E is a semistable elliptic curve, and (b) the Modular Lifting Conjecture for p = 3 and 5 already implies the Taniyama-Shimura-Weil Conjecture (that E is modular).

Our modest goal is to explain how the theory of automorphic forms is used to prove that for p = 3, the second hypothesis of the Modular Lifting Conjecture automatically follows from the first, i.e., if ρ̄_{E,3} is irreducible, then it is modular.

1.3. The Theorem of Langlands-Tunnell

The crucial ingredient in proving the modularity of ρ̄_{E,3} is the following:

Theorem 1.3. (cf.[La] and [Tu]). Suppose

σ : G_{ℚ} ⟶ GL_{2}(ℂ)

is a continuous, irreducible two dimensional representation whose image in P GL_{2}(ℂ) is a solvable group. Suppose moreover that σ is “odd” in the sense that

det(σ(τ)) = −1

(τ is an automorphism in G_{ℚ} defined by complex conjugation). Then there exists a (normalized)

g(z) = ∑_{n=1}^{∞} b_{n}e^{2πinz} ∈ S_{1}(Γ_{0}(N), ψ)

(for some N and ψ), such that f is an eigenform for all the Hecke operators, and

(1.3.1)  b_{q} = trace(σ(F r_{q})

for almost all primes q.

Remarks

(1) Because any continuous representation

σ : G_{ℚ} ⟶ GL_{2}(ℂ)

factors through some finite Galois group Gal(K/ℚ), its image in GL_{2}(ℂ) is finite, and its image in P GL_{2}(ℂ) is just one of the symmetry groups of a regular polyhedron in ℝ^{3}. (Indeed P GL_{2}(ℂ) is isomorphic to the group of proper rotations in ℝ^{3}, and its finite subgroups are all known; cf. §13 of [Shaf].) From this it is deduced that the image of any irreducible σ in P GL_{2}(ℂ) is either A_{5} (the icosahedral case), S_{4} (the octahedral case), A_{4} (the tetrahedral case), or D_{2n} (the dihedral case). As we shall recall in §5.3, in the dihedral case the existence of the required weight one from g(z) above is essentially due to much earlier work of Hecke and Maass. Hence in dealing with “solvable” σ, the theorem of Langlands and Tunnell is ultimately concerned with “just” the tetrahedral and octahedral possibilities.

(2) The relevant theorems of [La] and [Tu] do not actually produce the required modular form g(z), but rather a certain automorphic representation π(σ). Using the fact that det(σ(τ)) = −1, we shall explain in §4.2 of Lecture II how this automorphic representation produces g(z) itself. In the meantime, we take the above theorem as given, and use it to prove the modularity of ρ̄_{E,3}.

1.4. Proof of the Modularity of ρ̄_{E,3}

More precisely, we need to prove:

Proposition 1.4. If ρ̄_{E,3} is irreducible, then it is modular, i.e., there exists a normalized eigenform

f(z) = ∑_{n=1}^{∞} a_{n}e^{2πinz}

of weight one, and a prime λ of ℚ̄ containing 3, such that

a_{q} ≡ q + 1 − #(E(𝔽_{q})) (mod λ)

for almost all primes q.

The strategy of proof is simple. First one “lifts” ρ̄_{E,3} to a complex representation σ : G_{ℚ} ⟶ GL_{2}(ℂ) to which the Theorem of Langlands-Tunnell is applicable; this produces a modular form g(z) of weight one whose Fourier coefficients b_{q} are almost everywhere equal to trace(F r_{q}). Then one multiplies g by an Eisenstein series of weight one, whose (nontrivial) Fourier coefficients are all congruent to 0 mod 3; this essentially produces the required form of weight two whose Fourier coefficients are congruent to trace(F r_{q}) mod some divisor of 3 (and hence also congruent to q + 1 − #(E(𝔽_{q})), by virtue of (1.1.2)).

Because of the importance of Proposition 1.4, we shall go through its proof carefully (expanding on the single paragraph allotted it in Chapter V of [W1]). We note that the idea of applying “Langlands-Tunnell” in this context goes back to Serre (cf. [Se], §5.3, p.220).

Step 1. Extend ρ̄_{E,3} : G_{ℚ} ⟶ GL_{2}(𝔽_{3}) to a complex representation

σ : G_{ℚ} ⟶ GL_{2}(ℂ)

by composing ρ̄_{E,3} with specific (injective) homomorphism

Ψ : GL_{2}(𝔽_{3}) ↪ GL_{2}(ℤ(\sqrt{−2})) ⊂ GL_{2}(ℂ)

described below.

Following [RuSi], we introduce Ψ directly through the formulas

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

and

Ψ ( \begin{matrix}1 & −1 \\ 1 & 1 \end{matrix} ) = ( \begin{matrix}1 & −1 \\ −\sqrt{−2} & −1 + \sqrt{−2} \end{matrix} ) .

Here α = ( \begin{matrix}−1 & 1 \\ −1 & 0 \end{matrix} ) and β = ( \begin{matrix}1 & −1 \\ 1 & 1 \end{matrix} ) are two convenient gen-

erators of GL_{2}(𝔽_{3}). Once it is checked that the above formulas indeed preserve the required relations, it is immediately seen that the resulting homomorphism

Ψ : GL_{2}(𝔽_{3}) ⟶ GL_{2}(ℤ(\sqrt{−2})) ⊂ GL_{2}(ℂ)

is the identity upon reduction mod(1 + \sqrt{−2}). In particular,

(1.4.1)  tr(Ψ(g)) ≡ tr(g)(mod(1 + \sqrt{−2}))

and

(1.4.2)  det(Ψ(g)) ≡ det(g)(mod 3 = (1 + \sqrt{−2})(1 − \sqrt{−2})) .

N.B. This representation

Ψ : GL_{2}(𝔽_{3}) ⟶ GL_{2}(ℂ)

is really just one of the (three) so-called cuspidal representations of GL_{2}(𝔽_{3}); compare, for example, [PS1] §10.

Step 2. Check that

σ : Ψ ∘ ρ̄_{E,3} : G_{ℚ} ⟶ GL_{2}(ℂ)

is “odd”, irreducible and solvable.

Let us first check that ρ̄_{E,3} itself has odd determinant. On the one hand, (1.1) implies

det(ρ_{E,3}(τ)) = ω_{3}(τ) ,

and it is clear that ω_{3}(τ) = −1. On the other hand, det σ(τ) is apriori ±1, since τ^{2} = 1, and (1.4.2) implies det(σ(τ)) ≡ det ρ̄_{E,3}(τ) (mod 3). So since −1 ≢ 1 (mods), we must have det(σ(τ)) = −1, as required. As for the “solvable” assertion, just recall that

P GL_{2}(𝔽_{3}) ≈ S_{4} ;

this says that the image of σ = Ψ · ρ̄_{E,3} in P GL_{2}(ℂ) is a subgroup of S_{4}, hence itself solvable.

Now what about irreducibility? From the fact that det ρ̄_{E,3}(τ) = −1, it follows that ρ̄_{E,3} has distinct eigenvalues in 𝔽_{3} (namely 1 and −1). We claim this implies ρ̄_{E,3} is absolutely irreducible, i.e., irreducible over \overline{𝔽}_{3} as well as 𝔽_{3}. Indeed, the only matrices in M_{2}(\overline{𝔽}_{3}) which can commute

with ρ̄_{E,3}(G_{ℚ}) (in particular ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) and some non-diagonal matrix

ρ̄_{E,3}(g)) are the scalar matrices λI themselves. Hence by Schur’s Lemma, ρ̄_{E,3} is absolutely irreducible.

Now suppose that the complex representation σ = Ψ · ρ̄_{E,3} is not irreducible. We claim this implies its image in GL_{2}(ℂ) must be abelian. Indeed, any complex representation of a finite (or compact) group is completely reducible. In the case of σ, this means σ is the sum of two characters, and this clearly implies that its image in GL_{2}(ℂ) is abelian.

On the other hand, as ρ̄_{E,3} is absolutely irreducible, the only matrices commuting with its image in GL_{2}(\overline{𝔽}_{3}) must be scalar ones (again by Schur’s Lemma). So pulling back through the embedding Ψ, we conclude ρ̄_{E,3} has both an abelian and irreducible image in GL_{2}(\overline{𝔽}_{3}), an obvious contradiction. Thus σ = Ψ ∘ ρ̄_{E,3} must after all be irreducible.

Step 3. Apply Theorem 1.3 to get a normalized eigenform

g(z) = ∑_{n=1}^{∞} b_{n}e^{2πinz} in some S_{1}(Γ_{0}(N_{1}), ε_{1})

with

(1.4.3)  b_{q} = trace(σ(F r_{q}))

for almost all primes q.

Remark. Recall that for any normalized new form of weight k ≥ 1 (and character ψ), the Fourier coefficients a_{n} (together with the values ψ(n)) lie in the ring of integers O_{K} of some number field K (of finite degree over ℚ). In particular, for our form g(z) above, it makes sense to discuss congruence conditions on the coefficients b_{n} module a prime ideal 𝔭 of (the appropriate) O_{K}.

Now recall that (1.4.1) implies

\begin{aligned}trace(σ(F r_{q})) & = trace(Ψ · σ(F r_{q})) \\ & ≡ trace(ρ̄_{E,3}(F r_{q})) \end{aligned}

modulo (1 + \sqrt{−2}). So by (1.1.2), (1.4.3), and the above Remark, we have

(1.4.4)  b_{q} ≡ q + 1 − #(E(𝔽_{q})) (mod 𝔭)

for almost every q, and 𝔭 some prime of ℚ̄ containing (1+\sqrt{−2}) (and hence 3). In other words, we’ve proven that ρ̄_{E,3} is modular, but with the hitch

that ∑^{∞} ; n=1b_{n}e^{2πinz} is of weight one instead of two! To remedy this, we

follow two ideas, going back respectively to Shimura, and Deligne-Serre.

Step 4. Pick a modular (non-cuspidal) form E of weight 1, such that E ≡ 1 (3); the product

g(z)E(z) = ∑_{n=1}^{∞} c_{n}e^{2πinz}

is of weight 2 (for some level N and character ψ), and

c_{n} ≡ b_{n} (mod 𝔭)

(for 𝔭 a prime of ℚ̄ lying above 3...).

More explicitly, take

E(z) = E_{1,χ}(z) = 1 + 6 ∑_{n=1}^{∞} ∑_{d|n} χ(d)e^{2πinz}

where

χ(d) = \{ \begin{matrix}0 & if d ≡ 0 (mod 3) \\ 1 & if d ≡ 1 (3) \\ −1 & if d ≡ −1 (3) \end{matrix}

is an odd Dirichlet character mod 3 (i.e., χ(−1) = −1). Then E_{1,χ} ∈ M_{1}(Γ_{0}(3), χ), and gE_{1,χ} belongs to S_{2}(Γ_{0}(3N_{1}, ε_{1}χ). Furthermore, each “non-constant” Fourier coefficient of E_{1,χ} is divisible by 3 (in fact 6), so it easily follows that

c_{n} ≡ b_{n}(mod 𝔭) .

Note that g(z)E(z) is the product of a normalized eigenform with an Eisenstein series, but not itself such an eigenform. If it were, we would (by (1.4.4)) have completed our task of proving ρ̄_{E,3} modular. To finish the job, we need the following result of Deligne and Serre:

Lemma (cf. §6.10 of [DS]). Suppose

f_{1}(z) = ∑_{n=1}^{∞} c_{n}e^{2πinz}

is a normalized element of S_{k}(Γ_{0}(N), ψ), and K is a finite extension of ℚ whose ring of integers contains the coefficients c_{n} and ψ(n). Suppose 𝔭 is a prime ideal of O_{K} containing −3, and that f, is a mod 𝔭 eigenform, i.e., there exists b_{n} such that T_{n}f_{1} − b_{n}f_{1} ≡ 0 (mod 𝔭) for all n. Then there exists an f in S_{k}(Γ_{0}(N), ψ), and d_{n}, such that for all n,

T nf = d_{n}f ,

and

d_{n} ≡ c_{n} (mod 𝔭′)

for some prime 𝔭′ dividing 𝔭.

We want to apply this Lemma to our modular form g(z)E_{1}(z) of weight 2. Since the constant term in the Fourier expansion of E_{1}(z) is 1, this gE is indeed normalized, i.e., c_{1} = 1. Let K be its corresponding number field with prime ideal 𝔭 (dividing 3). Since E ≡ 1 (mod 3), we have T_{n}f_{1} ≡ b_{n}f_{1} (mod 𝔭) (since T_{n}g = b_{n}g) for all n. Thus the Lemma is indeed applicable (to g(z)E_{1}(z) in place of f_{1}), and it produces a normalized form f in S_{2}(Γ_{0}(N), ψ) such that T_{p}f = a_{p}f for all p, and a_{p} ≡ c_{p} ≡ b_{p} (mod p). In particular, for almost all q,

a_{q} ≡ q + 1 − #E(𝔽_{q}) (mod 𝔭′) ,

as required.

§2. Automorphic Representations of Weight One

2.1. For σ an irreducible, “odd”, solvable two-dimensional representation of G_{ℚ}, the relevant results of Langlands and Tunnell do not directly produce the required modular form g(z), but rather - as already suggested - a certain automorphic cuspidal representation π(σ), which is shown to correspond to such a form g(z). An honest explanation of how this works requires (at least) the exposition of representation theory given in the pages below. Roughly speaking, a classical eigen-cusp form g, of weight one, and fixed level and character, generates a collection of irreducible representations

{π_{p}}_{p≤∞}

of the “local” groups GL_{2}(ℚ_{p}) - each of which is uniquely determined by the data attached to g(z); moreover this collection comprises what is called an automorphic cuspidal representation π of the adele group GL_{2}(𝔸_{ℚ}).

Once the notion of “automorphic form” is liberated from its classical (upper half-plane) setting, it seems completely natural to take the further step of replacing ℚ by an arbitrary number field F, and GL_{2} by any GL_{n}, n ≥ 1. Thus one ultimately views “Dirichlet characters mod n”, Hecke characters of a number field, classical modular forms, and even “Maass cusp forms” (cf. Remark 2.5.5) as manifestations of one and the same kind of global object, namely, an automorphic representation of GL_{n} over a number field. It is this language which Langlands used to formulate the following general Langlands Reciprocity Conjecture (LRC): for any n-dimensional representation σ of Gal(F̄/F) (or more generally the Weil group W_{F}), there is a corresponding automorphic representation

π(σ) = ⊗_{v}π_{v}

of GL_{n}(𝔸_{F}) such that for almost all the primes v of F,

trace σ(F r_{v}) = trace(t_{π_{v}})

(with t_{π_{v}} the Langlands class of π_{v} in GL_{n}(ℂ)); moreover, if σ is irreducible, then π(σ) is cuspidal.

Our main goal in Lectures II and III will be to describe the ideas behind statement and the proof of this “Strong Artin Conjecture” in case n = 2 and σ solvable, namely:

Theorem 2.1. To each irreducible

σ : W_{F} ⟶ GL_{2}(ℂ)

with solvable image in P GL_{2}(ℂ), there corresponds an automorphic cuspidal representation π(σ) = ⊗π_{v} of GL_{2}(𝔸_{F}) with the property that (its central character equals det σ and)

trace(t_{π_{v}}) = trace σ(F r_{v})

for almost every prime v of F.

Our more modest goal in the rest of this Lecture is to explain how Theorem 2.1 gives the required classical result (Theorem 1.3) when F = ℚ, σ factors through the Galois group G_{ℚ}, and det(σ) is odd. One step is to show that for such σ (as just underlined), the corresponding π(σ)’s are “automorphic representations of GL_{2}(𝔸_{ℚ}) of weight one”. This step will actually be postponed until we discuss the correspondence σ ⟶ π(σ) in earnest in Lecture II; cf. Prop. 4.2. A second step is to show there is a one-one correspondence between these “automorphic representations of weight one” and the classical new forms of weight one required in Theorem 1.3 (cf. Prop. 2.5).

2.2. Archimedean Representation Theory (GL_{2})

Let G = G_{∞} denote GL_{2}(ℝ), and 𝔤 its complexified Lie algebra. Let K = K_{∞} = O(2, ℝ) denote the real 2 × 2 orthogonal group, a maximal compact subgroup of G.

Definition. An admissible representation of G on a Hilbert space H is a homomorphism

π : G ⟶ GL(H)

such that (i) the map (g, v) ⟶ π(g)v from G × H ⟶ H is continuous, and (ii) (“admissibility”) the restriction of π to K contains each irreducible unitary representation of K with finite multiplicity (recall that each such representation is automatically finite dimensional).

Remark. For an admissible representation π : G ⟶ Aut(H), let V = V_{K} ⊂ H denote the subspace of K-finite vectors, i.e., those v in H whose translates under π(K) span a finite-dimensional space. Such vectors are not preserved by the action of π(G), but they are by the corresponding (differentiated) action of the Lie algebra 𝔤 of G. In fact, as a representation space jointly for the action of 𝔤 and K, V enjoys certain “compatibility” properties which ensure that it is (what’s called) a (𝔤, K) − module. The advantage of these modules is that they are “algebraic” linear objects as opposed to “analytic” Lie group theoretic objects, and yet they accurately reflect the nature and properties of π. For example, π is irreducible in the usual sense (that H has no Hilbert space subspaces invariant under π(G)) if and only if V is irreducible in the algebraic sense that it has no vector space subspaces invariant under both π(K) and dπ(𝔤)). This leads to an equivalence between the natural categories of irreducible admissible representations of G and irreducible (𝔤, K) modules. In particular, whenever convenient, we allow ourselves to confuse π with the (𝔤, K) module V.

Example 2.2. Let μ_{1} and μ_{2} denote two characters of ℝ^{x} (i.e., not necessarily “unitary” homomorphisms from ℝ^{x} to ℂ^{x}). Let H = H(μ_{1}, μ_{2}) denote the Hilbert space of functions f : G ⟶ ℂ such that

f (( \begin{matrix}a_{1} & x \\ 0 & a_{2} \end{matrix} ) g) = \frac{a_{1}}{a_{2}} ^{1/2}μ_{1}(a_{1})μ_{2}(a_{2})f(g)  (8)

for all ( \begin{matrix}a_{1} & x \\ 0 & a_{2} \end{matrix} ) in the Borel subgroup B consisting of upper triangular

matrices of G, and such that

∥f∥^{2} = ∫_{K} |f(k)|^{2}dk < ∞ .

(Strictly speaking, one first looks at continuous such f, and then defines H(μ_{1}, μ_{2}) to be the closure of such functions with respect to the norm ∥f∥...). Then the right regular action of G on functions f in H(μ_{1}, μ_{2}) defines an admissible representation of G which is denoted π(μ_{1}, μ_{2}), and called the representation of G induced from the character μ_{1}μ_{2} (of B). The corresponding (𝔤, K) module V_{K} consists of finite linear combinations of smooth functions ϕ_{k} defined by

ϕ_{k}(g) = μ_{1}(a_{1})μ_{2}(a_{2})e^{−ikθ}

if g has “Iwasawa decomposition”

g = ( \begin{matrix}a_{1} & x \\ 0 & a_{2} \end{matrix} ) r(θ) .

(Here r(θ) denotes the rotation element ( \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} ) in K, and

there is an obvious compatibility condition coming from the elements ±I in B ∩K, namely μ_{1}μ_{2}(−1) = e^{−ikπ}.) From this picture it is a simple matter to prove that π(μ_{1}, μ_{2}) is irreducible if and only if μ_{1}μ_{2}^{−1}(x) ≠ x^{p}sgn(x) with p a non-zero integer. Slightly less transparent is the proof of the following:

Fact. Every irreducible admissible representation π of G is (equivalent to) a π(μ_{1}, μ_{2}), or an irreducible subrepresentation thereof. For example, suppose μ_{1}μ_{2}^{−1}(x) = x^{p}sgn(x) with p a positive integer. Then H(μ_{1}, μ_{2}) contains exactly one invariant subspace, namely

{...ϕ_{−p−3}, ϕ_{−p−1}, ϕ_{p+1}, ϕ_{p+3} ⋯ } ,

and the restriction of π(μ_{1}, μ_{2}) to this subspace realizes an irreducible discrete series representation of “lowest weight p + 1”.

Concluding Remarks.

(a) All the above notions, and most of the results, hold for an arbitrary semisimple or reductive Lie group; in particular, every irreducible admissible representation of such a group is still realizable as a subrepresentation of the analogous induced representation. (The theory for GL_{2}(ℝ) is essentially Bargmann’s, and the general theory essentially due to Harish-Chandra; for many more details, see the recent survey paper of Knapp’s [Kn].)

(b) For any irreducible admissible representation π, an analogue of Schur’s Lemma implies that there is a character of ℝ^{x}, denoted ω_{π} and called the central character of π, such that

π ( \begin{matrix}r & 0 \\ 0 & r \end{matrix} ) = ω_{π}(r)I for all r ∈ ℝ^{x} .

2.3. P -adic Representation Theory (GL_{2})

In this Section, F is a p-adic field with a ring of integers O_{F}, and G = GL_{2}(F). As in the real case, most of the notions and facts we review below extend not only to GL_{n}, but to an arbitrary reductive p-adic group; however, we recall here only those facts (even for GL_{2}) which are really needed in the sequel.

Definition 2.3.1. An admissible representation of G on a vector space V is a homomorphism

π : G ⟶ GL(V)

such that (i) the stabilizer in G of any v in V is open, and (ii) (“admissibility”) for any compact open subgroup K^{0} ⊂ G, the space {v ∈ V : π(k)v = v ∀ k ∈ K^{0}} is finite-dimensional.

Remark. Suppose π is an irreducible unitary representation of G in some Hilbert space H (same definition as in case of Lie groups), and V_{K} is its subspace of K-finite vectors (for any compact open K, say K = GL_{2}(O_{F})). Then (by a Theorem of J. Bernstein) the restriction of π(G) to V = V_{K} produces an admissible representation of G in the above sense. Thus the p-adic notion of an admissible representation (on a vector space V) is a natural analogue of the archimedean notion of a (𝔤, K) module. What’s special in the p-adic case is that G itself acts on V_{K} (and there is no need to go to the Lie algebra...).

Example (of π(μ_{1}, μ_{2})). For each pair of characters μ_{i} : F^{x} ⟶ ℂ^{x}, the induced representations π(μ_{1}, μ_{2}) in H(μ_{1}, μ_{2}) are defined just as in the archimedean case. But now reducibility is possible only if μ_{1}μ_{2}^{−1}(x) = |x|^{±1}, i.e., there is no room for many “discrete series” representations to appear as subrepresentations of π(μ_{1}, μ_{2}). This reflects the fact that there are now representations which are absolutely cuspidal, i.e., they cannot be constructed in this simple way.)

Fortunately, we shall not need to discuss the cuspidal representations in these Lectures, but rather only those representations which are as far from being cuspidal as possible!

Definition 2.3.2. An irreducible admissible representation π of G is unramified (or of conductor zero) if its space of K = GL_{2}(O_{F}) fixed vectors is non-empty.

In this case, it is known that the space of K-fixed vectors is onedimensional, and that π is either one-dimensional (of the form χ ∘ det, for some unramified character χ of F^{x}), or an irreducible π(μ_{1}, μ_{2}) with μ_{1} and μ_{2} unramified characters of F^{x}. As we shall recall in the next paragraph, it is these latter representations which typically play a crucial role in the adelic theory of automorphic representations.

Concluding Remark. Even if some π does not have “conductor zero”, there will still exist a smallest (positive) integer N (called the conductor of π) such that the space

\{v ∈ V : π ( \begin{matrix}a & b \\ c & d \end{matrix} ) v = ω_{π}(a)v ∀ k ∈ K^{N}\}

is non-empty. (Here K^{N} denotes the Hecke congruence subgroup of K

consisting of ( \begin{matrix}a & b \\ c & d \end{matrix} ) with c ≡ 0 (mod 𝔭^{N}).) This fact was first

proved in [Cas], and then generalized to GL_{n} in [J-PS-S1]; if N=conductor (π), then it is also known (as in the case N = 0) that the space V^{K^{N}} is automatically one-dimensional.

2.4. Adelic Representations (GL_{2})

This is mostly a matter of putting together the local representation theory.

Suppose we are given a collection of irreducible admissible representations {π_{p}} of the local groups G_{v} = GL_{2}(ℚ_{p}), such that π_{p} is unramified for all but finitely many p. Then the restricted tensor product

⨂_{p≤∞} π_{p}

makes sense as a representation of the restricted product GL_{2}(𝔸_{ℚ}) = \begin{matrix}∏ & ′ \\ p≤∞ \end{matrix}GL_{2}(ℚ_{p}), and defines an irreducible “admissible” representation π in

a sense that can be made precise; cf. §4.c of [Ge] or §9 of [JL] for more details. Conversely, any irreducible “admissible” (and in particular any irreducible unitary) representation π of GL_{2}(𝔸_{ℚ}) is uniquely factorizable as

π = ⊗π_{p} ,

with almost every π_{p} unramified. Moreover, the obvious analogues of these statements hold for an arbitrary number field F.

Example: G = GL_{1}. In this case, we consider a collection of characters χ_{v} of F_{v}^{x}, one for each prime of F, such that almost all the χ_{v}’s are unramified, i.e., trivial on O_{v}^{x}. Then

χ = ∏_{v} χ_{v}

defines a nice (continuous) character of the ideles 𝔸_{F}^{x} = GL_{1}(𝔸_{F}), and every such character thus arises (cf. Prop. 7-1-12 of [Gold]). Of course, in number theory one is primarily concerned with characters χ of 𝔸_{F}^{x} which are trivial on F^{x}, i.e., with “grossencharacters”, or characters of the idele class group

GL_{1}(F)⧹GL_{1}(𝔸^{x}) = F_{x}⧹𝔸_{F}^{x} .

These are the “automorphic representations” of GL(1) over F.

Special Example. Since Q has class number one,

(∗)  𝔸_{ℚ}^{x} = ℚ^{x} · ℝ^{+} · ∏_{p<∞} ℤ_{p}^{x} .

Thus a Dirichlet character ψ : (ℤ/Nℤ)^{x} ⟶ ℂ^{x} determines a grossencharacter χ_{ψ} as follows: for any p < ∞, ψ can be pulled back to a character χ_{p} of ℤ_{p}^{x} through the canonical homomorphism from ℤ_{p}^{x} to (ℤ/Nℤ)^{x}; the product ∏_{p<∞}χ_{p} then defines a character of ∏_{p<∞}ℤ_{p}^{x}, and hence by (∗) a

character χ_{ψ} of 𝔸_{ℚ}^{x} trivial on ℝ^{+} as well as ℚ^{x}. In this way one obtains a grossencharacter of 𝔸_{ℚ}^{x} of finite order, and all such grossencharacters arise in this way for suitably large N. (Note that χ_{p} is unramified for the primes p not dividing N.)

Following this lead for GL_{1}, it is clear that one should define automorphic representations for G = GL_{n} in terms of the quotient space

GL_{n}(F) \ GL_{n}(𝔸) .

For simplicity, we give a precise definition only for G = GL_{2} and cuspidal automorphic representations.

Definition. Fix a grossencharacter ω of F, and let L_{0}^{2}(G(F) \ G(𝔸), ω) denote the (closure of the) space of all continuous φ : G(𝔸) ⟶ ℂ such that

(i) φ(γgz) = ω(z)φ(g) for all γ ∈ G(F) and z ∈ Z(𝔸) = \{( \begin{matrix}z & 0 \\ 0 & z \end{matrix} )\}

(the center...);

(ii) ∫_{Z(𝔸)G(F)⧹G(𝔸)} |φ(g)|^{2}dg < ∞ ;  and

(iii) φ is cuspidal, i.e., for any g in G(𝔸),

∫_{F⧹𝔸} φ (( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) g) dx = 0 .

Then an irreducible admissible (necessarily unitary) representation π = ⊗π_{v} of G(𝔸) is called automorphic cuspidal if it is equivalent to an irreducible summand of the right regular representation of G(𝔸) in L_{0}^{2} (G(F)\ G(𝔸), ω) for some ω.

Remarks. (i) In a completely similar way, the notion of an automorphic cuspidal representation π can be defined for G = GL_{n}, and more generally, for an arbitrary reductive algebraic group G. In this generality, it is a Theorem of Gelfand and Piatetski-Shapiro (cf. [GGPS]) that the right regular representation R_{0} of G(𝔸) in L_{0}^{2}(G(F) \ G(𝔸), ω) decomposes discretely, and with finite multiplicities, i.e.,

R_{0} = ⨁ m_{π}π, with m_{π} < ∞ .

For G = GL_{n}, it is actually known that this multiplicity is one (cf. [JL], [GK], and [Shal]), but for other groups this need no longer hold. For general G, it is at least still true that any automorphic cuspidal representation has a factorization of the type

π = ⊗π_{v} discussed above, with π_{v} almost everywhere an unramified representation of G(F_{v}).

(ii) There is also the notion of an automorphic (not necessarily cuspidal) representation of G(𝔸), but as we shall not focus on these in the sequel, we refrain from giving a precise definition. Suffice it to say that such π are the irreducible admissible representations of G(𝔸) which are built out cuspidal representations of the Levi components of G by way of induction (and taking of quotients); for example, for GL_{2}, the automorphic (noncuspidal) representations are the quotients of the induced representations π(μ_{1}, μ_{2}) with μ_{1} and μ_{2} grossencharacters of F (viewed as “cuspidal representations” of the diagonal subgroup of GL_{2}). For a thorough discussion, see [BoJa] and [La2].

2.5. A Dictionary (Between the Classical and Modern Theories for GL_{2})

In the last paragraph, we recalled how classical Dirichlet characters correspond to certain automorphic representations of GL(1), namely those grossencharacters which are of finite order. We now describe an analogous result for GL(2).

Proposition. There is a 1-1 correspondence between normalized new forms

f(z) = ∑_{n=1}^{∞} a_{n}e^{2πinz}

in S_{k} (Γ_{0}(N), ψ), and irreducible automorphic cuspidal representations

π = ⊗_{p}π_{p}

of G(𝔸) such that:

(a) the central character of π is χ_{ψ} ;

(b) π_{p} is unramified for all p ∤ N, and

(c) π_{∞} has “lowest weight k”. Moreover, if N = ∏ p_{i}^{α_{i}}, and p|N, then π_{p} has conductor α_{i}; but if p ∤ N, then π_{p} = π(μ_{1}, μ_{2}), with μ_{1}, μ_{2} unramified characters of ℚ_{p}^{x} such that

(2.5.1)  p^{\frac{k−1}{2}} (μ · (p) + μ_{2}(p)) = a_{p} .

Remark 2.5.2. In case k > 1, π_{∞} is a discrete series representation of “lowest weight k”, sitting inside some π_{∞}(μ_{1}, μ_{2}) (cf. Fact 2.2); this discrete series representation has central character trivial on ℝ^{+}, and is denoted D_{k}. However, if k = 1, then π_{∞} is not a subrepresentation of any π_{∞}(μ_{1}, μ_{2}), but rather equals the full induced (“principal series”) representation π_{∞} (1, sgn) (see the proof below). In this case, the corresponding representation π = ⊗π_{p} (with π_{∞} = π_{∞} (1, sgn)) is called an automorphic cuspidal representation of weight one.

Corollary of the Proposition. A normalized new form in S_{1}(Γ_{0}(N), ψ), with eigenvalues {a_{p}}, is one and the same thing as an automorphic cuspidal representation ⊗π_{p} of weight one such that (cf. (2.5.1))

a_{p} = μ_{1}(p) + μ_{2}(p), for all p ∤ N .

N.B. Here μ_{1} and μ_{2} are the two unramified characters of ℚ_{p}^{x} inducing the unramified representation π_{p} = π_{p}(μ_{1}, μ_{2}) of GL_{2}(ℚ_{p}). From now on, we rewrite the above relation in the more suggestive form

(2.5.3)  a_{p} = trace (t_{π_{p}})

with

t_{π_{p}} = ( \begin{matrix}μ_{1}(p) & 0 \\ 0 & μ_{2}(p) \end{matrix} ) in GL_{2}(ℂ)

the so-called Langlands class of π_{p}.

Sketch of the Proof (of the Proposition)

The first step is at the level of functions. Using the decomposition

G(𝔸_{ℚ}) = GL_{2}(ℚ)GL_{2}^{∗}(ℝ) ∏_{p<∞} K_{p}^{N} ,

(analogous to (∗) for GL_{1}), one defines for any f in S_{k}(Γ_{0}(N), ψ) a function φ_{f} on G(𝔸_{ℚ}) by

(2.5.4)  φ_{f}(γg_{∞}k_{0}) = f(g_{∞}(i))j(g_{∞}, i)^{−k}χ_{ψ}(k_{0}) ;

here K_{p}^{N} = \{\binom{a b}{c a} ∈ K_{p} = GL_{2}(ℤ_{p}) : c ≡ σ(N)\}, and j(g_{∞}, z) = (cz +

d)(det g_{∞})^{− \frac{1}{2}} if g_{∞} = \binom{a b}{c a}. It is now a standard matter (cf. [Cas] and [De]) to check that this map f ⟶ φ_{f} is an isomorphism from S_{k}(Γ_{0}(N), ψ) to the space of smooth functions {φ} on GL_{2}(𝔸_{ℚ}) such that

(i) φ(γg) = φ(g) ∀ γ ∈ GL_{2}(ℚ) ; (ii) φ(gk) = (χ_{ψ})(k)φ ∀ k ∈ K_{p}^{N} ;

(iii) φ(gr(θ)) = e^{−ikθ}φ(g) ∀ r(θ) = \binom{cos θ −sin θ}{sin θ cos θ} ∈ G_{∞} ; (iv) φ(zg) = χ_{ψ}(z)φ(g) ∀ z ∈ Z(𝔸) ;

(v) If X denotes the “pushing down” differential operator corresponding to the element \binom{i 1}{1 −i} of 𝔤, then

X · φ = 0

(this is the condition reflecting the holomorphy of f); (vi) φ is of “moderate growth” on G(𝔸) (relecting the holomorphy of f “at the cusps”); and (vii) φ is cuspidal, i.e.,

∫_{𝔽⧹𝔸}φ (\binom{1 x}{0 1}) dx ≡ 0

(reflecting the cuspidality of f...).

Conditions (i) - (vii) imply that φ_{f} belongs to L_{0}^{2}(G(ℚ)⧹G(𝔸), χ_{ψ}). The second step of the proof is then to show that φ_{f} generates (under right translation by G(𝔸)) an irreducibly invariant subspace π_{f} (of L_{0}^{2}(χ_{ψ})) whose corresponding representation π_{f} = ⊗π_{p} is as claimed; one must also show that all such representations are thus obtained. These arguments are explained in §5 of [Ge], but only really for the case k > 1, where π_{∞} is the discrete series representation D_{k}. In case k = 1, one must argue as follows (in order to identify π_{∞}).

Suppose μ_{i}(x) = |x|^{s_{i}}sgn(x)^{ε_{i}}, and π_{∞} = π(μ_{1}, μ_{2}). Then H(μ_{1}, μ_{2}) consists only of functions ϕ_{k} with k of the same parity as ε_{1} + ε_{2}. (In particular, π(1, sgn) consists only of “odd” functions...). A straightforward computation (á la Bargmann...) also shows that

X · ϕ_{k} = (\frac{s_{1} − s_{2} + 1}{2} − \frac{k}{2} ) ϕ_{k}

and

ϕ_{k} (g \binom{r 0}{0 r}) = r^{s_{1}+s_{2}}ϕ_{k}(g), for r > 0 .

This means that if ω_{π_{∞}} is to be trivial on ℝ^{+}, we must have s_{1} = −s_{2} = s, and if X · ϕ_{1} is to be 0, we must have s_{1} = s_{2} = s = 0, i.e., π_{∞} = π(1, sgn) as claimed.

Concerning the converse direction, we recall the following. Suppose π = ⊗π_{p} is an irreducible subrepresentation of L_{0}^{2}(G(ℚ \ G(𝔸_{ℚ}), χ_{ψ}) (for some grossencharacter χ_{ψ} of finite order), and π_{∞} is “of weight k” (k ≥ 1). If the conductor c(π) = ∏_{p<∞}c(π_{p}) of π is N, let φ_{π} denote the function in

the space H_{π} of π which is right K_{p}^{N} invariant for all p. (This function is uniquely determined up to a scalar; cf. the Concluding Remark of §2.3.) Then via the correspondence φ_{π} ⟶ f_{π} (inverse to (2.5.4)), we obtain from π the required new form of weight k.

Remark 2.5.5 The fact that the principal series π(1, sgn) leads to a holomorphic form f(z) (of weight one...) relies on the critical confluence of conditions

Xϕ = 0

and

ϕ(gr(θ)) = e^{−iθ}ϕ(g) .

Indeed, these two conditions together imply that the function f defined by

f(z) = ϕ(g_{∞}(i))j(g_{∞}, i)

will be holomorphic on the upper half-plane 𝔥. The delicacy of this point can be appreciated by examining what would happen if we took π_{∞} to be the principal representation π(1, 1) (or π(sgn, sgn)) in place of π(1, sgn). In this case, only ϕ_{k}’s of even parity occur in H(μ_{1}, μ_{2}), X ·ϕ_{k} = (\frac{1−k}{2} ) ϕ_{k} is never zero, and no holomorphic f’s arise in 𝔥. The crucial point now is that ϕ_{0} will be K_{∞} − invariant, and hence directly define a function f(z) on 𝔥, with the property that

Δf = −y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) f = 1/4f .

Indeed, let D denote the standard Casimir operator in the center of the universal enveloping algebra of 𝔤, which for K_{∞}-invariant functions corresponds exactly to the Laplace-Beltrami operator Δ above. Then the action of D in H(μ_{1}, μ_{2}) (with μ_{2}(x) = |x|^{s_{i}}sgn(x)^{ε_{i}}) is given by the formula

D · ϕ_{k} = \frac{(s_{1} − s_{2})^{2} − 1}{4} ϕ_{k}

(cf. Lemma 5.6 of [JL], p.166, keeping in mind that our Casimir is 1/2 theirs). Thus the same reasoning as used above (to show that an automorphic cuspidal representation ⊗π_{p} with π_{∞} = π(1, sgn) corresponds to a classical cusp form of weight 1) shows also that a cuspidal representation ⊗π_{p} of GL_{2}(𝔸_{ℚ}) with π_{∞} = π(1, 1) (or π(sgn, sgn)) corresponds to a Maass cusp form of “eigenvalue 1/4”.

2.6. Reformulation of Theorem 1.3 Suppose

σ : G_{ℚ} ⟶ GL_{2}(ℂ)

is a continuous, irreducible, two dimensional “odd” representation whose image in P GL_{2}(ℂ) is solvable. Then there exists an automorphic cuspidal representation π(σ) = ⊗_{p}π_{p} of GL_{2}(𝔸_{ℚ}) which is of weight one, central character det σ, and such that for almost all p, π_{p} = π(μ_{1}, μ_{2}) is unramified with

(2.6.1)  trace σ(F r_{p}) = trace(t_{π_{p}}) = μ_{1}(p) + μ_{2}(p) .

Remarks. (2.6.2). Recall that the matrix

( \begin{matrix}μ_{1}(p) & 0 \\ 0 & μ_{2}(p) \end{matrix} ) = t_{π_{p}}

is the Langlands class in GL_{2}(ℂ) attached to the unramified representation π_{p}. (2.6.3). According to Corollary 2.5, the existence of an automorphic cuspidal π = ⊗π_{p} as above implies the existence of a new form f = ∑ a_{n}e^{2πinz} some S_{1}(Γ_{0}(N), ψ) with

a_{p} = trace σ(F r_{p})

for almost all p. Thus this representation theoretic reformulation of Theorem 1.3 indeed implies Theorem 1.3. (2.6.4). In the next few lectures we shall explain how the more general Theorem 2.1 is proved; this will imply Theorem 2.6, in case F = ℚ, σ factors through G_{ℚ}, and det σ is odd, for it is a simple matter to see that π(σ) is then actually of weight one, i.e., π_{∞} = π(1, sgn) (cf. Prop. 4.2 of Lecture II). (2.6.5). “Strong Multiplicity One” for GL(2) asserts that two automorphic cuspidal representations π and π′ are equivalent as soon as they are equivalent almost everywhere, i.e.,

π_{p} ≅ π′_{p} for almost all p .

This fact is explicitly used, together with multiplicity one for GL(2), in the proof that a new form f generates an irreducible subspace π_{f} of L_{0}^{2}. In the statement of Theorem 2.6 (or 2.1), it also implies that π(σ) (if it exists at all) is unique. Indeed, once the central character is fixed, condition (2.6.1), which holds almost everywhere, uniquely determines π_{p}. Similarly, in the classical version (Theorem 1.3) of Langlands-Tunnell, the new form g(z) is uniquely determined by the condition (1.3.1) which fixes its eigenvalues almost everywhere; this reflects the fact that the theory of new forms is one and the same thing as the strong multiplicity one result coupled with the notion of conductors! (See [Cas] or [Ge] for a further explanation of this point.)

Lecture II The Langlands Program: Some Results and Methods

We start this Lecture by describing the Local Langlands Conjecture (LLC) for GL(n) over a local field F. In case n is a prime, this is a Theorem over any field F, known as the “Local Langlands Correspondence”. Thus we can (and will) describe the resulting correspondence in some detail for n = 2, and apply it to refine (and generalize) the two-dimensional Langlands Reciprocity Conjecture (LRC) as follows:

To each continuous irreducible representation

σ : W_{F} ⟶ GL_{2}(ℂ)

of the Weil group of a number field F, there is associated an automorphic cuspidal representation π(σ) = ⊗π_{v} of GL_{2}(𝔸_{F}) with the property that

π_{v} ⟷ σ_{v}

for every place v. (Here σ_{v}, a two-dimensional representation of the local Weil group W_{F_{v}}, is the “Langlands parameter” of the corresponding representation π_{v} of GL_{2}(F_{v}); for almost every v, σ_{v} and π_{v} are unramified, and the relation σ_{v} ⟷ π_{v} reduces to the more familiar relation σ_{v}(F r_{v}) ∼ t_{π_{v}} in GL_{2}(ℂ).)

As we shall see, the “classical version” of the Langlands-Tunnell Theorem (Theorem 1.3 of 2.6) follows immediately from the proof of the general Reciprocity Conjecture in the solvable case; indeed, when F = ℚ, and σ factors through G_{ℚ} and is “odd”, π(σ) must be automorphic of weight one, i.e. π(σ_{∞}) = π_{∞}(1, sgn) (cf. Prop. 4.2).

In the second half of this Lecture, we also begin to collect the automorphic results required for proving the global LRC in the two-dimensional solvable case. As we shall see, all these results, as well as the LRC itself, are but special realizations of a “Principle of Functoriality with respect to the L-group”, namely:

Langlands’ Functoriality Conjecture Given two reductive F -groups G and G′ (with G′ quasi-split), and a

morphism

ρ : ^{L}G ⟶ ^{L}G′

between their L-groups, there is a corresponding mapping of automorphic representations

π ⟶ π(ρ) = ⊗π′_{v}

from G to G′, such that for almost every v, ρ takes the Langlands class t_{π_{v}} in ^{L}G to the Langlands class t_{π′_{v}} = t_{π_{v}(ρ)} in ^{L}G′.

§3. The Local Langlands Correspondence for GL(2)

3.1. The Archimedean Case

We assume F = ℝ. (For the simpler case of F = ℂ, which we do not need, see Remark 3.1.2 below). In this case, the Weil Group W_{F} is an extension of ℂ^{x} by ℤ/2ℤ given by

W_{ℝ} = ℂ^{x} ∪ jℂ^{x}

where j^{2} = −1, and jcj^{−1} = c̄, and the natural surjection

φ : W_{ℝ} ⟶ Gal(ℂ/ℝ)

is given by φ(ℂ^{x}) = 1 and φ(jℂ^{x}) = τ (complex conjugation). We are interested in the set of equivalence classes Φ(GL_{n}/ℝ) of n-dimensional complex representations σ of W_{ℝ} whose images consist of semisimple elements in GL_{n}(ℂ).

Example 3.1.1. The one-dimensional representations of W_{ℝ} are of the form μ ∼ (t, ε), taking z in ℂ^{x} to |z|_{ℂ}^{t}, t ∈ ℂ, and j to ε = ±1. (Indeed, if μ(j) = w, then on ℂ^{x}, μ(z̄) = μ(jzj^{−1}) = wμ(z)w^{−1} = μ(z) = z^{t}z̄^{s} with t = s, i.e., μ(z) = r^{2t} = |z|^{t}; also μ(−1) = 1 = μ(j^{2}) = w^{2} ⟹ w = ±1). On the other hand, the two-dimensional irreducible representations of W_{ℝ} are all induced from some character

z ⟶ |z|_{ℂ}^{t} (\frac{z}{|z|})^{m}

of ℂ^{x}, with t arbitrary in ℂ, and m ≥ 1 an integer. Clearly these representations are “semisimple”. It is also easy to show that every n-dimensional semisimple representation σ of W_{ℝ} is a direct sum of these one and twodimensional irreducible representations.

Theorem. The Local Langlands Correspondence for GL_{n}(ℝ). There is a well defined bijection

σ ⟺ π(σ)

between Φ(GL_{n}/ℝ), the set of classes of n-dimensional semisimple complex representations σ of W_{ℝ}, and ∏(GL_{n}/ℝ), the set of classes of irreducible admissible representations π of GL_{n}(ℝ); moreover, the L and ε factors assigned to σ and π are preserved by this correspondence.

Remarks. The existence of this correspondence, formulated and proved more generally by Langlands for an arbitrary reductive Lie group, is the subject matter of [La3]; the fact that L and ε factors may be defined for σ and π in the context of GL_{n}, and then preserved by this correspondence, is discussed in [Ja].

Example of GL_{2}(ℝ)

Suppose first that σ is the sum of two one-dimensional representations (i.e., characters) μ_{i} ∼ (t_{1}, ε_{1}) as in Example (3.1.1). Then π(σ) is taken to be the unique irreducible quotient of the (induced representation) π(μ_{1}, μ_{2}), where μ_{i}(x) = |x|_{ℝ}^{t_{1}}(sgn(x))^{ε_{1}}, and the order of t_{1}, t_{2} is arranged so that Re(t_{1}) ≥ Re(t_{2}). For example, if σ = (\frac{1}{2}, 0) ⊕ (−\frac{1}{2}, 0), then π(σ) is the trivial representation, whereas if σ = (0, 0)⊕(0, 1) (resp. (0, 0)⊕(0, 0)) then π(σ) is the irreducible principle series representation π(1, sgn) (of “lowest weight 1”) (resp. the class 1 principle series representation π(1, 1), with Casimir eigenvalue λ = −\frac{1}{4}). On the other hand, suppose now that σ is the irreducible two-dimensional representation of Ex. 3.1.1, with parameters t and m ≥ 1. Then π(σ) is taken to be the discrete series representation D_{m+1} ⊗ |det( )|_{ℝ}^{t}, with D_{m+1} of lowest weight m + 1 and trivial central character.

Remark 3.1.2. In case F = ℂ, the Weil group is just ℂ^{x} and each n dimensional semisimple representation σ is just a sum of characters μ_{i} of the form (\frac{z}{z̄})^{m_{i}} |z|_{ℂ}^{t_{i}} with m_{i} ∈ ℤ. In this case, there are no discrete series, and to each σ as above, the corresponding π(σ) is just the unique irreducible quotient of Ind μ_{1}μ_{2} ⋯ μ_{n}, with the μ_{i}’s so arranged that Re(t_{1}) ≥ ⋯ ≥ Re(t_{n}).

3.2. The p-adic case

In case F is a p-adic field, its Weil group W_{F} is a dense subgroup of Gal(F̄/F), equipped with an isomorphism F^{x}⟵^{∼}W_{F}^{ab}. In particular, the one-dimensional (complex) representations of W_{F} are again identified with the irreducible admissible representations (i.e. characters) of F^{x} = GL(1, F), just as in the archimedean case. However, unlike in the archimedean case, there are now irreducible representations of W_{F} of arbitrary dimension (reflecting the existence of extensions of F of arbitrary degree...). This fact considerably complicates the representation theory - and concommitent local Langlands correspondence - for GL(n). Fortunately, for our purposes, we don’t need to describe the full Langlands correspondence; instead, we need only the following:

Theorem 3.2. For each two-dimensional “semisimple” representation σ of W_{F}, there is exactly one irreducible admissible representation π of GL_{2}(F) with

(3.2.1)  ω_{π}(α) =^{:} π \binom{α 0}{0 α} = det σ(α)I

and such that for all characters χ of F^{x}, and ψ_{F} of F,

L(s, π ⊗ χ) = L(s, σ ⊗ χ)
L(s, π̃ ⊗ χ^{−1}) = L(s, σ̃ ⊗ χ^{−1})
ε(s, π ⊗ χ, ψ_{F}) = ε(s, σ ⊗ χ, ψ_{F}) .

Moreover, all irreducible admissible π thus arise. (Here the L and ε factors on the left-hand side are those of Jacquet-Langlands, and those on the right-hand side the local factors of [La4]; “∼” denotes the contragredient representation.)

Remark 3.2.2. The existence parts and exhaustion of the Theorem are easy, except for the case of irreducible σ (which is due - for arbitrary F - to Kutzko [Ku]). The uniqueness part is Corollary 2.19 of [JL], and the resulting bijection

σ ⟷ π(σ)

amounts to the Langlands correspondence for GL(2).

Caution: Missing in the image of the map

σ ⟶ π(σ)

just described are the “special representations” of GL_{2}(F). Although they can be obtained by considering representations of the Weil-Deligne group W′_{F} in place of W_{F} (see, for example, [Ta] or [Kud]), we prefer to ignore these representations as they play no crucial role in the sequel. In fact, for the global applications we have in mind to the Reciprocity Conjecture, it is crucial to make explicit only the following unramified part of the correspondence.

Example 3.2.3. Recall that if k denotes the residue field of F, then W_{F} consists of those elements of Gal(F̄/F) whose image in Gal(k̄/k) is an integer power of the Frobenius automorphism generator of Gal(k̄/k). Thus the inertia subgroup I of Gal(F̄/F) is contained in W_{F}, and a representation σ : W_{F} ⟶ GL_{2}(ℂ) is called unramified if it is trivial on I. In this case, since I \W_{F} ≅ ℤ (integral powers of the generator of Gal(F̄/F)), σ is completely determined by where it takes the (class of a) Frobenius element F r of W_{F}. So suppose (after conjugation, if necessary) that

σ(F r) = (^{p^{−s_{i}}} _{p^{−s_{2}}} ) in GL_{2}(ℂ) ,

with s_{1}, s_{2} ∈ ℂ. Then the corresponding representation π(σ) of GL_{2}(F) will be the unramified induced representation π(μ_{1}, μ_{2}), with μ_{1}(x) = |x|^{s_{i}}, i.e., the Langlands class

t_{π(σ)} = ( \begin{matrix}μ_{1}(p) & 0 \\ 0 & μ_{2}(p) \end{matrix} )

will be conjugate to σ(F r). (More precisely if π(μ_{1}, μ_{2}) is itself reducible, then π(σ) will be the unique irreducible unramified quotient of π(μ_{1}, μ_{2}), perforce one-dimensional...).

§4. The Langlands Reciprocity Conjecture (LRC)

4.1. Reformulation of Theorem 2.6 (“Langlands-Tunnell”) For F a global field, the Weil group W_{F} maps surjectively onto Gal(F̄/F),

and there is a canonical isomorphism

W_{F}^{ab} ≅ _{F^{x}}⧹𝔸x F .

For each place v of F, there is also an injection W_{F_{v}} ⟶ W_{F}, defining a map

σ ⟶ σ_{v}

from the two dimensional semisimple representations of W_{F} to those of W_{F_{v}} (cf. [Ta] for background). For a given σ, almost all the resulting σ_{v}’s will be unramified, and these unramified uniquely determine σ. (This is “strong multiplicity one” on the “Galois side”.)

Using the local Langlands correspondence for GL(2), we can now attach to any nice σ : W_{F} ⟶ GL_{2}(ℂ) a global representation π(σ) of GL_{2}(𝔸_{F}), namely

π(σ) = ⊗π(σ_{v}) .

The thrust of the Conjecture below is that this π(σ) must be automorphic. Conjecture (LRC). Suppose

σ : W_{F} ⟶ GL_{2}(ℂ)

is irreducible. Then there exists an automorphic cuspidal representation π = ⊗π_{v} of GL_{2}(𝔸_{F}) such that

π_{v} = π(σ_{v}) for all v.

(In particular, the Hecke-Jacquet-Langlands L-function

L(s, π) = ∏_{v} L(s, π_{v})

attached to π - which is known by [JL] to be entire - will equal the Artin L-function L(s, σ) = ∏_{v} L(s, σ_{v}).)

Remarks. (1) This conjecture is actually equivalent (via the “converse theorem” for L-functions on GL(2)) to Artin’s conjecture for two dimensional irreducible σ; cf. 5.3.1 below. Thus this LRC is sometimes called the “Strong Artin Conjecture”.

(2) If σ is reducible and the sum of two grossencharacters μ_{1} and μ_{2}, then there is easily seen to be an automorphic (non-cuspidal) representation π = ⊗π_{v} of GL(2, 𝔸_{F}) with π_{v} = π(σ_{v}) for all v, namely the induced “Eisensteinian” representation π(μ_{1}, μ_{2}) (or appropriate irreducible quotient thereof).

(3) When F = ℚ, and σ : W_{ℚ} ⟶ GL_{2}(ℂ) factors through G_{ℚ}, the above form of the LRC clearly implies the “almost everywhere” version which we stated in §2.6. According to the Proposition below, these two forms of the LRC are actually equivalent!

Proposition 4.1. Suppose σ is a two-dimensional representation of W_{F}, and π is a cuspidal representation of GL_{2}(𝔸_{F}). Then

π_{v} = π(σ_{v}) for all v

if and only if

tr(t_{π_{v}}) = trace(σ_{v}(F r_{v})

for almost all v (where both π_{v} and σ_{v} are unramified).

Note that this last condition really says that, for the unramified places, π_{v} = π_{v}(σ_{v}) (cf. Example 3.2.3 above). Thus this proposition essentially amounts to “strong multiplicity one” for GL(2); for further discussion, see pages 23 - 24 of [La].

4.2. Relations with Classical Forms

Proposition. Suppose σ : G_{ℚ} ⟶ GL_{2}(ℂ) is irreducible, and “odd”, and let π(σ) denote the corresponding automorphic cuspidal representation of GL_{2}(𝔸_{ℚ}) (assuming it exists!). Then π(σ) corresponds (via the correspondence f ⟷ π_{f} already described) to a normalized new form

f(z) = ∑_{n=1}^{∞} a_{n}e^{2πinz} ∈ S_{1}(Γ_{0}(N), ψ)

with N = conductor(σ), ψ determined by the central character of π(σ), and

a_{p} = trace(σ(F r_{p}))

for almost every p.

Proof. By Prop. 2.5, it suffices to check that π_{∞} = π_{∞}(σ_{∞}) is of the form π(μ_{1}, μ_{2}), with μ_{1} ≡ 1 and μ_{2} = sgn( ). Equivalently, we must check that σ_{∞} is a sum of these two characters. But when viewed as a representation of W_{ℝ}, σ_{∞} is clearly trivial on ℂ^{x}. This means σ_{∞} cannot be induced from a non-trivial character of ℂ^{x}. Thus σ_{∞} must be reducible (cf. Example 3.1.1), say the sum of two characters μ_{i}, with μ_{i} ∼ (t_{i}, ε_{i}). Since σ_{∞} is trivial on ℂ^{x}, it follows that each t_{i} = 0. On the other hand, the assumption det σ(τ) = −1 implies that σ(τ) is not a scalar; hence

σ(τ) ∼ ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) ,

which means π_{∞} = Ind 1 · sgn, as claimed.

Concluding Remarks.

(1) If det σ is even, then by the same reasoning as above, σ_{∞} is the sum of two characters, but now either both trivial or both the sgn character. Thus one concludes π_{∞} = π(1, 1) or π(sgn, sgn), and from Remark 2.5.5 it follows that π_{σ} corresponds to a cuspidal Maass eigenform of eigenvalue 1/4 for Δ.

(2) In [DS], Deligne and Serre associated to each normalized new

form f(z) = ∑^{∞} ; n=1a_{n}e^{2πinz} in S_{1}(Γ_{0}(N), ψ) an irreducible two-dimensional representation σ of G_{ℚ}, of conductor N and (odd) determined ψ, such that

trace σ(F r_{p}) = a_{p}

for almost all primes p. Taken together with the Langlands reciprocity conjecture for F = ℚ and “odd” σ (or equivalently, Artin’s conjecture for such σ), their result says that new forms of weight one are one and the same thing as irreducible, odd two-dimensional representations of G_{ℚ} (satisfying Artin’s conjecture...).

(3) One expects an analogue of Deligne-Serre to hold for cuspidal Maass-eigenforms (of eigenvalue 1/4), i.e., that to each such form there should correspond an irreducible two-dimensional, even representation of G_{ℚ} (satisfying Artin’s conjecture), with L(σ, s) = L(f, s). But this remains an open problem; cf. 4.3 below.

4.3. Representations of W_{F} vs. “Arithmetic” Automorphic Representations of GL(2)

For further reference, it will be convenient to repeat in a more precise form the classification of two-dimensional “semisimple” representations

σ : W_{F} ⟶ GL_{2}(ℂ)

over a number field F.

Proposition. Each σ as above is classified according to its image in P GL_{2}(ℂ), called the “type” of the representation:

(i) Cycle type μ ⊕ ν : σ is the direct sum of the two one-dimensional representations defined by Hecke characters μ and ν.

(ii) Dihedral type: σ is irreducible of the form Ind_{W_{E}}^{W_{F}}θ, with θ a character of E^{x} \ 𝔸_{E}^{x}, E a quadratic extension of F, and θ ≠ θ^{τ} for τ ≠ 1 in Gal(E/F). (Such representations are also called monomial.)

(iii) Exceptional type: the image of σ in P GL_{2}(ℂ) is A_{4}, S_{4} or A_{5}. Now lets assume that the LRC holds for all irreducible σ : W_{F} ⟶

GL_{2}(ℂ), and ask which automorphic cuspidal representation of GL_{2}(𝔸_{F}) are of the form π(σ) for some σ? A necessary condition is clearly the following.

Definition. Given an irreducible admissible representation π = ⊗π_{v}, and a real place v, let σ_{v} : W_{ℂ/ℝ} ⟶ GL_{2}(ℂ) denote the Langlands parameter of π_{v}. Then π is called of type A_{00} (resp. A_{0}) if the restriction of σ_{v} to ℂ^{x} is trivial (resp. the sum of characters of the form z → z^{a}z̄^{b}, a, b, ∈ ℤ). Alternately, π of type A_{00} (resp. A_{0}) is called arithmetic (resp. of Galois type).

Example 4.3.1. If σ : W_{ℚ} ⟶ GL_{2}(ℂ) actually factors through Gal(ℚ̄/ℚ), then the corresponding cuspidal π(σ) (if it exists) will be of type A_{0} (cf. Prop. 4.2 and the Remarks immediately following it). Conjecturally, one expects that all cuspidal π of GL_{2}(𝔸_{ℚ}) of type A_{00} are “motivic”, i.e. arise in this way; in case det(σ) is odd, this is the result of Deligne-Serre.

On the other hand, cuspidal π(σ) of type A_{0} are related to ℓ-adic representations of G_{ℚ} or W_{ℚ} (or the L-series attached to ℓ-adic cohomology spaces of varieties over ℚ). This is the subject matter of [Ant] (really a representation theoretic reformulation and strengthening of “Eichler-Shimura” theory). For example, if σ_{∞} is induced from the character z → z^{−n}z̄^{−m} of ℂ, with n > m ≥ 0, let D_{k} denote the discrete series representation of GL_{2}(ℝ) of lowest weight k = n − m + 1 (and appropriate central character). Then Langlands in [La5] associates to π = ⊗π_{p} of type A_{0} (with π_{∞} = π(σ_{∞}) = D_{k}) a two-dimensional ℓ-adic representation σ of Gℚ̄ whose local L and ε factors are (eventually) shown to agree with those of π_{p} for all p; cf. [Car].

§5. The Langlands Functoriality Principle Theory and Results

All the automorphic results used to prove the Reciprocity Conjecture in the two-dimensional solvable case, as well as the LRC itself, are but special realizations of what Langlands calls “functoriality of automorphic forms with respect to the L-group”. Hence it seems worthwhile to review some of the necessary background on “functoriality” in this Section.

5.1. L groups and L-factors

Recall that for GL_{2}, an unramified representation π_{p} = π_{p}(μ_{1}, μ_{2}) is parametrized by a semisimple conjugacy class in GL_{2}(ℂ), namely the Langlands class

t_{π_{p}} = ( \begin{matrix}μ_{1}(p) & 0 \\ 0 & μ_{2}(p) \end{matrix} ) .

More generally, an arbitrary irreducible admissible representation π_{p} is parametrized by a Langlands parameter

σ_{p} : W_{ℚ_{p}} ⟶ GL_{2}(ℂ) ,

and the “local Langlands Conjecture” says that the same should hold for GL_{n}, over any local field F_{v}. Namely, each nice representation π of GL_{n}(F_{v}) should be attached to a parameter σ_{v} : W_{F_{v}} ⟶ GL_{n}(ℂ).

For an arbitrary reductive group G, over a local field F_{v}, Langlands introduced the notion of the L − group ^{L}G to take the place of GL_{n}(ℂ) in parametrizing the irreducible admissible representations of G(F_{v}). Roughly speaking, each nice representation of G(F_{v}) should be attached to a “semisimple” homomorphism

φ : W_{F_{v}} ⟶ ^{L}G ,

and in the case of unramified representations, this should amount to fixing a certain semisimple conjugacy class in ^{L}G (again called the Langlands class t_{π_{v}} attached to π_{v}).

In general, if G is defined over a local or global field F, then ^{L}G is a group of the form

Ĝ⋊Gal(F̄/F) .

Here Ĝ is the complex Lie group “dual” to the root datum of G(ℂ), and the action of Gal(F̄/F) on Ĝ is trivial if and only if G is split over F; cf. §2 of [Bo] for details. It is sometimes convenient to replace Gal(F̄/F) by Gal(E/F), where E is any Galois extension of F over which G splits. Indeed, since Gal(F̄/E) acts trivially on Ĝ, we can take ^{L}G to be Ĝ⋊Gal(E/F) (now a complex reductive Lie group). For example, for G = GL(n), and F local or global, we can take

LG = GL_{n}(ℂ) or GL_{n}(ℂ) × Gal(E/F)

for any E. It is also convenient to define a semi-direct product

LG = Ĝ⋊ ∑

for ∑ any group endowed with a homomorphism into Gal(F̄/F), for example, the Weil group W_{F}. Henceforth, we deal almost exclusively with this “W eil” form” of ^{L}G.

For the moment, let us also assume that G is unramified over F, i.e, quasi-split over the local field F, and split over an unramified extension E. For such a group (like GL_{n}(F)), an irreducible admissible representation is called unramified if its restriction to a very special maximal compact subgroup (like GL_{n}(O_{F}) contains the identity representation (and then just once). If F_{r} denotes a Frobenius generation for Gal(E/F), then the unramified representation π of G(F) are in one - one correspondence with the semisimple ^{L}G^{0}-conjugacy classes t_{π} in ^{L}G^{0}⋊F r ⊂ ^{L}G. The resulting bijection

π ⟷ t_{π}

attaches to each unramified π its Langlands class t_{π} = g⋊F r in ^{L}G, and it is in terms of these classes, and their matrix realizations, that the general Langlands L-factors are defined.

Definition. By a representation of ^{L}G is meant a continuous homomorphism r : ^{L}G ⟶ GL_{m}(ℂ) whose restriction to Ĝ is a morphism of complex Lie groups. Given such a r, and an unramified representation π, one sets

(5.1.1)  L(s, π, r) = det(1 − r(t_{π})q^{−s})^{−1} ,

where q is the order of the residue class field of F.

Example. If F = ℚ_{p}, π = π_{p} is the local component of a cuspidal representation of GL(2, 𝔸_{ℚ}) associated to a new form in S_{k}(SL_{2}(ℤ)), and r : GL_{2}(ℂ) ⟶ GL_{2}(ℂ) is the “standard” representation taking g to s, then

\begin{aligned}L(s, π, r) & = [(1 − μ_{1}(p)p^{−s})(1 − μ_{2}(p)p^{−s})]^{−1} \\ & = (1 − a_{p}p^{−s− \frac{k−1}{2}} + p^{−2s})^{−1} , \end{aligned}

with a_{p} = p^{k− \frac{1}{2}} (μ_{1}(p) + μ_{2}(p)).

The question remains: what L-factors can be assigned to irreducible admissible π which are not unramified? In the case of GL_{n} (and some other classical groups as well now), the local representation theory of G may be used to directly define L (and ε-factors L(s, π, r) (and ε(s, π, r)), at least for r sufficiently close to the standard embedding of ^{L}G in some GL_{d}(ℂ). For example, for GL_{n}, and r : GL_{n}(ℂ) ⟶ GL_{n}(ℂ) the identity,

[GoJa] constructs such L and ε factors, and in [Ja] they are related (modulo the local Langlands conjecture for GL(n)) to the L and ε factors of their corresponding Langlands parameter.

In general, a typical Langlands parameter φ for G is a continuous homomorphism

φ : W_{F} ⟶ ^{L}G(= Ĝ × W_{F})

such that φ(w) is “semisimple” for each w in W_{F}, and such that the composition with projection onto W_{F} induces the identity map. Then to each representation r : ^{L}G ⟶ GL_{d}(ℂ), and class of parameters φ (modulo conjugation by Ĝ), one can attach the Langlands factor

(5.1.2)  L(s, φ, r) = det(I − q^{−s}r(φ(F r)|_{V^{I}})^{−1}.

Here F r is a Frobenius element in W_{F}, and V^{I} is the subspace of ℂ^{d} invariant under the action of the inertia subgroup. Note that when φ is unramified, i.e. trivial on I, then φ is determined by the semisimple element φ(F r) = g⋊F r in ^{L}G and L(s, φ, r) reduces to the Langlands L-function L(s, π, r), with π the unramified representation of G(F) such that t_{π} is conjugate to φ(F r). In general, a form of the “local Langlands conjecture” for G asserts that any irreducible admissible π is associated to some parameter φ : W_{F} ⟶ ^{L}G, and then L(s, π, r) should be L(s, φ, r), for any r.

N.B. If G is not quasi-split, it’s Langlands parameters must satisfy certain additional “rationality” condition; cf. 8.2(ii) of [Bo].

5.2. Statement of the Functoriality Principle

A homomorphism between L-groups

ρ : ^{L}G ⟶ ^{L}G′

is called an L−morphism if it commutes with the natural projections onto W_{F}. Such a morphism clearly gives rise to a map

φ ⟶ ρ ∘ φ

between the Langlands parameters of G and G′, hence (conjecturally) also between representations of G and G′ (by the local Langlands conjecture).

Now suppose F is global, and π = ⊗π_{v} is an automorphic cuspidal representation of G(𝔸_{F}). For almost every place v of F, G_{v} and π_{v} will be unramified, and the corresponding Langlands class t_{π} in ^{L}G_{v} defined. Thus the (partial Langlands) L-function L^{S}(s, π, r) can be defined for any representation r : ^{L}G → GL_{α}(ℂ) through the formula

L^{S}(s, π, r) = ∏_{{v \atop “unramified”}} L(s, π_{v}, r_{v}) .

(Here each r_{v} arises through composition of the natural embedding of LG_{v} = Ĝ⋊W_{F_{v}} ↪ ^{L}G = Ĝ⋊W_{F}.) Langlands has shown that this Euler product converges in some half-plane, and conjectured that it admits a meromorphic continuation to ℂ, with only finitely many poles in Re(s) ≥ 0.

N.B. (1) If one accepts the local Langlands Conjecture, one can also introduce L(s, π_{v}, r_{v}) at the remaining places (as in (5.1.2)), and define the “completed” functions L(s, π, r) (and ε(s, π, r).

(2) If G = GL_{n}, and r : ^{L}G → GL_{n}(ℂ) is the standard representation, then L^{S}(s, π, r) is simply denoted L^{S}(s, π).

The Functoriality Principle. Suppose that G′ is quasi-split, and ρ : ^{L}G → ^{L}G′ is a morphism of L-groups. For each v, consider the corresponding commutative diagram

LG_{v}  ⟶^{ρ_{v}}  ^{L}G′_{v}
↓  ↓
LG  ⟶^{ρ}  ^{L}G′

Then to each automorphic cuspidal representation π = ⊗π_{v} of G(𝔸_{F}) there corresponds an automorphic representation π′ = ⊗π′_{v} of G′(𝔸_{F}) such that for almost all v (where both π_{v} and π′_{v} are unramified)

t_{π′_{v}} = ρ_{v}(t_{π_{v}}).

In particular, for any representation r′ : ^{L}G′ → GL_{d}(ℂ),

L^{S}(s, π′, r′) = L^{S}(s, π, r′ ∘ ρ) .

Moreover, if one accepts the local Langlands Conjecture for G, then the Langlands parameter of π′_{v} should be the image (under ρ_{v}) of the Langlands parameter of π_{v} for every v.

Example. Take G = {1}, and G′ = GL(n). In this case, a morphism

ρ : ^{L}G ⟶ ^{L}G′ = GL(n, ℂ) × W_{F}

must be of the form ρ(1, w) = σ(w) × w, with

σ : W_{F} ⟶ GL_{n}(ℂ)

a continuous representation (and conversely, any Artin representation σ : W_{F} ⟶ GL_{n}(ℂ) determines a morphism ρ_{σ} through this formula). Since G = {1}, its only automorphic representation π is the trivial one, with Langlands class 1⋊F r_{v} for every (finite) v. Thus the Functoriality Conjecture in this case asserts that (for any given σ) there is an automorphic representation π(σ) = ⊗π_{p} of GL_{n}(𝔸_{F}) such that

t_{π_{p}} = σ(F r_{r})

for almost every p.

This example shows that the general Reciprocity Conjecture is but a special instance of the Functoriality Principle. Hence it is clear that this Principle is more a guiding light than a problem to be solved in the near future!

5.3. Established Examples of Functoriality

We collect here some instances of “functoriality” which are required for the proof of Langlands-Tunnell.

(A) Automorphic Induction

This is a generalization of the classical construction of Hecke and Maass, whereby an automorphic cuspidal representation of GL_{2}(ℚ) (a modular form, or Maass form, in their language) is attached to each Hecke character of a quadratic extension of ℚ (which is purely imaginary or real, respectively).

For a general formulation, fix a number field F, and K a cyclic Galois extension of F of degree n. Let G denote the group Res_{K/F} GL_{1} (defined by “restriction of scalars” from K to F) and let G′ denote the group GL_{n}. Then G is isomorphic to a maximal F -torus of G′, and

LG = (GL_{1}(ℂ)x ⋯ xGL_{1}(ℂ))⋊W_{F} ,

with W_{F} acting on Ĝ through its projection onto Gal(K/F), and (the generator of) Gal(K/F) acting through cyclic permutation of the n GL_{1}(ℂ) factors of Ĝ. Let ρ_{1} be the natural homomorphism of ^{L}G into the normalizer of a maximal torus of Ĝ′ = GL(n, ℂ), and define a morphism of L-groups

ρ : ^{L}G ⟶ ^{L}G′

by ρ(g) = (ρ_{1}(g), ρ_{2}(g)), where ρ_{2} : ^{L}G → W_{F} is the canonical projection. Note that an automorphic form on G is the same thing as a grossencharacter χ of K, since G(F) = K^{x}; and when v splits (completely) in K, the representation π_{v} = π_{v}(χ) (induced from the χ_{v}’s above v) satisfies t_{π_{v}} = ρ(t_{χ_{v}}) if χ is unramified at v. Thus the principle of functoriality suggests the following:

Theorem 5.3.1. For each grossencharacter χ of K there is an automorphic representation π(χ) of GL_{n}(𝔸_{F}) whose L-function L^{S}(s, π) equals the Hecke-L-function L^{S}(s, χ); moreover, L^{S}(s, π(χ)) is entire (and hence π(χ) is cuspidal automorphic) if χ does not factor through the norm map N_{K/F} (equivalently χ is not fixed by the natural action of the Galois group Gal(K/F)).

For n = 2 and F = ℚ, this Theorem follows essentially from the classical work of Hecke and Maass. For n = 2 and F arbitrary, it is proved in [JL] (using L-functions), [LL] (using the “stable trace formula”), and [ST] (using theta-series); it also follows from Jacquet’s “relative trace formula” (cf. §VIII 4 of [Ge2]). For n = 3 it is proved in [J-PS-S2] (using Lfunctions) and for arbitrary n in [AC] (using the trace formula). The only cases needed in the sequel are n = 2 or 3, and here it is simplest to (follow [JL] and [J-PS-S2] and) appeal to the so-called “Converse Theorem to Hecke Theory”.

For this, suppose that π = ⊗π_{v} is an irreducible admissible representation of GL_{n}(𝔸_{F}) whose central character is invariant under F^{x}. If π is actually automorphic, then it is known from “Hecke theory” (cf. [GoJa]) that π is “nice” relative to any idele class character ω of F, i.e., L(s, π ⊗ω) and L(s, π̃ ⊗ ω^{−1}) are absolutely convergent in some half-place, admit analytic continuations to the whole s-plane which are bounded in vertical strips, and have a functorial equation relating s to 1 − s; moreover, if π is cuspidal, then these analytic continuations are also entire. For n = 2 or 3, the Converse Theorem (cf. [JL] and [J-PS-S2]) simply says that the converse to these statements are also true.

Remarks 5.3.1.

(a) In real life situations, like the application to proving π(χ) automorphic in Theorem 5.3.1, the situation is complicated by the fact that the representation we are trying to prove automorphic may not be easily defined at every place, but rather only at almost all places; thus, in fact, a more complicated “almost everywhere” version of the converse theorem is needed; cf. §§13-14 of [J-PS-S2].

(b) In this paper [Co-PS], Cogdell and Piatetski-Shapiro conjecture that the Converse Theorem should also hold for any n, with the additional caveat that for n ≥ 4, π need only be almost everywhere equivalent to some automorphic representation of GL_{n}(𝔸_{F}); cf. [He].

(c) If χ (in Theorem 5.3.1) is not fixed by any non-trivial element of Gal(K/F), then Ind_{W_{K}}^{W_{F}} χ = σ is an irreducible n-dimensional representation of W_{F} with L(s, σ) = L(s, χ) (a Hecke L-series with grossencharacter χ over K). Hence Theorem 5.3.1 may be viewed as an affirmation of the Langlands Reciprocity Conjecture for monomial representations.

(d) Note that “on the Galois side”, induction brings a Langlands parameter for GL_{1} (over K), namely χ : W_{K} ⟶ ℂ^{x}, to a Langlands parameter for GL_{2} over F, namely σ = Indχ : W_{F} ⟶ GL_{2}(ℂ). “On the automorphic side”, this map is reflected by the correspondence χ ⟶ π(χ) = π(σ) (hence the aptness of the terminology “automorphic induction”).

(e) Finally, we note that the “converse theorem approach” to Theorem 5.3.1 in case n = 2 or 3 does not depend on K being a normal (Galois) extension of F. This will be crucial in the application to the Reciprocity Conjecture in the Octahedral case; cf. §7.2.

(B) The Symmetric Square Lifting

Let A denote the three dimensional representation of P GL_{2}(ℂ) determined by the adjoint action of P GL_{2}(ℂ) on the Lie algebra of SL(2, ℂ), and denote the resulting (three-dimensional) representation

GL_{2}(ℂ  ⟶^{Ad}  GL_{3}(ℂ)
↘  A ↗
P GL_{2}(ℂ)

of GL(2, ℂ) by Ad. This representation Ad may be viewed as a natural morphism between the L-groups of GL(2) and (GL(3)

Theorem 5.3.2. (The “Symmetric Square” Lift from GL(2) to GL(3); cf.[GeJa]). (i) To each cuspidal automorphic representation π of GL_{2}(𝔸_{F}), there exists an automorphic representation ∏ of GL_{3}(𝔸_{F}) such that for almost all v,

Π_{v} = Π_{v}(Ad(σ_{v}))

whenever π_{v} = π_{v}(σ_{v}); equivalently,

t_{Π_{v}} = Ad(t_{π_{v}}).

(ii) This left of π to GL(3) is cuspidal automorphic unless π is monomial, i.e., of the form π(σ), with σ induced from a Hecke character of some quadratic extension K.

Method of Proof. The “converse theorem for GL(3)” says that L^{S}(s, π, Ad) will be the L-function of an automorphic representation Π of GL(3) (with t_{Π_{v}} = Ad(t_{π_{v}}) ⋯) as soon as L^{S}(s, π, Ad) is shown to have the expected analytic properties; moreover, this Π will be cuspidal if and only if all L^{S}(s, Π ⊗ ω)’s are entire. To establish the required analytic properties, it is shown (following [Sh]) that

L^{S}(s, π, Ad) = A_{S}(s) ∫_{SL_{2}(F)} ⧹_{SL_{2}(𝔸)} φ_{π}(g)Θ(g)E(g, s)dg .

Hence φ_{π} belongs to the space of π, Θ(g) is a theta-function on Weil’s metaplectic group, E(g, s) is an Eisenstein series of half-integral weight which is real analytic in g and meromorphic in s, and A_{S}(s) is a meromorphic function which at the possible poles of E(g, s) can be chosen non-zero.

N.B. The idea of using the integral of an automorphic form to derive analytic properties of its L-function of course goes back to Hecke, and even Riemann. But the idea of mixing automorphic forms in the integral with Eisenstein series was first systematically developed by Rankin and Selberg, and is now a flourishing industry; cf. below.

(c) Rankin-Selberg Products (Especially GL(3) × GL(3)) Underlying this work is the following instance of Langlands functorial-

ity. Viewing GL(k, ℂ) as the L-group of GL(k), and GL(n, ℂ) × GL(m, ℂ) as that of GL(n) × GL(m), consider the natural L-group morphism

ρ : GL_{n}(ℂ) × GL_{m}(ℂ) ⟶^{⊗} GL_{nm}(ℂ)

given by the tensor product map.

So far, there seems no hope of establishing Langlands functoriality in this case, i.e., of proving the existence of an automorphic ∏ on GL_{nm} such that t_{Π_{v}} = t_{π_{v}} ⊗t_{π′_{v}} for two given cuspidal representations π′ on GL_{n} and π on GL_{n}. Indeed, this is an important open problem, whose solution would play a crucial role in finding “the” group whose irreducible representations are expected to parametrize all the automorphic cuspidal representations of GL_{n} (not just those “arithmetic” ones coming from representations of W_{F}); cf. [Ram] for further discussion along these lines. A big first step, however, was taken by Jacquet and Shalika:

Theorem 5.3.3 (cf. [JaSh1,2] and [Mo-Wald]. Given cuspidal representations π on GL_{n} and π′ on GL_{m}, let L^{S}(s, π × π′) denote the partial L-function

∏_{vεS}[det(I − (t_{π_{v}} ⊗ t_{π′_{v}})q^{−s}]^{−1} .

Then (i) L^{S}(s, π × π′), originally defined only in some right half-plane, extends to a meromorphic function in all of ℂ, with functional equation relating s to 1-s; and (ii) L^{S}(s, π × π′) may be “computed” to an Euler product L(s, π×π′) = ∏_{all v} L(s, π_{v}×π′_{v}) which is holomorphic on Re(s) ≥ 1 if m ≠ n, and otherwise has a pole at s with Re(s) = 1 if and only if |det( )|^{s−1} ⊗ π ≅ π̃ (the contragredient of π′).

As already suggested, the proof of this result constitutes a non-trivial representation-theoretic generalization of the classical integral representations of Rankin and Selberg; see [Ja] for the case of GL(2) × GL(2). In the sequel, we need only the case n = m = 3.

Concluding Remarks.

(1) There is one more example of functoriality needed for the proof of Langlands-Tunnell, namely the theory of base-change of Saito, Shintani and Langlands. However, since that theory is so intimately tied up with Artin’s conjecture, and its proof relies on the trace formula rather than L-functions, it seems convenient to postpone discussion of it until the last Lecture.

(2) There are of course large aspects of the Langlands Program which we have not seriously broached here because they have no immediate bearing on Wiles’ work. Perhaps the most obvious such topic is the (conjectured) relation between Hasse-Weil zeta-functions of algebraic varieties (“motive” L-functions) and automorphic L-functions of type L(s, π, r). For example, in [La6] the zeta-functions of certain Shimura varieties are related to automorphic L-functions of degree 2^{n}. This “program” represents the beginnings of a higher dimensional analogue of the theory of Eichler-Shimura and has greatly influenced much of the work during the last twenty years in representation theory and the theory of automorphic forms. Among other things, it pushed to the forefront the need to refine and generalize the “Selberg trace formula”; more about this in the next Lecture. It also brought into representation theory such crucial but different concepts as “L-indistinguishability”, “endoscopy”, “L-packets”, etc, and encouraged the use of new algebra-geometric methods for counting points on these varieties.

(3) Finally, one should say a few words about the relation between the Langlands Program and the Shimura-Taniyama-Weil Conjecture itself. Personally, I do not think that it is so significant that the Langlands Program actually includes the S-T-W conjecture as a special “example” (and that’s why I haven’t bothered to broach the topic here). After all, Taniyama obviously made his Conjecture - and Shimura and Weil understood its importance - before the Langlands Program was conceived. Also, from the other point of view, its equally clear that including the S-T-W Conjecture inside the Langlands Program is more incidental than crucial to the Program. Rather the crux of the Program is two pronged: its overall vision relating motives of all kinds to automorphic representations, and its methods which push representation theory to the forefront, and infuse the subject with a seemingly endless string of challenging problems. It is these aspects of the Langlands Program which (albeit indirectly) play a role in the proof of Fermat’s Last Theorem.

Lecture III Proof of the Langlands-Tunnell Theorem

Abstract

Our task is to describe the proof of the following:

Theorem. Suppose F is a number field and the irreducible representation

σ : W_{F} ⟶ GL_{2}(ℂ)

has a solvable image in P GL_{2}(ℂ). Then there exists a (unique) irreducible automorphic cuspidal representation π(σ) = ⊗π_{v} of GL_{2}(𝔸_{F}) such that

trace(σ(F r_{v})) = trace(t_{π_{v}})

for almost every v.

The crucial instance of the Functoriality Conjecture required in the proof of this Theorem is the theory of “Base Change” as developed in [La1]. This we describe in §6, along with its proof, which relies heavily on trace formula methods. The application of base change to the Langlands Reciprocity theorem is explained in §7, the proof of the actual theorem proceeding in two steps: first the base change (trace formula) methods are exploited to produce the best possible candidate for π(σ) (which is called π_{ps}(σ)); then the results from the theory of L-functions (recalled in §5) are used to prove that π_{ps}(σ) actually equals π(σ).

§6. Base Change Theory

(6.1). Fix E a cyclic extension of the number field F, of prime degree ℓ.

Roughly speaking, the theory of “base change” describes the correspondence between automorphic representations of the groups GL_{n}(𝔸_{F}) and GL_{n}(𝔸_{E}) which reflects the operation of restriction of Galois representations of W_{F} to W_{E}. The first results on base change for automorphic forms (or representations) used the theory of L functions, and were restricted to the case of quadratic E and GL_{2}. The introduction of the trace formula is due to H. Saito, who dealt with GL_{2} and arbitrary cyclic E using the classical language of automorphic forms; cf. [Sai]. Immediately after that, Shintani reformulated Saito’s results using group representations, and gave the correct local definition of base change lifting; cf. [Shin]. Finally, Langlands saw the connection with Artin’s conjecture, and reshaped the trace formula proof for GL_{2} in a form suitable for the later generalization to GL_{n} developed by Arthur and Clozel; see [La1] and [AC] for a more detailed history. Since only the case n = 2 is required here, we restrict ourselves henceforth to this case.

Definition. Suppose π = ⊗π_{v} is an automorphic cuspidal representation of GL_{2}(𝔸_{F}), and ∏ = ⊗_{w} ∏_{w} is an automorphic representation of GL_{2}(𝔸_{E}). Then ∏ is a base change lift of π, devoted BC_{E/F} (π), if for each place v of F, and ω/v, the Langlands parameter attached to ∏_{w} equals the restriction to W_{E_{w}} of the Langlands parameter σ_{v} : W_{F_{v}} ⟶ GL_{2}(ℂ) of π_{v}.

Remarks.

(i) The above (essentially local) definition of base-change lifting is at the level of Langlands parameters rather than representations. The key idea of [Shin] is to define the lift of π_{v} on GL_{2}(F_{v}) directly in terms of a character identity between π_{v} and the extension of ∏_{w} to the group GL_{2}(E_{w})⋊Gal(E_{w}/F_{v}). Implicit here is the fact that ∏_{w} is Gal(E_{w}/F_{v}) invariant and hence this extension, call it ˜∏_{w} exists. If τ is a generator of Gal(E_{w}/F_{v}) the character this identity reads

χ∏^{}_{˜ w}(g⋊τ) = χ_{π_{v}}(x)

whenever N_{E/F,σ}(g) = g^{τ^{ℓ−1}} ⋯ g^{τ} g is conjugate in GL_{2}(E_{w}) to a regular semisimple element x of G(F_{v}).

(ii) Functoriality. Let G = GL_{2} and set G′ = Res_{E/F} (G). As recalled in §5.3, Ĝ′ is then a product of ℓ copies of GL_{2}(ℂ) indexed and permuted by Gal(E/F). So let

ρ : ^{L}G = GL_{2}(ℂ) × W_{F} ⟶ ^{L}G′ = Ĝ′⋊W_{F}

be the natural morphism which takes g × w in ^{L}G to (g, ⋯, g)⋊w in ^{L}G′. The Functoriality Principle suggests the existence of a map taking automorphic cuspidal representations π of GL_{2}(𝔸_{F}) to automorphic representations ∏ of GL_{2}(𝔸_{E}) ≈ G′(𝔸_{F}) such that for π_{v} and ∏_{w} unramified,

(∗)  t_{Π_{w}} = ρ_{v}(t_{π_{v}}) .

Using either definition of lifting given above, it is easy to check that if π_{v} = π(μ_{1}, μ_{2}) (with μ_{i} an unramified character of F_{v}) then BC_{E/F} (π_{v}) = ∏(ν_{1}, ν_{2}), with ν_{i} = μ_{i} ∘ N_{E_{w}/F_{v}}. From this it follows that (∗) holds, i.e. Base Change is functorial.

N.B.. In verifying that (∗) holds, one must keep in mind that π(ν_{1}, ν_{2}) (viewed as a representation of GL_{2}(E_{w})) corresponds first to the Langlands class g × σ in GL_{2}(ℂ) × σ, with

g = ( \begin{matrix}μ_{1} ∘ N(ω̃_{w}) & 0 \\ 0 & μ_{2} ∘ N(ω̃_{w}) \end{matrix} )

= ( \begin{matrix}μ_{1}(ω̃_{v})^{2} & 0 \\ 0 & μ_{2}(ω̃_{v})^{2} \end{matrix} );

but the corresponding class in^{L}G′ = GL_{2}(ℂ)×GL_{2}(ℂ)⋊σ (viewing π(ν_{1}, ν_{2}) as a representation of G′(F_{v}) ≈ GL_{2}(E_{w})) is

(( \begin{matrix}μ_{1}(ω̃_{v}) & 0 \\ 0 & μ_{2}(ω̃_{v}) \end{matrix} ) , ( \begin{matrix}μ_{1}(ω̃_{v} & 0 \\ 0 & μ_{2}(ω̃_{v}) \end{matrix} ) , σ) ,

i.e. just ρ(t_{π_{v}}). (Indeed, the Hecke algebras of GL_{2}(E_{w}) and G′(F_{v}) are the same, and if f_{w} and f′_{w} represent the same element in this algebra H_{w}, then (f′_{w})^{v}(g_{1}, ⋯, g_{ℓ} × σ) = f_{w}^{v}(g_{e} ⋯ g_{2}g_{1}); see 6.3 below for definitions of H_{w} and the Satake isomorphism (f′)^{v}.

(iii) Because we assuming E over F cyclic of prime degree, each v of F either remains inert or splits completely. In the later case, it is clear that E_{w} ≈ F_{v} for any w/v, and the base change lift of π_{v} is just ∏_{w} ≈ π_{v}. This case being trivial, we usually assume (as above) that we are dealing with the inert local case.

Theorem (cf. [La1].

(a) Every cuspidal representation π of GL_{2}(𝔸_{F}) has a unique base change lift to GL_{2}(𝔸_{E}); the lift is itself cuspidal (as opposed to “just” automorphic), unless E is quadratic over F, and π is monomial (or dihedral) of the form π(σ), with σ = Ind_{W_{E}}^{W_{F}}θ;

(b) If two cuspidal representations π and π′ have the same base change lift to E, then π′ ≈ π ⊗ ω for some character ω of F^{x}N_{E/F} (𝔸_{E}^{x}) \ 𝔸_{F}^{x}; and

(c) A cuspidal representation ∏ of GL_{2}(𝔸_{E}) equals BC_{E/F} (π) for some cuspidal π on GL_{2}(𝔸_{F}) if and only if ∏ is invariant under the natural action of Gal(E/F).

In some ways, the proof of Base Change is as interesting as the result itself. Since it involves a form of the trace formula which should (and does) generalize, and apply to other instances of functoriality, we devote some time to it below.

(6.2.). The Trace Formula of Arthur-Selberg

Recall that the right regular representation R_{0} of G(𝔸_{F}) in the space of cusp forms L_{0}^{2}(G(F) \ G(𝔸_{F}), ω) decomposes discretely as

R_{0} = ∑ m_{π}π ,

and it is the cuspidal constituents π which are the building blocks of the theory of automorphic forms on G. What the “trace” in “the trace formula” refers to is the distributional trace of R_{0}. More precisely, suppose f(g) is any nice compactly supported “test function” on G(𝔸), and define the operator R_{0}(f) on L_{0}^{2}(G(F) \ G(𝔸), ω) through the formula

R_{0}(f) = ∫_{Z(𝔸)}⧹_{G(𝔸)}f(g)R_{0}(g)dg .

(For simplicity, assume that the central character ω of R_{0} is trivial.) Then clearly

trace R_{0}(f) = ∑ m_{π}tr(π(f)) ;

but as we know next to nothing about the π’s which occur in R_{0}, we also know next to nothing about tr(R_{0}(f)). The original idea of the trace formula was to give an alternate formula for trace R_{0}(f), which ultimately gives some of the sought after information about R_{0} and its constituents π.

The original trace formula was introduced by Selberg, in the context of a semisimple Lie group G and discrete subgroup Γ (in place of our G(𝔸) and G(F)). In his famous 1956 paper [Sel], Selberg first of all described a general formula for the case of compact Γ \ G (equivalently G(F) \ G(𝔸)); it took the form

(6.2.1)  tr(R_{0}(f) = ∑_{π} m_{π}tr π(f) = ∑_{{γ}} m_{γ}Φ_{f}(γ)

with {γ} running over the conjugacy classes in G(F), and each Φ_{f}(γ) an

“orbital integral” ∫_{G_{γ}(𝔸)}⧹_{G(𝔸)}f(g^{−1}γg)dg. Secondly, Selberg treated in

detail certain non-compact quotient cases such as SL_{2}(ℤ) \ SL_{2}(ℝ), which already required the analytic continuation of Eisenstein series to handle the continuous spectrum of L^{2} outside L_{0}^{2}.

Subsequently, in the 1960’s and 70’s, Langlands developed a general theory of Eisenstein series valid for any reductive group G, and Arthur used it to develop a general trace formula in the context of not necessarily compact quotients G(F) \ G(𝔸). The resulting trace formula of Arthur takes the form

(6.2.2)  ∑ ; 𝔬 J𝔬(f) = ∑_{χ} J_{χ}(f) .

Here the left (or geometric) side of Arthur’s formula is a sum of special types of equivalence classes in G(F) (generalizing the ordinary notion of equivalence in the case of compact quotient), and the sum over the right (or spectral) side is over certain classes of automorphic cuspidal representations of “Levi subgroups” of G (as opposed to just the cuspidal representations of G(𝔸) itself in the case of compact quotient). Although it looks like the “trace” has been lost in Arthur’s trace formula, this is not really so: certain of the spectral terms J_{χ}(f) add up to exactly trace R_{0}(f), and so making (6.2.2) explicit still (ultimately) gives us the information we seek about trace R_{0}(f).

Now instead of focusing efforts on finding an explicit, new formula for trace R_{0}(f), it was the idea of Langlands to compare the trace formulas for two different groups G and G′, in order to find relations (presumably “functorial”) between their automorphic representations. For example, this is exactly the strategy exploited in §16 of [JL] to establish the correspondence between automorphic cuspidal representations of G = D^{x} and G′ = GL_{2}, D a division quaternion algebra over F. In the case of G = GL_{2} and G′ = Res_{E/F} G this strategy brings us back to the proof of Theorem 6.1 which we now explain.

(6.3). The Proof of Base Change

If τ denotes a generator of Gal(E/F), then τ acts naturally on G′(𝔸) ≈ GL_{2}(𝔸_{E}), and hence on L^{2}(G′(F) \ G′(𝔸_{F})) through the rule

(τ · φ)(g) = φ(g^{τ}) .

We can also define a twisted regular representation R^{τ} through the composition of R with τ. Then for any nice f′ on G′(𝔸_{F}), there is a “twisted” version of the trace formula of the form

(6.3.1)  \begin{matrix}∑ \\ 𝔬 & ′ \end{matrix} J𝔬′^{τ}(f′) = ∑_{χ′} J_{χ′}^{τ}(f′) ,

with the “twisted” trace (R_{0}^{τ}(f′)) hidden inside the right side of (6.3.1), and “twisted” orbital integrals Φ_{f′}^{τ}(γ′) = ∫ f′(g^{−τ} γg)dg on the left. The significance of working with a twisted formula for G′ is that then only Galois invariant cuspidal representations ∏ will contribute (to tr(R_{0}^{τ}(f′))) ∗; hence we might indeed establish the desired base change map π ⟶ ∏ between G and G′ by relating (6.3.1) to (6.2.2), and ultimately tr R_{0}(f) to tr(R_{0}^{τ}(f′)).

The first step is to prove that the left-hand (i.e., geometric) sides of the trace formulas for G and G′ coincide, at least for certain “matching” f and f′. This matching is a non-trivial local step, which first of all requires that the orbitals integrals Φ_{f_{v}}(Nγ′) on G_{v} match the twisted orbital integrals Φ_{f′_{w}}^{τ}(γ′) on G′_{w}. Moreover, it must be shown that this matching f′ ⟶ f is compatible with “base change at the unramified places”, in the following sense.

If H_{v} denotes the Hecke algebra of bi-K_{v}-invariant (compactly supported smooth) functions on G_{v}, each f_{v} in H_{v} may be viewed as a function on ^{L}G_{v} through the formula

f_{v}^{V} (t) = tr π_{v}(f_{v})

whenever t_{π_{v}} = t. (This is the Satake isomorphism f_{v} ⟶ f_{v}^{V}, defined analogously for the Hecke algebra H′_{w} of G′_{w}.) Then the base change map of Hecke algebras, dual to the base change morphism ρ : ^{L}G_{v} ⟶ ^{L}G′_{w}, is defined by

ρ^{V} : (f′_{w})^{V} ⟶ f′_{v}(g) =^{·} (f′_{w})^{V} (ρ(g)) ,

∗ This is because τ permutes the constituents ∏ of R_{0}, and a permutation matrix without fixed points has zero trace...

46 Modularity and the Langlands Reciprocity Conjecture

and the compatibility condition mentioned above is that f′_{w} will match its image ρ^{V} (f′_{w}) = f′_{v} in the above sense, for any f′_{w} in H′_{w}. (This is what is known as the fundamental lemma, in the context of “base change”.)

The next step, which takes a great deal more work on the spectral sides of (6.2.2) and (6.3.1), is to conclude from the equality of the geometric trace formulas that

trace R_{0}(f) (essentially) = trace R_{0}^{τ}(f′)

for such matching f and f′. Equivalently,

(6.3.2)  ∑_{{π \atop cuspidal}} tr π(f) = ∑_{{π′ cuspidal \atop (π′)^{τ} ≈π}} tr(τ ∘ π′)(f′)

with the sum on the right only over Galois fixed π′.

Now for π_{v}, π′_{w}, f_{v}, f′_{w} all “unramified”, we will have π′_{w} equal to the base change lift of π_{v} if and only if

tr(τ ∘ π′_{w})(f′_{w}) = tr π_{v}(f_{v})

for any f_{v} the base change image of f′_{w} as above. (Indeed, in terms of Satake transforms, this last identity reads (f′_{w})^{V} (t_{Π_{w}}) = f_{v}^{V} (t_{π_{v}}) = (f′_{w})^{V} (ρ(t_{π_{v}})), i.e. t_{Π_{w}} = ρ(t_{π_{v}}) as required.)

In this way, with a “linear independence of characters” argument, (6.3.2) ultimately implies that for a given π occurring on the left-hand side, there must be a π′ on the right-hand side which is Galois invariant, and almost everywhere the base change lift of π (and, conversely, all such Galois invariant π′ thus arise). Thus the required correspondence is established.

Remark. Whenever the trace formula can be used to establish an instance of functoriality (like base change above), it offers the additional bonus of characterizing the image of the automorphic representations in question. This is not so for the method of L-functions (witness the example of the lifting from GL(2) to GL(3), where the image is left uncharacterized, or Prop. 7.2 below giving base change for non-Galois cubic E).

§7. Application to Artin’s Conjecture

The idea of applying “base change” to attack Artin’s Conjecture arises from the following observation. Suppose that for any σ : W_{F} ⟶ GL_{2}(ℂ) there really is a corresponding cuspidal representation π(σ) of GL_{2}(𝔸_{F}). Then it follows from the original definition of base change lifting that BC_{E/F} (π(σ)) = π(Res σ|_{W_{E}}) for any cyclic extension E of F. This means that if we start with σ, and want to find candidates for π(σ), then the thing to do is to pick an E such that π(Res σ|_{W_{E}}) is already known to exist, and look among the cuspidal π’s such that BC_{E/F} (π) = π(Res σ|_{W_{E}}). In this way the following “obvious” strategy unfolds: Among the possible candidates for π(σ), pick a “best possible” one, call it π_{ps}(σ) (for π_{pseudo}(σ)), and then prove that π_{ps}(σ) must equal π(σ). Roughly speaking, the first step uses the trace formula (via base change), while the second uses L-functions.

Convention. Henceforth, if we are given σ : W_{F} ⟶ GL_{2}(ℂ), and any field E over F, then by σ_{E} we denote the restriction of σ to W_{E}.

(7.1). The Tetrahedral Case

(a) Choosing π_{ps}(σ)

We are given an irreducible representation

σ : W_{F} ⟶ GL_{2}(ℂ)

whose image in P GL_{2}(ℂ) is isomorphic to A_{4}. This group is solvable, with composition series

A_{4} ▷ D_{2} ▷ {e} .

(In general, D_{n} will denote the dihedral group of 2n elements; in this case, D_{2} is the Klein 4-group). Since A_{4}/D_{2} ≅ A_{3} ≅ ℤ_{3}, the inverse image of D_{2} in W_{F} under the map

W_{F} ⟶ A_{4} ⊂ P GL_{2}(ℂ)

is a (normal) subgroup of index 3, hence the Weil group of a cubic extension of F_{1} call it E. Pictorially:

1 ⟶ W_{E} ⟶ W_{F} ⟶ Gal(E/F) ⟶ 1
↓  ↓  ↓ ≀
1 ⟶ D_{2} ⟶ A_{4} ⟶  ℤ_{3}  ⟶ 1

Thus the resulting representation σ_{E} : W_{E} ⟶ GL_{2}(ℂ) is “monomial” in the sense of Prop. 4.3.

Now let π(σ_{E}) denote the automorphic cuspidal representation of GL_{2}(𝔸_{E}) attached to this monomial representation by Theorem 5.3.1. This representation of GL_{2}(𝔸_{E}) is clearly invariant under the action of Gal(E/F); indeed, π(σ_{E})^{τ} = π(σ_{E}^{τ}) = π(σ_{E}). So by (the Base Change) Theorem 6.1, π(σ_{E}) will be the base change lift of exactly three classes of irreducible cuspidal representations π_{i} of GL_{2}(𝔸_{F}), each one related to the other by a twist ω ∘ det for some character ω of F^{x}N_{E/F} (𝔸_{E}^{x}) \ 𝔸_{F}^{x} , i.e.

π_{i} = π_{j} ⊗ ω ∘ det .

These π_{i}’s are our natural candidates for π(σ).

Recall that the central character of π(σ) is to be det σ. On the other hand, the central character ω_{i} of each π_{i} above “base change lifts” to the central character of π(σ_{E}), which is det σ_{E} = (det σ) ∘ N_{E/F}. Since each ω_{i} = ω_{j}ω^{2} if π_{i} = π_{j} ⊗ ω ∘ det, it is clear that exactly one of these π_{i}’s has central character det σ, and this is the one we choose to be π_{ps}(σ).

(b) Proving π_{ps}(σ) = π(σ)

Write π_{ps}(σ) = ⊗π_{v}. Then for each v, π_{v} = π_{v}(σ′_{v}) for some σ′_{v} : W_{F_{v}} ⟶ GL_{2}(ℂ), and what we must prove is that

(7.1.1)  σ′_{v} = σ_{v}

for almost every v.

Note that the restriction of σ′_{v} to W_{E_{w}} (for w/v) is by construction the same as the restriction of σ_{v} to W_{E_{w}}. Thus there is nothing to prove in case v splits (completely) in E, and we henceforth assume E_{w} cubic and unramified over F_{v}.

If F r_{v} denotes a Frobenius element of Gal(E_{w}/F_{v}) we can suppose

σ_{v}(F r_{v}) = ( \begin{matrix}a_{v} & 0 \\ 0 & v_{v} \end{matrix} )

and

σ′_{v}(F r_{v}) = ( \begin{matrix}c_{v} & 0 \\ 0 & d_{v} \end{matrix} )

for some a_{v}, b_{v}, c_{v}, d_{v} in ℂ^{x}. Then to prove (7.1.1) it will suffice to prove

that ( \begin{matrix}a_{v} & 0 \\ 0 & b_{v} \end{matrix} ) is conjugate to ( \begin{matrix}c_{v} & 0 \\ 0 & d_{v} \end{matrix} ). But the fact that σ_{v}

and σ′_{v} have the same restrictions to W_{E_{w}} means that σ_{v}(F r_{v})^{3} is conjugate to σ′_{v}(F r_{v})^{3} (since F r_{v}^{3} belongs to W_{E_{w}}). Thus

( \begin{matrix}a_{v}^{3} & 0 \\ 0 & b_{v}^{3} \end{matrix} ) is conjugate to  ( \begin{matrix}c_{v}^{3} & 0 \\ 0 & d_{v}^{3} \end{matrix} ) .

In particular, for some pair of cube roots of 1, say ξ and ξ′,

\begin{aligned}c_{v} & = ξa_{v} and d_{v} = ξ′b_{v} \\ c_{v} & = ξb_{v} and d_{v} = ξ′a_{v} . \end{aligned}

We claim now that ξ′ = ξ^{2}. Indeed π_{ps}(σ) was chosen so that ω_{π_{ps}(σ)} = det(σ). Since this implies det σ′_{v} = det σ_{v}, we must have ξ, ξ′ = 1, i.e., ξ′ = ξ^{2}. So to prove (7.1.1) it will suffice to prove

(7.1.2)  ξ = 1

To continue, let us assume (for the moment) that

(7.1.3)  Ad ∘ σ′_{v} = Ad ∘ σ_{v} .

Since the kernel of Ad : GL_{2}(ℂ) ⟶ GL_{3}(ℂ) is precisely the group of scalar matrices

\{( \begin{matrix}λ & 0 \\ 0 & λ \end{matrix} )\}, it follows from (7.1.3) that σ_{v}(F r_{v}) and σ′_{v}(F r_{v}) must

differ by some scalar λ ≠ 0. Thus

( \begin{matrix}ξa_{v} & 0 \\ 0 & ξ^{2}b_{v} \end{matrix} ) is conjugate to ( \begin{matrix}λa_{v} & 0 \\ 0 & λb_{v} \end{matrix} ) ,

and it suffices to prove

λ = 1 .

If λa_{v} = ξa_{v} and λG_{v} = ξ^{2}b_{v} then λ = ξ = ξ^{2} = 1 for the trivial reason that ξ is a cube root of 1. On the other hand, if λa_{v} = ξ^{2}b_{v} and λb_{v} = ξa_{v}, then λ^{2} = 1 (since a_{v} = ξ^{2}/λb_{v} = \frac{λ}{ξ} b_{v}). If λ = −1, this means that the image of

σ_{v}(F r_{v}) = ( \begin{matrix}a_{v} & 0 \\ 0 & b_{v} \end{matrix} ) = ( \begin{matrix}a_{v} & 0 \\ 0 & a_{v} \end{matrix} ) ( \begin{matrix}1 & 0 \\ 0 & ξλ \end{matrix} )

in P GL_{2}(ℂ) is of order 6 (since ξλ will then have order 6). But as A_{4} has no elements of order 6, this means we are done.

It remains to prove (7.1.3). For this, we note (following Serre) that Ad ∘ σ : W_{F} ⟶ GL_{3}(ℂ) is a monomial representation. In particular, there is a character θ of W_{E} (not invariant by Gal(E/F)) such that Ad ∘ σ = Ind_{W_{E}}^{W_{F}}θ. This means (again by Theorem 5.3.1, this time with n = 3) that there is associated to this irreducible representation Ad ∘ σ a cuspidal automorphic representation of GL_{3}(𝔸_{F}), call it ∏_{1}. On the other hand, by the “symmetric square lift” (Theorem 5.3.2) π_{ps}(σ) has a lift to GL_{3}(𝔸_{F}), call it ∏_{1}^{∗}, which is almost everywhere associated to the Langlands parameter Ad ∘ σ′_{v}. Thus to prove (7.1.3), it clearly suffices to prove that

(7.1.4)  ∏_{1} ≈^{∘} ∏_{1}^{∗} .

N.B. The automorphic representation ∏_{1}^{∗} will be cuspidal automorphic (by Theorem 5.3.2) if and only if π_{ps}(σ) is not monomial. But if π_{ps}(σ) were equal to π(σ′) for any irreducible two dimensional (let alone monomial) representation of W_{F} we would have to conclude that σ′ = σ (which is impossible, since σ is tetrahedral, not monomial). Therefore ∏_{1}^{∗} is also cuspidal, and the proof of 7.1.4 reduces to the following:

Lemma. The Rankin-Selberg L-functions L(s, ∏_{1}^{∗} × ˜∏1) on GL(3)× GL(3) has a pole at s = 1 (and so - by Theorem 5.3.3 - ∏_{1}^{∗} is indeed isomorphic to ∏_{1}).

Proof. By definition

L(s, ∏_{1}^{∗} × ˜∏_{1}) = ∏_{v}L(s, ∏_{1}^{∗})_{v} × ( ˜∏_{1})_{v}) ,

where for almost every v (namely the “unramified” v),

L(s, (∏_{1}^{∗})_{v} × ( ˜∏_{1})_{v}) = L(s, (Ad ∘ σ′_{v}) ⊗ (Ad ∘ σ̃_{v})) .

Keeping in mind that Ad ∘ σ is monomial, it is possible to check that we also have

(7.1.5)  L(s, (∏_{1}^{∗})_{v} × ( ˜∏_{1})_{v}) = L(s, (∏_{1})_{v} × ( ˜∏_{1})_{v})

(again for almost every v). Indeed, since Ad ∘ σ is induced from θ on E, we have

Ad ∘ (σ̃_{v}) = ⨁_{w|v} Ind_{W_{Ew}}^{W_{Fv}} θ_{w}^{−1} .

Hence

Ad(σ′_{v}) ⊗ Ad(σ̃_{v}) = ⨁_{w|v} Ind_{W_{Ew}}^{W_{Fv}} (θ_{w}^{−1} ⊗ ∑_{w}^{1})

if ∑_{w} (resp, ∑′_{w}) denotes the restriction of Ad(σ_{v}) (resp. Ad(σ′_{v})) to W_{E_{w}}. (Here we are using the fact that for σ (resp. ∑) a representation of some group G (resp. a subgroup H), σ ⊗ Ind_{H}^{G} ∑ ≅ Ind_{H}^{G}(∑ ⊗Res σ|_{H}).) Similarly we have

Ad(σ_{v}) ⊗ Ad(σ̃_{v}) = ⨁_{w|v} Ind_{W_{Ew}}^{W_{Fv}} (θ_{w}^{−1} ⊗ ∑_{w}) .

So since ∑_{w} ≅ ∑′_{w} almost everywhere (by construction), we indeed have

L(s, (∏_{1}^{∗})_{v} × ( ˜∏_{1})_{v}) = L(s, Ad(σ′_{v}) ⊗ Ad(σ̃_{v}))
= L(s, Ad(σ_{v}) ⊗ Ad(σ̃_{v}))
= L(s, (∏_{1})_{v} × ( ˜∏_{1})_{v})

for almost every v.

Using (7.1.5), it remains to show that ∏_{1}^{∗} = ∏_{1}. So suppose (7.1.5) holds for all v outside the finite set S. Then

\begin{aligned}L(s, ∏_{1}^{∗} × ˜∏_{1}) & = (∏_{v∈S} \frac{L(s, (∏_{1}^{∗})_{v} × ( ˜∏_{1})_{v})}{L(s, (∏_{1})_{v} × ( ˜∏_{1})_{v})}) \\ & × L(s, ∏_{1} × ˜∏_{1}) . \end{aligned}

But by Theorem 5.3.3, L(s, ∏_{1} × ˜∏_{1}) has a pole at s = 1; moreover, the quotient expression in parentheses above is non-zero at s = 1. Therefore L(s, ∏_{1}^{∗} × ˜∏_{1}) also has a pole at s = 1, as asserted, and this in turn implies (by the same Theorem 5.3.3) that ∏_{1}^{∗} ≅ ∏_{1}.

(7.2). The Octahedral Case

(a) Choosing π_{ps}(σ)

In this case, the image of σ(W_{F}) in P GL_{2}(ℂ) is S_{4}, and the pull-back of the normal subgroup A_{4} ⊂ S_{4} is the Weil group W_{E} of a quadratic extension E of F.

Pictorially:

1 ⟶ W_{E} ⟶ W_{F} ⟶ Gal(E/F) ⟶ 1
↓  ↓  ↓ ≀
1 ⟶ A_{4} ⟶ S_{4} ⟶  ℤ_{2}  ⟶ 1

Since σ_{E} = Res σ |_{W_{E}} is now of tetrahedral type, we know π(σ_{E}) exists as an irreducible cuspidal representation of GL_{2}(𝔸_{E}) (by the results of the last paragraph). Moreover, we again have π(σ_{E}) invariant under the action of Gal(E/F). So again by Theorem 6.1, we conclude that π(σ_{E}) must equal BC_{E/F} (π_{i}) for (this time) two irreducible cuspidal representations π_{i} of GL_{2}(𝔸_{F}). The problem now is that we can no longer distinguish these π_{i}’s by their central characters. Indeed, π_{1} now equals π_{2} ⊗ ω for a quadratic character of F^{x} \ 𝔸^{x}; hence ω_{π_{1}} = ω_{π_{2}}ω^{2} = ω_{π_{2}}!

Tunnell’s contribution to the “Langlands-Tunnell Theorem” was to get around this problem by appealing to a new kind of base-change which appeared only after the publication of [La1], namely the following result:

Proposition (cf. [J-PS-S3]). If L is a cubic not necessarily Galois extension of F, then each automorphic cuspidal representation π of GL_{2}(𝔸_{F}) has a base change lift ∏ on GL_{2}(𝔸_{L}), i.e. ∏ = BC_{L/F} (π) is automorphic, and for almost every place v of F, and place w of L dividing v, π_{v} = π_{v}(σ_{v}) implies ∏_{w} = π(Res_{L_{w}/F_{v}}(σ_{v})).

The proof of [J-PS-S3] uses the theory of L-functions for the groups GL(3) and GL(2) × GL(3) (and is entirely analogous to Jacquet’s original proof of base change for GL_{2} over a quadratic extension in [Ja]). The idea is to introduce the representation ∏ on GL_{2}(𝔸_{L}) through the formula

L(s, ∏ ×χ) = L(s, π × π(χ)) ;

here χ is any Hecke character of L, π(χ) is the corresponding automorphic representation of GL_{3}(𝔸_{F}) (whose existence is assured by Theorem 5.3.1 in the non-Galois case - recall Remark 5.3.1 (e)), and L(s, π × π(χ)) is the Rankin-Selberg L-function on GL(2) × GL(3). Then one shows that L(s, ∏ ×χ) has the analytic properties required by the Converse Theorem to ensure that ∏ is automorphic. (The fact that each ∏_{w} is the base change lift of π_{v} is relatively easy to check, from the definitions.)

N.B. The trace formula methods of [La1] fail in this context precisely because there may not be any Galois group attached to L over F (hence no way to define the twisted trace R_{0}^{τ} ⋯). On the other hand, because L-function methods are used, there is no way to characterize the image of this base change map; fortunately, as we shall now see, there is also no need for this in the application Tunnell found for this result.

What Tunnell did in [Tu] is introduce L/F as the cubic (non-normal) subextension of K/F fixed by the 2-Sylow subgroup (of order 8) of S_{4}. (More precisely, L is the cubic subextension fixed by all elements of Gal(K/F) mapping to this chosen Sylow subgroup.) Then if M is the composition in K of L and E (the quadratic Galois extension chosen above), we have the diagram

K
D_{4}[  |
M  ] A_{4}
ℤ_{2} / | ] ℤ_{3}  ] S_{4}
S_{3}[  L E
\ |
F

and the crucial:

Lemma (cf. [Tu], p.174). There is a unique i = 1, 2 such that BC_{L/F} (π_{i}) = π(σ_{L}) (and this is the π to be designated as π_{ps}(σ)).

Proof. Note first that π(ρ) actually exists, since the 2-Sylow subgroup used to define L is just D_{4}, and therefore σ_{L} is monomial; similarly, BC_{L/F} (π_{i}) exists for i = 1, 2 by the Base Change Theorem quoted above. To prove the Lemma, one appeals to the identity

BC_{M/L}(BC_{L/F} (π_{i})) = π(σ_{M}) for i = 1, 2 .

(This is “transitivity of base change”; it follows immediately from the definition of base change.) Since BC_{L/F} (π_{2}) and BC_{L/F} (π_{1}) have the same (quadratic) base change to M, it follows that BC_{L/F} (π_{2}) ≈ BC_{L/F} (π_{1}) ⊗ ω_{M/L}.

Now we claim that the representations BC_{L/F} (π_{i}) are distinct for i = 1, 2. Indeed, if they were not, we would have BC_{L/F} (π_{1}) ≈ BC_{L/F} (π_{1}) ⊗ ω_{M/L}, which by Lemma 11.7 of [La1] implies π_{1} is “monomial”. By part (b) of Theorem 1, this would then imply BC_{M/L}(BC_{L/F} (π_{1})) = π(σ_{M}) is not cuspidal. But the image of σ_{M} in P GL_{2}(ℂ) is S_{3} ≈ D_{3}, which means that σ_{M} itself is monomial and irreducible, i.e., π(σ_{M}) is cuspidal. This contradiction establishes that BC_{L/F} (π_{1}) and BC_{L/F} (π_{2}) are the two (distinct) cuspidal representations of GL_{2}(𝔸_{L}) yielding π(σ_{M}) upon base change to M. Since we also have B_{M/L}(σ_{L}) = π(σ_{M}), it must be that π(σ_{L}) = BC_{L/F} (π_{i}) for (exactly) one i, as required.

(b) Proving π_{ps}(σ) = π(σ).

Write π_{ps}(σ) = ⊗π_{v}(σ′_{v}) as before. Then one proves exactly as in the tetrahedral case (but without having to take a lift to GL_{3}(ℂ)) that the non-existence of an element of order 6 in S_{4} implies σ_{v} ≅ σ′_{v} for almost all v. Since no new ideas are involved, we simply refer the reader to [Tu] for details.

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