LECTURES IN HANGZHOU, AUGUST 2005

Henry H. Kim

Department of Mathematics University of Toronto Toronto, Ontario M5S 3G3 CANADA henrykim@math.toronto.edu

The main reference is [CKM].

References

[CKM] J.W. Cogdell, H. Kim, and R. Murty, Lectures on Automorphic L-functions, American Mathematical Society, 2004.

[CKPSS] J.W. Cogdell, H. Kim, I.I. Piatetski–Shapiro, and F. Shahidi, Functoriality for the classical groups, Publ. Math. IHES 99 (2004), 163-233.

[Ki1] H. Kim, Functoriality for the exterior square of GL_{4} and the symmetric fourth of GL_{2}, J. of AMS 16 (2003), 139-183.

[Ki2] , On local L-functions and normalized intertwining operators, Can. J. Math 57 (2005), 535–597.

[Ki-Sa] H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selberg conjectures, appendix 2 to [Ki1].

[Ki-Sh] H. Kim and F. Shahidi, Functorial products for GL_{2} ×GL_{3} and functorial symmetric cube for GL_{2}, Ann. of Math. 155 (2002), 837-893.

[La] R.P. Langlands, Euler Products, Yale University Press, 1971.

[Sh1] F. Shahidi, On the Ramanujan conjecture and finiteness of poles for certain Lfunctions, Ann. of Math. 127 (1988), 547–584.

[Sh2] , A proof of Langlands conjecture on Plancherel measures; complementary series for p-adic groups, Annals of Math. 132 (1990), 273–330.

Typeset by AMS-TEX

Lecture 1. Langlands functoriality conjecture.

Let’s start with a simple question: For which primes p does x^{2} − x − 1 ≡ 0 (mod p) have 2 solutions?

This is equivalent to the following question: Let L = ℚ(\sqrt{5}). Then for which prime p splits completely in L?

The answer is given as follows: Let P (L/ℚ) = {p| p splits completely in L}. Then by the definition of Legendre symbol, P (L/ℚ) = \{p| (\frac{5}{p}) = 1\}. By computer calculation, P (L/ℚ) contains 11, 19, 29, 31, 41... If we look at the list carefully, we can see that they are primes congruent to 1, 4 modulo 5. Indeed, by quadratic reciprocity law, (\frac{5}{p}) = (\frac{p}{5}). Hence P (L/ℚ) = \{p| p ≡ 1, 4 (mod 5)\}.

More generally, let L/K be a Galois extension and let P (L/K) be the set of prime ideals in K which split completely in L.

Fact. P (L/K) determines L completely.

The goal of class field theory is to describe the Galois extension L in terms of data in K, namely, determine P (L/K) in terms of data in K. When L/K is abelian, the answer is given completely. For example, P (ℚ(ζ_{m})/ℚ) = {p ≡ 1 (mod m)}. But when L/K is not abelian, not much is known.

Example 1 (due to Serre). Consider f(x) = x^{3} − x − 1. Here the discriminant is -23. Let L be the splitting field of f. Then Gal(L/ℚ) ≃ S_{3}. If p is unramified, f(x) ≡ 0 (mod p) has 0,1,3 solutions. Then P (L/ℚ) = {p| f(x) ≡ 0 (mod p) has 3 solutions.}. By computer calculation, we see that P (L/ℚ) = {59, 101, 167, 173, ...}. On the other hand, f(x) ≡ 0 (mod p) has 0 solutions when p = 2, 3, 13, 29, 31, 41, ... It is hard to see the pattern. The pattern comes from modular forms. Let ρ : S_{3} ⟶ GL_{2}(ℂ) be the 2-dimensional representation of S_{3}. Then we have the Artin L-function L(s, ρ, L/ℚ). It is given by the Euler product

L(s, ρ, L/ℚ) = ∏_{p}(1 − a_{p}p^{−s} + (\frac{p}{23}) p^{−2s})^{−1},

where a_{p} = N_{f}(p) − 1, and N_{f}(p) is the number of solutions for f(x) ≡ 0 (mod p). Here L(s, ρ, L/ℚ) = \frac{ζ_{E}(s)}{ζ(s)}, where E = ℚ(α), and α is a root of f(x). This comes from the fact that Ind_{H}^{S_{3}}1 = 1 + ρ. Here H is the Galois group of L/E, and H ≃ ℤ/2ℤ.

Langlands functoriality. (1) L(s, ρ, L/ℚ) is the L-function attached to a modular form of weight 1, level 23. More precisely,

F = q ∏_{k=1}^{∞}(1 − q^{k})(1 − q^{23k}) = ∑_{n=1}^{∞} a_{n}q^{n}, q = e^{2πiτ} , τ = x + iy.

(2) P (L/ℚ) = {p| a_{p} = 2}.

Example 2 (due to Serre). f(x) = x^{4} −x−1. The Galois group is S_{4}. Let L be the splitting field of f and E = ℚ(α), where α is a root of f. Let H = Gal(L/E). Then H ≃ S_{3}, and Ind_{H}^{S_{4}}1 = 1 + ρ, where ρ : S_{4} ⟶ GL_{3}(ℂ) be a 3-dimensional representation. We have the Artin L-function L(s, ρ, L/ℚ) = \frac{ζ_{E}(s)}{ζ(s)}.

In order to attach a modular form to ρ, we need a 2-dimensional representation. So consider an extension L̃/L such that Gal(L̃/ℚ) = GL(2, 𝔽_{3}). Here (GL(2, 𝔽_{3}) : S_{4}) = 2. Then we have a 2-dimensional representation σ : GL(2, 𝔽_{3}) ⟶ GL_{2}(ℂ) such that ρ = Sym^{2}(σ). Langlands functoriality shows that σ is associated to F, a modular form of weight 1. Write F = ∑_{n=1}^{∞} a_{n}q^{n}. Then P (L/ℚ) = {p| a_{p} = ±2}.

Example 3. f(x) = x^{5} − x − 1. The Galois group is S_{5}. It is not solvable. Let L be the splitting field of f and E = ℚ(α), where α is a root of f. Let H = Gal(L/E). Then H ≃ S_{4}, and Ind_{H}^{S_{5}}1 = 1 + ρ, where ρ : S_{5} ⟶ GL_{4}(ℂ) be a 4dimensional representation. We have the Artin L-function L(s, ρ, L/ℚ) = \frac{ζ_{E}(s)}{ζ(s)}. We do not have the answer for P (L/ℚ). We only know its size. Chebotarev density theorem says that P (L/ℚ) has Dirichlet density \frac{1}{120}.

Langlands functoriality conjecture. There exists a cuspidal representation π = ⊗π_{p} of GL_{4} such that L(s, ρ, L/ℚ) = L(s, π), and P (L/ℚ) = {p| Satake parameter of π_{p} is diag(1, 1, 1, 1)}.

Weaker assertion (Artin conjecture). L(s, ρ, L/ℚ) is entire.

We only know that L(s, ρ, L/ℚ) has meromorphic continuation to all of ℂ and satisfies a functional equation.

More generally, Langlands conjectured that given an irreducible representation ρ : Gal(\overline{ℚ}/ℚ) ⟶ GL_{n}(ℂ), there exists a cuspidal representation π = ⊗π_{p} of GL_{n} such that ρ(F rob_{p}) =Satake parameter of π_{p}. It is usually referred to as the strong Artin conjecture. A weaker assertion, known as Artin conjecture, claims that the Artin L-function L(s, ρ) is entire.

Much effort has been made when n = 2. Let ρ̄ : Gal(\overline{ℚ}/ℚ) ⟶ P GL_{2}(ℂ) ≃ SO_{3}(ℂ). Then Im(ρ̄) is D_{2n} (dihedral), A_{4} (tetrahedral), S_{4} (octahedral), A_{5} (icosahedral). The first three groups are solvable. It is a theorem of Langlands and Tunnell that for the case of A_{4}, S_{4}, the strong Artin conjecture is true, and it has been used by Andrew Wiles in his proof of Fermat’s last theorem.

Cuspidal representations generalize classical modular forms (holomorphic and Maass forms). They can be understood as “direct summands” of the right regular representation of G(𝔸) on the Hilbert space L^{2}(G(F)\G(𝔸)), where 𝔸 is the ring of adeles. If π is a cuspidal representation, then we have a tensor product decomposition π = ⊗ π_{v}, where v runs through all places of F; π_{v} is an irreducible, unitary representation of G(F_{v}) for all v; π_{v} is spherical for almost all v, namely, it has the Satake parameter (semi-simple conjugacy class) {t_{v}} in ^{L}G (the L-group of G).

When G = GL_{2}, F = ℚ, there are two types of cuspidal representations: First, cuspidal representations attached to holomorphic cusp forms of weight k with respect to a congruence subgroup of SL_{2}(ℤ); π = π_{f}, where f(τ) = ∑_{n=1}^{∞} a_{n}n^{\frac{k−1}{2}} e^{2πinτ} , τ = x + iy. Then π_{f} = ⊗π_{p}, and the Satake parameter of π_{p} is diag(α_{p}, β_{p}), where a_{1} = 1, a_{p} = α_{p} + β_{p}.

Second, cuspidal representations attached to Maass cusp forms. They are eigenfunctions of the Laplacian; y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) f + (\frac{1}{4} − t^{2})f = 0, where t ∈ iℝ

or t ∈ ℝ, 0 < t < \frac{1}{2}. Then f(τ) = ∑_{n≠0} a_{n}|n|^{− \frac{1}{2}} W (nτ), where W (τ) = y^{\frac{1}{2}} K_{t}(2πy)e^{2πix} and K_{t} is the K-Bessel function.

Ramanujan conjecture. |α_{p}| = |β_{p}| = 1.

Selberg conjecture. t ∈ iℝ, or \frac{1}{4} − t^{2} ≥ \frac{1}{4}.

Theorem (Deligne, 1973). Ramanujan conjecture is true for holomorphic cusp forms.

Most general form of Langlands functoriality conjecture: Let H, G be two reductive groups. To each homomorphism of L-groups, r : ^{L}H ⟶ ^{L}G, there is associated a lift (transfer) of automorphic representations of H to automorphic representations of G which satisfy canonical properties.

Example 1. H = \{e\}, G = GL_{n} over ℚ. Then ^{L}H = Gal(\overline{ℚ}/ℚ) and ^{L}G = GL_{n}(ℂ). Then Langlands functoriality conjecture is the strong Artin conjecture.

Example 2. H = GL_{2}, G = GL_{m+1}. Let Sym^{m} : GL_{2}(ℂ) ⟶ GL_{m+1}(ℂ) be the mth symmetric power representation. We can prove the functoriality for m ≤ 4. We will explain this in Lecture 3.

Example 3. H = SO_{2n+1}, SO_{2n}, Sp_{2n}, G = GL_{N}, where N = 2n or 2n + 1. Then ^{L}H = Sp_{2n}(ℂ), SO_{2n}(ℂ), SO_{2n+1}(ℂ), and r : ^{L}H ⟶ ^{L}G is the embedding. This case is explained in detail by J. Cogdell.

Let us consider very particular Galois representations, namely, monomial representations: Given a 1-dimensional character χ of H < G, we can form an induced representation Ind_{H}^{G}χ. Let (G : H) = n. Then ρ = Ind_{H}^{G}χ : G ⟶ GL_{n}(ℂ). It is called monomial representation. We assume that ρ is irreducible. Let G = Gal(L/ℚ), H = Gal(L/E). Then by class field theory, χ can be considered as a grössencharacter of E of finite order, and L(s, Ind_{H}^{G}χ, L/ℚ) = L(s, χ), Hecke L-function. Hence L(s, Ind_{H}^{G}χ, L/ℚ) is entire, and Artin conjecture is true. However, the strong Artin conjecture is open. Namely, it is expected that there exists a cuspidal representation π of GL_{n} such that L(s, Ind_{H}^{G}χ, L/ℚ) = L(s, π). We call π automorphic induction, we denote it by I_{E}^{ℚ}χ.

Fact. Langlands functoriality for monomial representations implies Langlands functoriality for general Galois representations.

Known cases: (1) E/ℚ is cyclic (solvable Galois). It is a special case of Arthur-Clozel. (2) E/ℚ is non-normal cubic extension. It is a result of Jacquet, Piatetski-Shapiro, and Shalika, a consequence of the converse theorem for GL_{3}. (3) The Galois closure of E/ℚ is solvable and χ is certain algebraic character. It is a result of M. Harris.

I will give two examples of non-normal E/ℚ whose Galois closure is not solvable; one is non-normal quintic extension, and the other, non-normal sextic extension.

Consider f, irreducible quintic polynomial with integer coefficients such that the Galois group is A_{5}. (J. Buhler’s example: f(x) = x^{5} +10x^{3} −10x^{2} +35x−18) Let L be the splitting field of f, E = ℚ(α), where α is a root of f. Then H = Gal(L/E) ≃ A_{4}. Let K/E be such that Gal(L/K) be the Klein 4-group and Gal(K/E) is a cyclic group of order 3. Let χ be the grössencharacter of E attached to K/E by class field theory. It is considered as a character of H.

Theorem. Under certain conditions, Ind_{H}^{A_{5}}χ is automorphic, namely, there exists a cuspidal representation π = I_{E}^{ℚ}χ of GL_{5} such that L(s, Ind_{H}^{A_{5}}χ, E/ℚ) = L(s, π).

Note that A_{5} does not have a 2-dimensional representation. But there exists a central extension L′/ℚ such that G′ = Gal(L′/ℚ) has a 2-dimensional representation. Let ρ : G′ ⟶ GL_{2}(ℂ). Such ρ factors through Ã_{5} ≃ SL(2, 𝔽_{5}), where Gal(L̃/ℚ) = Ã_{5}. In many cases, ρ is modular, namely, there exists a modular form f of weight 1 such that L(s, ρ) = L(s, f). For example, R. Taylor showed that f(x) = x^{5} − 2x^{3} + 2x^{2} + 5x + 6 gives rise to such ρ. The condition in the above theorem is that ρ is modular.

Here Ind_{H}^{A_{5}}χ is equivalent to Sym^{4}(ρ), twisted by a character. Let ρ give rise to a cuspidal representation π of GL_{2}. Then Sym^{4}ρ gives rise to Sym^{4}π. It is a cuspidal representation of GL_{5} (the result of Kim, Kim-Shahidi. We will explain this in Lecture 3).

Second example (This is a new result which has not been published elsewhere.): Let N be the normalizer of a 5-Sylow subgroup in Ã_{5}. It is a group of order 20, isomorphic to ℤ/2ℤ × D_{10} ≃ D_{20}. Let L̃/ℚ be such that Gal(L̃/ℚ) = Ã_{5}, and Gal(L̃/F) = N. Then F/ℚ is a non-normal sextic extension. Take a quadratic character χ of N, non-trivial on its 2 component. Namely, let F̃/F be a quadratic extension such that Gal(L̃/F̃) ≃ D_{10}. Then χ is the quadratic character attached to F̃/F by class field theory. Then the induced character IndÃ_{5} ; Nχ is equivalent to Sym^{5}(ρ), twisted by a character. (It is due to Serre.) If ρ gives rise to a cuspidal representation π of GL_{2}, Sym^{5}ρ gives rise to Sym^{5}π. It is a cuspidal representation of GL_{6} (the result of Kim-Shahidi, Wang). Hence we have

Theorem. Suppose ρ is modular. Then IndÃ_{5} ; Nχ is automorphic, namely, there exists a cuspidal representation π = I_{F}^{ℚ}χ of GL_{6} such that L(s, IndÃ_{5} ; Nχ, F/ℚ) = L(s, π).

Another consequence is that Sym^{m}(π) is automorphic for all m.

Irreducible representations of G′ are equivalent to a twist of irreducible representations of Ã_{5}. We know all irreducible representations of Ã_{5}. Let σ, σ^{τ} be two 2-dimensional representations of Ã_{5}, where τ ∈ Aut(ℂ) sends \sqrt{5} to −\sqrt{5}. Then irreducible representations of Ã_{5} are

(1) trivial (1 dim)

(2) σ, σ^{τ} (2 dim)

(3) Sym^{2}(σ), Sym^{2}(σ^{τ}) (3 dim)

(4) Sym^{3}(σ) ≃ Sym^{3}(σ^{τ}), σ ⊗ σ^{τ} (4 dim)

(5) Ind_{H}^{A_{5}} χ ≃ Sym^{4}(σ) ≃ Sym^{4}(σ^{τ}) (5 dim)

(6) Sym^{2}(σ) ⊗ σ^{τ} ≃ σ ⊗ Sym^{2}(σ^{τ}) ≃ Sym^{5}(σ) ≃ Sym^{5}(σ^{τ}) (6 dim)

Kim-Shahidi proved that Sym^{5}(π) is an automorphic representation of GL_{6}. S. Wang proved that it is cuspidal. Now Sym^{m}(ρ) is equivalent to a direct sum of irreducible representations of Ã_{5} twisted by a character. Each direct summands is automorphic. Hence Sym^{m}(π) is automorphic.

Lecture 2. Automorphic L-functions and Langlands-Shahidi method.

An L-function is a very special type of meromorphic functions of one complex variable. On the surface it is not clear why the L-functions play decisive roles. It was Riemann who introduced his zeta function in order to study the distribution of prime numbers: The prime number theorem says that π(x) = ∑_{p≤x} 1 ∼ \frac{x}{ln x}. It is equivalent to the fact that ζ(1 + it) ≠ 0 for t ∈ ℝ. Dirichlet’s theorem on arithmetic progression says that there are infinitely many primes in the arithmetic progression an + b, where (a, b) = 1, n = 1, 2, ... It comes from the fact that L(1, χ) ≠ 0 for a Dirichlet character χ mod a.

An L-function is associated to the set A; arithmetic-geometric objects such as Galois groups, elliptic curves, and Shimura varieties. It is also associated to the set B; automorphic forms and representations. Langlands conjecture is that the set B contains the set A. The L-functions in the set B are called automorphic L-functions. Special case of such relationship for elliptic curves is called Taniyama-Shimura-Wiles theorem, i.e., elliptic curves over ℚ are modular.

Over ℚ, an L-function which is associated to an object F takes the form of Euler product over all primes p, L(s, F) = ∏_{p} L_{p}(s, F), L_{p}(s, F) = ∏_{j=1}^{m}(1 − α_{j}(p)p^{−s})^{−1} for almost all primes, where α_{j}(p) ∈ ℂ. As a function of s ∈ ℂ, this product converges absolutely for Re(s) >> 0 and we can multiply out to get a series L(s, F) = ∑ a(n)n^{−s}.

Example (1) Riemann zeta function ζ(s) = ∏_{p}(1 − p^{−s})^{−1} = ∑_{n=1}^{∞} n^{−s}.

(2) Dirichlet L-function L(s, χ) = ∏_{p}(1−χ(p)p^{−s})^{−1} = ∑_{n=1}^{∞} χ(n)n^{−s}, where χ is a character of (ℤ/qℤ)^{×}.

(3) E : y^{2} = x^{3} + ax + b, a, b ∈ ℤ. Let N_{E}(p) be the number of solutions mod p. Let a_{E}(p) = p − N_{E}(p). Then L_{p}(s, E) = (1 − a_{E}(p)p^{−s} + p^{−2s})^{−1} for p non-singular.

More explicitly, let E : y^{2} = x^{3} − 4x^{2} + 16. Then we have some numerical calculations (due to Silverman):

\begin{matrix}p & 2 & 3 & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 97 & 239 & 1327 & 3733 & 4817 \\ N_{E}(p) & 2 & 4 & 4 & 9 & 10 & 9 & 19 & 19 & 24 & 104 & 269 & 1259 & 3619 & 4949 \\ a_{E}(p) & 0 & −1 & 1 & −2 & 1 & 4 & −2 & 0 & −1 & −7 & −30 & 68 & 114 & −132 \end{matrix}

Let F = q ∏_{k=1}^{∞}((1 − q^{k})(1 − q^{11k}))^{2} = ∑_{n=1}^{∞} b_{n}q^{n}. Here F is a modular form of weight 2. Consider the L-function L(s, F) = ∑_{n=1}^{∞} b_{n}n^{−s}. Then

L(s, F) = ∏_{p≠11}(1 − b_{p}p^{−s} + p^{−2s})^{−1}(1 − b_{11}11^{−s})^{−1}.

We have a_{E}(p) = b_{p} for all p ≠ 2.

(4) Automorphic L-functions. Let π = ⊗π_{v} be a cuspidal representation of G(𝔸), where G is a split reductive group. Let ^{L}G be the L-group of G. Let r : ^{L}G ⟶ GL_{N}(ℂ) be a finite dimensional representation of ^{L}G. For v ∉ S, π_{v} is spherical and it gives rise to a Satake parameter (semi-simple conjugacy class) {t_{v}}, t_{v} ∈ ^{L}G. Form the local L-function L(s, π_{v}, r) = det(I − r(t_{v})q_{v}^{−s})^{−1}. Let L_{S}(s, π, r) = ∏_{v∉S} L(s, π_{v}, r).

For example, let π = ⊗π_{v} be a cuspidal representation of GL_{2}(𝔸). Let diag(α_{v}, β_{v}) be the Satake parameter of π_{v}. Let Sym^{m} : GL_{2}(ℂ) ⟶ GL_{m+1}(ℂ) be the mth symmetric power representation. Then

L(s, π_{v}, Sym^{m}) = ∏_{j=0}^{m}(1 − α_{v}^{m−j}β_{v}^{j}q_{v}^{−s})^{−1}.

Central problems: (1) L(s, F) has a meromorphic continuation to all of ℂ and satisfies a function equation of the form: Let Λ(s, F) = L(s, F) (some γ-factors and factors at bad places). Then Λ(s, F) = ϵ(s, F)Λ(1 − s, F′). For example, Λ(s) = ζ(s)π^{− \frac{s}{2}} Γ(\frac{s}{2}) = Λ(1 − s).

(2) Λ(s, F) is bounded in vertical strips.

(3) Grand Riemann Hypothesis; non-trivial zeros of L(s, F) are all on Re(s) = \frac{1}{2}.

(4) Generalized Ramanujan conjecture: |α_{j}(p)| = 1.

(5) Birch, Swinnerton-Dyer conjecture: Let E/ℚ be an elliptic curve. The order of vanishing of L(s, E) at s = 1 (center of symmetry) is equal to the rank of the group of rational points on E.

Here (3) and (5) are two of seven one million dollar prize problems of Clay Math. Institute. There are other problems such as Siegel zeros (real zeros close to 1), and special value problems. For example, L(1, χ) contains a class number of a quadratic extension K/ℚ, and χ is the non-trivial character of Gal(K/ℚ).

Here even meromorphic continuation is not obvious. For example, it is clear that ∏_{p≡1 (mod 4)}(1 − p^{−s})^{−1} converges for Re(s) > 1. We can continue up to Re(s) > 0. But it is known that it has no meromorphic continuation to all of ℂ.

Let π = ⊗π_{v} be a cuspidal representation of GL_{2}(𝔸). The L-function L(s, π, Sym^{m}) was introduced by Langlands to solve Ramanujan and Sato-Tate conjecture. For example, if we know that L(s, π, Sym^{m}) is absolutely convergent for Re(s) > 1 for

all m, then |α_{v}^{m}| ≤ q_{v}, |β_{v}^{m}| ≤ q_{v} for all m. This implies that |α_{v}| ≤ qv^{\frac{1}{m}} , |β_{v}| ≤ qv^{\frac{1}{m}}

for all m. Since |α_{v}β_{v}| = 1, we have |α_{v}| = |β_{v}| = 1. Kim-Shahidi proved meromorphic continuation and functional equation for m ≤ 9.

Let π = ⊗π_{v} be a cuspidal representation of M(𝔸), where π_{v} is spherical for v ∉ S. Let ψ = ⊗ψ_{v} be a fixed character of 𝔸/F.

Conjecture (Langlands). For v ∈ S, we can define a local factor L(s, π_{v}, r) (of the form P_{v}(q_{v}^{−s})^{−1}, where P_{v} is a polynomial with constant 1), and a local ϵ-factor ϵ(s, π_{v}, r, ψ_{v}) (monomial in q_{v}^{−s}) such that L(s, π, r) = ∏_{all v} L(s, π_{v}, r) has a meromorphic continuation to all of ℂ and satisfies a functional equation L(s, π, r) = ϵ(s, π, r)L(1 − s, π, r̃), where r̃(g) = ^{t}r(g)^{−1}.

There are two ways of studying automorphic L-functions:

(1) method of Rankin-Selberg (integral representations); expresses L-functions as integrals of Eisenstein series, theta functions, etc. For example, Riemann proved that

π^{− \frac{s}{2}} Γ(\frac{s}{2})ζ(s) = ∫_{0}^{∞} x^{\frac{s}{2}} \frac{1}{2}(θ(x) − 1) \frac{dx}{x}, θ(x) = ∑_{−∞}^{∞} e^{−n^{2}πx}.

The Poisson summation formula gives rise to the functional equation of the theta function, θ(x^{−1}) = x^{\frac{1}{2}} θ(x), and the functional equation of the Riemann zeta function follows.

(2) Langlands-Shahidi method; uses Eisenstein series attached to maximal parabolic subgroups.

We will briefly explain Langlands-Shahidi method. Let P = MN be a maximal parabolic subgroup of G. Let π be a cuspidal representation of M(𝔸). Then we can form an induced representation, for s ∈ ℂ,

I(s, π) = Ind_{P}^{G}π ⊗ exp(sα̃, H_{P} ( )),

where α̃ is the fundamental weight corresponding to α, and α is a simple root such that P is associated to Δ − {α}. (Δ is the set of simple roots) For example, if P = MN ⊂ Sp_{2n}, M ≃ GL_{n} (Siegel parabolic subgroup), then I(s, π) = Ind_{P}^{G}π ⊗ |det|^{s}.

Given f_{s} ∈ I(s, π), we define an Eisenstein series

E(s, f_{s}, g) = ∑_{γ∈P (F)\G(F)} f_{s}(γg).

Let E_{0}(s, f_{s}, g) = ∫_{N(F)\N(𝔸)} E(s, f_{s}, ng) dn. It is called constant term. If P is self-conjugate (most cases),

E_{0}(s, f_{s}, g) = f_{s}(g) + M(s, π)f_{s}(g), M(s, π)f_{s}(g) = ∫_{N(𝔸)} f_{s}(w_{0}^{−1}ng) dn,

where w_{0} is a Weyl group element. M(s, π) is called global intertwining operator from I(s, π) to I(−s, w_{0}(π)).

Langlands proved that the poles of E(s, f_{s}, g) and M(s, π) are the same and they have meromorphic continuation to all of ℂ and satisfy a functional equation E(−s, M(s, π)f_{s}, g) = E(s, f_{s}, g).

Let f_{s} = ⊗f_{v} and I(s, π) = ⊗I(s, π_{v}). Then M(s, π) = ⊗A(s, π_{v}, w_{0}), where

A(s, π_{v}, w_{0})f_{v}(g) = ∫_{N(F_{v})} f_{v}(w_{0}^{−1}ng) dn.

It is called local intertwining operator. For almost all v, f_{v} is the unique K_{v}-fixed vector in I(s, π_{v}) such that f_{v}(e) = 1. Then Langlands proved (1971),

A(s, π_{v}, w_{0})f_{v} = ∏_{i=1}^{m} \frac{L(is, π_{v}, r_{i})}{L(1 + is, π_{v}, r_{i})}f̃_{v},

where f̃_{v} is the unique K_{v}-fixed vector in I(−s, w_{0}(π_{v})) and r_{i} is certain irreducible finite-dimensional representation of ^{L}M. Hence

M(s, π) = ∏_{i=1}^{m} \frac{L_{S}(is, π, r_{i})}{L_{S}(1 + is, π, r_{i})} ⊗_{v∉S} f̃_{v} ⊗ ⊗_{v∈S}A(s, π_{v}, w_{0})f_{v}.

By induction, this gives a meromorphic continuation of L_{S}(s, π, r_{i}) for each i. But it does not give the desired functional equation.

At the suggestion of Langlands, Shahidi calculated ψ-Fourier coefficients of E(s, f_{s}, g) for globally generic cuspidal representations, where ψ is a generic character of U. Here U is a maximal unipotent subgroup such that B = T U is a Borel subgroup. Then ψ_{M} = ψ|_{U_{M}} is a generic character of U_{M} = U ∩ M. We say that π = ⊗π_{v} is ψ_{M}-globally generic if ∫_{U_{M} (F)\U_{M} (𝔸)} φ(ug)\overline{ψ_{M}(u)} du ≠ 0 for a cuspidal function φ in the space of π. This implies that each π_{v} is locally generic, i.e., has a Whittaker model. If λ_{ψ_{v}}(s, π_{v}) is the Whittaker functional for the space of I(s, π_{v}), then by the uniqueness of Whittaker functional up to a constant,

λ_{ψ_{v}}(s, π_{v}) = C_{ψ_{v}}(s, π_{v}, w_{0})λ_{ψ_{v}}(−s, w_{0}(π_{v}))A(s, π_{v}, w_{0}),

for some constant C_{ψ_{v}}(s, π_{v}, w_{0}) ∈ ℂ.

Consider ψ-Fourier coefficient of Eisenstein series

E_{ψ}(s, f_{s}, g) = ∫_{U(F)\U(𝔸)} E(s, f_{s}, ug)\overline{ψ(u)} du.

Then Shahidi showed that

E_{ψ}(s, f_{s}, e) = \frac{∏_{v∈S} W_{v}(e_{v})}{∏_{i=1}^{m} L_{S}(1 + is, π, r_{i})}, W_{v}(e_{v}) = λ_{ψ_{v}}(s, π_{v})(f_{v}).

Functional equation of Eisenstein series implies

∏_{i=1}^{m} L_{s}(is, π, r_{i}) = ∏_{v∈S} C_{ψ_{v}}(s, π_{v}, w_{0}) ∏_{i=1}^{m} L_{S}(1 − is, π, r̃_{i}).

Induction on i and detailed analysis of C_{ψ_{v}}(s, π_{v}, w_{0}) lead to the definition of local factors L(s, π_{v}, r_{i}), ϵ(s, π_{v}, r_{i}, ψ_{v}) for v ∈ S, and the functional equation L(s, π, r_{i}) = ϵ(s, π, r_{i})L(1 − s, π, r̃_{i}) (Shahidi 1990).

But it had been thought that the location of poles is hard to obtain from Langlands-Shahidi method. Things changed by using spectral theory:

Langlands’ Lemma. Consider the residual spectrum L^{2} = L_{dis}^{2}(G(F)\G(𝔸))_{(M,π)}. (1) L^{2} = {0} unless w_{0}π ≃ π.

(2) L^{2} is spanned by the residues of E(s, f_{s}, g) for s > 0.

(3) Suppose E(s, f_{s}, g) has a pole at s = s_{0}. Then L^{2} = {⊗_{v}J(s, π_{v})}, where J(s, π_{v}) is the image of N(s, π_{v}, w_{0}), the normalized local intertwining operator, namely,

A(s, π_{v}, w_{0}) = ∏_{i=1}^{m} \frac{L(is, π_{v}, r_{i})}{L(1 + is, π_{v}, r_{i})ϵ(is, π_{v}, r_{i})}N(s, π_{v}, w_{0}).

Then

(*)  M(s, π) = ∏_{i=1}^{m} \frac{L(is, π, r_{i})}{L(1 + is, π, r_{i})ϵ(is, π, r_{i})} ⊗_{v} N(s, π_{v}, w_{0}).

(1) and (2) imply that M(s, π) is holomorphic for s > 0 unless w_{0}π ≃ π.

Trick. w_{0}(π ⊗ χ) ≄ π ⊗ χ if χ is a grössencharacter which is highly ramified at one finite place.

We denote π ⊗ χ by π. Then M(s, π) is holomorphic for s > 0.

We have to show that each local operator N(s, π_{v}, w_{0}) is holomorphic and non-vanishing for Re(s) ≥ \frac{1}{2}. This requires the study of representations of p-adic groups. The main ingredients are the following standard module conjecture and classification of discrete series representations.

Standard module conjecture. Given a non-tempered generic representations π_{v}, there is a tempered data π_{0} and a complex parameter Λ_{0} which is in the corresponding positive Weyl chamber so that π_{v} = I_{M_{0}}(Λ_{0}, π_{0}) = Ind_{M_{0}}^{M}(π_{0} ⊗

q_{v}^{<Λ_{0},H_{P_{0}}^{M} ( )>}).

According to Langlands, any non-tempered representation π_{v} can be written as a Langlands’ quotient, namely, the quotient of I_{M_{0}}(Λ_{0}, π_{0}). The above conjecture claims that if π_{v} is generic, π_{v} is I_{M_{0}}(Λ_{0}, π_{0}) itself.

Theorem. Except for certain cases (for the exact list, see [Ki2].) the local normalized intertwining operators N(s, π_{v}, w_{0}) are holomorphic and non-vanishing for Re(s) ≥ \frac{1}{2}.

Let us illustrate it with an example of GL_{m} × SO_{2n+1}. Let σ_{v}, π_{v} be generic unitary non-tempered representations of GL_{m}(F_{v}), SO_{2n+1}(F_{v}), resp. By the standard module conjecture proved by Muić, σ_{v}, π_{v} can be written as follows:

σ_{v} = Ind (|det|^{α_{1}}σ_{1} ⊗ ⋯ ⊗ |det|^{α_{p}}σ_{p} ⊗ σ_{p+1} ⊗ |det|^{−α_{p}}σ_{p} ⊗ ⋯ ⊗ |det|^{−α_{1}}σ_{1}),
π_{v} = Ind (|det|^{β_{1}}τ_{1} ⊗ ⋯ ⊗ |det|^{β_{q}}τ_{q} ⊗ τ_{0}),

where 0 < α_{p} ≤ ⋯ ≤ α_{1} < \frac{1}{2}, 0 < β_{q} ≤ ⋯ ≤ β_{1}, and σ_{i}, τ_{j}, i = 1, ⋯, p, j = 1, ⋯, q, are discrete series representations of GL(F_{v}), τ_{0} (resp. σ_{p+1}) is a generic tempered representation of SO_{2r+1}(F_{v}) (resp. GL(F_{v})).

First we need to prove that β_{1} < 1. Next we have

I(s, σ_{v} ⊗ π_{v}) = Ind (|det|^{s+α_{1}}σ_{1} ⊗ ⋯ ⊗ |det|^{s+α_{p}}σ_{p} ⊗ |det|^{s}σ_{p+1}⊗

|det|^{s−α_{p}}σ_{p} ⊗ ⋯ ⊗ |det|^{s−α_{1}}σ_{1} ⊗ |det|^{β_{1}}τ_{1} ⊗ ⋯ ⊗ |det|^{β_{q}}τ_{q} ⊗ τ_{0}).

Then N(s, σ_{v} ⊗ π_{v}, w_{0}) is a product of the rank-one operators

N(s ± α_{i} ± β_{j}, σ_{i} ⊗ τ′_{j}), N(2s ± α_{i} ± α_{j}, σ_{i} ⊗ σ′_{j}), N(s ± α_{i}, σ_{i} ⊗ τ_{0}),

where τ′_{j} is either τ_{j} or τ̃_{j} and σ′_{j} is either σ_{j} or σ̃_{j}. Here, for the sake of simplicity of notation, we have dropped the dependence of the local normalized operators on the Weyl group elements since these elements do not play any role in the argument.

For all s such that Re(s) ≥ \frac{1}{2}, since 0 < α_{p} ≤ ⋯ ≤ α_{1} < \frac{1}{2} and 0 < β_{q} ≤ ⋯ ≤ β_{1} < 1, we have Re(s ± α_{i} ± β_{j}) > −1 and Re(2s ± α_{i} ± α_{j}) > −1. Hence the operators N(s ± α_{i} ± β_{j}, σ_{i} ⊗ τ′_{j}) and N(2s ± α_{i} ± α_{j}, σ_{i} ⊗ σ′_{j}), both corresponding to the case GL_{k} × GL_{l} ⊂ GL_{k+l}, are holomorphic for Re(s) ≥ \frac{1}{2}. For this result, one needs a classification of discrete series representations of GL_{n}. Next, we consider the operator N(s ± α_{i}, σ_{i} ⊗ τ_{0}) which corresponds to the case GL_{a} ×SO_{2r+1} ⊂ SO_{2a+2r+1}. Since σ_{i} ⊗τ_{0} is tempered, the operator N(s, σ_{i} ⊗τ_{0}) is holomorphic for Re(s) ≥ 0. Consequently N(s ± α_{i}, σ_{i} ⊗ τ_{0}) is holomorphic for Re(s) ≥ \frac{1}{2}.

Thus we conclude N(s, σ_{v} ⊗ π_{v}, w_{0}) is holomorphic for Re(s) ≥ \frac{1}{2}. Applying Zhang’s lemma, we see that N(s, σ_{v} ⊗ π_{v}, w_{0}) is non-zero for Re(s) ≥ \frac{1}{2}.

The result on the local intertwining operators together with (∗) implies that ∏_{i=1}^{m} L(is, π, r_{i}) is holomorphic for Re(s) ≥ \frac{1}{2}.

Next, from E_{ψ}(s, f_{s}, e) = \frac{∏_{v∈S} W_{v}(e_{v})}{∏_{i=1}^{m} L_{S}(1+is,π,r_{i})}, we see that ∏_{i=1}^{m} L(1 + is, π, r_{i}) has no zeros for Re(s) ≥ 0.

By using induction on i, we see that L(s, π, r_{i}) is holomorphic for Re(s) ≥ \frac{1}{2} and has no zeros for Re(s) ≥ 1. By the functional equation, we conclude that L(s, π, r_{i}) is entire.

When w_{0}(π) ≃ π, we cannot conclude that L(s, π, r_{i}) is entire. However, we can show

Theorem (Kim-Shahidi). Suppose m = 1. Then L_{S}(s, π, r_{1}) has at most a simple pole at s = 1. Suppose m ≥ 2. Under the condition L_{S}(2, π, r_{2}) ≠ 0, L_{S}(s, π, r_{1}) has at most a simple pole at s = 1.

Idea of Proof. It follows from the local-global principle: If M(s, π) has a pole at s = s_{0}, then the quotient of I(s_{0}, π_{v}) is unitary for all v. [If M(s, π) has a pole at s = s_{0}, then the Eisenstein series has a pole at s = s_{0}, and its residues form a residual spectrum, that is, a direct summand of L^{2}(G(F)\G(𝔸)). The local component of the residual spectrum is a quotient of I(s, π_{v}), which is unitary.]

So if we can show that the quotient of I(s, π_{v}) is not unitary at s_{0}, then M(s, π) has no pole at s_{0}. Take π_{v} to be a spherical representation.

Lemma. I(s, π_{v}) is irreducible for Re(s) ≥ 2.

Proof. J.S. Li showed that I(s, π_{v}) is irreducible at s = s_{0} if and only if ∏_{i=1}^{m} L(1− is, π_{v}, r_{i}) has no poles at s = s_{0}. Here L(s, π_{v}, r_{i}) = ∏_{j}(1 − α_{j}q_{v}^{−s})^{−1}. Shahidi showed that |α_{j}|, |α_{j}|^{−1} < q_{v}. So if Re(s) ≥ 2, L(s, π_{v}, r_{i}) has no poles.

Lemma. I(s, π_{v}) is not unitary for Re(s) ≥ 2.

Proof. I(s, π_{v}) is not unitary for Re(s) >> 0. Unitarity is preserved up to a point of reducibility. So I(s, π_{v}) is not unitary for Re(s) ≥ 2.

We have proved that M(s, π) is holomorphic for Re(s) ≥ 2. Since A(s, π_{v}, w_{0}) is non-vanishing, ∏_{i=1}^{m} \frac{L_{S}(is,π,r_{i})}{L_{S}(1+is,π,r_{i})} is holomorphic for Re(s) ≥ 2. Shahidi showed that L_{S}(s, π, r_{i}) is absolutely convergent for Re(s) > 2. So L_{S}(s, π, r_{1}) is holomorphic for Re(s) ≥ 2. Since M(s, π) has at most a simple pole at s = 1, ∏_{i=1}^{m} \frac{L_{S}(is,π,r_{i})}{L_{S}(1+is,π,r_{i})} has at most a simple pole at s = 1. So L_{S}(s, π, r_{1})L_{S}(2s, π, r_{2}) has at most a simple pole at s = 1. So under the assumption L_{S}(2, π, r_{2}) ≠ 0, L_{S}(s, π, r_{1}) has at most a simple pole at s = 1.

Corollary. Let π be a cuspidal representation of GL_{6}(𝔸_{F}). Then the exterior cube L-function L(s, π, ∧^{3}) (degree 20 L-function) has at most a simple pole at s = 1.

Proof. Apply the theory to M ⊂ E_{6}, where the derived group of M is SL_{6}.

We conclude with a discussion of some observation: Suppose m ≥ 2.

Conjecture. ∏_{i=1}^{m} L_{S}(1 + is, π, r_{i}) has at most a simple pole at s = 0.

Since L(s, π, r_{i}), i ≥ 3, is holomorphic for Re(s) > 0, this means that only one of L(s, π, r_{1}), L(s, π, r_{2}) can have a pole at s = 1.

Examples: (1) Consider GL_{m} × SO_{2n+1} ⊂ SO_{2(m+n)+1}. The L-functions in the constant term of the Eisenstein series are

L(s, σ × π)L(2s, σ, Sym^{2}).

If L(s, σ ×π) has a pole at s = 1, then σ ≃ σ̃ and L(s, σ, Sym^{2}) is holomorphic at s = 1 by the conjecture. Since L(s, σ × σ) = L(s, σ, Sym^{2})L(s, σ, ∧^{2}), L(s, σ, ∧^{2}) has a pole at s = 1. This has been proved by Ginzburg-Rallis-Soudry.

(2) Consider M = GL_{2} ⊂ G_{2} (the maximal Levi subgroup attached to the long simple root in the exceptional group of type G_{2}). The L-functions in the constant term of the Eisenstein series are

L(s, π, Sym^{3} ⊗ ω_{π}^{−1})L(2s, ω_{π}).

If L(s, π, Sym^{3} ⊗ ω_{π}^{−1}) has a pole at s = 1, π is monomial and hence ω_{π} ≠ 1. If ω_{π} = 1, then L(s, π, Sym^{3} ⊗ ω_{π}^{−1}) is holomorphic at s = 1 (the result of Ikeda).

Some questions in the Langlands-Shahidi method:

(1) Prove the above conjecture.

(2) Unitarity criterion for I(s, π_{v}) when π_{v} is a spherical representation. This will give the location of poles of L(s, π, r_{i}).

(3) Find the criterion for the pole of L(s, π, r_{1}) at s = 1. Goldberg-Shahidi have been studying the criterion for the pole of the local L-function at s = 0 in terms of twisted endoscopy. We want a global analogue. For example, we want to find a criterion of the pole in terms of some period integrals. For example, Flicker showed that L(s, σ, r_{A}) has a pole at s = 1 precisely when σ is GL_{k}(𝔸_{F})- distinguished, namely, the central character ω_{σ} satisfies the condition ω_{σ}|_{𝔸_{F}^{×}} = 1 and there exists a cuspidal function φ such that

∫_{GL_{k}(F)Z(𝔸_{F})\GL_{k}(𝔸_{F})} φ(h) dh ≠ 0,

where Z is the center of GL_{k}.

(4) Develop the theory for Kac-Moody groups (infinite-dimensional groups).

Lecture 3. Functoriality of symmetric powers of cuspidal representations of GL_{2}.

Let π = ⊗π_{v} be a cuspidal representation of GL_{2} over a number field F. Let Sym^{m} : GL_{2}(ℂ) ⟶ GL_{m+1}(ℂ) be the mth symmetric power representation. The local Langlands correspondence, proved by Langlands for archimedean places, and by Harris-Taylor, Henniart for p-adic places, says that π_{v} is parametrized by ϕ_{v} : W_{F_{v}} ×SL_{2}(ℂ) ⟶ GL_{2}(ℂ). Consider the composition Sym^{m}(ϕ_{v}) : W_{F_{v}} × SL_{2}(ℂ) ⟶ GL_{m+1}(ℂ). By the local Langlands correspondence, Sym^{m}(ϕ_{v}) is associated to an irreducible, admissible representation Sym^{m}(π_{v}) of GL_{m+1}(F_{v}). Let Sym^{m}(π) = ⊗Sym^{m}(π_{v}). It is an irreducible, admissible representation of GL_{m+1}(𝔸).

Langlands functoriality conjecture. Sym^{m}(π) is automorphic.

Theorem. (1) (Gelbart-Jacquet) Sym^{2}(π) is an automorphic representation of GL_{3}. It is not cuspidal if and only if π is of dihedral type, i.e., π ≃ π ⊗ χ for χ ≠ 1.

(2) (Kim-Shahidi) Sym^{3}(π) is an automorphic representation of GL_{4}. It is not cuspidal if and only if π is either of dihedral type, or of tetrahedral type, i.e., Sym^{2}(π) is cuspidal and Sym^{2}(π) ≃ Sym^{2}(π) ⊗ χ for χ ≠ 1.

(3) (Kim) Sym^{4}(π) is an automorphic representation of GL_{5}.

(4) (Kim-Shahidi) Sym^{4}(π) is not cuspidal if and only if π is either of dihedral type, or of tetrahedral type, or of octahedral type, i.e., Sym^{3}(π) is cuspidal and Sym^{3}(π) ≃ Sym^{3}(π) ⊗ χ for χ ≠ 1.

3.1 Proof of functoriality of Sym^{2}(π).

In order to illustrate our method, let us prove the functoriality of Sym^{2}(π) using our method. Let ϕ = Ad : GL_{2}(ℂ) ⟶ SO_{3}(ℂ) ⊂ GL_{3}(ℂ) be the adjoint representation. Let π = ⊗_{v}π_{v} be a cuspidal representation of GL_{2}(ℂ). By the local Langlands’ correspondence, we have Ad(π) = ⊗_{v}Ad(π_{v}), irreducible admissible representation of GL_{3}(𝔸). In this case, note that, for a grössencharacter χ,

L(s, π × π̃) = L(s, Ad(π))L(s, 1), L(s, (π ⊗ χ) × π̃) = L(s, Ad(π) ⊗ χ)L(s, χ).

Hence L(s, Ad(π_{v})⊗χ_{v}) and γ(s, Ad(π_{v})⊗χ_{v}, ψ_{v}) are Artin L and γ-factors. Here L(s, Ad(π)⊗χ) appears in the Langlands-Shahidi method for P = MN ⊂ Sp(4), where M ≃ GL_{1} × SL_{2}: Take π_{0} to be any irreducible constituent of π|_{SL_{2}(𝔸)} and consider Σ = χ ⊗ π_{0}. Then (M, Σ) gives rise to the fact that L(s, Ad(π) ⊗ χ) is entire if χ^{2} ≠ 1. We recall

Converse theorem. Suppose Π = ⊗Π_{v} is an irreducible, admissible representation of GL_{N} such that ω_{Π} = ⊗ω_{Π_{v}} is a grössencharacter. Let S be a finite set of finite places and let T^{S}(m) be a set of cuspidal representations of GL_{m} that are unramified at S. Suppose L(s, σ × Π) is nice (entire, functional equation, bounded in vertical strips) for all σ ∈ T^{S}(m) ⊗ χ, m = 1, ..., N − 2, where χ is a grössencharacter which is highly ramified at S. Then there exists an automorphic representation Π′ of GL_{N} such that Π′_{v} ≃ Π_{v} for v ∉ S.

Apply the converse theorem twice to Ad(π) = ⊗_{v}Ad(π_{v}), and S_{1} = {v_{1}}, S_{2} = {v_{2}}, where v_{1}, v_{2} are any finite places. Then we obtain Π_{1}, Π_{2} automorphic representations of GL_{3}(𝔸) such that Π_{1v} ≃ Ad(π_{v}) for all v ≠ v_{1}, and Π_{2v} ≃ Ad(π_{v}) for all v ≠ v_{2}. So by the strong multiplicity one, Π_{1} ≃ Π_{2} ≃ Ad(π). Therefore Ad(π) is an automorphic representation. By the classification of automorphic representations of GL_{n}, Ad(π) is equivalent to a subquotient of

Ind σ_{1}|det|^{r_{1}} ⊗ ⋯ ⊗ σ_{k}|det|^{r_{k}},

where σ_{i}’s are (unitary) cuspidal representations of GL_{n_{i}} and r_{i} ∈ ℝ.

We use the weak Ramanujan property: Let π = ⊗π_{v} be a cuspidal representation of GL_{n}, and π_{v} is spherical with the Satake parameter diag(α_{1v}, ..., α_{nv}). We say π satisfies the weak Ramanujan property if given ϵ > 0, there exists T_{ϵ}, a density zero set such that max_{i}{|α_{iv}|, |α_{iv}|^{−1}} ≤ q_{v}^{ϵ} for v ∉ T_{ϵ}.

Lemma. Cuspidal representations of GL_{2}, GL_{3} satisfy the weak Ramanujan property.

Since π satisfies the weak Ramanujan, so does Ad(π). Hence r_{1} = ⋯ = r_{k} = 0. From the relation L(s, (π ⊗ χ) × π̃) = L(s, Ad(π) ⊗ χ)L(s, χ), the left hand side has a pole at s = 1 if and only if π ⊗ χ ≃ π, i.e., π is monomial. Then the right hand side has a pole. So Ad(π) ≃ χ ⊞ π′, where π′ is an automorphic representation of GL_{2}. Hence Ad(π) is not cuspidal if and only if π is of dihedral type.

3.2 Proof of functoriality of Sym^{3}(π).

This is obtained indirectly from the functorial product associated with the tensor product map

GL_{2}(ℂ) × GL_{3}(ℂ) ⟶ GL_{6}(ℂ).

Let π_{1}, π_{2} be cuspidal representations of GL_{2}, GL_{3}, resp. For each place v, π_{iv}, i = 1, 2, are parametrized by ϕ_{iv} : W_{F_{v}} × SL_{2}(ℂ) ⟶ GL_{i+1}(ℂ), i = 1, 2. Then ϕ_{1v} ⊗ ϕ_{2v} : W_{F_{v}} × SL_{2}(ℂ) ⟶ GL_{6}(ℂ). By the local Langlands correspondence, ϕ_{1v} ⊗ ϕ_{2v} gives rise to an irreducible, admissible representation π_{1v} ⊠ π_{2v} of GL_{6}(F_{v}). Let π_{1} ⊠ π_{2} = ⊗(π_{1v} ⊠ π_{2v}). It is an irreducible, admissible representation of GL_{6}(𝔸).

Theorem (Kim-Shahidi). π_{1} ⊠ π_{2} is automorphic.

Let π be a cuspidal representation of GL_{2}, and let Ad(π) be the adjoint square, i.e., Ad(π) = Sym^{2}(π)⊗ω_{π}^{−1}, where ω_{π} is the central character. Then π⊠Ad(π) = (Sym^{3}(π) ⊗ ω_{π}^{−1}) ⊞ π. Here ⊞ is the isobaric sum, and it denotes the unitary induction Ind_{GL_{4}×GL_{2}}^{GL_{6}}(Sym^{3}(π) ⊗ ω_{π}^{−1}) ⊗ π.

Proof of the theorem. We apply the converse theorem to π_{1} ⊠ π_{2}. Let S be a finite set of finite places such that π_{1v}, π_{2v} are spherical for v ∉ S, v < ∞. We need the triple product L-functions

L(s, σ × (π_{1} ⊠ π_{2})) = L(s, σ × π_{1} × π_{2}),

where σ is a cuspidal representation of GL_{m}, m = 1, 2, 3, 4. These are available from Langlands-Shahidi method:

(1) m = 1: Rankin-Selberg L-function of GL_{2} × GL_{3}

(2) m = 2: D_{5} − 2 case. Use Spin(10).

(3) m = 3: E_{6} − 1 case. Use simply connected E_{6}.

(4) m = 4: E_{7} − 1 case. Use simply connected E_{7}.

Functional equation: due to Shahidi (1990)

Bounded in vertical strips: due to Gelbart and Shahidi (2001)

Entire: trick to use χ which is highly ramified at one finite place

By applying the converse theorem, we obtain an automorphic representation Π = ⊗Π_{v} of GL_{6} such that Π_{v} ≃ π_{1v} ⊠ π_{2v} for v ∉ S. By classification of automorphic representations of GL_{N} (due to Jacquet-Shalika), Π is equivalent to a subquotient of Ind|det|^{r_{1}}σ_{1} ⊗ ⋯ ⊗ |det|^{r_{k}}σ_{k}, where r_{1}, ..., r_{k} ∈ ℝ and σ_{i}’s are cuspidal representations of GL_{n_{i}}.

We need: r_{1} = ⋯ = r_{k} = 0.

We use the weak Ramanujan property. Let diag(α_{1}, β_{1}), diag(α_{2}, β_{2}, γ_{2}) be Satake parameters for π_{1v}, π_{2v}, resp. Then the Satake parameter of Π_{v} is diag(α_{1}α_{2}, α_{1}β_{2}, α_{1}γ_{2}, β_{1}α_{2}, β_{1}β_{2}, β_{1}γ_{2}). This implies that Π satisfies the weak Ramanujan property. Hence r_{1} = ⋯ = r_{k} = 0.

Proving that Π_{v} ≃ π_{1v} ⊠ π_{2v} for v ∈ S requires extra work using base change. The most difficult case is when v|2, π_{1v} is an extraordinary supercuspidal representation of GL_{2}(F_{v}), and π_{2v} is a supercuspidal representation of GL_{3}(F_{v}) attached to a non-normal cubic extension K/F_{v}, namely, π_{2v} is associated to Ind_{W_{K}}^{W_{Fv}} χ, where χ is a character of W_{K}. This has been done in the appendix by Bushnell and Henniart.

3.3 Proof of functoriality of Sym^{4}(π).

This is obtained indirectly from the functoriality of the exterior square of GL_{4}: Let π be a cuspidal representation of GL_{4}, and let ∧^{2} : GL_{4}(ℂ) ⟶ GL_{6}(ℂ) be the map given by the exterior square. We can define, by the local Langlands correspondence, the irreducible admissible representation ∧^{2}π = ⊗ ∧^{2} π_{v} of GL_{6}(𝔸).

Theorem (Kim). Let T be a set of places such that v|2, 3 and π_{v} is a supercuspidal representation of GL_{4}(F_{v}). Then there exists an automorphic representation Π = ⊗Π_{v} of GL_{6} such that Π_{v} ≃ ∧^{2}π_{v} for v ∉ T.

Let π be a cuspidal representation of GL_{2}. Then we can prove

∧^{2}(Sym^{3}(π) ⊗ ω_{π}^{−1}) = (Sym^{4}(π) ⊗ ω_{π}^{−1}) ⊞ ω_{π}.

Proof of the theorem. We apply the converse theorem to ∧^{2}π. We need the L-functions

L(s, σ × ∧^{2}π) = L(s, σ ⊗ π, ρ_{m} ⊗ ∧^{2}ρ_{4}),

where ρ_{m} : GL_{m}(ℂ) ⟶ GL_{m}(ℂ) is the standard representation and σ is a cuspidal representation of GL_{m}, m = 1, 2, 3, 4.

These L-functions show up in D_{n} − 3 case for n = 4, 5, 6, 7. The reason there is an exceptional set T is because we could not prove the identity

L(s, η_{v} ⊗ ∧^{2}π_{v}) = L(s, π_{v}, ∧^{2} ⊗ η_{v}),

γ(s, η_{v} ⊗ ∧^{2}π_{v}, ψ_{v}) = γ(s, π_{v}, ∧^{2} ⊗ η_{v}, ψ_{v}), (**)

for v ∈ T. The left hand sides are defined by Rankin-Selberg method. The right hand sides are defined by Langlands-Shahidi method as normalizing factors of intertwining operators.

It is conjectured that when η_{v} is highly ramified, we have the following

Stability of γ-factors. Let π_{1v}, π_{2v} be two irreducible admissible representations of GL_{4}(F_{v}) with the same central character. Then for every highly ramified character η_{v},

L(s, π_{1v}, ∧^{2} ⊗ η_{v}) = L(s, π_{2v}, ∧^{2} ⊗ η_{v}) ≡ 1,
γ(s, π_{1v}, ∧^{2} ⊗ η_{v}, ψ_{v}) = γ(s, π_{2v}, ∧^{2} ⊗ η_{v}, ψ_{v}).

Since it is not available, we have to use the descent method of Ramakrishnan. Namely, we first look at a good case when T is empty. When T is not empty, we use the observation of Henniart that a supercuspidal representation of GL_{n}(F_{v}) becomes a principal series after a solvable base change.

Theorem (Ramakrishnan). Fix n, p ∈ ℕ with p prime. Let F a number field, {K_{j}|j ∈ ℕ} a family of cyclic extensions of F with [K_{j} : F] = p, and for each j ∈ ℕ, π_{j} a cuspidal automorphic representation of GL_{n}(𝔸_{K_{j}}). Suppose that, given j ∈ ℕ,

(π_{j})_{K_{j}K_{r}} ≃ (π_{r})_{K_{j}K_{r}},

for almost all r ∈ ℕ. Then there exists a unique cuspidal automorphic representation π of GL_{n}(𝔸_{F}) such that

(π)_{K_{j}} ≃ π_{j},

for all but a finite number of j.

For the functoriality of the symmetric fourth, we can prove (∗∗) directly and hence we do not need the descent method of Ramakrishnan.

Lecture 4. Applications.

4.1 Properties of symmetric power L-functions.

For simplicity, let π be a cuspidal representation of GL_{2} with the trivial central character and suppose that Sym^{4}(π) is cuspidal. Then we have

(1) L(s, π, Sym^{m}), m ≤ 9, has a meromorphic continuation to all of ℂ and satisfies a functional equation.

(2) L(s, π, Sym^{m}), m ≤ 8, is invertible for Re(s) ≥ 1 (no poles and no zeros).

(3) L(s, π_{v}, Sym^{9}) is holomorphic for Re(s) ≥ 1 for any v. If π_{v} is spherical with diag(α_{v}, β_{v}) as the Satake parameter, then q_{v}^{− \frac{1}{9}} < |α_{v}|, |β_{v}| < qv^{\frac{1}{9}} .

(4) L(s, π, Sym^{3}), L(s, π, Sym^{4}) are entire.

It is conjectured that the multiplicity of poles of L(s, π, Sym^{m}) at s = 1 determines π. For example, if π is a cuspidal representation of icosahedral type, L(s, π, Sym^{m}) has a simple pole at s = 1 when m = 12.

Let us show how the properties of L(s, π, Sym^{9}) are proved in order to indicate the limitation of our method: Consider the case E_{8} − 2. Let M be a maximal Levi subgroup in the exceptional group of type E_{8} whose derived group M_{D} is SL_{4} × SL_{5}. Let π_{i}, i = 1, 2, be cuspidal representations of GL_{4}(𝔸) and GL_{5}(𝔸) with central characters ω_{i}, i = 1, 2, resp. Let π_{i0}, i = 1, 2, be irreducible constituents of π_{1}|_{SL_{4}(𝔸)} and π_{2}|_{SL_{5}(𝔸)}, resp. Then Σ = ω_{1}^{5}ω_{2}^{8} ⊗ π_{10} ⊗ π_{20} can be considered as a cuspidal representation of M(𝔸). Applying Langlands-Shahidi method, we then get the L-function L(s, π_{1}⊗π_{2}, ρ_{4}⊗∧^{2}ρ_{5}) as our first L–function. In fact, there are five L-functions in the constant term of the Eisenstein series; namely

L_{S}(s, Σ, r_{1}) = L_{S}(s, π_{1} ⊗ π_{2}, ρ_{4} ⊗ ∧^{2}ρ_{5});
L_{S}(s, Σ, r_{2}) = L_{S}(s, π_{1} ⊗ (π̃_{2} ⊗ ω_{2}), ∧^{2}ρ_{4} ⊗ ρ_{5});
L_{S}(s, Σ, r_{3}) = L_{S}(s, π̃_{1} × (π_{2} ⊗ ω_{1}ω_{2}));
L_{S}(s, Σ, r_{4}) = L_{S}(s, π̃_{2}, ∧^{2}ρ_{5} ⊗ ω_{1}ω_{2}^{2});
L_{S}(s, Σ, r_{5}) = L_{S}(s, π_{1} ⊗ ω_{1}ω_{2}^{2}).

Each of the L-functions, especially, L_{S}(s, Σ, r_{1}) has a meromorphic continuation and satisfies a standard functional equation.

We apply the above to π_{1} = A^{3}(π) and π_{2} = Sym^{4}(π). By standard calculations, we have

\begin{matrix}L_{S}(s, π_{1} ⊗ π_{2}, ρ_{4} ⊗ ∧^{2}ρ_{5}) = L_{S}(s, π, Sym^{9})L_{S}(s, π, Sym^{7} ⊗ ω_{π})L_{S}(s, π, Sym^{5} ⊗ ω_{π}^{2})^{2} \\ L_{S}(s, Sym^{3}(π) ⊗ ω_{π}^{3})^{2}L_{S}(s, π ⊗ ω_{π}^{4}). \end{matrix}

Meromorphic continuation and functional equation of L_{S}(s, π, Sym^{9}) now follow from those of L_{S}(s, π_{1} ⊗ π_{2}, ρ_{4} ⊗ ∧^{2}ρ_{5}).

The group of type E_{8} is the largest finite dimensional exceptional group. The groups of type E_{n}, n ≥ 9, are not finite dimensional, and they are called Kac-Moody groups.

4.2 Refined Ramanujan and Selberg bounds.

Let π = ⊗ π_{p} be a cuspidal automorphic representation for GL_{2}(𝔸_{ℚ}). For p a prime at which π_{p} is spherical, let diag(α_{p}, β_{p}) be the corresponding Satake parameter. Let Sym^{4}(π) be the symmetric fourth, the cuspidal representation of GL_{5}. (We assume that it is cuspidal since if not, π is associated to a Galois representation, and it satisfies the Ramanujan conjecture.) Then from the fact that L(s, Sym^{4}(π), Sym^{2}) is absolutely convergent for Re(s) > 1, we can show that

Theorem (Kim-Sarnak). p^{− \frac{7}{64}} ≤ |α_{p}| ≤ p^{\frac{7}{64}} . Let λ_{1}(Γ) be the smallest (nonzero) eigenvalue of the Laplacian on Γ\ℍ, where Γ is a congruence subgroup of SL_{2}(ℤ). Then

λ_{1}(Γ) ≥ \frac{975}{4096} ≈ 0.238...

4.3 Density of tempered places.

Let π be a cuspidal representation of GL_{2}. Let S(π) be the set of places where π_{v} is tempered. Ramanujan conjecture predicts that S(π) is the set of all places. Ramakrishnan proved, using the functoriality of symmetric square, that S(π) has lower Dirichlet density≥ \frac{9}{10}. By using the functoriality of symmetric cube and symmetric fourth, we can show

\underline{δ}(S(π)) ≥ \frac{34}{35}.

Here \underline{δ}(S) is the lower Dirichlet density, defined by

\underline{δ}(S) = \underline{lim}_{s→1+} \frac{∑_{v∈S} q_{v}^{−s}}{− log(s − 1)}.

4.4 Special case of Sato-Tate conjecture.

Let π = ⊗π_{v} be a cuspidal representation of GL_{2} with the trivial central character. We also assume the Ramanujan conjecture. Let diag(α_{v}, β_{v}) be the Satake parameter for the spherical π_{v}. Let a_{v}(π) = α_{v} + β_{v}. Since |a_{v}(π)| ≤ 2, write a_{v}(π) = 2 cos θ_{v}, 0 ≤ θ_{v} < π. Sato and Tate conjectured that θ_{v} is distributed in the following way:

#\{v : q_{v} ≤ x, θ_{v} ∈ (a, b)\} ∼ (\frac{2}{π} ∫_{a}^{b} sin^{2} θ dθ) π_{F} (x),

as x → ∞. Here π_{F} (x) = #\{v : q_{v} ≤ x\} ∼ \frac{x}{ln x}. The weaker version of Sato-Tate conjecture claims that the set #{v : θ_{v} ∈ (a, b)} has positive lower density for any a < b.

Theorem. For every ϵ > 0, there exist T^{+}, T^{−} of positive lower density such that a_{v}(π) > 2 cos \frac{2π}{11} − ϵ for v ∈ T^{+} and a_{v}(π) < −2 cos \frac{2π}{11} + ϵ for v ∈ T^{−}.

Here 2 cos \frac{2π}{11} = 1.68...

4.5 Weak Ramanujan property.

We can prove that cuspidal representations of GL_{4} satisfy the weak Ramanujan property. Let π = ⊗_{v}π_{v} be a cuspidal representation of GL_{4}(𝔸). Let π_{v} be an unramified component with the trace a_{v}, i.e., a_{v} = α_{1} + α_{2} + α_{3} + α_{4}, where the Satake parameter of π_{v} is given by diag(α_{1}, α_{2}, α_{3}, α_{4}). Then given ϵ > 0, the set of places where |a_{v}| ≥ q_{v}^{ϵ} has density zero.

Note that at a place where π_{v} is non-tempered, the trace a_{v} has one of the following three forms: Here u_{1}, u_{2}, u_{3} are complex numbers with absolute value one. We suppress the dependence of all the factors on v for the simplicity of notation, except a_{v}.

S_{1}; a_{v} = u_{1}q^{a} + u_{2}q^{a} + u_{1}q^{−a} + u_{2}q^{−a}, where 0 < a < \frac{1}{2};

S_{2}; a_{v} = u_{1}q^{a} + u_{2} + u_{3} + u_{1}q^{−a}, where 0 < a < \frac{1}{2};
S_{3}; a_{v} = u_{1}q^{a_{1}} + u_{2}q^{a_{2}} + u_{1}q^{−a_{1}} + u_{2}q^{−a_{2}}, where 0 < a_{2} < a_{1} < \frac{1}{2}.

Fix ϵ > 0. Then inside S_{2}, the set of places where |a_{v}| > q^{ϵ} has density zero. It means the set of places where a > ϵ has density zero.

Suppose S_{1} has a subset S′ of positive density where q^{a} > q^{ϵ} for v ∈ S′. Then consider the lift ∧^{2}π. For v ∈ S_{1}, the trace of ∧^{2}π_{v} has the form

b_{v} = u_{1}u_{2}q^{2a} + u_{1}^{2} + u_{1}u_{2} + u_{2}^{2} + u_{1}u_{2}q^{−2a}.

Then |b_{v}| > q^{ϵ} for v ∈ S′. This is a contradiction.

Suppose S_{3} has a subset S′ of positive density where q^{a_{1}} > q^{ϵ} for v ∈ S′. Then consider the lift ∧^{2}π. For v ∈ S_{3}, the trace of ∧^{2}π_{v} has the form

b_{v} = u_{1}u_{2}q^{a_{1}+a_{2}} + u_{1}u_{2}q^{a_{1}−a_{2}} + u_{1}^{2} + u_{2}^{2} + u_{1}u_{2}q^{−a_{1}+a_{2}} + u_{1}u_{2}q^{−a_{1}−a_{2}}.

Then |b_{v}| > q^{ϵ} for v ∈ S′. This is a contradiction.

4.6 Hypothesis H.

Let π = ⊗ π_{p} be a cuspidal automorphic representation for GL_{m}(𝔸_{ℚ}). For p a prime at which π_{p} is spherical, let diag(α_{1,p}, ..., α_{m,p}) be the corresponding Satake parameter. For a positive integer k, let a_{π}(p^{k}) = α_{1,p}^{k} + ⋯ + α_{m,p}^{k}. Then Rudnick and Sarnak made the following

Hypothesis H. For any fixed k ≥ 2,

∑_{p} \frac{(log p)^{2}|a_{π}(p^{k})|^{2}}{p^{k}} < ∞.

It has an important application to distribution of zeros of L-functions. Rudnick and Sarnak proved Hypothesis H for m = 2, 3. Using the functoriality of the exterior square of GL_{4}, we can prove Hypothesis H for m = 4.

Consider the exterior square ∧^{2}π. There exists an automorphic representation Π of GL_{6}(𝔸_{ℚ}) such that Π_{p} ≃ ∧^{2}π_{p} except possibly for p = 2, 3, and Π is an isobaric sum of (unitary) cuspidal representations of GL_{n_{i}}. If π_{p} is non-tempered, the Satake parameter of ∧^{2}π_{p} is as follows:

S_{1} : diag(u_{1}u_{2}p^{2a}, u_{1}u_{2}, u_{1}^{2}, u_{2}^{2}, u_{1}u_{2}, u_{1}u_{2}p^{−2a})

S_{2} : diag(u_{1}u_{2}p^{a}, u_{1}u_{3}p^{a}, u_{1}^{2}, u_{2}u_{3}, u_{1}u_{2}p^{−a}, u_{1}u_{3}p^{−a})
S_{3} : diag(u_{1}u_{2}p^{a_{1}+a_{2}}, u_{1}u_{2}p^{a_{1}−a_{2}}, u_{1}^{2}, u_{2}^{2}, u_{1}u_{2}p^{−a_{1}+a_{2}}, u_{1}u_{2}p^{−a_{1}−a_{2}}).

Hence by applying Luo-Rudnick-Sarnak to Π, we see that if p ∈ S_{1}, 2a ≤ \frac{1}{2} − \frac{1}{37}; if p ∈ S_{2}, a ≤ \frac{1}{2} − \frac{1}{17}; if p ∈ S_{3}, 0 < a_{2} < a_{1} ≤ \frac{1}{2} − \frac{1}{17} and a_{1} + a_{2} ≤ \frac{1}{2} − \frac{1}{37}.

If p ∈ S_{1}, a ≤ \frac{1}{4} − \frac{1}{74}. Hence

∑_{p∈S_{1}} \frac{(log p)^{2}|a_{π}(p^{k})|^{2}}{p^{k}} < ∞.

If p ∈ S_{2}, we apply the same technique in Rudnick-Sarnak. Namely, note that |u_{1}p^{a} + u_{2} + u_{3} + u_{1}p^{−a}| ≥ p^{a} + p^{−a} − 2. Hence

|α_{1,p}| + |α_{2,p}| + |α_{3,p}| + |α_{4,p}| ≤ |α_{1,p} + α_{2,p} + α_{3,p} + α_{4,p}| + 4.

Hence

|a_{π}(p^{k})|^{2} << |a_{π}(p)|^{2k}.

Since |a_{π}(p)| << p^{\frac{1}{2} − \frac{1}{17}} ,

∑_{p∈S_{2}} \frac{(log p)^{2}|a_{π}(p^{k})|^{2}}{p^{k}} << ∑_{p∈S_{2}} \frac{(log p)^{2}|a_{π}(p)|^{2}}{p^{1+ \frac{2}{17} (k−1)}} .

Since k > 1, we apply the fact that L(s, π × π̃) converges absolutely for Re s > 1, and hence

∑_{p∈S_{2}} \frac{(log p)^{2}|a_{π}(p^{k})|^{2}}{p^{k}} < ∞.

If p ∈ S_{3}, note that 2a_{2} ≤ \frac{1}{2} − \frac{1}{37}. Hence a_{2} ≤ \frac{1}{4} − \frac{1}{74}. Note also that |u_{1}(p^{a_{1}} + p^{−a_{1}}) + u_{2}(p^{a_{2}} + p^{−a_{2}})| ≥ p^{a_{1}} + p^{−a_{1}} − (p^{a_{2}} + p^{−a_{2}}). Hence

|α_{1,p}| + |α_{2,p}| + |α_{3,p}| + |α_{4,p}| ≤ |α_{1,p} + α_{2,p} + α_{3,p} + α_{4,p}| + 2p^{a_{2}}.

Hence

|a_{π}(p^{k})|^{2} << |a_{π}(p)|^{2k} + p^{2ka_{2}}.

We again obtain

∑_{p∈S_{3}} \frac{(log p)^{2}|a_{π}(p^{k})|^{2}}{p^{k}} < ∞.