EISENSTEIN SERIES AND THEIR APPLICATIONS. In Lecture one, we look at Eisenstein series as functions on the upper half plane.

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1 EISENSTEIN SERIES AND THEIR APPLICATIONS HENRY H. KIM Eisenstein series provide very concrete examples of modular forms. Langlands was led to his functoriality principle by studying Eisenstein series and their constant terms. We describe Eisenstein series, both classical and adelic, and their applications. See [1] for the details and references therein. 1. Lecture One: Classical Eisenstein series In Lecture one, we look at Eisenstein series as functions on the upper half plane classical holomorphic Eisenstein series. We follow [7], Chapter VII. Let H = {z = x + iy, y > 0} be the upper half plane. Let, for k > 1 positive integer, E k (z) = 1 (c,d)=1 (cz + d) k. { ( ) } 1 n Here the sum (c, d) = 1 is the same as γ Γ \Γ, where Γ = SL (Z), and Γ = ± n Z. 0 1 The series converges absolutely and uniformly ( if ) k > 1, and E k is a modular form of weight a b k, i.e., E k (γz) = (cz + d) k E k (z) for γ =. We have the Fourier expansion c d E k (z) = 1 + γ k n=1 σ k 1 (n)q n, where q = e πiz, σ s (n) = d n ds, and γ k = ( 1) k 4k B k, and B k are the Bernoulli numbers x e x 1 = 1 x + ( 1) k+1 x k B k (k)!. k=1 Date: November 015. partially supported by an NSERC grant. 1

2 HENRY KIM Let (z) = E3 4 E = n=1 τ(n)qn, where τ(n) is the Ramanujan τ-function. Then is a cusp form of weight 1, and we have an infinite product expansion: (z) = q (1 q n ) 4. n=1 We have many remarkable congruences which have natural explanation from Galois representations: Here the prime 691 comes from E 1 (z) = τ(n) n σ 7 (n) mod 3 3 τ(n) nσ 3 (n) mod 7 τ(n) σ 11 (n) mod 691 σ 11 (n)q n, 6550 = n=1 Conjecture 1.1. (Lehmer s Conjecture) τ(n) 0 for all n 1. Since τ(n) is multiplicative, i.e., τ(mn) = τ(m)τ(n) for (m, n) = 1 and τ(p k+1 ) = τ(p)τ(p k ) p 11 τ(p k 1 ) for p prime and k > 1, it can be shown that it is enough to prove τ(p) 0 for all prime p. Let j(z) = E3 4. Then j is a modular function of weight 0, and it plays an important role: j(z) = 1 q c(n)q n = 1 q q q q n=1 It is a remarkable fact that c(n) s are related to dimensions of irreducible representations of the Monster M, the largest of the sporadic simple groups, with group order M = John McKay observed first = , and Thompson found = = = , where the numbers on the right are dimensions of irreducible representations of the Monster.

3 EISENSTEIN SERIES AND THEIR APPLICATIONS classical non-holomorphic Eisenstein series. We follow [1], p. 15. Let E(z, s) = 1 The normalized Eisenstein series is (c,d)=1 y s cz + d s = 1 E (z, s) = 1 π s Γ(s)ζ(s)E(z, s) = 1 π s Γ(s) γ Γ \Γ Im(γz) s. (m,n) (0,0) y s mz + n s. The series converges for Re(s) > 1, and it becomes an eigenfunction for the Laplacian = y ( x + y ), i.e., E(z, s) = s(1 s)e(z, s). We have the Fourier expansion E (z, s) = π s Γ(s)ζ(s)y s +π s 1 Γ(1 s)ζ( s)y 1 s + r 0 r s 1 σ1 s ( r ) yk s 1 (π r y)e πirx, 0 e where K s (y) = 1 E (z, 1 s) = E (z, s). t+t 1 y t s dt t is the K-Bessel function. This show the functional equation Non-vanishing of ζ(1 + it). Eisenstein series can be used to show that ζ(1 + it) 0 for all t R, which is equivalent to the prime number theorem, π(x) = p<x 1 that E(z, s) = y s + πs 1 Γ(1 s)ζ( s) Γ(s)ζ(s) y 1 s + πs Γ(s)ζ(s) r 0 x log x as x. Note r s 1 σ1 s ( r ) yk s 1 (π r y)e πirx. Then 1 0 E(z, s)eπirx dx = r s 1 σ 1 s ( r ) yk s 1 (π r y). The numerator is the Whittaker function which is entire. Now the holomorphy of E(z, s) on Re(s) = 1 π s Γ(s)ζ(s) implies the non-vanishing of ζ(s) on Re(s) = 1. Namely, the information of Eisenstein series give rise to the information on L-function. This is the beginning observation of Langlands-Shahidi method. Sarnak [6] refined this more to obtain ζ(1 + it) 1 (log t) Rankin-Selberg method. Let f(z) = n=1 a(n)qn be a cusp form of weight k for SL (Z). Let L(s, f) = k 1 n=1 a(n)n n s be the L-function. We can show easily that L(s, f) has an analytic continuation to all of C, and satisfies the functional equation Λ(s, f) = (π) s Γ(s + k 1 )L(s, f) = ( 1)k Λ(1 s, f). Now let L(s) = n=1 a(n) n s. Here L(s) is related to the

4 4 HENRY KIM symmetric square L-function L(s, Sym f) = ζ(s) n=1 a(n )n s k+1. In order to study the analytic property of L(s), we need Rankin-Selberg method: (4π) s k+1 Γ(s + k 1) a(n) n s k+1 = n=1 Γ\H y k f(z) E(z, s) dxdy y. Here note that y f(z) is Γ-invariant, and dxdy is Γ-invariant measure on H. The analytic continuation and functional equation of E(z, s) give rise to those of L(s). Let ψ(s, f) = π s k+1 s y Γ(s)Γ(s+ k 1)ζ(s) n=1 a(n) n s k+1. Here L(s, f f) = ζ(s) a(n) n s k+1, n=1 ζ(s)l(s, Sym f) = L(s, f f). Then ψ(s, f) has a meromorphic continuation to all of C, with a simple pole at s = 1, and the residue is 3 π (f, f), and satisfies the functional equation ψ(1 s, f) = ψ(s, f) Roelcke-Selberg Spectral decomposition. Let L (Γ\H) = L dis (Γ\H) L cont(γ\h), be the spectral decomposition of the Laplacian into a discrete part and continuous part. Let L cusp(γ\h) be the space of cusp forms, i.e., those f such that 1 0 f(x + iy) dx = 0 for almost all y. Then L dis (Γ\H) = L cusp(γ\h) C 1. Now the Eisenstein series E(z, s) is not in L (Γ\H) for any s C, but E(z, 1 + it) spans a continuous spectrum, i.e., the map L (R 0 ) L (Γ\H), h Eh = h(it)e(z, 1 + it) dt, is an isometry onto L cont(γ\h), and (Eh) = E(( 1 4 t )h). This means that any f L (Γ\H) has a spectral decomposition f(z) = j=0 (f, u j )u j (z) + 1 4π (f, E(, 1 + it))e(z, 1 + it) dt, where the first sum runs over an orthonormal basis of the discrete part Volume of the fundamental domain. Langlands used Eisenstein series to compute the volume of the fundamental domain. Let s see it in our special case: Now E(z, s) / L 1 (Γ\H) for Re(s) > 1, but E(z, s) L 1 (Γ\H) if 0 < Re(s) < 1, so that I(s) = Γ\H E(z, s) dz makes sense there. For any h Cc (R >0 ), consider θ h (z) = 1 γ Γ \Γ f(im γz). This is a finite sum, and

5 EISENSTEIN SERIES AND THEIR APPLICATIONS 5 θ h is compactly supported. Let ĥ(s) = R >0 h(y)y s dy y be the Mellin transform. By Mellin inversion, h(y) = 1 πi Re(s)=s 0 ĥ(s)y s ds. Then for s 0 > 1, θ h (z) = 1 1 ĥ(s)(im γz) s ds = 1 ĥ(s)e(z, s) ds. πi Re(s)=s 0 πi Re(s)=s 0 γ Γ \Γ We compute I = Γ\H θ h(z) dz in two different ways: On the one hand, move the contour to Re(s) = 1 : I = vol(γ\h) 3 π ĥ(1 ) + 1 ĥ(s)i(s) ds. πi Re(s)= 1 On the other hand, we can compute I directly using the definition of θ h by the unfolding method: I = Γ \H h(im z) dz = h(y) dy R >0 y = ĥ(1 ). Hence vol(γ\h) = π 3, and I(s) 0. We have used the following lemma: Lemma 1.. Suppose that I(t) is bounded, and R >0 ĥ(it)i(t) dt = aĥ(1) for all h C c (R >0 ). Then I 0 and a = 0. Proof. By taking h 1 = yh h, we have ĥ1 = (s 1)ĥ, and hence R >0 ĥ(it)i 1 (t) dt = 0 for I 1 (t) = (it 1)I(t). Since this is true for all h, I 1 0, and hence I Siegel s Eisenstein series. Siegel generalized the classical holomorphic Eisenstein series to Siegel domain. H g = {Z = X +iy : Y > 0}, where X, Y are g g symmetric matrices, and Y is ( ) A B positive definite. Let Γ = Sp g (Z) = {γ = : AB t = BA t, CD t = DC t, AD t BC t = I}. C D ( ) A B Then we have the action γz = (AZ + B)(CZ + D) 1. Let Γ = { Γ}, and define 0 D the Siegel Eisenstein series E k (Z) = Then E k (Z) has the Fourier expansion γ Γ \Γ det(cz + D) k. E k (Z) = T 0 a(t )e πit r(t Z),

6 6 HENRY KIM where a(t ) satisfies, when g =, a ( ) n r = r m d (n,r,m), d>0 d k 1 a ( 1 r This means that E k satisfies the Maass relations. The Maass subspace S k (Γ ) is a subspace of cusp forms generated by Saito-Kurokawa lift. They satisfy the Maass relations. We have an isomorphism S + k 1 (Kohnen plus space) S k (Γ ), given by g = n 1 n 3,4 mod 4 r d d mn d ) c(n)q n T >0 a(t )e πitr(t Z),. where a(t ) = d (n,r,m), d>0 ( ) ( ) 4mn r d k 1 n r c d, T =. r m Ikeda lift. By using the Eisenstein series, Ikeda constructed cusp form F of weight k + g on H g attached to an elliptic cusp form f of weight k such that k g (mod ). When g = 1, it is Saito-Kurokawa lift. Let E k+g (Z) = T 0 a(t )eπit r(t Z). When T > 0, a(t ) is given by a(t ) = ( )L(1 k, χ T )f k 1 T p D T Fp (T ; p k 1 ), where D T = det(t ) = d T f T, i.e., d T is the discriminant of Q[ ( 1) k D T ], and χ T is the quadratic character. Here F p (T ; X) is a Laurent polynomial which satisfies F p (T ; X 1 ) = F p (T ; X). Now let f(z) = n=1 a(n)qn. Then a(p) = p k 1 (α p + αp 1 ), where α p = 1. Let h(z) = ( 1) k n 0,1(mod 4) c(n)qn be the weight k + 1 Hecke eigenform which corresponds to f by the Shimura correspondence. Note that the Cohen Eisenstein series H k+ 1 (z) corresponds to the ordinary Eisenstein series E k (z) by the Shimura correspondence, and the d T th Fourier coefficient of H k+ 1 Then is the Ikeda lift. is L(1 k, χ T ). F (Z) = T >0 c(d T )f k 1 T F p (T ; α p )e πit r(t Z), p

7 EISENSTEIN SERIES AND THEIR APPLICATIONS 7. Lecture Two: Adelic Eisenstein series We follow [1], p. 19. In Lecture Two, we define adelic Eisenstein series and describe their role in representation theory..1. Eisenstein series in adelic setting. First, we reformulate Siegel s Eisenstein series in terms of adelic language and induced representation. We can take Γ to be any congruence subgroup such as Γ(N) = {γ Sp n (Z) : γ I (mod N)}. We can also consider vector-valued Eisenstein series. Let G = Sp n, and A = R A f be the ring of adeles and A f = pq p be its finite part. Then we have the strong approximation theorem: G(A) = G(Q)G(R)K, where K is any compact open subgroup of G(A f ). Usually we take K to be a subgroup of finite index in K = p K p, where K p = Sp n (Z p ). Then Γ = G(Q) G(R)K = Sp n (Z), and Γ = G(Q) G(R)K. ( ) a 0 Let P be the Siegel parabolic subgroup, i.e., P = MN, M = { } GL n, and 0 t a 1 ( ) I S N = { : S = t S}. Note that P (Q) Γ = Γ. Then P (Q)\G(Q) Γ \Γ. 0 I Hence the sum over Γ \Γ can be translated over the sum over P (Q)\G(Q). This gives rise to the concept of induced representation. A special case is called the degenerate principal series: Let χ be a grössencharacter. Let I(s, χ) be the space of all smooth K-finite functions Φ(g, s) on ( ) a 0 G(A) such that Φ(nm(a)g, s) = χ(det(a)) det(a) s+ρn Φ(g, s), where n N, m(a) =, 0 t a 1 and ρ n = n+1, which is for normalization. Then I(s, χ) = vi v (s, χ v ), where I v (s, χ v ) is the local induced representation. We can take Φ(g, s) = v Φ v (g, s), and the restriction of Φ(g, s) to K is independent of s. Note that Φ(g, s) is determined by its restriction to K. For any Φ(g, s) I(s, χ), the Siegel Eisenstein series is defined by E(g, s, Φ) = Φ(γg, s). γ P (Q)\G(Q) Then E(g, s, Φ) has a meromorphic continuation to the whole C, and satisfies the functional equation E(g, s, Φ) = E(g, s, M(s)Φ), where M(s) is the global intertwining operator M(s) : I(s, χ) I( s, χ 1 ), defined by M(s)Φ(g, s) = Φ(ng, s) dn. N(A)

8 8 HENRY KIM Notice that we have shifted the line of symmetry to Re(s) = 0. Special case: Let Φ(g, s) = Φ (s) p Φ p(s). Take Φ p to be the normalized spherical section for all p. (Φ p (k, s) = 1 for all k K.) For a positive integer l, let Φ (k, s) = (det k) l, where ( ) a b k U(n) corresponds to k K by K = { : a + bi U(n)}. Here note that b a det l is the simplest representation of U(n). Any representation of U(n) is indexed by (l 1,..., l n ), l 1 l n. One can take an arbitrary representation φ of U(n). Let z = x + iy = x + i t vv = g(i I n ) H n, where g = n(x)m(v) P (R). Then we can show that E(g, s, Φ) = det(y) 1 (s+ρn+l) det(cz + d) l det(cz + d) s ρn+l. γ Γ \Γ So when s = l ρ n, we obtain the classical Siegel Eisenstein series of weight l. The advantage of this construction is that we can generalize for any compact subgroup K, and for any representation of φ of U(n). Also we can choose Φ p, different from spherical section to allow ramification. We can also generalize it to an arbitrary number field... Eisenstein series attached to induced representation. The above construction can be generalized to any reductive group and any parabolic subgroup. Let (π, W ) be an irreducible admissible representation of M(F ), where P = MN G is a maximal parabolic subgroup. Here F is a local or global field. Let I(s α, π) be the induced representation of G(F ); I(s α, π) = Ind G P π e s α,h P ( ) 1, where α is the fundamental weight corresponding to α, namely, α = ρ P, α 1 ρ P. The representation space V = V (s α, π) is the vector space of all smooth functions f : G(F ) W such that f(mng) = π(m)e s α+ρ P,H P (m) f(g), for all m M(F ), n N(F ), g G(F ). The action is (I(s α, π)(g)f)(h) = f(hg), for all f V and g, h G(F ). The reason we add ρ P is to normalize I(s α, π) so that if π is unitary, then I(s α, π) is unitary for s ir. The Eisenstein series is defined by, for f s I(s α, π), E(s, π, f s, g) = γ P (Q)\G(Q) f s(γg).

9 EISENSTEIN SERIES AND THEIR APPLICATIONS 9 More generally, let P = MN G be an arbitrary parabolic subgroup. Given an irreducible representation of M(F ) and a certain parameter Λ, one can define the induced representation I(Λ, π), and Eisenstein series E(g, Λ, π)..3. Spectral decomposition. As in GL case, Eisenstein series provide the continuous spectrum of L (G(Q)\G(A)). It has the orthogonal decomposition; L (G(Q)\G(A)) = (M,π) L (G(Q)\G(A)) (M,π), where M is a Levi subgroup of G, and π is a cuspidal representation of M(A) taken modulo conjugacy (Here we normalize π so that the action of the maximal split torus in the center of G at the archimedean place is trivial.). Here L (G(Q)\G(A)) (M,π) is generated by the pseudo- Eisenstein series ˆφ(g) = 1 (πi) dima E(g, φ, Λ) dλ, Re(Λ)=Λ 0 where A is the maximal split torus of M. Here the subspace (G,π) L dis (G(Q)\G(A)) (G,π), is the space of cuspidal representations L cusp(g(q)\g(a)). It belongs to the discrete spectrum L dis (G(Q)\G(A)). Moreover, the residues of the Eisenstein series span a part of the discrete spectrum L dis (G(Q)\G(A)) (M,π). Hence we have L (G(F )\G(A)) = L dis (G(F )\G(A)) L cont(g(q)\g(a)); L dis (G(Q)\G(A)) = L dis (G(Q)\G(A)) (M,π) = L cusp(g(q)\g(a)) L res(g(q)\g(a)). (M,π) Its orthogonal complement of the cuspidal spectrum in L dis (G(Q)\G(A)) is called the residual spectrum. Write L cusp = Π m cusp (Π) Π, L res = Π m res (Π) Π, where m cusp (Π), m res (Π) are called multiplicity of Π. Residual representations occur with multiplicity one (if the inducing cuspidal representation has multiplicity one) in L res. But cuspidal representations can occur with arbitrarily large multiplicity in some groups such as G.

10 10 HENRY KIM.4. Residual spectrum. We give several examples of residual spectrum. The trace formula cannot distinguish cuspidal spectrum from residual spectrum. The trace formula gives a formula in terms of Arthur packets like sum over all representations of Arthur packets = sum of some orbital integrals. Here Arthur packet contains both cuspidal representations and residual representations. We compute the residue of Eisenstein series attached to the trivial character of Borel subgroup and everywhere unramified for G = Sp 8 and G. Here G = Sp 8 is the smallest symplectic group which has non-trivial residues. Let G = Sp 8. Let E(g, Λ, Φ) = γ B(Q)\G(Q) f(γg) be the Eisenstein series attached to I(Λ), where Λ = (λ 1,..., λ 4 ) C 4. Let E 0 (g, Λ, f) be the constant term along B = T U, namely, E 0 (g, Λ, f) = E(ug, Λ, f) du = M(w, Λ)f(g), U(Q)\U(A) w W where M(w, Λ)f(g) = U wuw f(w 1 ug) du. Here W is the Weyl group of Sp 1 8, namely, W S 4 Z 4. Then the iterated residue is square integrable only when Λ = (4, 3,, 1), in which case we get the constant, and Λ = (, 1, 0, 1), (1, 0,, 1), in which case we get the non-trivial residue. Let G = G. This is done by Langlands [4]. We follow the notation from Moeglin-Waldspurger [5]: Simple roots are β 1 (long), β 6 (short). Let β = β 1 + β 6, β 3 = β 1 + 3β 6, β 4 = β 1 + β 6, β 5 = β 1 + 3β 6. Then {β 1,..., β 6 } are positive roots. The Weyl group W D 1, the dihedral group of order 1. Then the constant term of E(g, Λ, f) is E 0 (g, Λ, f) = w W M(w, Λ)f(g). If f = f v, unramified everywhere, M(w, Λ)f is a quotient of product of the completed Riemann zeta functions; Let ξ(s) = π s Γ( s )ζ(s). Then M(w, Λ)f = Let Λ = xβ 3 + yβ 4 with x, y C. Then α>0, wα<0 ξ( Λ, α ) ξ( Λ, α + 1) f.

11 EISENSTEIN SERIES AND THEIR APPLICATIONS 11 E 0 (e, Λ, f) = 1 + ξ(x) ξ(x + 1) + ξ(y) ξ(y + 1) + ξ(x)ξ(3x + y) ξ(x + 1)ξ(3x + y + 1) + ξ(y)ξ(x + y) ξ(y + 1)ξ(x + y + 1) ξ(x)ξ(3x + y)ξ(x + y) + ξ(x + 1)ξ(3x + y + 1)ξ(x + y + 1) + ξ(y)ξ(x + y)ξ(3x + y) ξ(y + 1)ξ(x + y + 1)ξ(3x + y + 1) ξ(x)ξ(3x + y)ξ(x + y)ξ(3x + y) + ξ(x + 1)ξ(3x + y + 1)ξ(x + y + 1)ξ(3x + y + 1) ξ(y)ξ(x + y)ξ(x + y)ξ(3x + y) + ξ(y + 1)ξ(x + y + 1)ξ(x + y + 1)ξ(3x + y + 1) ξ(x)ξ(x + y)ξ(3x + y)ξ(x + y)ξ(3x + y) + ξ(x + 1)ξ(x + y + 1)ξ(3x + y)ξ(x + y + 1)ξ(3x + y + 1) ξ(y)ξ(x + y)ξ(x + y)ξ(3x + y)ξ(3x + y) + ξ(y + 1)ξ(x + y + 1)ξ(x + y + 1)ξ(3x + y + 1)ξ(3x + y + 1) ξ(x)ξ(y)ξ(x + y)ξ(x + y)ξ(3x + y)ξ(3x + y) + ξ(x + 1)ξ(y + 1)ξ(x + y + 1)ξ(x + y + 1)ξ(3x + y + 1)ξ(3x + y + 1). Then the iterated residue is square integrable only when Λ = β 3 + β 4, in which case we get the constant, and when Λ = β, in which case we get non-trivial residue. If we allow ramification, then it becomes a little more complicated: Namely, we allow f = f v to be arbitrary. Then f p is spherical for almost all p. When f p is not spherical, then the residue is the image (1 + 1 E)R(ρ, β )I(β ), where R(ρ, β ) is the normalized intertwining operator, and f p, if f p π 1p R(ρ, β )I(β ) = π 1p π p, and E(f p ) =. f p, if f p π p Let π S = v / S π 1v v S π v. Then π S occurs in L (G(Q)\G(A)) if and only if S 1. Here π S belongs to an Arthur packet. Its multiplicity in the discrete spectrum is bigger than 1 6 (#S + ( 1) #S )..5. Volume of the fundamental domain. We follow [3]. For T 0, let T be the Arthur truncation operator for φ on G(Q)\G(A). For the constant function 1, T 1 is the characteristic function of a compact subset F(T ) of G(Q)\G(Q) 1, where G(A) 1 = {g G(A) : H G (g) = 0} and H G is a certain homomorphism. For example, if G = GL n, G(A) 1 = {g GL n (A) : det(g) = 1}. Then as T, we obtain the fundamental domain F. We fix a Haar measure on G(Q)\G(A) 1 so that K gets measure 1. We want to find the volume of the fundamental domain of G(Q)\G(A) 1.

12 1 HENRY KIM We use the formula by Jacquet-Lapid-Rogawski []: Let E(g, λ, 1) be the Eisenstein series associated to the trivial character of the Borel subgroup. Then E(g, λ, 1) dg = v e wλ ρ,t F(T ) w W α wλ ρ, α where v = V ol({ α a αα : 0 a α < 1}. α>0, wα<0 ξ( λ, α ) ξ( λ, α + 1), We look at the residue at λ = ρ, the half sum of positive roots. We know that the residue of E(g, λ, 1) at λ = ρ is a constant, which is i>1 ξ(i)n i, where n i = #{α > 0 : ρ, α = i} #{α > 0 : ρ, α = i 1}. From this, we get V ol(f(t )) = v ξ(i) n i i>1 J ( 1) rankp J i>1 ξ(i)n i,j α w J J (1 w Jρ, α ) e w J ρ ρ,t, where P J is the parabolic subgroup corresponding to J, and w J is the element of the maximal length in D J, distinguished coset representatives of W/W J. Also n i,j = #{α > 0, w J α < 0 : ρ, α = i} #{α > 0, w J α < 0 : ρ, α = i 1}. As T, only the term corresponding to w = 1 survives, and hence V ol(f) = v i>1 ξ(i) n i. 3. Lecture Three: Langlands-Shahidi method In Lecture Three, we describe Langlands-Shahidi method. We follow [1]. Langlands observed that the constant term of Eisenstein series associated to cuspidal representation of maximal parabolic subgroups contains many new automorphic L-functions. The meromorphic continuation and functional equation of Eisenstein series give rise to the same for the automorphic L-functions. Let π = v π v be a cuspidal representation of M(A), and let r : L M GL N (C) be a finitedimensional representation. For almost all v (say, v / S), π v is spherical. So it is uniquely determined by a semi-simple conjugacy class {t v } L T. We form a local Langlands L-function 0. L(s, π v, r) = det(i r(t v )q s v ) 1. Let L S (s, π, r) = v / S L(s, π v, r) be a partial L-function. It converges absolutely for Re(s) Conjecture 3.1. (Langlands) L S (s, π, r) has a meromorphic continuation to all of C. Fix an additive character ψ = ψ v of A/Q. For each v S, we can define a local L-function L(s, π v, r) and a local root number ɛ(s, π v, r, ψ v ) such that the completed L-function L(s, π, r) = all v L(s, π v, r)

13 EISENSTEIN SERIES AND THEIR APPLICATIONS 13 has a meromorphic continuation and satisfies a functional equation L(s, π, r) = ɛ(s, π, r)l(1 s, π, r), where ɛ(s, π, r) = v ɛ(s, π v, r, ψ v ), and r(g) = t r(g) 1. Even the meromorphic continuation is not obvious: For example, it is known that the Euler product p 1(4) (1 p s ) 1 has a natural boundry at Re(s) = 0. Let π = π v be a cuspidal representation of GL. Let diag(α p, β p ) be a semi-simple conjugacy class of π p. Let Sym m : GL (C) GL m+1 (C) be the mth symmetric power representation. Then L(s, π p, Sym m ) = m (1 αp m i βpp i s ) 1. i=0 Holomorphic continuation and functional equations of m-th symmetric power L-functions are outstanding open problems. They provide test cases of Langlands functoriality: Let ψ p : W Qp SL (C) GL (C) be the parametrization of π p. Then we have Sym m ψ p : W Qp SL (C) GL m+1 (C). By the local Langlands correspondence proved by Harris-Taylor and Henniart, we have an irreducible admissible representation Sym m π p corresponding to Sym m ψ p. Let Sym m π = Sym m π p. It is an irreducible admissible representation of GL m+1 (A). Conjecture 3.. (Langlands functoriality conjecture) Sym m π = Sym m π p is an automorphic representation of GL m+1 (A). If we know that Sym m π is an automorphic representation of GL m+1, then by Jacquet-Shalika, α m p < p 1. So α p < p 1 m. If it is true for all m, then α p = 1. It is the Ramanujan conjecture. The current best bound is α p p If λ 1 is the first positive eigenvalue of the Laplacian on the corresponding hyperbolic space, we would have λ 1 > 1 4 (1 ( m ) ) for all m. So λ This is called the Selberg conjecture. The current best bound is λ (1 ( 7 3 ) ) = = It also implies that the L-functions L(s, π, Sym m ) is non-vanishing for all m. It implies the Sato-Tate conjecture; write α p = e iθp with 0 θ p π. Sato-Tate conjecture says that {θ p } is equidistributed with respect to the measure π sin θdθ. Let (a, b) [0, π]. Then lim x 1 b π(x) #{p x, θ p (a, b)} = a π sin θ dθ. Currently, we only know for m =, 3, 4 for arbitrary cusp form. When π comes from a holomorphic cusp form, we have a recent potential modularity result due to Taylor and others.

14 14 HENRY KIM It gives rise meromorphic continuation to all of C, and non-vanishing for Re(s) 1. It is enough to prove Sato-Tate conjecture. Langlands proved meromorphic continuation for many L-functions, and Shahidi computed nonconstant terms for globally generic cuspidal representations and proved the functional equations. Let G be a Chevalley group, and be the set of simple roots. Let P = P θ = MN be a maximal parabolic subgroup of G, where θ = {α}. Let w 0 be the unique element in W such that w 0 (θ) and w 0 (α) < 0. Then P is called self-conjugate if P = P = P w0 (θ). Let π be a cuspidal representation of M(A) and I(s, π) be the induced representation. For f s I(s, π), let E(s, π, f s, g) = γ P (Q)\G(Q) f s(γg) be the Eisenstein series. For a parabolic subgroup Q = M Q N Q, let E Q (s, π, f s, g) = N Q (Q)\N Q (A) E(s, π, f s, ng) dn. Theorem 3.3. Unless Q = P, P, E Q (s, π, f s, g) = 0. If P is self-conjugate, then If P is not self-conjugate, then E P (s, π, f s, g) = f s + M(s, π)f s. E P (s, π, f s, g) = f s, E P (s, π, f s, g) = M(s, π)f s. Let M(s, π) = A(s, π v, w 0 ), and f = f v, where f v is spherical for almost all v. When f v is spherical, we have Gindikin-Karpelevich formula: Let π v I(χ v ), where χ v is a unitary character. A(s, π v, w 0 )f v (e) = β>0, w 0 β<0 L(s α, β, χ v β ) L(1 + s α, β, χ v β ). Langlands observed that α, β = i for i = 1,..., m; Let V i be the subspace of L n, generated by E β such that α, β = i. Here L n is the Lie algebra of L N. For each i, the adjoint action of L M leaves V i stable. Let r be the adjoint representation of L M on L n, and r i = r Vi. Then Theorem 3.4. (Langlands-Shahidi) r i is irreducible for each i, and the weights of r i are the roots β in L n which restricts to iα in L A. Therefore, we have A(s, π v, w 0 )f v (e) = where L(s, π v, r i ) = β>0, α,β =i L(s, χ v β ). m i=1 L(is, π v, r i ) L(1 + is, π v, r i ),

15 EISENSTEIN SERIES AND THEIR APPLICATIONS 15 ( ) I X Example 1: Let G = GL k+l and M GL k GL l, and N = { }. Then L M 0 I GL k (C) GL l (C), and r(diag(g 1, g ))X = g 1 Xg 1. So r is irreducible. Hence m = 1. Let π 1 = π(µ 1,..., µ k ) and π = π(ν 1,..., ν l ) be spherical representations of GL k, GL l, resp. Then L(s, π 1 π, r) = L(s, π 1 π ) = i,j L(s, µ i ν 1 j ). Example : Let G = G, and P be attached to {β 1 }. Then P = MN, and M GL. In this case, m =, and r 1 = Sym 3 (ρ ) ( ρ ) 1 and r = ρ, where ρ : GL (C) GL (C). We obtain the symmetric cube L-function. Example 3: Let G = E 8, and P = MN, where the derived group of M is SL 3 SL SL 5. Then m = 6 (maximum length), and r 1 gives rise to the triple product L-function of GL 3 GL GL List of L-functions via Langlands-Shahidi method (split reductive groups) A n case. GL m GL n GL m+n gives the Rankin-Selberg L-function L(s, π 1 π ), where π i s are cuspidal representations of GL m, GL n, resp B n case. B n 1: m = ; r 1 gives the Rankin-Selberg L-function of GL k GSpin(l + 1); r gives the twisted symmetric square L-function of GL k. If G = SO(n + 1), r 1 gives the Rankin-Selberg L-function of GL k SO(l + 1); r gives the symmetric square L-function of GL k C n case. C n 1: m = ; r 1 gives the Rankin-Selberg L-function of GL k Sp(l); r gives the exterior square L-function of GL k, if k 1 (If k = 1, then m = 1) D n case. D n 1: m = ; r 1 gives the Rankin-Selberg L-function of GL k GSpin(l); r gives the twisted exterior square L-function of GL k. If G = SO(n), r 1 gives the Rankin-Selberg L-function of GL k SO(l); r gives the exterior square L-function of GL k (If k = 1, then m = 1). D n : m = ; r 1 gives the triple L-function of GL n GL GL ; r gives the twisted exterior square L-function of GL n. D n 3: m = ; r 1 gives L(s, σ τ, ρ n 3 ρ 4 ); r gives the twisted exterior square L-function of GL n 3.

16 16 HENRY KIM F 4 case. F 4 1: m = 4; r 1 gives L(s, σ τ) which is entire; r gives L(s, σ τ, Sym ρ ρ 3 ) which has a pole at s = 1 when τ Sym σ. F 4 : m = 3; r 1 gives L(s, σ τ, Sym ρ 3 ρ ); r gives L(s, σ, Sym ρ 3 ω τ ) which has a pole at s = 1 always. (xviii): m = ; M = GSpin(7) F 4 ; dim r = 1; r 1 is the 14-dim l irreducible representation of Sp 6 (C), called spherical harmonic. (xxii): m = ; M = GSp 6 F 4 ; r 1 =8-dim l spin representation of Spin(7, C); r gives the standard L-function of SO 7 (C) (7-dimensional) E 6 case. E 6 1: m = 3; r 1 gives the triple L-function of GL 3 GL GL 3 ; r gives the standard L-function of GL 3 GL 3. E 6 : m = ; r 1 = ρ 5 ρ ; r gives the Rankin-Selberg L-function of GL 5 GL which is entire. (x): m = ; dim r = 1; r 1 gives the exterior cube L-function of GL 6 (C) (0 dim l irreducible representation of GL 6 (C)) (xxiv): m = 1; r 1 =16-dimensional half-spin representation of Spin(10) E 7 case. E 7 1: m = 4; r 1 gives the triple L-function of GL 3 GL GL 4 ; r comes from D 6 3 case. E 7 : m = 3; r 1 = ρ 5 ρ 3 ; r gives the Rankin-Selberg L-function of GL 5 GL 3 which is entire. E 7 3: m = ; r 1 gives the L-function L(s, σ τ, Spin 16 ρ ), where σ, τ are cuspidal representations of GSpin 10, GL, resp. and Spin 16 is the 16-dimensional half-spin representation of Spin 16 (C). E 7 4: m = 3; r 1 gives the L-function L(s, σ τ, ρ 6 ρ ), where σ, τ are cuspidal representations of GL 6, GL, resp. (xi): m = ; r 1 gives the exterior cube L-function of GL 7 (C) (35-dimensional representation of GL 7 (C)); r gives the standard L-function of GL 7 (C) which is entire. (xxvi): m = ; dim r = 1 and r 1 gives the degree 3 = 5 spin L-function of Spin 1 (xxx): m = 1; r 1 gives the standard L-function of E E 8 case. E 8 1: m = 6; r 1 gives the triple L-function of GL 3 GL GL 5 ; r comes from E 7 case.

17 EISENSTEIN SERIES AND THEIR APPLICATIONS 17 E 8 : m = 5; r 1 gives the L-function L(s, σ τ, ρ 5 ρ 4 ), where σ, τ are cuspidal representations of GL 5, GL 4, resp. E 8 3: m = 4; r 1 gives the L-function L(s, σ τ, Spin 16 ρ 3 ), where σ, τ are cuspidal representations of GSpin 10, GL 3, resp. and Spin 16 is the16-dimensional half-spin representation of Spin 16 (C). E 8 4: m = 3; r 1 gives the standard L-function of E 6 GL ; r gives the standard L-function of E 6 ((xxx) case). E 8 5: m = 4; r 1 gives the L-function L(s, σ τ, ρ 7 ρ ), where σ, τ are cuspidal representations of GL 7, GL, resp. (xiii): m = 3; r 1 gives the degree 56 exterior cube L-function of GL 8 (xxviii): m = ; r 1 gives the degree 64 = 6 spin L-function of Spin 14 (xxxii): m = ; dim r = 1; r 1 gives the standard L-function of E G case. (xv) (attached to the maximal parabolic subgroup generated whose unipotent radical contains the long simple root subgroup) : m = ; dim r = 1, r 1 gives the third symmetric power L-function of GL (xvi) (attached to the maximal parabolic subgroup generated whose unipotent radical contains the short simple root subgroup): m = 3; dim r = 1, r 1 gives the standard L-function of GL We have M(s, π)f = m i=1 L S (is, π, r i ) L S (1 + is, π, r i ) v / S f 0 v v S A(s, π v, w 0 )f v. The meromorphic continuation of the Eisenstein series gives rise to the same for M(s, π). By induction on m, it gives rise to the meromorphic continuation of L S (s, π, r i ). For induction, we show that r i, i, appears at r 1 for some other group. However, it does not give the desired functional equation. We only get the functional equation of the quotient L S (s, π, r i ) L S (1 + s, π, r i ) = v S µ i (s, π v ) L S(1 s, π, r i ) L S ( s, π, r i ), where µ i is some meromorphic function. In order to obtain the functional equation, we need to isolate L S (s, π, r i ). Shahidi computed non-constant term of Eisenstein series for globally generic cuspidal representations. Let B = T U. Then U/[U, U] α U α. Hence if ψ is a character of U(Q)\U(A), ψ = α ψ α. We say that ψ is generic or non-degenerate if each ψ α is non-trivial. Let π be a

18 18 HENRY KIM cuspidal representation of M(A), and let ϕ be a function in the space of π. Let W ϕ (g) = ϕ(ug)ψ(u) du. U(Q)\U(A) W ϕ is called the Whittaker function. We say that π is globally generic if W ϕ 0 for some ϕ. Then W ϕ = W ϕv, i.e., π v is locally generic for all v. Now let E ψ (s, π, f s, g) = E(s, π, f s, ug)ψ(u) du. U(Q)\U(A) It is called ψ-th Fourier coefficient (or non-constant term) of E(s, π, f s, g). Then by Casselman- Shalika formula, Theorem 3.5. Suppose f s = f v, where f v = fv 0, spherical for v / S. Then v S E ψ (s, π, f s, e) = W f v (e) m i=1 L S(1 + is, π, r i ). By induction, this gives rise to functional equation of the form L S (s, π, r i ) = v S γ(s, π v, r i, ψ v )L S (1 s, π, r i ). By the theory of local coefficients, Shahidi was able to refine this to get the desired functional equation. Also from the fact that the Eisenstein series is holomorphic for Re(s) = 0, we obtain the fact that m i=1 L S(1+is, π, r i ) is nonvanishing for Re(s) = 0. For example, for cuspidal representations π 1, π of GL k, GL l, resp., L S (s, π 1 π ) has no zeros for Re(s) 1. By Sarnak s method, Gelbart- Lapid was able to get zero-free region of L S (s, π, r i ). 3.. Application to functoriality. The key theorem is Theorem 3.6. Suppose P is a maximal parabolic subgroup. Unless P is self-conjugate and w 0 π π, L dis (G(Q)\G(A)) (M,π) = 0, i.e., the Eisenstein series has no poles for Re(s) 0, and M(s, π) is holomorphic for Re(s) 0. We can choose a grössencharacter χ where χ p is highly ramified for some p such that w 0 (π χ) π χ. Then M(s, π χ) is holomorphic for Re(s) 0. We normalize the intertwining operator A(s, π p, w 0 ) as A(s, π p, w 0 ) = m i=1 L(is, π p, r i ) L(1 + is, π p, r i )ɛ(is, π p, r i ) N(s, π p, w 0 ).

19 Then M(s, π χ) = EISENSTEIN SERIES AND THEIR APPLICATIONS 19 m i=1 L(is, π χ, r i ) L(1 + is, π χ, r i )ɛ(is, π χ, r i ) p N(s, π p χ p, w 0 ). We prove that N(s, π p, w 0 ) is holomorphic and non-vanishing for Re(s) 1. By induction, we prove that L(s, π χ, r i ) is holomorphic for Re(s) 1, and non-vanishing for Re(s) 1. By the functional equation, we prove that L(s, π χ, r i ) is entire. We also have Theorem 3.7. (Gelbart-Shahidi) Suppose L(s, π, r i ) is entire. strips. Then it is bounded in vertical We can now apply the converse theorem of Cogdell and Piatetski-Shapiro. Theorem 3.8. Suppose Π = Π p is an irreducible admission representation of GL N (A) such that ω Π = ω Πp is a grössencharacter. Let S be a finite set of primes and let T S (m) be the set of cuspidal representations of GL m (A) that are unramified at all primes p S. Suppose L(s, σ Π) is entire, bounded in vertical strips, and satisfies a functional equation for all σ T S (m), m < n 1. Then there exists an automorphic representation Π of GL N (A) such that Π p Π p for all p / S Functoriality of Sym 3 π. Functoriality of Sym 3 π follows from the functoriality of the tensor product π Sym π. Consider φ : GL (C) GL 3 (C) GL 6 (C) be the tensor product. Then given cuspidal representations π 1, π of GL (A), GL 3 (A), resp, one can define an irreducible admission representation π 1 π = (π 1p π p ) by the local Langlands correspondence. Langlands functoriality conjecture says that π 1 π is an automorphic representation of GL 6 (A). In order to prove it by the converse theorem, we need the triple product L-functions L(s, σ π 1 π ), where σ is a cuspidal representation of GL m (A), m = 1,, 3, 4. They are available from Langlands-Shahidi method: (1) D 5 case: G = Spin(10), the derived group of M is SL SL SL 3 ; () E 6 1 case: G = E 6, the derived group of M is SL 3 SL SL 3 ; (3) E 7 case: G = E 7, the derived group of M is SL 4 SL SL 3. Here Sym π is the symmetric square, or Gelbart-Jacquet lift. It is a cuspidal representation of GL 3 (A) if it is not of dihedral type. Then when π has the trivial central character, π Sym π = π Sym 3 π Functoriality of Sym 4 π. Functoriality of Sym 4 π follows from the functoriality of the exterior square Π, where Π is a cuspidal representation of GL 4 (A); Consider : GL 4 (C)

20 0 HENRY KIM GL 6 (C) be the exterior square. Then given a cuspidal representation Π of GL 4 (A), one can define an irreducible admission representation Π by the local Langlands correspondence. Langlands functoriality conjecture says that Π is an automorphic representation of GL 6 (A). In order to prove it by the converse theorem, we need the L-functions L(s, σ Π, ρ m ρ 4 ), where σ is a cuspidal representation of GL m (A), m = 1,, 3, 4, and ρ m is the standard representation of GL m (C). They are available from Langlands-Shahidi method: D n 3 case, n = 4, 5, 6, 7: G = Spin(n), the derived group of M is SL n 3 SL 4. Then when π has the trivial central character, (Sym 3 π) = 1 Sym 4 π. In this way, we prove that Sym 4 π is an automorphic representation of GL 5 (A) Speculation on Kac-Moody groups. It is evident that Langlands-Shahidi method gives rise to limited cases of L-functions due to the fact that there are only finitely many exceptional Lie groups. On the other hand, we can construct Kac-Moody groups attached to extended Dynkin diagrams. It is tempting to generalize the theory of Eisenstein series to Kac-Moody groups and Langlands-Shahidi method. However, the theory of Eisenstein series, in particular, the ones attached to maximal parabolic subgroups, is still in its infancy. Furthermore, if G is an infinite dimensional group, and P is a finite dimensional maximal parabolic subgroup, Shahidi [8] showed that P can never be self-conjugate. It means that the constant term of the Eisenstein series, even if it may be defined, may be trivial. Let G = E (1) 6, the affine Kac-Moody group, attached to the affine Dynkin diagram o α1 o α3 o α4 o α5 o α6 o α o α7 Consider P = P θ, where θ = {α 4 }. Then P = MN, and the derived group of M is SL 3 SL 3 SL 3. Shahidi [8] calculated the adjoint action of m C on n C. Then the following representations appear infinitely many times: r 1 = δ 1 + δ 7 + δ 6, r = δ 3 + δ + δ 5, r 3 = (δ 1 + δ 3 ) (δ + δ 7 ) (δ 5 + δ 6 ),

21 EISENSTEIN SERIES AND THEIR APPLICATIONS 1 where δ i s denote the fundamental weights attached to α i s. So r 1 is the triple tensor product of standard representations of sl 3 (C), and r = r 1, and r 3 is the direct sum of three 8-dimensional adjoint representations δ 1 + δ 3, δ + δ 7, δ 5 + δ 6 of these sl 3 (C). So if π 1, π, π 3, are cuspidal representations of GL 3, we obtain the triple product L-function L(s, π 1 π π 3 ). Similarly, if we take G is the Kac-Moody group of type T p,q,r, and P = MN, maximal parabolic subgroup such that the derived group of M is SL p SL q SL r, then we would get the triple product L-function L(s, π 1 π π 3 ), where π i, i = 1,, 3, is a cuspidal representation of GL p, GL q, GL r, resp. Then the converse theorem would imply the functoriality of π 1 π as an automorphic representation of GL pq. References [1] J. Cogdell, H. Kim and R. Murty, Automorphic L-functions, AMS Fields Monograph 0, 004. [] H. Jacquet, E. Lapid and J. Rogawski, Periods of automorphic forms, J. of AMS, 1 (1999), [3] H. Kim and L. Weng, Volume of truncated fundamental domains, Proc. Amer. Math. Soc. 135 (007), no. 6, [4] R. Langlands, On the functional equations satisfied by Eisenstein series. Lecture Notes in Mathematics, Vol Springer-Verlag, Berlin-New York, [5] C. Moeglin and Waldspurger, Spectral decomposition and Eisenstein series. Cambridge Tracts in Mathematics, 113, Cambridge University Press, Cambridge, [6] P. Sarnak, Nonvanishing of L-functions on Re(s) = 1, Contributions to automorphic forms, geometry, and number theory, , Johns Hopkins Univ. Press, Baltimore, 004. [7] J.P. Serre, A Course in Arithmetic, Springer-Verlag, [8] F. Shahidi, Infinite dimensional groups and automorphic L-functions, Pure and Applied Math. Quarterly, 1 (005), Department of mathematics, University of Toronto, Toronto, Ontario M5S E4, CANADA, and Korea Institute for Advanced Study, Seoul, KOREA address: henrykim@math.toronto.edu