1. Preliminaries Let F be a number field. For each place v of F , let - PDF document

16 on the space of π , called a Whittaker func- tional, which is continuous with respect to the seminorm topology defined by the Hilbert space norm on the space H ( π ) of π if F is archimedean (cf. [S,Sh4,Sh5]) (continu- ous with respect to the trivial locally convex topology on H ( π ) for which every seminorm is continuous if F is non–archimedean) and satisfies (2.6) � π ( u ) x, λ � = χ ( u ) � x, λ � , u ∈ U 0 , x ∈ H ( π ) ∞ , the subspace of C ∞ – vectors. By a theorem of Shalika [S], the space of all the Whittaker functionals on H ( π ) is at most one–dimensional. Changing

17 the splitting, we may assume χ is defined by (2.4) and (2.5). Now, assume F is global. Let ψ = ⊗ v ψ v be a non–trivial character of F \ A F . Then each ψ v is non–trivial. Moreover, for al- most all v , ψ v is unramified, i.e., O v is the largest ideal on which ψ v is trivial. The map (2.4) is F –rational and therefore extends to a map from U = U ( A F ) into G a ( A F ), send- ing U ( F ) into G a ( F ). We then define a character χ of U ( F ) \ U by (2.5). Conse- quently, if χ v ( u v ) is defined by (2.5) and ψ v for each v , then χ ( u ) = � v χ v ( u v ), where u = ( u v ) v ∈ U . Now, let π = ⊗ v π v is a cuspidal represen-

18 tation of M = M ( A F ). Choose a function ϕ in the space of π and set � (2.7) W ϕ ( m ) = U 0 ( F ) \ U 0 ϕ ( um ) χ ( u ) du. We shall say π is (globally) χ –generic if W ϕ � = 0 for some ϕ . Then each π v is χ v –generic. If ϕ = ⊗ v ϕ v , ϕ v ∈ H ( π v ), then � W ϕ ( m ) = � π v ( m v ) ϕ v , λ v � v � (2.8) = W ϕ v ( m v ) v for some χ v –Whittaker functional λ v on H ( π v ). We finally point out that every generic character χ of U ( F )( U ( F ) \ U ( A F ) if F is

19 global) points to a F –splitting and conversely, each F –splitting defines a generic character, both by means of (2.4) and (2.5). Now assume F is local and π v is an ir- reducible admissible representation of M v . Let s ∈ C and denote by I ( s, π v ) = I ( s ˜ α, π v ), the induced representation (2.9) I ( s, π v ) = Ind M v N v ↑ G v π v ⊗ q � s ˜ α,H Mv ( ) � ⊗ 1 . v This is the right regular action of G v on the space of smooth functions f from G v to H ( π v ), satisfying: (2.10) f ( mng ) = π v ( m ) q � s ˜ α + ρ P ,H Mv ( m ) � f ( g ) . v If v = ∞ , smooth would mean that f is

20 infinitely many times differentiable. On the other hand for p < ∞ , it simply means that f is locally constant, i.e., f ( gk ) = f ( g ) for k in some open compact subgroup depending on f (cf. [Ca1,Car]). One can now define a Whittaker func- tional λ χ v ( s, π v ) for I ( s, π v ) as follows: (2.11) � � f ( w − 1 0 n ′ ) , λ � χ ( n ′ ) dn ′ , λ χ v ( s, π v )( f ) = N ′ v where λ is a fixed Whittaker functional on H ( π v ). The integral converges as a princi- pal value integral and stabilizes as N ′ v is ap- proached by an increasing sequence of open compact subgroups (cf. [CS,Sh3]), i.e., we

21 may replace N ′ v with a sufficiently large open compact subgroup depending on f . 3. Eisenstein Series and Intertwining Operators; The Constant Term Let π = ⊗ v π v be a cusp form on M . Given a K M –finite function ϕ in the space of π , we extend ϕ to a function ˜ ϕ on G as follows. The representation π is a subrepre- sentation of L 2 0 ( Z 0 M M ( F ) \ M, ρ ), where Z 0 M is the A F –points of the connected compo- nent Z 0 M of the center of M and ρ is a character of Z 0 M ( F ) \ Z 0 M . The function ϕ is then in this L 2 –space and being K M – finite, its right translations by elements in

22 K M = Π v K M,v , K M,v = K v ∩ M v , generate a finite dimensional representation τ of K M (cf. [Car]). We may assume ϕ is so chosen that τ is irreducible and write τ = ⊗ v τ v , where for almost all v, τ v is trivial. Next we will choose irreducible (finite dimensional) representations ˜ τ v of each K v , containing τ v . Moreover, we assume ˜ τ v is the triv- ial representation for almost all K v . Set τ = ⊗ v ˜ ˜ τ v (cf. [Sh6]). Let P τ be the projection on the space of τ and fix measures dk v on each K M,v whose total mass is 1. Let dk be the product mea- sure on K M . Set

23 � (3.1) ϕ ( m ) = ˜ ϕ ( mk )˜ τ ( k ) dk · P τ . K M ϕ ( mk − 1 ) = ˜ Observe that ˜ τ ( k ) ˜ ϕ ( m ). This is a ˜ τ –function on M in Harish–Chandra’s ter- minology [HC]. We extend ˜ τ to all of G by τ ( k − 1 ) ˜ ϕ ( nmk ) = ˜ ˜ ϕ ( m ). It is easily checked to be a well–defined operator valued func- tion on G ([Sh6]). Next, set ˜ (3.2) Φ s ( g ) = ˜ ϕ ( g ) exp � s ˜ α + ρ P , H P ( g ) � and let Φ s be a matrix coefficient of this operator valued function. (See (3.7) below.) The corresponding Eisenstein series is then

24 defined by � (3.3) E ( s, Φ s , g, P ) = Φ s ( γg ) . γ ∈ P ( F ) \ G ( F ) We will also define the operator valued Eisenstein series by (3.4) ˜ E ( s, ˜ ˜ � Φ s , g, P ) = Φ s ( γg ) . γ ∈ P ( F ) \ G ( F ) They both converge for Re( s ) >> 0 and have a finite number of simple poles for Re( s ) ≥ 0, none with Re( s ) = 0 (cf. [HC,La2,MW1]). Let I ( s, π ) = ⊗ v I ( s, π v ) be the represen- tation of G = G ( A F ) induced from (3.5) π ⊗ exp � s ˜ α + ρ P , H M ( ) � ⊗ 1 .

25 Let f ∈ V ( s, π ) = ⊗ v V ( s, π v ) be defined by (3.6) f ( n 0 m 0 k 0 ) = exp � s ˜ α + ρ P , H P ( m 0 ) �· � τ ( k − 1 � ˜ 0 ) τ ( k ) x, ˆ x � π ( m 0 k ) ϕdk, K M with m 0 ∈ M, n 0 ∈ N and k 0 ∈ K . Here x ∈ H ( τ ) and ˆ x ∈ H (ˆ τ ), where ˆ τ is the contragredient of τ . Moreover f ( n 0 m 0 k 0 )( e ) = � ˜ Φ s ( g ) x, ˆ x � (3.7) = Φ s ( g ) , where the left hand side is the value of the cusp form f ( n 0 m 0 k 0 ) at identity and g = n 0 m 0 k 0 (cf. [Sh6]).

26 Observe that (3.8) E ( s, Φ s , g, P ) = � ˜ E ( s, ˜ Φ s , g, P ) x, ˆ x � . Given f ∈ V ( s, π ) and Re( s ) >> 0, de- fine the global intertwining operator A ( s, π ) by � N ′ f ( w − 1 0 n ′ g ) dn ′ . (3.9) A ( s, π ) f ( g ) = Finally, if at each v we define a local inter- twining operator by (3.10) � f v ( w − 1 0 n ′ g ) dn ′ , A ( s, π v , w 0 ) f v ( g ) = N ′ v then (3.11) A ( s, π ) = ⊗ v A ( s, π v , w 0 ) .

27 Observe that (3.12) A ( s, π ): I ( s, π ) → I ( − s, w 0 ( π )) and (3.13) A ( s, π v , w 0 ): I ( s, π v ) → I ( − s, w 0 ( π v )) . Using (3.7), we now define (3.14) ( M ( s, π )Φ s )( g ) = A ( s, π ) f ( g )( e ) , where by the left hand side we understand the value of the cusp form A ( s, π ) f ( g ) at e . This is basically the Langlands’ M ( s, π ) in- troduced in [La2], or as denoted by Harish– Chandra in [HC], his function c ( s, π ).

28 Constant Term Theorem. The constant term (3.15) � N ′ ( F ) \ N ′ E ( s, Φ s , n ′ g, P ) dn ′ E P ′ ( s, Φ s , g, P ) = is equal to (3.16) E P ′ ( s, Φ s , g, P ) = δ M , M ′ Φ s ( g )+( M ( s, π )Φ s )( g ) . Here δ M , M ′ is the Kronecker δ –function. Its analytic properties and therefore those of M ( s, π ) are exactly the same as E ( s, Φ s , − , P ) . In the split case the proof of the first part is in [La1], as in elsewhere. But the rest is among the main properties of Eisen-

29 stein series and included in several places [La2,HC,MW1]. To prove the first part, one just substi- tutes (3.3) in (3.15) and expands and uses Bruhat decomposition and cuspidality of ϕ . We finally express the functional equa- tion of Eisenstein series by (cf. [HC,La2]), (3.17) E ( − s, M ( π, s )Φ s , g, P ′ ) = E ( s, Φ s , g, P ) . 4. Constant Term and Automorphic L –Functions It follows immediately from the Constant Term Theorem that ⊗ v A ( s, π v , w 0 ) is a mero- morphic function of s with a finite num-

30 ber of simple poles for Re( s ) > 0, none on Re( s ) = 0. Assume v is an unramified place for π , i.e., that π v is spherical. Take f 0 v ∈ V ( s, π v ) such that f 0 v ( k ) is a fixed vector invariant under M ( O v ) for all k ∈ G ( O v ). With no- m � tation as in Section 2, let r = r i be the i =1 adjoint action of L M on L n . We have Lemma 4.1 [La1]. Assume π v is unrami- fied. Then A ( s, π v , w 0 ) f 0 v ( e v ) m � r i ) f 0 = L ( is, π v , ˜ r i ) /L (1 + is, π v , ˜ v ( e v ) . i =1

31 With a clever induction [La1,La2,Sh3] the problem reduces to that of SL 2 , which we will now explain. ∗ Here G = SL 2 and M = T = G m = F v . We need to calculate �� � � �� � 0 1 1 x f 0 dx v − 1 0 0 1 F v (4.1.1) �� �� � 1 0 f 0 = dx, v x 1 F v � � 0 − 1 since R w f 0 v = f 0 v , where w = . 1 0 We then have � � � � � � 1 0 f 0 dx = dx v x 1 F v | x | v ≤ 1 (4.1.2) � � � � � 1 0 f 0 + dx. v x 1 | x | v > 1

32 We now write (4.1.3) x − 1 � � � � � � 1 0 1 0 − 1 = , x − 1 x 1 0 x 1 and therefore for | x | v > 1, which implies x − 1 ∈ O v , x − 1 � � � � � � � � 1 0 1 f 0 = f 0 v v x 1 0 x = η − 1 v ( x ) | x | − 1 − s , v where η v is the character of F ∗ (with cusp form η = ⊗ v η v on A ∗ F ) defining the induced representation. Then (4.1.2) equals

33 ∞ � � η v ( ̟ n v ) q − n − ns | x | v d ∗ x 1 + v ( ̟ − n ) − ( ̟ − n +1 ) n =1 v v ∞ � � η v ( ̟ n v ) q − n − ns | ̟ − n d ∗ x = 1 + v | v v O ∗ n =1 v ∞ � η v ( ̟ n v )(1 − q − 1 v ) q − ns = 1 + . v n =1 Write η v ( x ) = | x | µ v to get v (4.1.4) ∞ � q − nµ v − ns (1 − q − 1 1 + v ) v n =1 1 = 1 + (1 − q − 1 v ) · q − µ v − s · v 1 1 − q µv + s v 1 1 − q µv +1+ s = . v 1 1 − q µv + s v

34 Now if α ∨ is the standard coroot of SL 2 , then q µ v = α ∨ ( A v ) , v following our H T identification. (See the remark below). Here A v ∈ PGL 2 ( C ) repre- sents the semisimple conjugacy class parametriz- ing π v and α ∨ is the root of PGL 2 ( C ), or the coroot of SL 2 . Clearly α ∨ ( A v ) is the eigenvalue for the adjoint action of L T on L n , evaluated at A v . Therefore = α ∨ ( A v ) − 1 = ˜ q − µ v (4.1.5) r ( A v ) v We thus get that (4.1.1) equals (4.1.6) (1 − α ∨ ( A v ) − 1 q − s v ) − 1 / (1 − α ∨ ( A v ) − 1 q − s − 1 ) − 1 v

35 which equals (4.1.7) L ( s, η v , ˜ r 1 ) /L (1 + s, η v , ˜ r 1 ) since m = 1, i.e., r = r 1 . SL 2 ( R ). It is instructive to also compute the case of SL 2 ( R )( GL 2 ( R ), respectively). We again need to calculate � � � � � 1 0 f 0 (4.1.12) dx. v x 1 R Here K = SO 2 ( R )( O 2 ( R ), respectively) and we can write � � � � 1 0 a y = k ( θ ) , x 1 0 b where � � cos θ sin θ k ( θ ) = . − sin θ cos θ

36 We can then take tan θ = − x, b = a − 1 = √ √ x 2 + 1 and y = x/ x 2 + 1. We need to calculate � ∞ � s 1 � 1 x 2 + 1) s 2 ( x 2 +1) − 1 / 2 dx, � √ ( x 2 + 1 −∞ � � � � a 0 = | a | s 1 | b | s 2 , or where η 0 b � ∞ ( x 2 + 1) − ( s 1 − s 2 +1) / 2 dx. 2 0 Let s = s 1 − s 2 and set x = tan θ , we need to calculate � π/ 2 (cos θ ) s − 1 dθ. 2 0

37 Using the standard formula � π/ 2 � p + 1 � cos p θ sin q θdθ = Γ 2 0 � q + 1 � � p + q � · Γ / 2Γ + 1 , 2 2 where � ∞ t s e − t d ∗ t, Γ( s ) = 0 (4.1.12) then equals (4.1.13) Γ(1 / 2)Γ( s/ 2) / Γ(( s + 1) / 2) . Using Γ(1 / 2) = √ π , (4.1.13) equals π − s/ 2 Γ( s/ 2) /π − ( s +1 / 2) Γ( s + 1 / 2)

38 which is again L ( s ) /L ( s + 1), where L ( s ) is the archimedean Hecke–Tate L –function at- tached to the character | x | s , the R –component of our cusp form on T . The main result of Langlands in [La1] can be stated as follows. Let S be a finite set of places with the property that if v �∈ S, π v is an unramified representation. Every f ∈ V ( s, π ) is of the form � ⊗ v �∈ S { f 0 f ∈ ⊗ v ∈ S V ( s, π v ) v } , where f 0 v is K v –spherical for some S . To be precise, the decomposition π = ⊗ v π v de- pends on a choice of M ( O v )–invariant vec- tors { x v } for all the unramified places which

39 one fixes once for all (cf. [S]). The functions f 0 v must then satisfy f 0 v ( k v ) = x v for all k v ∈ K v and all v �∈ S . Assume further that f = ⊗ v f v , with f v = f 0 v for all v �∈ S . For each i , let � L S ( s, π, r i ) = (4.1) L ( s, π v , r i ) . v �∈ S Then by Lemma 4.1 m � L S ( is, π, ˜ r i ) /L S (1 + is, π, ˜ A ( s, π ) f ( e )=( r i )) i =1 (4.2) � ⊗ v �∈ S f 0 v ( e v ) ⊗ v ∈ S A ( s, π v , w 0 ) f v ( e v ) . It now follows from the properties of the constant term A ( s, π ) that

40 Theorem 4.2 (Langlands) [La1]. The product quotient m � L S ( is, π, ˜ r i ) /L S (1 + is, π, ˜ r i ) i =1 is meromorphic on all of C . Clearly one needs an induction to get this to lead to meromorphic continuation of each L –function in the product. We will soon discuss this induction. 5. Examples We shall now give a number of important examples of L –functions which appear in constant terms for appropriate pairs ( G , M ). We refer to [La1,Sh2] for the complete list.

41 5.1 . Let G = GL n + t , M = GL n × GL t , π = ⊗ v π v cusp form on GL n ( A F ) and π ′ = ⊗ v π ′ v one on GL t ( A F ). Then m = 1 π ′ ), the Rankin–Selberg and we get L ( s, π × ˜ π ′ ) product L –function for the pair ( π, ˜ (cf. [JPSS1] and [Sh10]). It will be discussed by Cogdell in more length. 5.2 . Let G be a classical group, split over F and let M = GL n × G ′ , where G ′ is a classical group of the same type, but lower rank. Let π and π ′ be cuspidal representa- tions of GL n ( A F ) and G ′ . Then m = 2. π ′ ) as its first L –function. One gets L ( s, π × ˜ For i = 2, we get L ( s, π, ρ ), where ρ = Λ 2 if L G is orthogonal and ρ = Sym 2 if L G is

42 symplectic (cf. [CKPSS1,2]. 5.3 . Let G = GSpin n + t , M = GL n × GSpin t , π and π ′ cusp forms on GL n ( A F ) and GSpin t ( A F ), respectively. Then m = π ′ ) as our first 2. Again we get L ( s, π × ˜ L –function. The second L –function is then an appropriate twist of either L ( s, π, Λ 2 ) or L ( s, π, Sym 2 ). 5.3.a . Let G = GSpin 5+2 n , M = GL n × Gpin 5 = GL n × GSp 4 , and let ( π, π ′ ) be a cusp form on GL n ( A F ) × GSp 4 ( A F ). Again π ′ ) as our first L –function. we get L ( s, π × ˜ This is very important. 5.3.b . G = GSpin 6+2 n , M = GL n ×

43 GSpin 6 . We have 0 → {± 1 } → GL 4 ( C ) → GSO 6 ( C ) → 0. Suppose π is on GL n ( A F ) and π ′ on GL 4 ( A F ). We then get L ( s, π ⊗ π ′ , ρ n ⊗ Λ 2 ρ 4 ) as our first L –function (cf. [K4]). ˜ 5.4 . Let G be a simply connected group of either type E 6 or E 7 . Choose M such that M D , the derived group of M , is ei- ther SL 3 × SL 2 × SL 3 or SL 3 × SL 2 × SL 4 , respectively. There exist F –rational injec- tions from M into GL 3 × GL 2 × GL t , t = 3 or 4, which are identity on SL 3 × SL 2 × SL t . Let π 1 ⊗ π 2 ⊗ σ be a cuspidal representation of GL 3 ( A F ) × GL 2 ( A F ) × GL t ( A F ). Then m = 3 or 4 according as if G = E 6 or E 7 . The first L –function is then L ( s, π 1 × π 2 × ˜ σ )

44 (cf. [KS2]). All these L –functions will be revisited later in connection with functoriality. 6. Local Coefficients, Non–constant Term and the Crude Functional Equation Changing the splitting in U , we may as- sume each χ v is defined by means of ψ v through equation (2.5). At each v , let λ χ v ( s, π v ) be the Whittaker functional defined by equation (2.11). Next, let A ( s, π v , w 0 ) be the local intertwining operator defined by (3.10). Finally, let λ χ v ( − s, w 0 ( π v )) be the corresponding functional defined for I ( − s, w 0 ( π v )) by means of (2.11). Using our

45 assumption on χ v , Rodier’s theorem points to the existence of a complex function C χ v ( s, π v ) of s , depending on π v , χ v and w 0 such that λ χ v ( s, π v ) = C χ v ( s, π v ) (6.1) λ χ v ( − s, w 0 ( π v )) · A ( s, π v , w 0 ) . This is what we call the “Local Coeffi- cient” attached to s, π v , χ v and w 0 (cf. [Sh2,Sh3]). The choice of w 0 is now specified by our fixed splitting as in [Sh4]. Now, let E χ ( s, Φ s , g, P ) (6.2) � = E ( s, Φ s , ug, P ) χ ( u ) du, U ( F ) \ U the χ –nonconstant term of E ( s, Φ s , g, P ) (cf. [Sh2,Sh3,Sh6]), where χ = ⊗ v χ v .

46 If we substitute for E ( s, Φ s , ug, P ) its def- inition in (3.3) and do some telescoping, we get, using orthogonality of χ , that � (6.3) E χ ( s, Φ s , e, f ) = λ χ v ( s, π v )( f v ) , v where f v is the local component of f defined by (3.6) in which ϕ = ⊗ v ϕ v is identified with W ϕ = ⊗ v W v , where W ϕ is defined by (2.7). As explained in [S], W v ( e v ) = 1 for almost all v . We now appeal to the following formula of Casselman–Shalika [CS]: Theorem 6.1(Casselman-Shalika [CS]). Assume π v and ψ v are both unramified and if f v ( e v ) defines a Whittaker function W v in

47 the Whittaker model of π v , assume W v ( e v ) = 1 . Observe that this is the case for almost all v . Then (6.4) m � r i ) − 1 . λ χ v ( s, π v )( f v ) = L (1 + is, π v , ˜ i =1 In fact, if W f v ( g v ) = λ χ v ( s, π v )( I v ( g v ) f v ) is the Whittaker function attached to f v , then (6.3) can be written as Theorem 6.2. One has � E χ ( s, Φ s , e, f ) = W f v ( e v ) v ∈ S (6.5) m � L S (1 + is, π, ˜ r i ) − 1 . i =1

48 Remark. As opposed to intertwining oper- ators, Whittaker functions are by no means multiplicative and therefore proof of Theo- rem 6.1 cannot be reduced to rank one cal- culations by means multiplicativity (cocycle relations). It should be pointed out that in the case of SL 2 , Theorem 6.1 is an easy ex- ercise. Corollary 6.3 [Sh3]. The product m � L S (1 + is, π, r i ) � = 0 i =1 for Re ( s ) = 0 . In particular, if π and π ′ are cusp forms on GL n ( A F ) and GL t ( A F ) , then L (1 , π × π ′ ) � = 0 ,

49 where local L –functions at every place are corresponding Artin ones ([HT], [He1]). Proof. Modulo non–vanishing of W f v ( e ) for Re( s ) = 1, which is highly non–trivial if v = ∞ (cf. Casselman–Wallach [Ca2,W]), this follows from unitarity (and therefore holomorphy) of M ( s, π ) for Re( s ) = 0, and Theorem 6.2. Observe that the integration in (6.2) is over a compact set. Now, computing the non–constant terms from the two sides of the functional equa- tion (3.17), Lemma 4.1 and Theorems 6.1 and 6.2, together with Definition (6.1) im- plies:

50 Theorem 6.4 (Crude Functional Equa- tion) [Sh3,Sh6]. We have m � � L S ( is, π, r i ) = C χ v ( s, ˜ π v ) i =1 v ∈ S (6.6) m � L S (1 − is, π, ˜ r i ) . i =1 We just point out that by Lemma 4.1, Theorem 6.1 and Definition (6.1) (6.7) m � C χ v ( s, ˜ π v ) = L (1 − is, π v , ˜ r i ) /L ( is, π v , r i ) , i =1 whenever π v is unramified.

51 7. The Main Induction, Functional Equations and Multiplicativity To prove the individual functional equa- tions, i.e., for each L ( s, π, r i ) with precise root numbers and L –functions , we appeal to the following induction statement (cf. [Sh1,Sh2]). It is crucial in all the results that we prove from now on. Proposition 7.1 [Sh1]. Given 1 < i ≤ m , there exists a split group G i over F , a maximal F –parabolic subgroup P i = M i N i and a cuspidal automorphic representation π ′ of M i = M i ( A F ) , unramified for every v �∈ S , such that if the adjoint action of

52 m ′ L M i on L n i decomposes as r ′ = r ′ � j , then j =1 L S ( s, π, r i ) = L S ( s, π ′ , r ′ 1 ) . Moreover m ′ < m . It was observed by Arthur [A], that each M i can be taken equal to M and π ′ = π . More precisely: Proposition 7.2 [A]. Given i, 1 < i ≤ m , there exists a split connected reductive F – group G i with M as a Levi subgroup and m ′ < m for which r ′ 1 = r i . Each G i can be taken to be an endoscopic group for G . (Its L –group is the centralizer of a semisimple element in L G .)

53 Next, we need the following variant of a result of Henniart and Vigneras. In it, we assume that the defining additive character ψ for χ is local component of a global one. Here we shift the ramification to archimedean places. Consequently, we need to use a re- sult of Dixmier–Malliavin on convolution al- gebras for real semisimple groups. Proposition 7.3 [Sh1]. Let σ be an ir- reducible χ –generic supercuspidal represen- tation of G = G ( F ) , where F is a non– archimedean local field and G is defined over F . Let B = TU be the Borel subgroup of G defining χ . Then there exists a num- ber field K with a ring integers O , a split

54 group H over K , a non–degenerate charac- ter ˜ χ = ⊗ v ˜ χ v of U H ( K ) \ U H ( A K ) , and a globally ˜ χ –generic cusp form π = ⊗ v π v on H = H ( A K ) such that: a) K v 0 = F for some place v 0 of K , b) ˜ χ v 0 = χ , c) as a group over F, H = G , d) π v 0 = σ , and finally e) for every other finite place v of K, v � = v 0 , π v is of class one with respect to a special maximal compact subgroup Q v of H ( K v ) . Here U H is the unipotent radical of a Borel subgroup of H for which U H as a group over F equals U .

55 Proposition 7.3 [Sh1,Sh4]. Assume ei- ther F v is archimedean or π v has a vector fixed by an Iwahori subgroup. Let ϕ v : W ′ F v → L M v be the homorphism of the Deligne–Weil group parametrizing π v . For each i , let L ( s, r i · ϕ v ) and ε ( s, r i · ϕ v , ψ v ) be the Artin L –function and root number attached to r i · ϕ v . Then (7.1) m r i · ϕ v , ψ v ) L (1 − is, r i · ϕ v ) � C χ v ( s, π v )= ε ( is, ˜ . L ( is, ˜ r i · ϕ v ) i =1 Remark. If G is quasisplit, but not split, then a product of Langlands λ –functions (Hilbert symbols) will also appear on the right hand side of (7.1).

56 Applying these propositions inductively and using the Crude Functional Equa- tions (6.6) then implies: Theorem 7.4 [Sh1]. Assume G is a split reductive algebraic group over a local field F of characteristic zero. Let P = MN , P ⊃ B , be a maximal parabolic subgroup as be- fore. Let χ be a generic character defined by the splitting and ψ F ∈ ˆ F . Given an irre- ducible admissible χ –generic representation σ of M = M ( F ) , these exist m complex functions γ ( s, σ, r i , ψ F ) , 1 ≤ i ≤ m , such that:

57 1) If F and σ satisfy the conditions of Propo- sition 7.3, then γ ( s, σ, r i , ψ F ) = ε ( s, r i · ϕ, ψ F ) (7.2) L (1 − s, ˜ r i · ϕ ) /L ( s, r i · ϕ ) . 2) Equation (7.1) holds (in the form m � γ ( is, σ, ˜ r i , ψ F ) ). i =1 3) γ ( s, σ, r i , ψ F ) is multiplicative under in- duction (to be discussed below). 4) Whenever σ becomes a local component of a globally generic cusp from, then γ ’s become the local factors needed in their functional equations. Moreover 1), 3) and 4) determine the γ –functions uniquely.

58 What is multiplicativity? We will discuss this only in examples since the gen- eral formulation is complicated. It simply says that the γ –functions are multiplicative under parabolic induction and is a conse- quence of multiplicativity of intertwining op- erators (3.10) under that (cf. [Sh3]). This is very deep from the point of view of Rankin– Selberg method and usually quite hard to prove. Here are some examples: Example 1 (cf. [Sh7]). Suppose G = Sp (2 n + 2 t ) and M = GL n × Sp (2 t ), where n and t are positive integers. Write σ = σ 1 ⊗ τ . Suppose M ′ = GL n 1 × . . . × GL n k × GL t 1 × . . . × GL t ℓ × Sp (2 a ), where

59 n 1 + . . . + n k = n and t 1 + . . . + t ℓ + a = t . By case C n of [Sh2], r 1 is equal to the tensor product of the standard representation of GL n ( C ) and SO 2 t +1 ( C ). k ℓ If σ ′ = ′′ σ ′ b ⊗ τ ′ , then multi- � � j ⊗ σ j =1 b =1 plicativity simply means that, if M ′ N ′ ↑ G σ ′ ⊗ 1 , σ ⊂ Ind then (7.3) k ℓ � � ′′ γ ( s, σ ′ γ ( s, σ 1 × τ, ψ F ) = j × σ b , ψ F ) j =1 b =1 k ′′ � γ ( s, σ ′ γ ( s, σ ′ j × τ ′ , ψ F ) . j × ˜ σ b , ψ F ) j =1

60 If ρ n is the standard representation of GL n ( C ), then r 2 = Λ 2 ρ n . With σ 1 and σ ′ j as before, multiplicativity for r 2 means k � γ ( s, σ 1 , Λ 2 ρ n , ψ F ) = γ ( s, σ ′ j , Λ 2 ρ n j , ψ F ) j =1 � γ ( s, σ ′ i × σ ′ j , ψ F ) . 1 ≤ i<j ≤ k No more r i beyond r 2 shows up and this is the case for all the classical groups. Equal- ity of the dimension on both sides of (7.4) simply means the following trivial identity: � k � k � 2 k � � � � � ( n 2 n i − n i = i − n i )+2 n i n j . i =1 i =1 i =1 1 ≤ i<j ≤ k

61 L –function and root number L –functions are now defined using γ –functions. When σ is tempered, we define L ( s, σ, r i ) as the inverse of the normalized polynomial P ( q − s ) in q − s satisfying P (0)= 1 and γ ( s, σ, r 1 , ψ F ) = ε ( s, σ, r i , ψ F ) (7.5) L (1 − s, σ, ˜ r i ) /L ( s, σ, r i ) . The L –function L ( s, σ, ˜ r i ) and the root number ε ( s, σ, r i , ψ F ) are also uniquely de- fined by (7.5). To proceed we need the fol- lowing theorem.

62 Theorem 7.5. Suppose σ is tempered. Then L ( s, σ, r i ) are all holomorphic for Re ( s ) > 0 . With this theorem in hand, L ( s, σ, r i ) are now also multiplicative if σ is tempered. (See below.) To define L ( s, σ, r i ) for any irreducible χ – generic representation, we appeal to Lang- lands classification [La3,Si]. We embed σ ⊂ Ind M ′ ( N ′ ∩ M ) ↑ M σ ′ ν ⊗ 1 , where σ ′ ν is quasi– tempered with a negative Langlands param- Then σ ′ eter ν . 0 is tempered. By multi- plicativity, we then write γ ( s, σ, r i , ψ F ) as a product of appropriate γ –functions γ ( s, σ ′ ν,j , r ij , ψ F ), where j runs over a finite

63 index set determined by M ′ and M , i.e., (7.6) � γ ( s, σ ′ γ ( s, σ, r i , ψ F ) = ν,j , r ij , ψ F ) . j More precisely, for each j , there exist Levi subgroups (not necessarily maximal) j and ˜ j ⊂ ˜ M ′ M j of G with T ⊂ M ′ M j as a maximal Levi subgroup. The representa- tion σ ′ ν,j is a quasi–tempered representation of M ′ j for which σ ′ 0 ,j is tempered. The repre- sentation r ij of L M ′ j is an irreducible consti- tutent of the action of L M ′ j on the Lie alge- bra of the L –group of ˜ M j ∩ N ′ j , where N ′ j ⊂ U is the unipotent radical of P ′ j = M ′ j N ′ j . Thus the γ –function γ ( s, σ ′ ν,j , r ij , ψ F ) is a

64 γ –function attached to the pair ( ˜ M j , M ′ j ). When ν = 0, by Conjecture 7.5, the prod- uct of the numerators of γ ( s, σ ′ 0 ,j , r ij , ψ F ) equals to the numerator of the product and 0 ,j , r ij ) − 1 denotes the normalized if L ( s, σ ′ numerator of γ ( s, σ ′ 0 ,j , r ij , ψ F ), i.e., the re- ciprocal of the tempered L –function attached 0 ,j and r ij by means of the pair ( ˜ to σ ′ M j , M ′ j ), we then use L ( s, σ ′ ν,j , r ij ) to denote its an- alytic continuation to ν . We now set � L ( s, σ ′ (7.7) L ( s, σ, r i ) = ν,j , r ij ) . j This agrees with the way Artin L –functions are defined [La3,KSh,Sh1,T]. Details are given in [Sh3]; also see the discussion in pages

65 862 and 863 of [KS2]. The root number is then defined uniquely to satisfy (7.5). We should point out that in Definition (7.7) we do not need to assume the validity of Con- jecture 7.5. But if valid, it implies that L – functions are also multiplicative, if the rep- resentations are tempered. Having defined our L –functions and root numbers everywhere, we set � L ( s, π, r i ) = L ( s, π v , r i ) v and � ε ( s, π, r i ) = ε ( s, π v , r i , ψ v ) . v We then have:

66 Theorem 7.6(Functional Equation [Sh1]). For each i, 1 ≤ i ≤ m , (7.8) L ( s, π, r i ) = ε ( s, π, r i ) L (1 − s, π, ˜ r i ) . Exercise 1. Use the pair ( G , M ), G = GL 3 and M = GL 2 × GL 1 , to get the stan- dard L –function for GL 2 . Determine L – functions using our method and show that they are equal to those of Jacquet–Langlands. Exercise 2. Let G = E sc and M be such 6 that M D = SL 3 × SL 2 × SL 3 . Fact 1. There exists a F –rational map (in- jection) f : M → GL 3 × GL 2 × GL 3 whose restriction to M D is identity.

67 Fact 2. m = 3 and if π 2 ⊗ π 1 ⊗ σ is an unram- ified representation of GL 3 ( F ) × GL 2 ( F ) × GL 3 ( F ), where F is a local field, then L ( s, π 2 × π 1 × ˜ σ ) = L ( s, ( π 2 ⊗ π 1 ⊗ σ ) · f, r 1 ) . Define γ ( s, π 2 × π 1 × ˜ σ ) to be γ ( s, ( π 2 ⊗ π 1 ⊗ σ ) · f, r 1 ), using our method for arbi- trary local representations π 2 ⊗ π 1 ⊗ σ . As- sume ˜ σ = ( F ∗ ) 3 × U ↑ GL 3 ( F ) ( µ 1 ⊗ µ 2 ⊗ µ 3 ) ⊗ 1 . Ind Show that multiplicativity implies: 3 � γ ( s, π 2 × π 1 × ˜ σ ) = γ ( s, π 2 × ( π 1 ⊗ µ j )) , j =1 where the γ –function on the right are those of Rankin–Selberg for GL 3 ( F ) × GL 2 ( F ).

68 (This is crucial to Kim–Shahidi’s proof of functoriality [KS2] of the inclusion GL 2 ( C ) ⊗ GL 3 ( C ) ֒ → GL 6 ( C ), to be discussed later.) Exercise 3. Let G = SO (2 m + 2 n + 1) and M = GL m × SO (2 n + 1). Let σ ⊗ π be an irreducible admissible χ –generic representa- tion of GL m ( F ) × SO 2 n +1 ( F ), where F is a m � local field. Assume σ ֒ → Ind µ j . ( F ∗ ) m × U ↑ GL m ( F ) j =1 Show that multiplicativity implies: m � γ ( s, σ × π ) = γ ( s, π ⊗ µ j ) . j =1 (This is crucial to Cogdell–Kim–Piatetski– Shapiro–Shahidi’s proof [CKPSS1] of func-

69 torial transfer from generic cusp forms on classical groups to GL 2 n ( A F ).) 8. Twists by Highly Ramified Characters Holomorphy and Boundedness Since our aim is to establish those an- alytic properties of L –functions from our method which are crucial in proving the strik- ing new cases of functoriality, we will limit our discussion on the issue of holomorphy of L –functions only to twists by highly rami- fied characters. In fact, as we explained in earlier sections, the functional equations for L –functions within our method are proved quite generally and multiplicativity and the

70 related machinary necessary for applying con- verse theorems to our L –functions are in perfect shape. Nothing that general can be said about the holomorphy and possible poles of these L –functions. On the other hand, there has recently been some remarkable new progress on the question of holomorphy of these L – functions, mainly due to Kim [K2,K3,KS1]. They rely on reducing the existence of the poles to that of existence of certain unitary automorphic forms, which in turn point to the existence of certain local unitary rep- resentations. One then disposes of these representations, and therefore the pole, by

71 checking the corresponding unitary dual of the local group. In view of the functional equation, this needs to be checked only for Re( s ) ≥ 1 / 2, if a certain local assumption on normalized local intertwining operators is valid. To explain, let A ( s, π v , w 0 ) be the local intertwining operator attached by equa- tion (3.10) to our inducing representation π v . We recall that we are dealing with a pair ( G , M ) and a χ –generic cuspidal rep- resentation π = ⊗ v π v of M = M ( A F ). Let, for each i, 1 ≤ i ≤ m, L ( s, π v , r i ) and ε ( s, π v , r i , ψ v ) be the corresponding L –function and root number specified earlier. We de-

72 fine a normalized operator N ( s, π v , w 0 ) by (8.1) N ( s, π v , w 0 ) = r ( s, π v , ψ v ) A ( s, π v , w 0 ) , where the normalizing factor is defined as [Sh1] m � r ( s, π v , ψ v ) = ε ( is, π v , ˜ r i ) (8.2) i =1 L (1 + is, π v , ˜ r i ) /L ( is, π v , ˜ r i ) . Theorem 8.1. The operator N ( s, π v , w 0 ) is holomorphic and non–zero for Re ( s ) ≥ 1 / 2 . It should be mentioned that it is a result of Yuanli Zhang [Z] that, if Theorem 7.5

73 is valid for the pair ( G , M ), then the non– vanishing of N ( s, π v , w 0 ) for Re( s ) ≥ 1 / 2 follows from its holomorphy over the same range. Arguments given in [CKPSS1,K4,KS2], then proceed under the validity of Theorem 8.1 for ( G , M ) as well as for all other related lower rank pairs (that come into the multi- plicativity), which consequently are verified in each of the cases in [CKPSS1,K4,KS2]. The main issue with this argument is that one cannot always get such a contradiction and rule out the pole. In fact, there are many unitary duals whose complementary series extend all the way to Re( s ) = 1, mak-

74 ing the results far from general. On the other hand, if one considers a highly ramified twist π η (see below) of π , then it can be shown quite generally that every L ( s, π η , r i ) is entire (cf. [Sh8] for its local analogue). In fact, if η is highly ram- ified, then by checking central characters, w 0 ( π η ) �≃ π η , whose negation is a necessary condition for M ( s, π η ) to have a pole, a ba- sic fact from Langlands spectral theory of Eisenstein series (Lemma 7.5 of [La2]). This lemma was used by Kim in [K2] and in view of the present powerful converse theorems of Cogdell and Piatetski–Shapiro [CPS1,2,3], that is all one needs to prove our cases of

75 functoriality [CKPSS1,K4,KS2]. We formal- ize this by quoting the following (Proposi- tion 2.1) from [KS2]. Theorem 8.2. There exists a rational char- acter ξ ∈ X ( M ) F = X ( M ) , with the follow- ing property. Let S be a non–empty finite set of finite places of F . For every globally generic cuspidal representation π of M = M ( A F ) , there exist non–negative integers f v , v ∈ S , such that for every gr¨ ossencharacter η = ⊗ v η v of F for which the conductor of η v , v ∈ S , is larger than or equal to f v , every L –function L ( s, π η , r i ) , i = 1 , . . . , m , is entire, where π η = π ⊗ ( η · ξ ) . The ra- tional character ξ can be simply taken to be

76 ξ ( m ) = det( Ad ( m ) | n ) , m ∈ M , where n is the Lie algebra of N . The last ingredient in applying converse theorems is that of boundedness of each L ( s, π, r i ) in every vertical strip of finite width, away from its finite number of poles. The finiteness of poles is again a consequence of the finiteness of the poles of M ( s, π ) for Re( s ) ≥ 0 and the functional equation satis- fied by each L ( s, π, r i ), but under the valid- ity of Assumption 8.1 (cf. [GS1]). In this full generality, the boundedness in finite vertical strips, away from their poles, were proved by Gelbart–Shahidi in [GS1], again using our method. The main difficulty in proving

77 this result is having to deal with reciprocals of each L ( s, π, r i ) , 2 ≤ i ≤ m , near and on the line Re( s ) = 1, the edge of critical strip, whenever m ≥ 2, which is unfortunately the case for each of the L –functions appearing in our cases of functoriality. We handle this by appealing to equation (6.5) (Theorem 6.2) and estimating the non–constant term (6.2) by means of Langlands [HC,La2] and M¨ uller [Mu], and a non–trivial result from complex function theory (Matsaev’s theo- rem). Here is the statement of the main re- sult of [GS1] as formulated for π η to avoid the issue of poles.

78 Theorem 8.3. Let ξ and η be as in The- orem 8.2. Assume η is sufficiently rami- fied so that each L ( s, π η , r i ) is entire. Then, given a finite real interval I , each L ( s, π η , r i ) remains bounded for all s with Re ( s ) ∈ I . Remark. Another proof of Theorem 8.3 is due to Gelbart and Lapid, following some ideas of Sarnak. 9. Examples of Functoriality with Applications Consider the embedding i : GL 2 ( C ) ⊗ GL 3 ( C ) ֒ → GL 6 ( C ) . This is a homomorphism from L GL 2 × L GL 3 into L GL 6 . Accordingly functoriality pre-

79 dicts a map Aut ( GL 2 ( A F )) × Aut ( GL 3 ( A F )) → Aut ( GL 6 ( A F )) . More precisely, let π 1 = ⊗ v π 1 v and π 2 = ⊗ v π 2 v be cusp forms on GL 2 ( A F ) and GL 3 ( A F ), respectively, with π 1 v given by t 1 v ∈ GL 2 ( C ) and π 2 v by t 2 v ∈ GL 3 ( C ) for almost all v . Let � v be the irreducible admissible spher- ical representation of GL 6 ( F v ) defined by { t 1 v ⊗ t 2 v } ⊂ GL 6 ( C ). Then we can state the functoriality in this case as: Functoriality. There exists an automor- phic representation � ′ = ⊗ v � ′ v of GL 6 ( A F ) such that � ′ v = � v for almost all v .

80 More precisely, let ρ iv : W ′ F v → GL i +1 ( C ), i = 1 , 2, parametrize π iv (Harris–Taylor [HT], Henniart [He1]) for all v . Let ρ 1 v ⊗ ρ 2 v be the six dimensional representation of W ′ F v , i.e., the homomorphism ρ 1 v ⊗ ρ 2 v : W ′ F v → GL 6 ( C ) . Denote by π 1 v ⊠ π 2 v the irreducible admis- sible representation of GL 6 ( F v ) attached to ρ 1 v ⊗ ρ 2 v . Let π 1 ⊠ π 2 = ⊗ v ( π 1 v ⊠ π 2 v ). Theorem 9.1 (Kim–Shahidi [KS2]). The irreducible admissible representation π 1 ⊠ π 2 of GL 6 ( A F ) is automorphic. Thus GL 2 ( C ) ⊗ GL 3 ( C ) ֒ → GL 6 ( C ) is functorial.

81 For the proof, one applies an appropriate version of the converse theorem [CPS2] to the following cases of our method. In each case G and M D , the derived group of M , are given as follows. 1) G = SL (5)( or GL(5)) , M D = SL 2 × SL 3 2) G = Spin(10), M D = SL 3 × SL 2 × SL 2 3) G = E sc 6 , M D = SL 3 × SL 2 × SL 3 4) G = E sc 7 , M D = SL 3 × SL 2 × SL 4 We then get the necessary analytic prop- erties of the highly ramified twisted L –functions L ( s, π 1 × π 2 × ( σ ⊗ η )), σ ⊗ η = σ ⊗ η · det,

82 η a highly ramified gr¨ ossencharacter, where σ ’s are appropriate cusp forms on GL j ( A F ), j = 1 , 2 , 3 , 4, respectively. Observe that L ( s, π 1 v × π 2 v × σ v ) = L ( s, ( π 1 v ⊠ π 2 v ) × σ v ) for almost all v . Similarly for root numbers. We in fact prove these equalities for all v . This is immediate from the fact that the local components of the weak transfer es- tablished through the converse theorem (The- orem 3.8 of [KS2]) is in fact π 1 v ⊠ π 2 v for each v (Theorem 5.1 of [KS2]). Proof of this is quite delicate and beside the tech- niques already discussed (functional equa- tions, multiplicativity,...), relies on several

83 other techniques such as base change, both normal [AC] and non–normal [JPSS2], as well as certain results from the theory of types [BH]. Next, let π = ⊗ v π v be a cusp form on GL 2 ( A F ). Let Ad( π ) be its Gelbart–Jacquet transfer [GJ]. Then π ⊠ Ad ( π ) = Sym 3 ( π ) ⊗ ω − 1 ⊞ π π implies that Sym 3 ( π ) is an automorphic rep- resentation of GL 4 ( A F ). More precisely, for every v let ρ v : W ′ F v → GL 2 ( C ) be the two dimensional represen- tation of the Deligne–Weil group W ′ F v at- tached to π v (cf. [Ku]).

84 Consider: Sym 3 · ρ v = Sym 3 ρ v : W ′ F v → GL 4 ( C ) . Let Sym 3 π v be the irreducible admissible representation of GL 4 ( F v ) attached to Sym 3 ρ v . Set Sym 3 π = ⊗ v Sym 3 π v . Theorem 9.2 (Kim–Shahidi [KS2]). Sym 3 π is an automorphic representation of GL 4 ( A F ) . It is cuspidal unless π is of di- hedral or tetrahedral type. Next, let � = ⊗ v � v be a cuspidal repre- sentation of GL 4 ( A F ). Let Λ 2 : GL 4 ( C ) → GL 6 ( C ) be the exterior square map. Let

85 ϕ v : W ′ F v → GL 4 ( C ) parametrize � v for all Then Λ 2 ϕ v : W ′ v [HT,He1]. F v → GL 6 ( C ) parametrizes Λ 2 � v , an irreducible admis- sible representation of GL 6 ( F v ). Theorem 9.3 (Kim [K4]). There exists an automorphic representation � ′ = ⊗ v � ′ v of GL 6 ( A F ) such that � ′ v = Λ 2 � v . Corollary 9.4 [K4]. The representation Sym 4 ( π ) is automorphic, where Sym 4 ( π ) = ⊗ v Sym 4 ( π v ) and π = ⊗ v π v is a cusp form on GL 2 ( A F ) . Proof. Apply Λ 2 to Sym 3 π ; Λ 2 (Sym 3 π ) = Sym 4 π ⊗ ω π ⊞ ω 3 π

86 Proposition 9.5 (Kim-Shahidi [KS3]). Sym 4 ( π ) is cuspidal unless π is of dihedral, tetrahedral or octahedral type. Theorem 9.3 is proved by using G =Spin 2 k +8 , M D = SL k +1 × SL 4 , k =0 , 1 , 2 , 3 . We get L ( s, π ⊗ σ, Λ 2 ρ 4 ⊗ ρ k +1 ) for each cusp form σ on GL k +1 ( A F ) , k = 0 , 1 , 2 , 3. If we use the isogeny SL 4 → SO 6 → 0, we note that functoriality established in Theorem 9.3 is a special case of functoriality for the embedding GSO 2 n ( C ) ֒ → GL 2 n ( C ) .

87 The result is the transfer Aut( GSpin 2 n ( A F )) � Aut( GL 2 n ( A F )) . This is now established by Asgari–Shahidi in Duke J. in the weak form. The full trans- fer is now being written. It needs general- ization of the descent of Ginzburg–Ralis– Soundry to GSpin groups. This is half– done by Hundley–Sayag. We have now com- pleted the rest of it and expect to prove results as strong as those of [CKPSS] for GSpin m , m = odd or even. We need some results on L S ( s, π, Λ 2 ⊗ χ ) and L S ( s, π, Sym 2 ⊗ χ ) when π is a cusp form on GL n ( A F ). (We need holomorphy for both for Re( s ) >

88 1. This implies non–vanishing for both for Re( s ) > 1. Much stronger results now seem to be available through the work of D. Belt for Λ 2 ⊗ χ and S. Takeda for Sym 2 ⊗ χ , us- ing Jacquet–Shalika and Bump–Ginzburg, respectively.) In the cases of Asgari–Shahidi we have G = GSpin 2( m + n ) , M = GL m × GSpin 2 n for 1 ≤ m ≤ 2 n − 1. The L –functions are product L –functions. Apply the converse theorem. Applications : Theorem 9.6 (Kim–Shahidi [KS3]). Let F be an arbitrary number field. Let π =

89 ⊗ v π v be a cusp form on GL 2 ( A F ) . Assume π v is parametrized by � � α v 0 t v = ∈ GL 2 ( C ) . 0 β v Then q − 1 / 9 < | α v | and | β v | < q 1 / 9 . v v Similar inequality holds at archimedean places ( λ > 0 . 23765432 ... ) . This is proved using the techniques of [Sh2] (which led to 1/5 using Sym 2 π and groups of either type F 4 or E 6 , by apply- ing a general theorem from [Sh2] which im- plies L ( s, π v , Sym 5 ( ρ 2 )) is holomorphic for

90 Re( s ) ≥ 1 for all such v ). When this general theorem is applied to a simply connected group of type E 8 with a Levi M for which M D = SL 5 × SL 4 , together with a represen- tation related to Sym 4 π ⊗ Sym 3 π leading to L ( s, π v , Sym 9 ( ρ 2 )), one gets 1/9. We refer to [CPSS] for an important application of Theorem 9.6. Next, we have Theorem 9.7 (Kim–Sarnak [KSa]). Suppose F = Q , then p − 7 / 64 ≤ | α p | and | β p | ≤ p 7 / 64 . At the archimedean places we get the esti- 975 mate λ ≥ 4096 = 0 . 2380371 for the first positive eigenvalue of ∆ .

91 Proof. This is proved by means of analytic methods of Duke–Iwaniec [DI] applied to L ( s, Sym 4 π, Sym 2 ), (cf. [BG]) along the lines of Bump–Duke–Hoffstein–Iwaniec [BDHI] which led to 5 / 28 + ε over Q , when applied to L ( s, Sym 2 π, Sym 2 ).

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