Explicit Trilinear Forms and the Triple Product L-Function

EXPLICIT TRILINEAR FORMS AND THE TRIPLE PRODUCT L-FUNCTION

MICHAEL WOODBURY

Contents 1. Introduction 1 1.1. Statement of results 2 1.2. Application to subconvexity 4 2. Global trilinear form 5 2.1. Ichino’s formula 5 2.2. Measures 6 3. Matrix coefficients: nonarchimedean places 7 3.1. Matrix coefficient associated to Iwahori fixed vectors 8 3.2. Application to unramified representations 10 3.3. Application to the special representations 13 4. Local trilinear forms 15 4.1. Local L-factors 15 4.2. Explicit computations: F nonarchimedean 16 4.3. Explicit computations: F = R 21 4.4. Generalizing Watson’s formula 22 5. Proof of Theorem 1.3 23 References 26

1. Introduction The theory of L-functions has played a central role in number theory from its origins in Dirichlet’s proof of the density of primes in arithmetic progressions to the present day. In this paper we consider the so-called triple product L-function. We have Π= π π π 1 ⊗ 2 ⊗ 3 where each πi is an automorphic representation of GL2(F ) for a number ﬁeld F , and we 1 consider the central value L( 2 , Π). (See Section 4.1 for the deﬁnition of L(s, Π).) 1 Understanding the value L( 2 , Π) began with work of Garrett[9] and was furthered by Gross-Kudla[11], Kudla-Harris[14],[13] and others. An especially beautiful formula was given by Watson[29] for many choices of πi. Recently, Ichino [15] gave a very general formula extending these results. Although his result is more general than Watson’s, it is less explicit. The results of the present paper can be interpreted as making explicit the formula of Ichino. We do this by computing certain local trilinear forms. As a consequence, we also obtain a generalization of Watson’s formula.

Date: September 28, 2010. 1 Another important application of the results of this paper, and the initial motivation for this work, is to prove a conjecture of Venkatesh [28, Hyp 11.1] which has appliction toward subconvexity for the triple product L-function. It is often the case that controlling the growth of the special values over a family of L-functions has number theoretic applications. The general principle of convexity gives bounds on these values. An improvement on this bound is often required for number theoretic applications. Bernstein and Reznikov, in [2], apply Watson’s formula to establish subconvexity in the “eigenvalue aspect.” They ﬁx two Maass forms and vary the eigenvalue of the third form. Venkatesh [28] dealt with subconvexity of the triple product L-function in “level aspect”. Using ergodic theory, he establishes bounds for the period integral

(1.1) J(ϕ1 ϕ2 ϕ3)= ϕ1(g)ϕ2(g)ϕ3(g)dg ⊗ ⊗ PGL2(F ) PGL2( ) ! \ AF for varying choices of ϕ π . In particular, he proved the following. i ∈ i Theorem 1.1 (Venkatesh). Let F be a number field and π1,π2 automorphic cuspidal representations of PGL2(AF ) with (finite) conductors n1, n2 respectively. Let π3 be another such representation with prime conductor p ! n1n2. Let # be a uniformizer of Fp. For given 1 !− ϕi πi (i =1, 2) fixed by GL2( p)), we put ϕ = ϕ1 ρ( 1 )ϕ2 ϕ3 where ϕ3 π3 and ρ is∈ the right regular action. ThenO ⊗ ⊗ ∈ " # (1 4α)(1 2α) " −4(3 4α−) (1.2) J(ϕ) ϕ 4 ϕ 4 ϕ 2 N(p) − − | |#",π1,π2 $ 1$L $ 2$L $ 3$L p where f Lp is the standard L -norm, and α is any bound towards Ramanujan for GL2 over F . (We$ can$ take α =1/9 by Kim-Shahidi[19].)

Venkatesh conjectured an upper bound for the central L-value in terms of J(ϕ) 2 which 1 | | would then give a subconvexity bound for the central L-value L( 2 , Π). (See Corollary 1.5.) To describe it properly, however, we need an analogy of the period integral J(ϕ) for a quaternion algebra B over F .

1.1. Statement of results. We begin by setting notations. Let πi,v denote irreducible B admissible representations of GL2(Fv) for i =1, 2, 3. Write πi,v for the corresponding representation of Bv, the division quaternion algebra over Fv, via Jacquet-Langlands. (This is zero if none exists.) Put Π = π π π . Globally, we let Π= π π π where v 1,v ⊗ 2,v ⊗ 3,v 1 ⊗ 2 ⊗ 3 each πi = " πi,v is a cuspidal automorphic representation, and if B is a quaternion algebra B over F , we define Π to be the corresponding automorphic representation of B× B× B×. $ × × For an algebraic group G over F we denote [G]=Z(G)(A)G(F ) G(A). \ Recall that a quaternion algebra B/F is determined by an even set of places Σ=ΣB where v Σif and only if B(F ) is a division algebra. We call v Σ ramified. ∈ v ∈ Given a global (or local) representation Π= " Πv (or Πv) as above, there exist certain holomorphic functions εv(s, Π,ψv) depending on a choice of additive character ψv : F C×. See [26] for details of their definition and properties.$ In particular, in the global case,→ if

ψ = ψv is trivial on F , the function ε(s, Π) := v εv(s, Π,ψ) is independent of ψ, and the L-function⊗ satisfies the functional equation % L(1 s, Π) = ε(s, Π)L(s, Π). − 2 We refer to Section 4.1 for the definition of the triple product L-function L(s, Π). Moreover, if the central character of Π is trivial (which we will always assume) then ε ( 1 , Π) = 1 is v v 2 ± independent of ψ . Denote the set v ε ( 1 , Π) = 1 by Σ(Π). This is a finite set. v { | v 2 − } Notice that J :Π C as given in (1.2) is a GL2(A)-invariant linear form. Prasad showed → that the existence of a GL2(A)-invariant form on Πis determined by the local epsilon factors. In [25] he proved the theorem below in almost all cases. Loke[22] completed the remaining cases.

Theorem 1.2 (Prasad,Loke). With the notation as above, the following is true.

Bv (1) dim HomGL2(Fv)(Πv, C) + dim HomBv (Πv , C)=1. 1 (2) dim HomGL2(Fv)(Πv, C)=1if and only if ε( 2 , Πv)=1. 1 (3) When ε( 2 , Π) = 1 the global quaternion algebra B = BΠ/F whose ramiﬁcation set is B Σ(Π) is the unique quaternion algebra for which HomB×(A)(Π , C) =0. 1 B ( (4) When ε( 2 , Π) = 1 one has HomB×(A))(Π , C)=0for all quaternion algebras B over F . −

We will prove the following in this paper.

Theorem 1.3. Fix π1,π2 cuspidal automorphic representations of GL2 over a number field F with conductors n1, n2 respectively. Fix an ideal n. Let π3 be cuspidal automorphic with conductor np for any fixed ideal n and a prime p ! n1n2n. Let # be a uniformizer of Fp. Let Sf = q gcd(n1, n2, n) , and S = v real . Then there is a quaternion algebra B over F { | } ∞ { } B B such that ΣB Sf S and there is a finite set of vectors i πi for i =1, 2 such that ⊂ ∪ ∞ F ⊂ 2 1 1+" ! 1 (1.3) L( ,π1 π2 π3) ",F,R N(p) ϕ1(b)ϕ2(b − )ϕ3(b)db 2 ⊗ ⊗ # [B×] 1 &' " # & & & B as N(p) and the Langlands parameters of&π3, remain bounded by R, for some& ϕi i (i =1, 2)→∞ and ϕ πB a new vector. ∞ ∈F 3 ∈ 3 We also obtain a generalization of Watson’s formula to totally real number fields. In particular, we prove the following. (See Theorem 4.7 for the more general statement.)

Theorem 1.4 (Watson when F = Q). Suppose that F is a totally real number field and let Π be a cuspidal automorphic representation as above of squarefree level N. Moreover, we assume that for each v , πi,v are discrete series representations such that the largest weight is the sum of the two|∞ smaller weights. Let B be any quaternion algebra such that B 1 Π =0, so dB = p Σ p divides N. Let εv = εv( 2 , Πv), and for finite v, and let qv be the ( ∈ B order of the residue field of Fv. Then % 2 ϕ(b)db ζ(2) L( 1 , Π) &![B×] & = 2 C 3& & 23 L(1, Π, Ad) v & &2 v & ϕi(b) & db ( | | i=1 ![B×] ( 3 where db is the Tamagawa measure on [B×] (defined in Section 2.2), ϕ = ϕ1 ϕ2 ϕ3 with B ⊗ ⊗ ϕi a new vector of πi (chosen so that at each real place the sum of the weights is zero), and 1 if v ! N 1 εv 1 ∞ − (1 ) if v d qv qv B Cv =  1+εv − 1 | (1 + ) if v N,v! dB  qv qv |  2 if v . |∞  The formula as stated in [29] appears slightly different only because it is presented in the language of classical modular forms. Note that this confirms Prasad’s theorem in this special case. 1.2. Application to subconvexity. From henceforth we will assume without loss of gen- 1 erality that ε( 2 , Π) = 1. Under this assumption, Theorems 1.1 and 1.3 imply the following. Corollary 1.5. Let Π= π π π where π are cuspidal automorphic representations of 1 ⊗ 2 ⊗ 3 i PGL2 over F with π1,π2 fixed and π3 has prime conductor p. For all π3 such that Σ(Π) = , one has the following subconvexity bound. ∅ 1 1 1 12 L( 2 , Π) π3, N(p) − # ∞ We describe the limitations of this corollary. We first remark that if v is a nonarchimedean 1 place then εv( 2 , Πv) = +1 whenever any one of the representations is unramified. In par- 1 ticular, this implies that εv( 2 ,πv) = +1 for all but finitely many v. Analogously, if v is 1 archimedean then ε( 2 , Πv) = +1 if any one of πi,v is not a discrete series. If all three are discrete series of weights k, l and m where k is the largest weight then ε( 1 , Π )= 1 if and 2 v − only if k < l + m. If this is the case, we say that Πv is balanced. Otherwise, we say Πv is unbalanced. Because Theorem 1.1 requires that n1n2 be relatively prime to the conductor of π3, which is required to be a prime p, we see that in this case the quaternion algebra B for which (1.3) is satisfied ramifies exactly at real primes v such that π1,v, π2,v and π3,v are discrete series and for which Πv is balanced. In other words, the only restriction on Πis that at each real place v either at least one of the representations is a principal series, or • if all three representations are discrete series, the weights are unbalanced. • If Theorem 1.1 could be generalized to arbitrary quaternion algebras, our Theorem 1.3, would be enough to make Corollary 1.5 unconditional. Remark. Michel and Venkatesh in [23] give a triple product formula when one of the representations is an Eisenstein series. It is possible that this formula could be adapted to the case that all three are cuspidal, and that it too would have applications to subconvexity. Theorems 1.3 and 1.4 are obtained via the formula of Ichino and the explicit calculation of certain local trilinear forms. The nonvanishing of the said forms was already known by the work of Gross-Prasad [12], but our contribution is the exact evaluation of each. We discuss Ichino’s result in Section 2. The local forms are constructed first via matrix coefficients, which are computed for particular choices of vectors in Section 3, and then finally we combine these results in Section 4 to evaluate the forms. Theorem 1.4 and further generalizations are discussed in Section 4.4. In the final section we complete the proof of Theorem 1.3. 4 2. Global trilinear form

In this section, F is a number field with ring of adeles A = AF , v a place of F and Fv the corresponding completion. Let G = GL with center Z and Π= π π π where 2 1 ⊗ 2 ⊗ 3 each πi is a unitary cuspidal automorphic representation of G over F . If ωi is the central 1 character of πi then we require that ω1ω2ω3 be trivial . We let L(s, Π) denote the triple 3 product L-function, L(s, Π, Ad) = i=1 L(s, πi, Ad) with L(s, πi, Ad) the adjoint L-function attached to πi and ζF (s) the zeta function of the number field F . See Section 4.1 for precise definitions. We use the subscript v%to represent the analogous local L and zeta functions. 2.1. Ichino’s formula. The central identity which we will use to relate the period on the right side of (1.3) to the L-value on the left is due to Ichino[15] and arises when studying the B space HomB×(A)(Π , C) of B×(A)-invariant linear forms where B× is embedded diagonally into B× B× B×. These are the so-called trilinear forms. Theorem 1.2 tells us that this space is at× most× 1-dimensional. It is more convenient to work with bilinear forms: B B (2.1) HomB ( ) B ( )(Π Π , C) × A × × A × This space is again at most 1-dimensional, and there is an obvious choice of invariant form: - (2.2) I(ϕ ϕ)= ϕ ϕ ϕ (b)db ϕ ϕ ϕ (b) db. ⊗ 1 2 3 1 2 3 ![B×] ![B×] B B This uniqueness follows from- the local fact that dim- Hom- - (Πv Πv , C) 1. In the Bv× ⊗ ≤ local case there is also an obvious choice of B×(Fv)-invariant form. Let - , v : πi,v" πi,v" C .· ·/ × → B B be the canonical B×(Fv)-invariant pairings. For ϕv = ϕ1,v ϕ2,v ϕ3,v Πv and ϕv Πv defined similarly, this gives a matrix coefficient - ⊗ ⊗ ∈ ∈ ΠB(g )ϕ , ϕ = πB (g )ϕ , ϕ πB (g )ϕ , ϕ πB (g )ϕ , ϕ - - . v v v v/v . 1,v v 1,v 1,v/v. 2,v v 2,v 2,v/v. 3,v v 3,v 3,v/v such that - - B - - Iv" (ϕv ϕv)= Πv (gv)ϕv, ϕv vdgv ⊗ F × B (Fv). / ! v \ × is B×(Fv)-invariant. The product- of these local forms - 2 Lv(1, Πv, Ad) (2.3) I (ϕ ϕ ) where I (ϕ ϕ )=ζ (2)− I" (ϕ ϕ ) v v ⊗ v v v ⊗ v Fv L ( 1 , Π ) v v ⊗ v v v 2 v ( is well defined because- the normalization is chosen- precisely so that for all but finitely- many v, Iv = 1. This product, like (2.2), is an element of (2.1), and since the space of such forms is at most 1 dimensional, they must differ by a constant. Ichino calculates this constant.

Theorem 2.1 (Ichino). Let I and Iv be deﬁned as above. Then 1 I(ϕ ϕ) C 2 L( 2 , Π) Iv(ϕv ϕv) (2.4) 3 ⊗ = 3 ζF (2) ⊗ j=1 ϕj(b)ϕj(b)db 2 · · L(1, Π, Ad) ϕv, ϕv v [B×] v . / - ( - 1 % ' In the sequel we will make the- stronger assumption that each ωi is trivial. - 5 whenever the denominators are nonzero for a constant C which depends only on the choice 1 of Haar measure. (For the choice given in Section 2.2 below, C = ζF (2)− .) Note that in the case that the central characters are trivial, Πis self dual. Using the composition of the injection Π * Π Πgiven by ϕ (ϕ, ϕ) with (2.2) yields a quadratic form, namely → × 0→ - 2 (2.5) I(ϕ) := I(ϕ ϕ)= ϕ (g)ϕ (g)ϕ (g)dg , ⊗ 1 2 3 &![G] & & & 2 & & such that I(ϕ)= J(ϕ) in the notation of& Theorem 1.1. Hence Ichino’s& result provides us with the necessary| tools| to prove Theorem 1.3. Indeed, we will derive exact formulas for Iv(ϕv) := Iv(ϕ ϕ) from which suﬃcient lower bounds will be obtained. (See, for example, Corollary 4.2.)⊗ 1 As described in the introduction, the functional equation implies that L( 2 , Π) must be 1 zero unless ε( 2 , Π) = 1. Thus Theorem 1.2, together with (2.4), implies one direction of the fact conjectured by Jacquet and proved by Harris and Kudla in [13] and [14]. We record their result here as it will be used in Section 5. Theorem 2.2 (Harris-Kudla). The central value L( 1 , Π) =0if and only if there exists some 2 ( B and some ϕ ΠB such that ϕ(b)db =0. ∈ [B×] (

By Theorem 1.2, the quaternion' algebra B is that for which ΣB =Σ(Π).

2.2. Measures. Let F be a p-adic ﬁeld with and q as above, and let B be a quaternion v Ov v algebra over Fv. If Bv× = GL2(Fv), we choose the (multiplicative) Haar measure dbv (or dgv) to be that for which the maximal compact subgroup K = GL ( ) has volume 1. We abuse v 2 Ov notation and also denote by dbv (or by dgv) the measure on PGL2(Fv) consistent with the choice above and the exact sequence

(2.6) 1 Fv× GL2(Fv) PGL2(Fv) 1. → → → → This means that the image of K in PGL2(Fv) again has volume 1. If Bv is division then it contains a unique maximal order Rv. We denote by dbv the Haar 1 measure on B× for which R× has measure (q 1)− . Again, we write db for the measure on v v − v PBv× compatible with the analogous exact sequence to (2.6). We remark that Rv× has index 2 2 in Fv× Bv×, and so vol(Fv× Bv×)= q 1 . \ \ − In the real case, let dx be the standard measure on R such that the volume of [0, 1] = 1. Deﬁne the subgroups

1 x a N = n(x) = ( 1 ) x R ,A= a(y) = ( 1 ) a R× , ∞ { | ∈ } ∞ { | ∈ } cos θ sin θ K = κθ = sin θ cos θ θ [0, 2π) ∞ { − | ∈ } of GL2(R) or PGL2(R). The we deﬁne the" Haar measure# via

1 dydx 1 2 (2.7) dθ, (1 y− )dθ1dydθ2 2π y 2 4π − | | in the N A K and K A K coordinate systems respectively. Note that in K A K coordinates∞ ∞ we take∞ a(y)∞ only∞ for∞ y 1. ∞ ∞ ∞ | |≥ 6 Although at first glance these choices may seem arbitrary, they are in fact consistent if one defines them in terms of the additive measure. Moreover, for these choices of local measures2, 1 if B is a global quaternion algebra over a number field F then db = ζF (2)− v dbv is the Tamagawa measure on A× B×(AF ). F \ % 3. Matrix coefficients: nonarchimedean places We fix the following notation for this section. Let F be a p-adic field with ring of integers F and uniformizer #. Let q be the order of the residue field F /(#). Let v = ord! be the O v(x) O additive valuation, and define x = q− . Let | | G = GL (F ),K= GL ( ),Z= ( z ) G , 2 2 OF { z ∈ } 1 b a1 P = NA where N = ( 1 ) G ,A= ( a2 ) G . !n 1 1 { ∈ } { ∈ } Let σn =( 1 ), e =( 1 ), w =(1 ).

We assume that χ1,χ2 : F × C× are unramified characters, meaning that χi × =1 → |OF for i =1, 2. The character χ1 χ2 gives a representation of P . We consider π = π(χ1,χ2) which is the (admissible) unitary⊗ induction from P to G of this representation. Hence π is the right regular action of G on the space of functions f : G C (i.e. [π(g)f](h)=f(hg)) such that → 1/2 a1 (3.1) f( a1 b g)=χ (a )χ (a ) f(g) a2 1 1 2 2 a & 2 & " # & & for all g G, and there exists a compact open subgroup& & L G which acts trivially on f. We will follow∈ the common abuse of notation in using& π &to denote⊂ both the space and the action. Note that π need not be irreducible. 1 1 The contragradient π of π is equal to π(χ1− ,χ2− ). Let dk be the Haar measure on G for which K has volume one. Then the pairing - (3.2) , : π π C f,f = f(k)f(k)dk .· ·/ × → . / !K is bilinear and G-invariant. (See [1], [5] for details.) Using this one defines the matrix - - - coefficient associated to f,f: Φ (g)= π(g)f,f . f,f . / Note that in this section we- do not necessarily assume that π is unitary. e We will compute explicitly these matrix coefficients- for particular choices of f and f— namely for those vectors that are fixed by K := K (#)= ( ab) K v(c) 1 . - 0 0 { cd ∈ | ≥ } This is a two dimensional subspace. To perform this calculation, we deriving general formulas in Section 3.1 and then apply these results to the unramified and special representations in Sections 3.2 and 3.3 respectively. First, we record some standard results regarding the decomposition of G and volumes of certain subsets. Recall that our measure on G (or on Z G) is that for which K (or its image) has volume 1. \ 2 If F has complex places or if Bv is definite we could define consistent measures as well, but since the explicit archimedean calculations of this paper involve only the real/indefinite case this is not necessary. 7 Lemma 3.1. Let Ω0 = K and Ωn = KσnK for n 1. Then Z G = n 0 Ωn and for n 1, ≥ \ ≥ ≥ . KσnK = K0σnK0 K0wσnK0 K0σnwK0 K0wσnwK0. Moreover, all of these unions are disjoint./ / / ab ab Let Xn = ( cd) K0 v(c)=n and Yn = ( cd) K0 v(b)=n . The following table is valid. { ∈ | } { ∈ | }

X K K0 Xn Yn KσnK K0σnK0 1 1 1 1 n 1 1 − q n − q n 1 n 1 q − vol(X) 1 (1 + q)− 1 q− 1 q− − q (1 + q ) 1 1+ q 1+ q 1+ q

X K0wσnK0 K0σnwK0 K0wσnwK0 n 2 n n 1 q − q q − vol(X) 1 1 1 1+ q 1+ q 1+ q

Table 1. Volumes of various subsets of GL2(F ), F a p-adic ﬁeld

3.1. Matrix coefficient associated to Iwahori fixed vectors. Recall that π = π(χ1,χ2) is the induced representation defined in (3.1). The Iwasawa decomposition G = NAK implies that a vector f π is uniquely determined by its restriction to K. With this in mind we ∈ define f0 and f1 to be the vectors whose restrictions to K are the characteristic function on K0 and K0wK0 respectively. Let f0 and f1 be the analogously defined vectors in π. K0 We call the space π of vectors fixed by K0 the space of Iwahori fixed vectors. Note that every such vector is a linear- combination- of f0 and f1 because K = K0 K-0wK0 and K0 ∪ K0wK0 = NK wK0. Moreover, if f π then f = af0 + bf1 where a = f(e) and b = f(w). Using the coset decompositions of∈ Lemma 3.1 it is immediate that

fi(kg)dk if j =0 K0 (3.3) Φi,j(g) := Φf ,f (g)= ! i j   q fi(wkg) if j =1. e !K0 Our goal is to determine Φ (g) for all g  G. By the G-invariance of the inner form if i,j ∈ f πK0 and f πK0 and k K then ∈ ∈ ∈ 0 1 Φ (kg)= π(kg)f,f = π(g)f,π− (k)f =Φ (g) and - f,-f . / . / f,f e e Φ (gk-)= π(g)π(k-)f,f =- Φ(g). f,f . / Hence Φ (g) depends only on the double coset K gK . So by Lemma 3.1 we need to perform i,j e 0 -0 the calculation for each of only four K0-double coset representatives.

n/2 q− Proposition 3.2. The values of Φi,j are given by 1 times the values in the following 1+ q table. (Note that the values for Φi,j(wσn) hold only when n>0. Otherwise, the table is valid for all n 0.) ≥ 8 g σn wσn σnw wσnw

n 1 n 1 n 1 α1 − α2 − 1 n 1 Φ0,0(g) α 1− 1 (1 ) 0 α 1 q α− α− q 2 q · 2 − 1 − ·

n n n n 1 α2 α1 1 Φ1,0(g) 0 α α − 1 (1 ) 2 1 q 1 α1α− q − 2 −

n n α1 α2 1 n n 1 Φ0,1(g) − 1 (1 ) α α 0 1 α− α2 q 1 2 q − 1 −

n+1 n+1 α α 1 Φ (g) αn 0 2 − 1 (1 ) αn 1,1 2 α2 α1 q 1 − −

Table 2. Values of matrix coeﬃcients of Iwahori ﬁxed vectors I

Proof. We prove the formula in the case that g K σ K . Let α = χ (#). ∈ 0 n 0 i i Let g, g" G. We define an equivalence relation such that g g" if there exists k K ∈ ∼ ∈ 0 such that g = g"k. Since the Iwahori fixed vectors are, by definition, invariant by K0, fi(g) depends only on the equivalence class of g. Realizing that K0 acts on G is by column operations it is easy to see that for ( ab) K , cd ∈ 0

ab a!n b !n b (3.4) ( ) σ =( n ) ( $ ) cd n c! d ∼ 01

3 where b and b" have the same valuation. Applying this to (3.3) (for j = 0) yields

n/2 ab !n b n n q− n 1 Φ0,0(σn)= f0(( cd) σn)dk = f0 (( 01)) dk = vol(K0)α1 q− = 1 α1 , ab K0 K0 1+ · q !( cd)∈ ! q

and

!n b Φ1,0(σn)= f1(kσn)dk = f1 (( 01)) dk =0. !K0 !K0

ab ab The equivalence class of an element of G of the form w ( cd) σn with ( cd) K0 . Using 1 01 x− 1 10 ∈ the relation ( )= 1 , it is not hard to show that 1 x − 0 x x− 1

" #" # n 1 b!− 1 ab wσnww 01 ( !n ) wK0 v(b) n (3.5) w ( ) σn 1 n ∈ n m ≥ cd b− ! 1 ! − ∼ wσnw − n ∗m K0 m = v(b) < n. 0 " b!# − ∈ ! " # " #

3 In the calculations below we will abuse notion by writing b instead of b! which diﬀers from b by a unit (and similarly for c, c!.) Since our character χ is unramiﬁed, this is harmless. 9 Hence, (using (3.3) in the case j = 1) we have

Φ0,1(σn)=q f0(wkσn)dk !K0 n 1 − n m ! − =q vol(Ym)f0( !m ) m=0 11 n 1 " # 1 − q m n m m m n/2 = − q− α − α q − 1+ 1 1 2 q m=0 1 n 1 n/2 − n/2 n n q− 1 n 1 m q− α1 α2 1 = 1 (1 )α1 (α1− α2) = 1 − 1 (1 ), 1+ − q 1+ − − q q m=0 q 1 α1 α2 1 − and

Φ1,1(σn)=q f1(wkσn)dk !K0 ∞ 1 =q vol(Ym)f1(( !n ) w) m=n 11 1 ∞ n/2 q n m n/2 q− n = − α q− q = α . 1+ 1 2 1+ 1 2 q m=n q 1 These results give the values in the first column of Table 2. The calculations for the other columns are similar. The only nuances are that the decompositions analogous to (3.4) and (3.5) may depend instead on v(c) and take a slightly different shape. We leave the details to the reader. ! 3.2. Application to unramified representations. Strictly speaking, the calculation of the local trilinear form could be carried out at this point using the results of Proposition 3.2 with respect to the basis f0,f1 of Iwahori fixed vectors. However, it turns out that the calculations are drastically{ simplified} by using a different basis. In this section we describe 1 1 this basis and the corresponding matrix coefficients in the case that χ χ− = ± . This 1 2 ( |·| implies that π = π(χ1,χ2) is irreducible. It is called an unramified principal series. The vector φ0 = f0 + f1, which we call the normalized new vector, is obviously fixed by K and, in fact, πK = φ . Note that φ (e) = 1. Using bilinearity, Φ = Φ . It is easy C 0 0 φ0,φo i,j i,j to see that Φφ ,φ is K-biinvariant, so using only the first (or any other) column of Table 2 0 0 e 2 above, we obtain the well-known formula of Macdonald. e Proposition 3.3. Let π = π(χ1,χ2) be an unramified admissible representation of GL2(F ). If φ , φ are the normalized new vectors of π and π respectively then the function Φ is 0 0 φ0,φ0 K-biinvariant and e 1 1 - α1− α-2 α1α2− q n/2 1 1 Φ (σ )= − αn − q + αn − q . φ0,φ0 n 1 1 1 2 1 1+  1 α− α 1 α α−  q − 1 2 − 1 2 e   for n 0 and αi = χi(#). ≥ 10 Recall that π = π(χ1,χ2). For π" = π(χ2,χ1), there is exists an intertwining operator

(3.6) M : π π". → The map M is given by the formula

1 x (3.7) (Mf)(g)= f(w ( 1 ) g)dx !F whenever the integral converges. It is deﬁned elsewhere by analytic continuation.

Lemma 3.4. Let π = π(χ1,χ2), and π" = π(χ2,χ1). Let φ0 (respectively φ0" ) be the nor- 1 malized new vector of π (resp. π") as above. Let φ = f f , and let φ" be the similarly 1 0 − q 1 1 deﬁned vector in πi". Then α α 1 α 1α 1 1 2− 1 1− 2 − q − q (3.8) Mφ0 = 1 φ0" and Mφ1 = 1 φ1" . 1 α α− −1 α− α − 1 2 − 1 2 Given this result we abuse notation and call φ0,φ1 eigenvectors of the M. Proof. As was remarked at the beginning of Section 3.1, if f V then ∈ Mf = af0" + bf1" where a = Mf(e) and b = Mf(w). 1 01 x− 1 10 Using the identity ( )= 1 one deduces that 1 x − 0 x x− 1

"1 x #" 0 # if x F (3.9) f0(w ( 01)) = 1 1 ∈O χ− ( x)χ (x) x − if x/ , 0 1 − 2 | | ∈OF and

1 x 1 if x # F (3.10) f0(w ( 01) w)= ∈ O 0 if x/# F . 0 ∈ O Therefore, by combining (3.7) and (3.9) we have

1 x Mf0(e)= f0(w ( 01)) !F 1 x = f0(w ( 1 ))dx n ! F× 1 ! O −∞ n 1 n = vol(# )(α− α ) OF 1 2 n= 1 1− ∞ 1 1 1 n 1 α1α2− =(1 ) (α1α2− ) = (1 ) 1 . − q − q − n=1 1 α1α2 1 − Similarly, (3.7) and (3.10) combine to give Mf (w) = vol(# )= 1 . So4 0 OF q 1 1 α1α2− 1 (3.11) Mf0 = (1 ) 1 f0" + f1" . − q 1 α α− q − 1 2 4 1 In all of these calculations, we see that the integral converges if and only if α1α2− < 1. However, via 1 | | analytic continuation, the results are the same for all χ χ− = 1. 1 2 ( 11 & & & & A similar calculation for f1 shows that 1 1 − q Mf1 = f0" + 1 f1" . 1 α α− − 1 2 Using the linearity of M, the result follows. ! We have now completed all of the computations necessary to prove the following extension of Proposition 3.3 using the basis φ ,φ of eigenvectors of M. { 0 1} 1 Proposition 3.5. Let π = π(χ1,χ2) such that χ1χ2− = . Let φ0,φ1 be the eigenvectors ( ± |·| 1 1 in the sense of Lemma 3.4, and φ0, φ1 the analogous vectors in π = π(χ1− ,χ2− ). The values n/2 q− of the matrix coeﬃcients Ψ (g)=Φ (g) are 1 times the values given by Table 3 in i,j φi,φj 1+ q 1 - - α1α− - 1 2 e − q which A = 1 is the eignevalue of φ0 and B is the eigenvalue of φ1. 1 α1α− − 2 −

(i, j) Φ (σ ) Φ (wσ ) Φ (σ w) Φ (wσ w) φi,φj n φi,φj n φi,φj n φi,φj n 1 n n n n 1 n n n n (0, 1) q (α1 e α2 )A (α1e α2 )A q (α1e α2 )A (αe1 α2 )A n − n − n − n 1 n− n −1 n− n (1, 0) (α1 α2 )B (α1 α2 )B q (α1 α2 )B q (α1 α2 )B 1 n − n n − n −1 n − n −1 n − n (1, 1) q (α1 A + α2 B) (α1 A + α2 B) q2 (α1 A + α2 B) q (α1 A + α2 B) n n − n n − n n n n (0, 0) α1 B + α2 A α1 B + α2 A α1 B + α2 A α1 B + α2 A Table 3. Values of matrix coeﬃcients of Iwahori ﬁxed vectors II

Proof. In the case of Ψ =Φ , we see that 1,0 φ1,φ0

1 Ψ (kg)= π(kg)φ , φe = π(g)φ , π(k− )φ = π(g)φ , φ =Ψ (g) 1,0 . 1 0/ . 1 0/ . 1 0/ 1,0 for all k K. So Φ(σn)=Φ(wσn) and Φ(wσnw)=Φ(σnw). Now we use the bilinearity of matrix coe∈ fficients to write - - - - 1 Φ =Φ0,0 +Φ0,1 (Φ1,0 +Φ1,1), φ1,φ0 − q e and calculate the value on σn using Table 2: 1 Ψ (σ )=Φ (σ )+Φ (σ ) (Φ (σ )+Φ (σ )) 1,0 n 0,0 n 0,1 n − q 1,0 n 1,1 n n n n 1 α1 α2 1 1 n = α1 + − 1 (1 ) (0 + α2 ) · q 1 α− α − q − q − 1 2 1 1 1 n n − q =(α1 α2 ) + 1 − q 1 α− α 7 − 1 2 8 α 1α ( 1 1− 2 ) + (1 1 ) n n q − q − q n n =(α1 α2 ) 1 =(α1 α2 )B. −  1 α− α  − − 1 2  12  For wσnw: Ψ (wσ w)=Φ (wσ w)+Φ (wσ w) 1 (Φ (wσ w)+Φ (wσ w)) 1,0 n 0,0 n 0,1 n − q 1,0 n 1,1 n n n n 1 1 α2 α1 1 n = α +0 − 1 (1 )+α 2 q q 1 α1α− q 1 · − − 2 − 9 1 1 : = 1 (αn αn) 1 − q = 1 (αn αn)B. q 1 2 1 α α 1 q 1 2 − − − 1 2− − − ; − < The computation of Ψ (σ )=Φ (σ ) is similar: 1,1 n φ1,φ1 n 1 1 Ψ1,1(σn) =Φ0,0(σn) eq (Φ0,1(σn)+Φ1,0(σn)) = q2 Φ1,1(σn) − n n 1 n 1 1 α1 α2 1 n = α (1 ) − 1 + 2 α q 1 q q 1 α− α2 q 2 − − − 1 1 1 1 1 1 n − q n 1 − q = α 1 1 + α + 1 q 1 1 α− α2 2 q 1 α− α2 7 7 − − 1 8 7 − 1 88 1 n n = q (α1 A + α2 B). The formulas for Ψ (wσ ),Ψ (σ w),Ψ (wσ w) and for Φ are derived in the same 1,1 n 1,1 n 1,1 n φ0,φ1 fashion. The final row is just a restatement of Proposition 3.3. ! e In order to prove Theorems 1.3 and 4.7 we will need to know something about the matrix 1 coefficientΨ =Φ where φ = !− φ . (In particular, see Corollary 1.5.) By way 2,2 φ2,φ2 2 1 0 of the following lemma and a messy calculation, one could give a formula for Ψ2,2 similar to those of Proposition 3.5.e " #

1 ! 1 Lemma 3.6. Let π = π(χ,χ− ), and α = χ(#). Let φ = − φ π where φ is as 2 1 0 ∈ 0 above. Then φ πK0 . More precisely, 2 ∈ " # 1 1 1/2 1 1/2 1/2 φ2 = 1 (α + α− )q− φ0 +(α− q αq− )φ1 1+ q − " # Proof. Let k =(ab) K . Then cd ∈ 0 1 1 ab !− a!− b φ2(gk)=φ0(g ( ) )=φ0(g 1 ) cd 1 c!− d ! 1 a b! ! 1 = φ0(g − 1 )=π( − )φp,0(g)=φp,2(g) "1 c!−# d " #1 a b! since 1 K and φp,0" is ﬁxed#" by K. # " # c!− d So φ = φ (e)∈f + φ (w)f . Writing this in terms of the basis φ ,φ and simplifying " 2 2 # 0 2 1 { 0 1} leads to the desired result. !

1 1 3.3. Application to the special representations. If χ1χ2− = ± then the resulting induced representation is reducible. Indeed, writing |·| 1/2 1/2 1 1/2 1 1/2 π = π(χ ,χ − ) and π = π(χ− − ,χ− ), |·| |·| |·| |·| this is reﬂected in the exact sequences - (3.12) 0 σχ π Cχ 0, −→ −→ −→ −→

1 1 (3.13) 0 Cχ− π σχ− 0 −→ −→ 13 −→ −→ - g σn wσn σnw wσnw

n n 1 n 1 n n 1 α q− − α (q− q− ) α q− Φ0,0(g) 1 −1 0 1 1+ q 1+ q 1+ q

n n n 1 n n α α q− − α (1 q− ) Φ1,0(g) 0 1 1 −1 1+ q 1+ q 1+ q

n 1 n 1 n n n 1 α (q− q− − ) α q− α q− Φ0,1(g) −1 1 1 0 1+ q 1+ q 1+ q

n n n 1 n n α α (1 q− − ) α q− Φ1,1(g) 1 0 − 1 1 1+ q 1+ q 1+ q

1/2 1/2 Table 4. Values of matrix coeﬃcients for π = π(χ ,χ − ) |·| |·| where Cµ is the 1-dimensional space on which G acts by µ(det g) and σµ is a special representation. 1/2 1/2 We write α = χ(#). So α1 = αq and α2 = αq− . Using these values of α1,α2, we n/2 q− rewrite Table 2 including the factor 1 as Table 4. 1+ q 1 Our standard model of σχ will be as a subset of π as in (3.12). We will see that σχ σχ− 1 1/2 1 1/2 3 which could be considered as a subset of π(χ− ,χ− − ). However, for the purposes |·| |·| of computing the inner form -

(3.14) , : σχ σχ C .· ·/ × → 1 it is better to consider σχ as the quotient π/Cχ− as in (3.13). As a matter of notation,

1 - given these models of σχ and σχ− we use letters f,φ to denote elements of π (or σχ), f,φ ¯ ¯ 1 to denote elements of π,- and f,φ to denote elements- of σχ− . The following is a well-known result. - - Lemma 3.7. The pairing- on π π deﬁned in (3.2) descends to a well deﬁned pairing on × σ σ 1 . That is to say, if f σ and f¯ σ 1 then χ × χ− ∈ χ ∈ χ− - f,f¯ = f(k)f(k)dk . / !K for any f π whose image in σ 1 is f¯. - ∈ χ− Note that φ = f + f π is K-invariant so it’s image in σ 1 must be zero. Therefore, - 0 0 1 χ− as a corollary- to Lemma 3.7,∈ we have that - - - - (3.15) f(k)dk = fφ (k)dk = f,0 =0 0 . / !K !K for all f σ . In other words, if f σ then f(k)dk = 0. In fact, the converse is also ∈ χ ∈ χ - K true. Namely if f π is such that f(k)dk = 0 then f σ . ∈ K ' ∈ χ The space σK0 is well known to be 1-dimensional. The vector φ = f 1 f π meets the χ ' 0 − q 1 ∈ criterion above, hence is an element of σχ. We call it the normalized new vector. In order to 14 1 choose a new vector in σχ− consistent with the choice of φ, note that by 1 1/2 1 1/2 1 1/2 1 1/2 M : π = π(χ− − ,χ− ) π(χ− ,χ− − )=π", |·| |·| → |·| |·| 1 σχ− is isomorphic to a subspace of π".

- ¯ 1 So we deﬁne the normalized new vector φ of σχ− to be the image of any vector φ for 1 1 1 which M(φ)=f " f " . By Lemma 3.4, M(f f ) = (1 + )φ . A simple calculation then 0 − q 1 0 − q 1 q 1 1 - 1 shows that (1 + q )φ1 and f0 have the same image in σχ− . We now- have all of the necessary ingredients- to prove- the generalization- of Proposition 3.3 to the case of π a special- representation.- Proposition 3.8. Let χ be an unramiﬁed character of F and α = χ(#). If φ, φ¯ are new

1 vectors of a special representation σχ and its contragradient σχ− respectively, the value of the matrix coeﬃcient Φ=Φ ¯ /Φ ¯ (e) is determined by the Table 5 in which n 1. φ1,φ1 φ1,φ1 ≥

g w σn wσn σnw wσnw 1 n n n 1 n n 1 n n n Φ(g) q− α q− α q − α q− − α q− − − − Table 5. Values of matrix coeﬃcient associated to the new vector of a special representation

Proof. We take φ, φ¯ as above. Then φ¯ is the image of f0. So, by Lemma 3.7,

1 1 Φ ¯ (g)= σ (g)φ, φ¯ = (f f )(kg)f (k)dk =Φ (g) Φ (g). φ1,φ1 . χ / 0 − q 1 0- 0,0 − q 1,0 !K The result can now be deduced from Table 4. - ! I(φ φ) Remark. For all of our applications we are interested in determining ⊗ . Since φ, φ = φ,φ . / ) *e Φφ,φ(e), it is convenient to normalize Φin this way. e - 4. Local trilinear forms

In this section we compute the term Iv(ϕv) appearing on the right hand side of (2.4). To this end, we tabulate the local L and zeta factors that appear in Ichino’s formula in Section 4.1. In Section 4.2 we calculate the forms Iv" (ϕv) using our results above. We now assume that all local representations are unitary and have trivial central character. 1 In the unramified case this means we only consider π(χ,χ− ) and that χ(#) is either complex λ with absolute value 1 or else equal to q− for some real λ which we can assume to be in 1 the interval (0, 2 ). (The latter case are the so-called complementary series.) In the case of 2 special representations σχ, we have χ = 1. 4.1. Local L-factors. Let F be a nonarchimedean local field with uniformizer #, and let q be the order of the residue field. We let χ,ν, µ : F × C× be unramified characters, and → s 1 denote γ = χ(#), β = ν(#) and α = µ(#). Then L (s, χ) = (1 γq− )− , and the local v − zeta function is ζFv (s)=Lv(s, 1). Corresponding to each irreducible admissible representation π of GL2(F ) there is a representation ρ : WF GL2(C) of the Weil group. Any representation ρ of the WF gives rise → 15 to an L-factor as in [27]. Let ρi correspond to πi in this manner. Let Π= π1 π2 π3. We define ⊗ ⊗ L(s, Π) = L(s, ρ ρ ρ ) and 1 ⊗ 2 ⊗ 3 3 3 L(s, Π, Ad) = L(s, Ad(ρ )) = L(s, Ad(ρ )) = L(s, π , Ad) ⊕i i i i i=1 i=1 ( ( where ρ1 ρ2 ρ3 : WF GL8(C) is the standard tensor product representation and ⊗ ⊗ → Ad(ρi):WF GL3(C) is the adjoint representation. For the triple→ product, we are especially interested in the following three cases: 1 1 1 2 1 3 Π = π(µ, µ− ) π(ν,ν− ) σ , Π = π(µ, µ− ) σ σ , Π = σ σ σ . ⊗ ⊗ χ ⊗ ν ⊗ χ µ ⊗ ν ⊗ χ The triple product L-function in each of these cases is

1 1 " δ 1 Lv(s, Π )= Lv(s + 2 ,µ ν χ)= " δ s 1/2 , 1 α β γq− − ",δ 1 ",δ 1 (∈{± } (∈{± } − 2 " " 1 Lv(s, Π )= Lv(s, µ νχ)Lv(s +1,µ νχ)= " s " s 1 , (1 α βγq− )(1 α βγq− − ) " 1 " 1 ∈({± } ∈({± } − − 1 L (s, Π3)=L (s + 3 , µνχ)L (s + 1 , µνχ)2 = . v v 2 v 2 (1 αβγq s 3/2)(1 αβγq s 1/2)2 − − − − − − The adjoint L-functions agree with that given by Gelbart-Jacquet [10]. For the cases at hand,

1 Lv(s, 1) L (s, π(χ,χ− ), Ad) = , v (1 γ2q s)(1 γ 2q s) − − − − − 1 L (s, σ , Ad) =L (s +1, 1) = . v χ v 1 q s 1 − − − Note that this is equal to the symmetric square L-function, which is so precisely because we are assuming that π has trivial central character. k For the discrete series representations πdis of GL2(R) of weight k we have the following L-functions. First, let s/2 s ζR(s)=π− Γ(s/2),ζC(s) = (2π)− Γ(s) where Γ(s) is the standard Γ-function. Then L(s, πk1 πk2 πk3 )=ζ (s +1/2)ζ (s + k 3/2)ζ (s + k 1/2)ζ (s + k 1/2), dis ⊗ dis ⊗ dis C C 1 − C 2 − C 3 − L(s, πk , Ad) = ζ (s + k 1)ζ (s + 1). dis C − R 4.2. Explicit computations: F nonarchimedean. Recall that we want to make Ichino’s formula (2.4) explicit, i.e. calculate Iv(ϕ) which is a product of L-factors times

Iv" (ϕ)= Π(g)ϕ, ϕ = Φ1(g)Φ2(g)Φ3(g)dg. Z(F ) G(F ). / Z(F ) G(F ) ! \ ! \ If all of the local data is unramiﬁed, Ichino calculated that the trilinear form I(ϕ) = 1—the normalization of (2.3) is chosen precisely to give this result. So in all of our computations at least one of the representations will be special. 16 We assume from now on that π3 = σχ, and we denote the normalized matrix coeﬃcient 3 associated to a new vector of σχ by Φ as in Proposition 3.8. So our object is to calculate

1 2 3 Iv" (ϕ1 ϕ2 ϕ3)= Φ (g)Φ (g)Φ (g)dg ⊗ ⊗ Z(F ) G(F ) ! \ i where Φ =Φϕi,ϕi for various choices of ϕi. From Tables 1 and 4 we read oﬀthat (for all n 0) ≥ γn vol(K0gK0)Φ(g)= 1 ±q(1 + q )

for g σn,wσn,σnw,wσnw with the minus sign in the cases σnw or wσn and the positive sign otherwise.∈{ } All of the matrix coeﬃcients we are considering are K0-biinvariant, so we combine this with Lemma 3.1 to obtain the formula

Φ1(e)Φ2(e) Φ1(w)Φ2(w) (4.1) Φ1(g)Φ2(g)Φ(g)dg = − Z(F ) G(F ) 1+q ! \ ∞ γn + Φ Φ (σ ) Φ Φ (wσ ) Φ Φ (σ w)+Φ Φ (wσ w) . 1+q 1 2 n − 1 2 n − 1 2 n 1 2 n n=1 1 9 : In computing the forms, we will use (many times) the following identities. Let

2 2 1 α 1 α− (4.2) A = − q and B = − q . 1 α2 1 α 2 − − − Then 2 2 1 α α− 1 α α− 1 1 α− + + α − q − q − q − q 1 (4.3) A + B = 2 + 2 = 1 = 1 + q , 1 α 1 α− α α− − 2 − 2 −2 2 1 α 1 α− 1+ α +1 α− 1 q q q q 1 1 − − − − − (4.4) Aα + Bα = 1 + 1 = 1 = q (α + α− ), α− α α α− α α− − − 1 − 1 α α− 2 1 2 1 α− α α− + + α 1 q q q q 1 − − − − − − (4.5) Aα + Bα = 2 + 2 = 1 = α + α 1 α 1 α− α α− 2 2 − 1 − 1 − 1 2 2 1 (4.6) Aα + Bα− =(Aα + Bα− )(α + α− ) (A + B)= (α + α− )+ 1 − q q − 2 2 1 1 2 2 1 (4.7) Aα− + Bα =(Aα− + Bα)(α + α− ) (A + B)=α + α− +1 . − − q 4.2.1. Only one of the representations is special. In this section we treat the case when two of the representations are unramified, and only one is special. For the application to subconvexity this is the case of most interest since it corresponds to the place p in Corollary 1.5. 1 1 Let π1 = π(µ, µ− ) and π2 = π(ν,ν− ) and π = σχ. Set α = µ(#), β = ν(#) and γ = χ(#). i i Finally, let φ0,φ1 be the basis of eigenvectors of πi as in Proposition 3.5. We combine{ the} results of the previous sections to prove the following. Proposition 4.1. Let Φ3 be the normalized matrix coefficient corresponding to a new vector 3 i i φ σχ as in Proposition 3.8, and define Ψj,k =Φφi ,φ for i =1, 2 as in Proposition 3.5. ∈ j k 17 Then

1 2 3 1 1 1 1 (4.8) Ψ0,0Ψj,kΦ (g)dg = L( 2 , Π ) q2 (1 q ) C Z G · − · ! \ where 2 2 1+ 1 α +α− if j = k =1, C = q2 − q 0 otherwise. 0 Proof. To show directly that C = 0 when j and k are not both 1 is not hard using (4.1). However, we give a more conceptual proof. Suppose that 4 : π1 π2 σχ C is any × × → G-invariant linear form. Note that if φ σχ then Lemma 3.7 implies K π(k)φdk = 0. 1 2 ∈ 1 2 We claim that 4(φ0,φ0,φ) = 0 for all φ σχ. Since φ0 and φ0 are K-invariant (and 4 is G-invariant), ∈ ' 1 2 1 1 1 2 1 2 4(φ0,φ0,π(k)φ)=4(π(k)− φ0,π(k)− φ0,φ)=4(φ0,φ0,φ) for all k K. Integrating over K, we find that ∈ 1 2 1 2 1 2 4(φ0,φ0,φ)=4(φ0,φ0, π(k)φdk)=4(φ0,φ0, 0) = 0. !K Since the matrix coefficients are bilinear forms, fixing the vectors in one of the coordinates gives a linear form in the other. For example, if we fix φ1 φ2 φ Πthen ⊗ ⊗ ∈ 1 2 4 : π1 π2 π C 4(φ ,φ ,φ)= -Φφ1,-φ1 Φφ2-,φ2 Φ-φ,φ(g)dg × × → Z G ! \ is a linear form on Π. From this point of view, it is immediatee thate ine each of the cases of Proposition 4.1 that are claimed to be zero, as a form on either Πor Π(or both) the claim applies. To complete the proof of Proposition 4.1, we need to do the calculation- of (4.8). Let Ai, Bi be the eigenvalues of the intertwining operator M on πi. By (4.1), the integral (4.8) − 1 is equal to q times

n 1 1 n αβγ α− β− γ ∞ B1A2 + A1B2 1 1 1 q q + (1 + q )− 1 n 1 n = q  9 : α− 9βγ : αβ− γ  n=1 + A1A2 + B1B2 1 q q  9 : 9 :  αβγ α 1β 1γ α 1βγ αβ 1γ B A A B − − A A − B B − 1 1 1 1 2 q 1 2 q 1 2 q 1 2 q = q + (1 + q )− αβγ + α 1β 1γ + α 1βγ + αβ 1γ 1 1 − − 1 − 1 − A − q − q − q − q B 1 Simplifying the expression inside the parentheses yields L( 2 , Π) times

4 γ 1 1 γ (A α− + B α)(A β + B β− ) (A + B )(A + B ) q 1 1 2 2 − q4 1 1 2 2 3 γ 1 1 1 1 + [(A + B )(A α− + B α)(β + β− )+(A α + B α− )(A β + B β− )] q3 2 2 1 1 1 1 2 2 2 γ 2 2 2 2 (A + B )(A + B )+(A + B )(A β + B β− )+(A + B )(A α− + B α ) − q2 1 1 2 2 1 1 2 2 2 2 1 1 C 18 D 1 1 1 Applying formulas (4.3)-(4.7), this becomes q (1 + q )L( 2 , Π) times

2 2 1 1 1 1 1 1 1 1 2 2 β +β− (4.9) (1 + )(1 + )+γ(α + α− )(β + β− )( + ) (α + α− + ) − q q q2 q q3 − q q 1 1 So, combining the above, we ﬁnd that (4.8) is equal to q2 L( 2 , Π) times

1 1 L( 2 , Π)− + (4.9). 1 1 Since L( 2 , Π)− is equal to 1 2 1 1 1 1 1 2 2 2 2 (4.10) (1 + ) (1 + )γ(α + α− )(β + β− )+ (α + α− + β + β− ), q2 − q q2 q2 the value of C in the statement of the proposition follows. ! 1 2 3 Corollary 4.2. Let the notation be as in Proposition 4.1, and set φ = φ0 φ2 φ where 1 2 !− 2 ⊗ ⊗ φ2 = 1 φ0 as in Lemma 3.6. Then

" # Iv(φ) 1 1 1 (4.11) = (1 + )− φ, φ q q . / In particular, Iv(φ) 1 where the implicit constant can be taken to be independent of Π and φ,φ q q. ) * 5 Proof. Apply Lemma 3.6 to write

1/2 1 1/2 1 1/2 1/2 2 2 2 βq− + β− q− β− q βq− φ2 = aφ0 + bφ1 where a = 1 ,b= −1 . 1+ q 1+ q

1 First, assume that β = β− . By the bilinearity of matrix coeﬃcients, 2 2 (4.12) Ψ2,2 =Φ = a Ψ1,1 + abΨ0,1 + abΨ1,0 + b Ψ1,1. aφ0+bφ1,aφ0+bφ1 | | | | 1 2 Lv(1,Π ,Ad) Recall that Iv = ζFv (2)− 1 1 Iv" . Therefore, (4.8) implies that Lv( 2 ,Π ) 1 Iv(φ) 2 Lv(1, Π , Ad) 1 2 3 =ζ (2)− Ψ Ψ Φ (g)dg Fv 1 1 0,0 2,2 φ, φ Lv( , Π ) Z G 2 ! \ . / 1 2 Lv(1, Π , Ad) 2 1 2 3 1 1 1 =ζ (2)− b Ψ Ψ Φ (g)dg = (1 ) . - Fv 1 1 0,0 1,1 q2 q − Lv( , Π ) | | Z G − 2 ! \ In the ﬁnal step we have used that

1 1+ 2 2 2 2 2 2 q 1 α +α− 1 1 β +β− 1 ζ (2)− L (1, Π , Ad) = (1 + )− (1 + )− . Fv v (1 1 ) q2 − q q2 − q − q λ Although the result is the same if π2 is a complementary series (i.e. β = q for some λ (0, 1 )) the method is slightly diﬀerent because the inner product is not the same. If ∈ 2 φ,φ" π , then ∈ 2 φ,φ" = φ(k)(Mφ")(k)dk. . / K ! 19 (Compare with (3.2).) Thus, since φ2 = φ2,

Ψ = π ( )φ ,φ = φ (k )(Mφ )(k)dk 2,2 . 2 · 2 2/ 2 · 2 !K =a2A Ψ + abA Ψ abB Ψ b2B Ψ . 2 0,0 2 1,0 − 2 0,1 − 2 1,1 The remainder of proof goes through as above, except one replaces b 2 with b2B /A . (We | | − 2 2 must divide by A2 because φ2, φ2 =Ψ2,2(e)=A2.) As it turns out, this is exactly the same . /2 expression in terms of β as was b . | | ! 4.2.2. Two of the representations are special. We now consider the case Π2 = π(µ, µ) σ ⊗ ν ⊗ σχ. The following result will be used in Theorem 4.7. Proposition 4.3. Let Π2 be as above, and denote by Φ2 and Φ3 the normalized matrix 2 3 coeﬃcients corresponding to the new vectors φ σν and φ σχ as in Proposition 3.8. Let 1 ∈ 1 ∈ 1 1 Ψ0,0 =Φφ1,φ be the matrix coeﬃcient corresponding to φ0 π(µ, µ− ) as in Proposition 3.5. 0 0 ∈ Then

2 2 1 2 3 1 1 1 2 α α− (4.13) Ψi,jΦ Φ (g)dg = q (1 q )L( 2 , Π )(1 q )(1 q ). Z G − − − ! \ Therefore, if we set φ = φ1 φ2 φ3 then Iv(φ) = 1 . 0 φ,φ q ⊗ ⊗ ) * Proof. The evaluation of (4.13) is obtained by a calculation analogous to the proof of (4.8). Since the method is identical, we leave the details to the reader. Note that 2 2 2 2 α 1 α− 1 1 1 ζ (2)− L (1, Π , Ad) = (1 )− (1 )− (1 )− . Fv v − q − q − q Thus, given (4.13), the conclusion that Iv(φ) = 1 is immediate. φ,φ q ! ) * 4.2.3. All three representations are special. This is the case that corresponds to the primes dividing N in Theorem 1.4. We denote ε = ε( 1 ,σ σ σ )= (χµν)(#)= αβγ. 2 χ ⊗ µ ⊗ ν − − Proposition 4.4. Let π1 = σµ and π2 = σν. Denote the matrix coeﬃcient associated to the new vector φi π as in Proposition 3.8 by Φi and set φ = φ1 φ2 φ3. Then ∈ i ⊗ ⊗ 1 2 3 1 3 1 1 ε 2 (4.14) Φ Φ Φ (g)dg = (1 + ε)Lv( 2 , Π ) q (1 q )(1 + q ) , Z G − ! \ Iv(φ) 1 ε 1 and = − (1 + ). φ,φ q q ) * Proof. Wee use Proposition 3.8 and apply (4.1):

1 2 1 n 1 2 ε 1 2 (1 q )(1 2 ) ∞ ε 1 2 (1 q )( 2 ) − q + − − q = − q 1 − q 1+q 1+q −q2 1+q − 1+ ε n=1 7 q2 8 1 ; < Equation (4.14) is immediate. The value of Iv(φ) now follows by multiplying by the appro- priate normalizing factors as given in Section 4.1. ! 20 B Proposition 4.5. Let Π= σµ σν σχ and Π the admissible representation of B associated to Π via Jacquet-Langlands where⊗ ⊗B is the unique quaternion division algebra over F . Let ε = (µνχ)(#). If φ ΠB and φ ΠB then − ∈ ∈ Iv(φ) 1 1 (4.15) - - = (1 ε) (1 ). φ, φ − q − q . / Proof. The Jacquet-Langlands lift of σχ is the character χB : B× C× given by χB(β)= χ N (β) where N is the reduced norm. Hence ΠB η where η→ = µ ν χ , and ◦ B B 3 B B B B B B Φφ,φ(β)dβ = Π (β)φ, φ dβ F B F B . / ! ×\ × ! ×\ ×

= ηB(β) φ, φ dβ F B . / ! ×\ × vol(F × B×) if η is trivial, = φ, φ \ B . / 0 otherwise. 0 2 Recall (see Section 2.2) that vol(F × B×)= q 1 . To obtain Iv, one multiplies this by the \ − 2 Lv(1,Πv,Ad) factor ζFv (2)− 1 . Using the values given in Section 4.1 and simplifying, the result Lv( 2 ,Πv) follows. ! 1 Our factor ε is indeed the local factor ε( 2 , Π). So Propositions 4.4 and 4.5 provide an explicit realization of Prasad’s result. More precisely, the integrated matrix coeﬃcient provides a trilinear form on Π(respectively ΠB) that is invariant by the diagonal action of G 1 2 3 (resp. B×.) It is nonzero on B× if and only if ε = 1. On G, φ = φ1 φ1 φ1 is a test vector if and only if ε = +1. − ⊗ ⊗

cos θ sin θ 4.3. Explicit computations: F = R. Let K = κθ = sin θ cos θ θ [0, 2π) , N = 1 x y { − | ∈ } n(x)=( 1 ) x R and A = a(y) = ( 1 ) y 1 . Then PGL2(R) can be given by the{ coordinates| KAK∈ .} Recall (2.7){ gives the Haar|| measure|≥ } " in this case.# k It is well known that for the weight k discrete series, which we denote by πdis, the matrix coeﬃcient corresponding to a new vector φ is given by (2√det g)k (4.16) Φ(g)= g =(ab) , det g>0 (a + d + i(b c))k cd − and Φ(g) = 0 if det g<0. In the KAK coordinates (and y 1) this is ≥ k/2 k 2πik(θ2 θ1) y (4.17) Φ(κ a(y)κ )=2 e − . θ1 θ2 (y + 1)k 1 k As a matter of notation, let φ− = π(( − ))φ. In particular, if φ π has weight m then 1 ∈ dis φ− has weight m. Moreover, if φ is a new vector then the matrix coeﬃcient associated to − φ− is Φ.

ki Proposition 4.6. For i =1, 2, 3, let πi = πdis be such that k = k1 = k2 + k3, and let φi πi k1 k2 k3 ∈ be a new vector. Set φ = φ1 φ2− φ3− Π = πdis πdis πdis. Then ⊗ ⊗ ∈ ∞ ⊗ ⊗ 2π Φφ,φ(g)dg = , PGL2(R) k 1 ! 21 − and Iv(φ) =2. φ,φ ) * Proof. Using the description above and the KAK coordinates, we have

k1+k2+k3 2 2π 2π (k1+k2+k3)/2 2 − ∞ y 2 Φφ,φ(g)dg = (1 y− )dθ1dydθ2 π (1 + y)k1+k2+k3 − !PGL2(R) !0 !1 !0 k k 2 ∞ y y − =22kπ − dy (1 + y)2k !1 yk 1 ∞ =22kπ − =2π/(k 1). −(k 1)(1 + y)2k 2 − E − − F1 2 Lv(1,Π ,Ad) Since φ,φ =Φ(e) = 1, to get Iv(φ) one multiplies this result by ζ (2)− 1 ∞ which R Lv( ,Π ) . / 2 ∞ is easily seen to be equal to (k 1)/π. Hence, Iv(φ) = 2. φ,φ ! − ) * 4.4. Generalizing Watson’s formula. We now work globally. Let F be a totally real number ﬁeld, and π1,π2,π3 cuspidal automorphic representations of PGL2(F ). It is clear from the description in Section 4.3 that if ϕ = ϕ ϕ ϕ for some ϕ = ϕ π then 1 ⊗ 2 ⊗ 3 i v i,v ∈ i I(ϕ) = 0 unless at each inﬁnite place the weights ki,v of ϕi sum to zero. $ We say that Π= π1 π2 π3 is almost balanced if at each real place v, the largest weight is the sum of the two smaller⊗ ⊗ weights.

Theorem 4.7. Let π1,π2,π3 cuspidal automorphic representations of PGL2 over a totally real number field F such that Π= π π π is almost balanced. Assume that the conductor 1 ⊗ 2 ⊗ 3 Ni of πi is squarefree for each i =1, 2, 3. Let N = gcd(N1, N2, N3). For j, k, l = 1, 2, 3 , write { } { } Nj = Nnjnjknjl where njk = nkj = gcd(Nj, Nk)/N. B Let M = N1N2N3. Let B be the global quaternion algebra such that dim HomB(Π , C)=1. (So the discriminant of B divides N.) Define the vector ϕ = ϕ ϕ ϕ ΠB as follows. For v , let k be the maximum 1 ⊗ 2 ⊗ 3 ∈ |∞ v weight, and let ϕi,v be twists of the normalized new vectors such that their weights add to zero as in Proposition 4.6. At a finite prime p nj, let ϕi,p be the normalized new vector for 1 | !− i = j, k, and the twists of the normalized new vector by 1 as in Lemma 3.6 (where it is denoted as φ .) In all other cases we take ϕ to be the normalized new vector. Let 2 i,v " # ϕi = vϕi,v. Under⊗ this choice of ϕ and for db the Tamagawa measure, 2 1 [B ] ϕ(b)db ζ (2) L( , Π) (4.18) × = F 2 C 3& & 2 23 L(1, Π, Ad) v i=1&' [B ] ϕi(b)& db v M & × | &| (|∞ where, for finite places,% ' 2 εp N(p) (1 + N(p) ) if p N 1 1 1 | C = (1 + )− if p nj p  N(p) N(p) |  1 if p n  N(p) | jk and for infinite places, Cv =2if all three representations πi,v are discrete series and Cv =1 1  otherwise. Here εp = εp( 2 , Πp). 22 Proof. This is proven by combining Ichino’s formula with local calculations above. The constant for p N is immediate given Propositions 4.4 and 4.5. The constants for p n or | | j njk follow from Corollary 4.2 and Proposition 4.3 respectively. In the real case, Proposition 4.6 gives the constant if all three local representations are discrete series. The remaining real cases follow from [15, Proposition 5.1.] together with the archimedean calculations of [29, Theorem 2]. ! Remark. Watson’s archimedean calculations of [29, Theorem 2] in the case that all three representations are principal series are taken directly from [17].

5. Proof of Theorem 1.3

We return to the notation of Section 2 in which F is any fixed number field and A = AF its ring of adeles. Let π1,π2 be cuspidal automorphic representations of GL2 over F with trivial central character and fixed conductors n1 and n2 respectively. Let π3 be a cuspidal autormorphic representation of GL2 over F with trivial central character and conductor np where p is prime and does not divide nn n . We denote Π= π π π . Let # be a 1 2 1 ⊗ 2 ⊗ 3 normalizer of Fp. Note that the Langlands parameters of π3, are allowed to vary (within a ∞ bounded set.) Moreover, although n is fixed, the local components π3,q for q n need not be. k | At the real places πi,v corresponds to either πdis,aweight k-discrete series with k 2 s s s ≥ even, or an irreducible (weight zero) principal series π = π( , − ) defined in the same R |·| |·| way as (3.1). At a complex place π is an irreducible principal series πs,k. i,v C Each of these is a (g,K)-module where g is the complexified Lie algebra of Gv = GL2(Fv) and K = O(2) or U(2) depending on whether v is real or complex respectively. The irreducible representations of K are called weights. In the real case these are integers and, as inθ is well-known they are given by the characters κθ e . In the complex case, the weights are nonnegative integers k which correspond to representations0→ of dimension k + 1. In the notation above for πk,s and πs , the minimal weight is encoded by the parameter k. The C R parameter s is a complex number. The assumption that π be restricted to a bounded set means that the s and k are bounded for each v . ∞ |∞ We will show that for the quaternion algebra B such that ΣB =Σ(Π) there exists a finite collection of vectors B πB and B πB such that for ϕ a new vector, F1 ⊂ 1 F2 ⊂ 2 3 2 1 1 1+" !− L( 2 , Π) ",F,π3, ,n N(p) ϕ1(b)ϕ2(b 1 )ϕ3(b)db # ∞ &![B×] & & " # & for some ϕ B. As a first step, we prove& the following. & i ∈Fi & & Proposition 5.1. Let Π= π1 π2 π3 be as in Theorem 1.3. Let B be the quaternion algebra such that Σ =Σ(Π). There⊗ exist⊗ vectors ϕ πB such that for ϕ = ϕ ϕ ϕ , B i ∈ i 1 ⊗ 2 ⊗ 3 1 2 − 1 1+" Iv(ϕv) (5.1) L( 2 , Π) N(p) ϕ(b)db . # ϕv,ϕv [B ] v n1n2n ;. /< &! × & | ( ∞ & & & & where the implied constant is dependant on 6, Π , N(n1n&2n) and ϕi,v& where i =1, 2 and v n n n. ∞ |∞ 1 2 Proof. We apply Theorem 2.1. Write πB = π . Clearly, we may assume that L( 1 , Π) = 0. i ⊗v i,v 2 ( Hence Theorem 2.2 guarantees that we may choose some ϕi = vϕi,v for i =1, 2, 3 such 23 ⊗ that, writing ϕ = ϕ ϕ ϕ , I(ϕ) = 0 and I (ϕ ) = 0 for all v. In particular, we may let 1 ⊗ 2 ⊗ 3 ( v v ( ϕi,v be the (normalized) new vector for all v ! n1n2n. For such places v, Iv(ϕv) = 1. Note 1 !− ∞ 1 that twisting ϕ2 by 1 means that Ip(ϕp) is as in Corollary 4.2, hence Ip(ϕp) N(p)− . To deal with the term ∼ " # 3 L(1, Π, Ad) L(1,πj, Ad) 3 2 = 2 j=1 [B ] ϕj(b) db j=1 [B ] ϕj(b) db × | | ( × | | in (2.4), we use the bound% of Iwaniec[18]' which says' that if ϕ π is a new vector then ∈ L(1,π , Ad) (5.2) i N(π )" ϕ(g) 2dg # i [G]| | as N(π ) . Here, N(π ) denotes' the conductor of π and the implied constant depends i →∞ i i continuously on the Langlands parameters of πi, . 1 ∞ After solving for L( 2 , Π) in (2.4) and applying (5.2), (5.1) follows. Since we are fixing " " n1, n2 and n, all of the terms in [N(π1)N(π2)N(π3)] except for N(p) can be absorbed into the implied constant. !

As a matter of terminology, we call ϕv a test vector if Iv(ϕv). Given test vectors for all v we can construct the global vector ϕ = vϕv such that I(ϕ) = 0. Therefore, given Proposition 5.1, Theorem 1.3 will follow provided⊗ we can show the following.(

Show that for each v n1n2n there exists a finite sets of vectors such that as π3, • | ∞ ∞ and π3,p vary a test vector can be chosen from the said finite sets. Show that the values of the corresponding linear forms is uniformly bounded. • We begin by bounding the terms I (ϕ )/ ϕ ,ϕ for archimedean primes v. v v . v v/ Lemma 5.2. Let πi, = πi,v be the archimedean parts of automorphic representations ∞ v |∞ 1 πi for i =1, 2, 3 such that π1 and π2 are fixed, and π3, is bounded. Assuming that L( , Π) = % ∞ 2 ( 0, let B be the quaternion algebra such that ΣB =Σ(Π). Then there exists a finite collection B B B of vectors i in πi, for i =1, 2 such that if ϕ3, π3, is a new vector then F ∞ ∞ ∈ ∞ I (ϕ ϕ ϕ ) δ v 1,v ⊗ 2,v ⊗ 2,v ≥ v (|∞ for some choice of ϕi, i and some δ> 0. ∞ ∈F k Proof. If v is real, for each of the finitely many choices k for which π3,v could be πdis, Theorem 1.2 guarantees we may chose vectors ϕ πB and ϕ πB such that ϕ = 1,v ∈ 1,v 2,v ∈ 2,v ϕ1,v ϕ2,v ϕ3,v is a test vector. Now⊗ assume⊗ that π = πs is a principal series. Loke[22] gives a choice of vectors ϕ π 3,v R i,v ∈ i,v (i =1, 2) independent of s such that for ϕ3,v equal to the normalized new vector, the unique (g,K)-invariant linear form, ϕ1,v ϕ2,v ϕ3,v is a test vector, hence the matrix coefficient is nonzero. ⊗ ⊗ Because the matrix coefficient attached to ϕ3,v is a continuous function with respect to its Langlands parameter5, under the boundedness condition there must be a lower bound on

5The local L-factors are also continuous since they are products of gamma functions. 24 the values of the matrix coefficient. More explicitly, given Loke’s choice of ϕ1,v,ϕ2,v we have a nonzero continuous map s λ :Ω C× s Iv(ϕ1,v ϕ2,v ϕ3,v) → 0→ ⊗ ⊗ where ϕs πs is the normalized new vector and s Ω . By the boundedness 3,v R C assumption∈ implies we may assume Ωis compact, hence∈ its image⊂ is uniformly bounded away from zero. If v is a complex place, Loke[22] again proved that there must be a test vector independent of s. The continuity argument of above again applies with Ωreplaced by Ω k , ,k ×{ 1 ··· m} where ki are the distinct nonnegative integers corresponding the possible weights. In this case, Loke does not give a test vector independent of s. However, a test vector for a given value s, by continuity, will also be a test vector for some open subset U Ω. Since Ωis s ⊂ compact, there exist finitely many values si such that iUsi =Ω. B ∪ The finite sets i, is obtained by taking the union of all of the vectors obtained above. ! F ∞ In each of Corollary 4.2, Proposition 4.3, Proposition 4.4 and Proposition 4.5 the value of Iv(ϕv)/ ϕv,ϕv is constant even if one of the representations is varied. That is to say, . / 2 2 suppose v = q is a finite prime such that q ! ni for i =1, 2 and q ! n. In particular, 1 π3,q = π(µ, µ− ) or σµ for some unramified character µ if v is finite. Hence there is a choice of ϕ such that I (ϕ )/ ϕ ,ϕ is nonzero and independent of µ. Hence, this completes the v v v . v v/ proof of Theorem 1.3 in the case that n1, n2, n are squarefree. Although we believe that this phenomenon–the existence a test vector such that Iv(ϕv) depends only on the level of π3,v–should hold more generally, it is not a priori evident. However, the following lemma allows us to conclude that n1, n2, n may be taken arbitrarily.

Lemma 5.3. Fix irreducible admissible representations π1,v,π2,v of GL2(Fv) that are local components of global automorphic representations. There exist finite collections π F1,v ⊂ 1,v and 2,v π2,v and a constant δ> 0 such that if π3,v is any other such representation of fixedF level⊂ such that Π = π π π admits a GL (F )-invariant linear form, v 1,v ⊗ 2,v ⊗ 3,v 2 v I (ϕ ϕ ϕ ) δ v 1,v ⊗ 2,v ⊗ 3,v ≥ for ϕ the new vector and some ϕ . 3,v i,v ∈Fi,v Proof. Let ωi be the central character associated to πi,v. A necessary condition for the existence of an invariant form is that ω1ω2ω3 be trivial, hence we may assume without loss of generality that πi,v has trivial central character for each i =1, 2, 3. Moreover, since πi,v is the local component of an automorphic representation it is unitary. 1 The level of a principal series representation π(µ, µ− ) is twice the level of the character µ. We conclude that principal series representations of fixed level and trivial central character are parametrized by characters χ : Fv× C with fixed level. → Let # be a uniformizer, qv the order of the residue field, and let i 1+u# u × if i 1 U (i) = { | ∈O } ≥ × if i =0. 0 O As is well known, for χ to be a character of level n means that χ(u#k)= # ks χ¯(u) 25| | (n) 1 whereχ ¯ is a character of the finite group ×/U . If π(χ,χ− ) comes from an automorphic O representations, we may assume s = sχ is a complex number with Re(s) [0,λ] where λ is any bound towards Ramanujan. Note we may also assume that Im(s) is bounded.∈ In other words, s is restricted to a compact set. 1 Let π π(χ ,χ− ) be such that I = 0, and denote the new vector by ϕ . Given 3,v 3 0 0 v ( 3,v ϕ1,v,ϕ2,v such that ϕ1,v ϕ2,v ϕ3,v is a test vector, as in the proof of Lemma 5.2, #1,v s ⊗ ⊗ 1 ⊗ ϕ ϕ is also a test vector for π π(µ, µ− ) whereµ ¯ =¯χ and s is in some open set 2,v ⊗ 3,v 3,v 3 µ containing sχ. Since the set of possible sχ is compact, and the number of charactersχ ¯ is finite, we find that a finite choice of choices ϕ1,v,ϕ2,v suffice to give a set of test vectors for π3,v any principal series representation. A completely analogous argument gives a finite set of choices for ϕ1,v,ϕ2,v in the case that π σ . In this case, however, the argument is simplified by the fact that χ must be 3,v 3 χ unitary, i.e. Re(sχ) = 0. Finally, if π3,v is a supercuspidal representation then (by [7]) its Kirillov model associated to a nontrivial additive character ψ is completely determined by the level of π3,v and the 1 epsilon factor ε( 2 ,π3,v,ψ). The assumption that ω3 is trivial forces the epsilon factor to be 1. So, at worst, two more choices for each of ϕ1,v and ϕ2,v will suffice. ± The existence of the constant δ is established in exactly the same fashion as in Lemma 5.2. Namely, values assumed by Iv on the given set of test vectors is the image of a compact set (n) (corresponding to the finite number of characters of ×/U and the admissible values of O sχ.) ! This completes the proof of Theorem 1.3. Remark. In [8] explicit test vectors are given for all cases except that where all three representations πi,v are supercuspidals. This can be used to make the choices of 1,v, 2,v in Lemma 5.3 explicit. However, note that the argument above is still needed toF ensureF the existence of δ.

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Department of Mathematics, University of Wisconsin–Madison, Madison, Wisconsin.