Triple Product Formula and the Subconvexity Bound of Triple Product L-Function in Level Aspect

TRIPLE PRODUCT FORMULA AND THE SUBCONVEXITY BOUND OF TRIPLE PRODUCT L-FUNCTION IN LEVEL ASPECT

YUEKE HU

Abstract. In this paper we derived a nice general formula for the local integrals of triple product formula whenever one of the representations has suciently higher level than the other two. As an application we generalized Venkatesh and Woodbury’s work on the subconvexity bound of triple product L-function in level aspect, allowing joint ramiﬁcations, higher ramiﬁcations, general unitary central characters and general special values of local epsilon factors.

1. introduction

1.1. Triple product formula. Let F be a number ﬁeld. Let ⇡i, i = 1, 2, 3 be three irreducible unitary cuspidal automorphic representations, such that the product of their central characters is trivial:

(1.1) w⇡i = 1. Yi Let ⇧=⇡ ⇡ ⇡ . Then one can define the triple product L-function L(⇧, s) associated to them. 1 ⌦ 2 ⌦ 3 It was first studied in [6] by Garrett in classical languages, where explicit integral representation was given. In particular the triple product L-function has analytic continuation and functional equation. Later on Shapiro and Rallis in [19] reformulated his work in adelic languages. In this paper we shall study the following integral representing the special value of triple product L function (see Section 2.2 for more details): ⇣2(2)L(⇧, 1/2) (1.2) f (g) f (g) f (g)dg 2 = F I0( f , f , f ), | 1 2 3 | ⇧, , v 1,v 2,v 3,v 8L( Ad 1) v ZAD (FZ) D (A) ⇤ \ ⇤ Y D D Here fi ⇡i for a specific quaternion algebra D, and ⇡i is the image of ⇡i under Jacquet-Langlands 2 0 correspondence. The local integral Iv is defined as L (⇧ , Ad, 1) 0 , , = v v , , , (1.3) Iv ( f1,v f2,v f3,v) 2 Iv( f1,v f2,v f3,v) ⇣v (2)Lv(⇧v, 1/2) where 3 D (1.4) Iv( f1,v, f2,v, f3,v) = <⇡i (g) fi,v, fi,v > dg. i=1 F DZ(F ) v⇤\ ⇤ v Y D Here < , > is a bilinear and D⇤(Fv) invariant unitary pairing for ⇡i,v. At unramified places, 0 · · Iv = 1. This is a special version of Ichino’s formula in [11]. Its local factors are, however, not explicit, and can not be used directly for applications. Later on, there are several works computing the local factors with ramifications. The work by Watson in [23], Nelson in [17] and Woodbury in [24] 1 covered all possible situations with square-free levels. The work of Nelson-Pitale-Saha in [18], however, is the only work that deal with higher ramifications, essentially with the assumption that ⇡1 = ⇡2 (and correspondingly f1 = f2) and ⇡3 is unramified. Such results are already very useful. The local computations by Woodbury together with the work of Venkatesh in [22] proved the subconvexity bound of Triple product L function for square- free levels. The local computations by Nelson-Pitale-Saha was used to prove the mass equidistribution on modular curve Y0(1). It is natural to expect that computing the local integral for more general situations would also generalize these applications. We remark here that the work in [18] is based on Lemma (3.4.2) of [16], which relates Iv to the local Rankin-Selberg integral. But this method can’t be generalized to the case when, for example, all the representations are supercuspidal, which is necessary for our consideration. Before I give the main result for local calculations in this paper, let me first give a very special and explicit corollary, which is already new. For any holomorphic new form f of weight k, suppose that it has Fourier expansion

a f (n) 2⇡inz (1.5) f (z) = 1 k e . > n 2 Xn 0 For two modular forms f1, f2 of same weight k and level N, deﬁne their Petersson inner product 1 dxdy < , > = k . (1.6) f1 f2 Pet f1(z) f 2(z)y 2 [SL2 : 0(N)] y (ZN) H 0 \ Corollary 1.1. Let fi be three holomorphic new forms of weight ki, such that k1 + k2 = k3. For simplicity assume that f f are level 1, and f is of level pn for n > 1. Let a (n) be the n th 1 2 3 fi Fourier coecient for fi as above. Then n 2 < f1 f2(p z), f3 >Pet L( f1 f2 f3, 1/2) (1.7) | | = ⌦ ⌦ C Cp, 3 8 L( fi, Ad, 1) 1 < fi, fi >Pet i=1 Q where Q

2, if ki , 0 for all i, (1.8) C = 1 81, otherwise. <> > 2 2 2 2 Lp( fi, Ad, 1) (( pp +:>1/ pp) a f1 (p) )(( pp + 1/ pp) a f2 (p) ) (1.9) C = . p ⇣2(2) (p + 1)3 pn 1 Q p n Note that it is necessary to use f2(p z) as a test function since if we simply use new forms, the integral will be zero trivially for the level reason. Scaling by pn is exactly the classical way to produce an old form, and in this case guarantees that the integral becomes nonzero. In general for any non-archimedean place v, let q be the size of the corresponding residue ﬁeld. We have the following result for the local integral.

Theorem 1.2. Let ci be the level of ⇡i at v, that is, there exists a unique up to constant new form ci in ⇡i which is invariant by K1($v ) (see (2.3)). Assume that c3 2 max c1, c2, 1 . Then the local integral { } (1 A)(1 B) = . (1.10) Iv c 1 (q + 1)q 3 2 Here the constants A, B can be explicitly given in terms of ⇡1, ⇡2 respectively, and are bounded and away from 1. In particular the local integral is nonzero. Note that the test vectors are normalized and chosen similarly as in the corollary above. The explicit values of A and B will also be given. See Theorem 4.1 for more details. Remark 1.3. The assumption that c 2 max c , c , 1 essentially means that one of the levels is 3 { 1 2 } suciently larger than the other two. We put 1 here to avoid the squarefree level case, which is already well studied before. If we further assume the central characters to be trivial, the assumption can be further improved to be that c > max c , c , 1 . 3 { 1 2 } The local integral becomes more complicated when, for example, c2 = c3 > c1. In a subsequent work in [10], we shall give an upper bound for the local integral in this case, and use this to prove mass equidistribution on Y0(N). In this paper, we shall use the theorem above to prove the subconvexity bound of triple product L function in level aspect. The tool developed in this paper can also be used to compute other local integrals, like Rankin- Selberg integral. We expect to see more applications of this tool in future research. 1.2. Subconvexity bound. We shall give the application of the above theorem to the subconvexity bound for the special value of triple product L-function L(⇧, 1/2). This topic can be approached in di↵erent ways. For example Bernstein and Reznikov in [1] gave the subconvexity bound in the eigenvalue aspect. What we seek is the subconvexity bound in the level aspect. In particular, we 4 will fix ⇡1 and ⇡2, let ⇡3 vary with finite conductor N. The finite conductor of ⇧ is roughly N . By the general principle for convexity bound, one can get the trivial bound (1.11) L(⇧, 1/2) << N1+✏. We’d like to establish the subconvexity bound 1 (1.12) L(⇧, 1/2) << N . as N . !1 The idea comes from Venkatesh’s work in [22]. One starts with the integral representation of the special value of triple product L-function (1.2). Suppose that the cusp forms and their local components are properly normalized. The idea is to give first an upper bound for the global integral 0 on the left-hand side of (1.2). Then a lower bound for the local integrals Iv will result in an upper bound for L(⇧, 1/2), which turn out to be a subconvexity bound in the level aspect. The local integrals basically gives the main term as in the trivial bound, and the power saving comes from the global upper bound. In particular, assume that ⇡3 is of prime conductor p. Venkatesh’s work together with Woodbury’s work on local integrals in [24] proved the following: 1 1/12 (1.13) L(⇧, 1/2) << N(p) . Their result, however, is based on the following conditions:

(1) ⇡i essentially have disjoint ramifications and ⇡3 has square-free finite conductor p. (2) All the central characters are trivial. (3) The special values of local epsilon factors ✏v(⇧, 1/2) = 1 for all places. (4) The infinity component of ⇡3 is bounded. In this paper, we will remove the first three conditions and prove a similar subconvexity bound. So we will allow high ramifications and joint ramifications, and general unitary central characters. 3 The third condition is related to Prasad’s thesis work on local trilinear forms, and turns out to be free to remove. This is because, as we will see later, all key calculations will be done on the GL2 side. We still assume the last condition to control the contribution from archimedean places. We shall prove the following main theorem:

Theorem 1.4. Let ⇡i, i = 1, 2, 3 be three unitary cuspidal automorphic representations of GL2, such that

(1.14) w⇡i = 1. Yi Fix ⇡ and ⇡ , and let ⇡ vary with changing finite conductor and N = Nm( ). Suppose that 1 2 3 N N the infinity component of ⇡3 is still bounded. Then 1 1/12 (1.15) L(⇡ ⇡ ⇡ , 1/2) N , as N . 1 ⌦ 2 ⌦ 3 ⌧ !1 The proof follows the same strategy. In particular we will generalized the power saving upper bound for the global integral, and the lower bound for local integrals directly comes from Theorem 1.2. We remark here that the proof for the global upper bound can be refined to slightly more general setting, that is, to allow the conductors of ⇡1 and ⇡2 also to change, but they have to be relatively small compared to N. In this way, one can obtain a hybrid type subconvexity bound in level aspect. See Remark 3.6 for more details.

1.3. Organizations of this paper. In Section 2 we will review necessary tools and results, as well as derive some new results which will be used in this paper. In particular we will discuss the Whittaker functional and matrix coecient for highly ramified representations of GL2, which is the key ingredient for the local integral of Triple product formula. In Section 3, we basically imitate Venkatesh’s proof and get a power saving upper bound for the global integral in more general setting. We will use amplification method and reduce the problem to a bound for global matrix coecient. In Section 4 we will prove Theorem 1.2, or Theorem 4.1. Despite its generality, its proof seems quite concise considering all possible combinations of cases. In Section 5 we will finish the proof of Theorem 1.4 by combining the results from the previous two sections. In the appendix we will prove the bound for the global matrix coecient which is used in the proof in Section 3. I would like to thank Tonghai Yang for encouraging me to work on this problem, and Freydoon Shahidi, Akshay Venkatesh and Michael Woodbury for helpful discussions.

2. Notations and preliminary results 2.1. Basic Notations and facts. Let F denote a number ﬁeld. Let G be a reductive algebraic F group. In this paper we will focus on G being GL or D⇤, where D is a quaternion algebra. Let 2 X = Z (A)G(F) G(A). Let L2(X) be the space of square integrable functions on X, and < , > be G \ · · the natural pairing on it given by

(2.1) < f1, f2 >= f1(g) f2(g)dg. ZX Any unitary cuspidal automorphic representation can be naturally embedded into L2(X) with the compatible unitary pairings. 4 Let Fv be the corresponding local ﬁeld of F at a place v. Let Kv denote the standard maximal compact subgroup of G(Fv), and

(2.2) K = Kv. Yv When v is a finite place, let $v denote a uniformizer of Fv and Ov denote the ring of integers at 1 v. Let q = $ . At places where G(F ) GL (F ), define for an integer c > 0: | v|v v ' 2 v c c (2.3) K ($ ) = k Kv k ⇤⇤ mod ($ ) . 1 v { 2 | ⌘ 01 v } ! We will call a function spherical if it is right invariant under Kv. For unramified representations, there is a unique up to constant spherical element. We will call a local representation to be of level c if there exists a unique up to constant element which is K ($c) invariant. Such an element is 1 v then called a new form. Now we record some basic facts about integrals on GL2(Fv) when v is finite. Lemma 2.1. For every positive integer c, 10 GL (F ) = B K ($c). 2 v $i 1 1 v 0 i c v a  ! Here B is the Borel subgroup of GL2.

We normalize the Haar measure on GL2(Fv) such that Kv has volume 1. Then we have the following easy result (see, for example, [9, Appendix A]). c Lemma 2.2. Locally let f be a K1($ ) invariant function, on which the center acts trivially. Then v 10 (2.4) f (g)dg = A f (b )db. i $i 1 0 i c v Fv⇤ GLZ2(Fv)   Fv⇤ ZB(Fv) ! \ X \ Here db is the left Haar measure on F⇤ B(F ), and v\ v q 1 q 1 = = = < < . A0 ,Ac c 1 , and Ai i for 0 i c q + 1 (q + 1)q (q + 1)q 2.2. Integral representation of special values of Triple product L function. The story begins with Prasad’s thesis work. For the triple product L-function L(⇧, s), there exist local epsilon factors ✏v(⇧v, v, s) and global epsilon factor ✏(⇧, s) = v ✏(⇧v, v, s), such that, (2.5) L(⇧, 1 s) = ✏(⇧, s)L(⇧ˇ , s). Q

With the assumption that i w⇡i = 1, we have ˇ Q ⇧ ⇧.

The special values of local epsilon factors ✏v(⇧v, v, 1/2) are actually independent of v and always take value 1. For simplicity, we will write ± ✏v(⇧v, 1/2) = ✏v(⇧v, v, 1/2).

For any place v, there is a unique (up to isomorphism) division algebra Dv. Then Prasad proved in [20] the following theorem about the dimension of the space of local trilinear forms:

Theorem 2.3. (1) dim HomGL2(Fv)(⇧v, C) 1, with the equality if and only if ✏v(⇧v, 1/2) = 1.  5 Dv (2) dim HomDv (⇧v , C) 1, with the equality if and only if ✏v(⇧v, 1/2) = 1. Dv  Here ⇧v is the image of ⇧v under Jacquet-Langlands correspondence. This motivated the following result which is conjectured by Jacquet and later on proved by Harris and Kudla in [7] and [8]: Theorem 2.4. there exist D and f ⇡D s.t. i 2 i L(⇧, 1/2) , 0 f (g) f (g) f (g)dg , 0 { }()8 1 2 3 9 > ZAD⇤(F) D⇤(A) > <> R\ => This result hints that > > :> ;> f1(g) f2(g) f3(g)dg

ZAD (FZ) D (A) ⇤ \ ⇤ could be a potential integral representation of special value of triple product L-function. Later on there are a lot of work on explicitly relating both sides. In particular one can see Ichino’s work in [11]. We only need a special version here (as in the introduction).

⇣2(2)L(⇧, 1/2) (2.6) f (g) f (g) f (g)dg 2 = F I0( f , f , f ), | 1 2 3 | ⇧, , v 1,v 2,v 3,v 8L( Ad 1) v ZAD (FZ) D (A) ⇤ \ ⇤ Y where L (⇧ , Ad, 1) 0 , , = v v , , , (2.7) Iv ( f1,v f2,v f3,v) 2 Iv( f1,v f2,v f3,v) ⇣v (2)Lv(⇧v, 1/2) and 3 D (2.8) Iv( f1,v, f2,v, f3,v) = <⇡i (g) fi,v, fi,v > dg. i=1 F DZ(F ) v⇤\ ⇤ v Y 2.3. Hecke operators. For most of this subsection we refer to [22]. Let f be a function on a group G and a compactly supported measure on G. Deﬁne the convolution of f with by

(2.9) f (x) = f (xg)d(g). ⇤ Zg If and are two compactly supported measures on G, we deﬁne the convolution to be 1 2 1 ⇤ 2 the pushforward to G of on G G, under the multiplication map 1 ⇥ 2 ⇥ (2.10) (g , g ) G G g g . 1 2 2 ⇥ 7! 1 2 Then one has the following compatibility relation (2.11) ( f ) = f ( ). ⇤ 2 ⇤ 1 ⇤ 1 ⇤ 2 Now we introduce the Hecke operators in this language. At a non-archimedean place v, let l be

the maximal prime ideal and r 0 be an integer. Define the measure µ⇤r on GL (F ) to be the l 2 v restriction of Haar measure to the set $r 0 K v K , v 01v 6 ! r 1 (q + 1)q , if r 1; so that the total mass of µl⇤r is 81, if r = 0. Define <> > :> 1 (2.12) µlr = µ⇤r 2k . qr/2 l 0 k r/2 X Via the natural inclusion of GL2(Fv) in GL2(A f ) where A f is the ring of finite adeles, we can regard rv µlr as a compactly supported measure on GL2(A f ). If n is an integral ideal v lv , define

rv (2.13) µn = µlv . Q Yv Convolution by µ can be thought of as n th Hecke operator. n For functions on which the center acts trivially, convolution with µn is a self-dual operator, that is,

(2.14) f ( f µ )dg = ( f µ ) f dg. 1 · 2 ⇤ n 1 ⇤ n · 2 ZX ZX One can also check for such functions that

1 (2.15) f (xg)dµn(g) = f (xg )dµn(g).

GZ(A) GZ(A) We have the following nice lemma about compositions of Hecke operators: Lemma 2.5. Let n, m be ideals. Let h be a function on G(A) that is spherical at all places v nm, and the center acts on h trivially. Then |

(2.16) h(xg)d(µ µ )(g) = h(xg)dµ 2 (g). n ⇤ m nmd G(A) d (n,m) G(A) Z X| Z We will also need to consider functions on which the center acts by a ﬁxed non-trivial unitary character w. From now on we will focus only on operators of form µl or µl2 . Letµ ˇl be the dual of µl in the sense of (2.14). Then one can easily check that 1 1 w($v ) (2.17)µ ˇl = w($ )µl = µ⇤. v pq l

Similarly letµ ˇl2 be the dual of µl2 . Then

1 2 (2.18)µ ˇ2 = (w($ )µ⇤ + µ⇤). l q v l2 o

When acting on spherical functions, µl⇤ and µl⇤2 are related as follows:

(2.19) µ⇤ µ⇤ = µ⇤ + (q + 1)w($ )µ⇤. l ⇤ l l2 v o Given a spherical function, let ˇl and ˇl2 be the eigenvalues ofµ ˇl andµ ˇl2 acting on it. Putting (2.17), (2.18) and (2.19) together, we have

2 1 q + 1 (2.20) ˇl2 = ˇl + (q ). qw($v) 7 Note that 1 q + 1 q 1. | qw($v)| Then one can easily check that, Lemma 2.6.

ˇ2 + ˇ 1. | l | | l| 2.4. Bounds for matrix coecient. If ⇡ is an irreducible unitary cuspidal automorphic representation, then its local component at v is also a unitary representation. At a non-archimedean place, it can be classified into one of the following four types: (1) supercuspidal representation; (2) ⇡(1,2) where i are unitary characters; 1/2 1/2 (3) special representation ( , ) where is unitary; ⌧ ⌧ |·| |·| (4) ⇡( , ), where is unitary and 0 <⌧<1/2. |·| |·| The first three types are tempered representations. The generalized Ramanujan Conjecture implies that only tempered representations can be the local component of a unitary cuspidal automorphic representation. What is known is a bound ↵ towards Ramanujan Conjecture. This means if type (4) ever happens, then ⌧<↵. The smaller ↵ is, the closer we are to the Ramanujan Conjecture for GL2. For our purpose, any ↵<1/4 would be enough to get a subconvexity bound. The current record is ↵ = 7/64. See [14], [2]. Using the bound towards Ramanujan Conjecture, one can bound the matrix coecient for the local component of a unitary cuspidal automorphic representation. 1 1 Locally for f ⇡v ⇡( , ), f ⇡ˇv ⇡( , ) in the standard model for the induced 1 2 1 2 2 2 1 2 representations, we can define the pairing by

(2.21) < f1, f2 >= f1(k) f2(k)dk.

KZv

We can deﬁne the matrix coecient of ⇡v associated to f1, f2 as

(2.22) (g) =<⇡v(g) f1, f2 >. See later subsections for the alternative definition and the definition when the representation is supercuspidal. We first record here the matrix coecient for spherical elements. (See for example, [3].) For simplicity, let i denote i($v) in the following formula if we don’t specify which element the characters are taking.

Lemma 2.7. Let ⇡ = ⇡(1,2) be an unramiﬁed unitary representation of GL2. Let be the matrix coecient associated to the spherical element in ⇡ such that (1) = 1. Then it’s bi-K invariant and

n n/2 n 1 n 1 $ 0 q (1 2q ) (2 1q ) (2.23) ( v ) = 1 2 . 01 1 + q 1 ! 1 2 Now we state the result for the bound of local matrix coecient for general elements.

8 Lemma 2.8. Let ⇡v be the local component of a unitary cuspidal automorhpic representation of GL at a finite place v. Let f ,f be two K finite elements in ⇡ , stabilized respectively by compact 2 1 2 v v open subgroups K and K . Then for any x F and ✏>0, 1,v 2,v 2 v x 0 1/2 1/2 (↵ 1/2+✏) v(x) (2.24) <⇡( ) f , f > F [K : K ] [K : K ] q | | f f . | 01 1 2 |⌧✏, v 1,v v 2,v || 1||v|| 2||v ! Proof. It follows from, for example, Lemma 9.1 of [22]. Here we briefly describe how to prove this result for induced representations at non-archimedean places. For spherical elements, one can use Lemma 2.7 above to check the inequality directly. More specifically if v(x) = n and f ’s are | | i spherical and normalized, then (2.25) n/2 n 1 n 1 x 0 q (1 2q ) (2 1q ) <⇡( ) f , f > = 1 2 | 01 1 2 | |1 + q 1 | ! 1 2 n/2 q n n 1 n 1 n 2 n 3 n 2 = + + + + + + 1 (( 1 1 2 2) q 1 2( 1 1 2 2 )) |1 + q ··· ··· | (↵ 1/2)n (n + 1)q .  The coecient (n + 1) will be essentially bounded by q✏n for any ✏>0, and the implicit constant can be taken to be 1 when q is large enough. When f1, f2 are not spherical, one can use the trick as in [5] to reduce the inequality to the spherical case. ⇤ Remark 2.9. This proof actually allow one to control the implicit constant. In particular one can take a product of the local inequalities and get a global inequality. Now we give a bound for the global matrix coecient. Let D be a global quaternion al- 2 gebra. Let ⇢ denote the right regular representation of D⇤(A) on L (ZAD⇤(F) D⇤(A)). Let F1, 2 \ F L (ZAD⇤(F) D⇤(A)) be two rapidly decreasing and K finite automorphic forms which don’t 2 2 \ have 1-dim components in their spectrum decompositions. Let S be a finite set of non-archimedean places. We assume that D is locally the matrix algebra at the places in S . Let KS = v S Kv and ev 2 Ki,S = v S Ki,v, where Ki,v stabilizes the local component of Fi at v. Let = v $v for ev 0, 2 N and N = Nm( ). Define the matrix Q N Q e Q $ v 0 a([ ]) = , N 01 Yv ! which can be naturally thought of as an element of D⇤(A). Proposition 2.10. With the setting as above, we have (2.26) 1/2 1/2 ↵ 1/2+✏ F (g)⇢(a([ ]))F (g)dg F [K : K ] [K : K ] N F 2 F 2 . | 1 N 2 |⌧✏, S 1,S S 2,S || 1||L || 2||L ZAD (FZ) D (A) ⇤ \ ⇤ We will prove this proposition in the appendix. Now the question is, for any given automorphic 2 form F L (ZAD⇤(F) D⇤(A)), how can we separate out the 1-dimensional components. Suppose 2 \ that in general the center acts on F by a unitary central character w. Then its 1-dimensional components can be given by the following projection:

(2.27) F F(x) = (x) f (y)(y)dy, 7! P 2 X X=w Z 9 where is a unitary Hecke character, and (x) = (det x). Then the remaining part F F doesn’t P have any 1-dimensional components.

2.5. Whittaker model for induced representations. Here we recall some basic results about the Whittaker model for induced representations at a finite place. This and the next subsection are purely local, so we will suppress the subscript v for all notations. Fix an additive character . Without loss of generality, we will always assume is unramified. Let ⇡ be a local irreducible (generic) representation of G. Then there is a unique realization of ⇡ in the space of functions W on G such that 1 n (2.28) W( g) = (n)W(g). 01 ! Locally for an induced representation of GL2, one can compute its Whittaker functional by the following formula: 1 m (2.29) W(g) = '(! g) ( m)dm, 01 mZF ! 2 01 where ' is an element of ⇡ in the model of induced representation and ! is the matrix . 10 When ⇡ is unitary, one can define a unitary pairing on ⇡ using the Whittaker model: !

↵ 0 ↵ 0 (2.30) < W , W >= W ( )W ( )d⇤↵. 1 2 1 01 2 01 ZF⇤ ! ! To get the Whittaker functional explicitly using (2.29), the ﬁrst step is to write 1 m ↵ 0 10 $i 1 ! = 01 01$i 1 ↵ m$i m ! ! ! ! 10 10 in form of B(F) K ($c) for 0 i, k c. Note that if k = c, then is absorbed into $k 1 1   $k 1 c ! ! K1($ ). Same for i. We record the following results from [9].

Lemma 2.11. (1) Suppose k = 0. i $ 1 10 c (1i) If i = 0, we need m < ↵( 1 + $OF) for B K ($ ); ↵ m$i m 2 $k 1 1 (1ii) If i > 0, we need v(m) v(↵). ! ! $i 1 Under above conditions we can write as ↵ m$i m ! ↵ $i + ↵ 101 1 + m ↵+m$i ↵+m$i ↵+m$i . 0 ↵ m$i 1101 ! ! ! (2) Suppose k = c. i c i (2i) If i < c, we need m ↵$ ( 1 + $ O ); 2 F (2ii) If i = c, we need v(m) v(↵) c.  10 $i 1 Under above conditions, we can write as ↵ m$i m ! ↵ m 1 10 ↵ . 0 m + $i 1 ! m ! (3) Suppose 0 < k < c. i k i (3i) If i < k, we need m ↵$ ( 1 + $ O⇤ ); 2 F (3ii) If i > k, we need v(m) = v(↵) k; k (3iii) If i = k, we need v(m) v(↵) k but m < ↵$ ( 1 + $OF).  $i 1 Under above conditions we can write as ↵ m$i m ! ↵$k ↵+m$i ↵+m$i 1 10 m$k 0 k . 0 m $ 1 01 ! ! ! Now let ⇡ be a unitary induced representation ⇡(µ1,µ2), where µ1 and µ2 are both ramiﬁed of level k1 and k2. Let c = k1 + k2 be the level of ⇡. Then by the classical results in [4], there exists a new form in the model of induced representation, which is right K ($c) invariant and supported 1 on 10 B K ($c), $k2 1 1 ! where B is the Borel subgroup. We shall consider the Whittaker function W associated to this new form. Let

k k k (2.31) C = µ ( $ 2 )µ ( $ 2 u) ( $ 2 u)du. 1 2 u ZO 2 ⇤F Denote ↵ 0 10 (2.32) W(i)(↵) = W( ). 01$i 1 ! ! Then the next lemma follows from (2.29) and (3) of the above lemma.

Lemma 2.12. (i) If i < k2, then i (i) 1 $ i k i i k i i k v(↵)/2 (2.33) W (↵) = C µ ( )µ (↵$ (1 $ 2 u)) (↵$ (1 $ 2 u))q 2 du. 1 u 2 u ZO 2 ⇤F Its integral in ↵ for fixed v(↵) is always 0. (ii) If k < i c, then 2  k2 (i) 1 $ k v(↵)/2 k ↵ = µ µ $ 2 ↵ $ 2 ↵ . (2.34) W ( ) C 1( i k ) 2( u)q ( u)du 1 + u$ 2 u ZO 2 ⇤F Its integral in ↵ is zero unless i = c or c 1. In particular 1, if v(↵) = 0; (2.35) W(c)(↵) = . 80, otherwise. <> 11 :> When i < c, 1 (i) q 1 , if i = c 1 and v(↵) = 0; (2.36) W (↵)d⇤↵ = 80, otherwise. v(↵)Zfixed <> > (iii) If i = k2, :> (2.37) $k2 $k2 (k2) 1 1/2 v(↵) W (↵) = C µ1( )µ2( ↵u) ( ↵u)q du. 1 + u$k2 |↵u(1 + u$k2 )| v(u) k ,u<$Zk2 ( 1+$O )  2 F (k2) The integral of W in ↵ for fixed v(↵) is always zero if either k1 > 1 or v(↵) , 0. When k = 1 = c i and v(↵) = 0, its integral against 1 is the same as expected from (2.36). 1 We shall also consider the case when ⇡ ⇡(µ1,µ2), where µ1 is unramified and µ2 is ramified of k level k. Then the level of ⇡ is k. In this case the new form is right K1($ ) invariant and supported k on BK1($ ). Then by (2) of Lemma 2.11, we have Lemma 2.13. (1) When i = k,

(k) ↵ 1 v(↵)+v(m) (2.38) W (↵) = µ ( )µ ( m) ( m)q 2 dm 1 m 2 v(m) Zv(↵) k  1 v(↵)+k 2 v(↵) k q µ1 ($) µ2( m) ( m)dm, if v(↵) 0, = v(m)= k 8 R >0, otherwise. <> (2) When i < k, > :> (i) i i k i i k i 1 v(↵) k+i (2.39) W (↵) = µ ($) µ (↵$ (1 $ u)) (↵$ (1 $ u))q 2 du. 1 2 u ZOF 2 Remark 2.14. In this lemma, the Whittaker functional is not normalized. But this turns out to be enough. 2.6. Kirillov model for supercuspidal representations. Now let’s consider supercuspidal representations. For the ﬁxed additive character , the Kirillov model of ⇡ is a unique realization on the space of Schwartz functions S (F⇤) such that

a1 m 1 1 (2.40) ⇡( )'(x) = w (a ) (ma x)'(a a x), 0 a ⇡ 2 2 1 2 2! where w⇡ is the central character for ⇡. Let W' be the Whittaker function associated to '. Then they are related by ↵ 0 '(↵) = W ( ), ' 01 ! W'(g) = ⇡(g)'(1). When ⇡ is unitary, one can deﬁne the G invariant unitary pairing on Kirillov model by

(2.41) < f1, f2 >= f1(x) f2(x)d⇤ x.

FZ⇤ 12 01 By Bruhat decomposition, one just has to know the action of ! = to understand the 10 whole group action. ! Deﬁne n ⌫(u), if x = u$ for u O⇤F; 1⌫,n(x) = 2 80, otherwise. > Roughly speaking, it’s the character ⌫<>supported at v(x) = n. We can then describe the action of > 01 :> ! = on explicitly according to [13]: 10 1⌫,n ! n 1 1 (2.42) ⇡(!)1⌫,n = C⌫w z0 1⌫ w0, n+n 1 . 0 ⌫ Here z0 = w($) and w0 = w⇡ O . n⌫ is an integer decided by the representation ⇡ and the character | ⇤F ⌫ (and independent of n). It’s well-known that n 2 for any ⌫. When we pick ⌫ to be the ⌫  trivial character, the number n is actually the level of this supercuspidal representation. Denote 1 c = n1. The local new form is simply 11,0. 10 The relation !2 = implies 01 ! n⌫ (2.43) n⌫ = n⌫ 1w 1 , C⌫C⌫ 1w 1 = w0( 1)z . 0 0 0 It is essentially proved in [9, Proposition B.3] that

Proposition 2.15. Suppose that c = n1 2 is the level of a supercuspidal representation ⇡ whose central character is unramified or level 1. If p , 2 and ⌫ is a level i character, then we have n = min c, 2i . ⌫ { } When p = 2 or the central character of ⇡ is highly ramified, we have the same statement, except when c 4 is an even integer and i = c/2. In that case, we only claim n c. ⌫ Remark 2.16. In [9, Proposition B.3], we didn’t prove the result for highly ramified central characters. But one can use the same method. If we denote the level of the central character by c(w⇡), the key observation is that c(w ) c/2. ⇡  Definition 2.17. We will say a function f (x) consists of level i components, or simply of level i, if we can write f (x) = a⌫,n1⌫,n, ⌫, Xn where a⌫,n , 0 only if ⌫ is of level i. Equivalently,

f (x)⌫(x)d⇤ x , 0 v(xZ)=n only if ⌫ is a level i character. i Let be an unramiﬁed additive character. By abuse of notation, we will say ($ x) is of level 1 i at v(x) = 0. But it should be understood that when i = 1, ($ x) at v(x) = 0 actually consists of level 1 and level 0 components. As a direct corollary to the above proposition, we have the following result about the Whittaker functional for supercuspidal representations: 13 (c) Corollary 2.18. (1) W (↵) = 11,0. (2) In general for 0 i < c, W(i)(↵) is supported only at v(↵) = min 0, 2i c , consisting of  { } level c i components and also level 0 components when i = c 1. (3) The exception happens when p = 2 or the central character is highly ramiﬁed, and c 4 is an even number and i = c/2. In that case, W(c/2) is supported at v(↵) 0, consisting of level c/2 components. Let’s see how the results above can be applied to the matrix coecient of a supercuspidal representation in general. Let

(g) =<⇡(g)F, F >= ⇡(g)F(x)F(x)d⇤ x,

FZ⇤ where F = 11,k S (F⇤) for an integer k 0, and ⇡ is supercuspidal of level c. One can easily 2 c+k c+k check that F is K1($ ) invariant, so (g) is bi-K1($ ) invariant. But we will only make use c+k of the right K1($ ) invariance now. am 10 By Lemma 2.1, to understand (g), it will be enough to understand ( ) for 01$i 1 0 i c + k. ! !   am 10 Proposition 2.19. (i) For c + k 1 i c + k, ( ) is supported on v(a) = 0   01$i 1 and v(m) k 1. On the support, we have ! ! 1, if v(m) k and i = c + k; am 10 (2.44) ( ) = 1 , if v(m) = k 1 and i = c + k; 01$i 1 8 q 1 ! ! > 1 > q 1 , if v(m) k and i = c + k 1. <> > > am 10 When v(m) = k 1 and i = c:+ k 1, ( c+k 1 ) consists of level 1 and 0 01$ 1 components as a function in a (or m), and ! !

am 10 1 k (2.45) ( c+k 1 )dm = q . 01$ 1 q 1 v(m)=Z k 1 ! ! am 10 (ii) For 0 i < c+k 1,i, c/2+k, ( ) is supported on v(a) = min 0, 2i c 2k ,  01$i 1 { } v(m) = i c 2k. It is of level c + k i! as a function! in a. (iii) When i = c/2 + k, the conclusion in (ii) still holds except when p = 2 or the central am 10 character is highly ramiﬁed, and c 4 is an even number. In that case, ( ) 01$i 1 is supported on v(a) 0,v(m) = i c 2k = c/2 k. It is of level c/2 in a. ! ! Proof. By deﬁnition,

(2.46) (g) = ⇡(g)F(x)d⇤ x.

v(xZ)=k To get a non-zero value for (g), we just need a level 0 component supported at v(x) = k for ⇡(g)F(x). We shall prove (ii) ﬁrst. Let 0 i < c + k 1, i , c/2 + k. According to Proposition  14 2.15, 10 1 $i ⇡( )11,k(x) = ⇡( ! !)11,k(x) $i 1 01 ! ! is supported at v(x) = min k, 2i c k , being a linear combination of all level c + k i characters. { } To see this, assume for simplicity that the central character is trivial. Then

(2.47) ⇡(!)11,k(x) = C111, k c(x). 1 $i The action of will then give a linear combination of all level c+k i characters supported 01 at v(x) = c k, according! to the classical results about Gauss sum. Then Proposition 2.15 will imply the claim above. Now by deﬁnition,

am 10 10 (2.48) ⇡( ) (x) = (mx)⇡( ) (ax), 01$i 1 11,k $i 1 11,k ! ! !

am 10 10 (2.49) ( ) = (mx)⇡( ) (ax)d⇤ x. 01$i 1 $i 1 11,k ! ! v(xZ)=k !

One can see that we need v(a) = min 0, 2i c 2k { } 10 to change the support of ⇡( ) (ax) to v(x) = k. $i 1 11,k When 0 i < c + k 1, we need! (mx) also to be of level c + k i at v(x) = k to get level 0  components from the product. So it’s supported at

v(m) = i c 2k. am 10 It’s clear now that ( ) as a function of a or m is of level c + k i. So (ii) is proved. 01$i 1 (iii) can be proved using! the same! method. When one uses the same method for (i), there will be two di↵erences which are worth noting. 10 The ﬁrst di↵erence is that when i = c + k 1, ⇡( )11,k(x) is a linear combination of level 1 $i 1 and also level 0 components. The second di↵erence is that! (mx) has level 0 component at v(x) = k when v(m) k 1. Now we will prove (2.45) and leave (2.44) to the readers, as the latter is actually much easier to check. 10 So suppose i = c + k 1, v(a) = 0 and v(m) = k 1. Then ⇡( c+k 1 )11,k(ax) and (mx) $ 1 will both be linear combinations of level 1 and level 0 characters. ! 15 am 10 10 ( c+k 1 )dm = (mx)⇡( c+k 1 )11,k(ax)d⇤ xdm 01$ 1 $ 1 v(m)=Z k 1 ! ! v(m)=Z k 1 v(xZ)=k ! 10 = (mx)⇡( c+k 1 )11,k(ax)dmd⇤ x $ 1 v(xZ)=k v(m)=Z k 1 ! k 10 = q ⇡( c+k 1 )11,k(ax)d⇤ x $ 1 v(xZ)=k !

10 1 The last step is to see that the level 0 component of ⇡( c+k 1 )11,k is 11,k. ⇤ $ 1 q 1 ! 3. Upper bound for the global period integral

From now on we take G = D⇤ as decided in Theorem 2.4. Denote X = ZAD⇤(F) D⇤(A). Let D \ ⇡i, i = 1, 2, 3 be three unitary automorphic cuspidal representations of GL2. Let ⇡i be the image of 2 ⇡i under Jacquet-Langlands correspondence. They are naturally embedded in L (X). We will fix ⇡1 and ⇡2 and let ⇡3 have varying finite conductor, but with bounded components at infinity.

Definition 3.1. At a finite place v, let ci denote the levels of ⇡i at v. Let S = v finite c 2 max c , c at v, c , 0 . { | 3 { 1 2} 3 } Let c c (c c ) = $ 3 2 , N = Nm( ) = $ 3 2 . N v N | v| v S v Y2 Y Remark 3.2. Note that we don’t take here to be exactly the conductor of ⇡ . But their di↵er- N 3 ence is controlled by the conductors of ⇡1 and ⇡2 which are fixed. In particular this di↵erence is negligible when we consider the asymptotic behavior. We claim without proof here that for v S , the local epsilon factor ✏ (⇧ , 1/2) = 1, so D is the 2 v v matrix algebra at these places. (We will prove this claim in Corollary 5.2. ) For this reason, the following definition makes sense: Definition 3.3. For defined as above, let N (c c ) $ 3 2 0 v , if v S ; a ([ ]) = 01 2 v N 80 1 >B1, C otherwise. <>@B AC and > a([:> ]) = a ([ ]). N v N Yv a([ ]) can be thought of as an element of D⇤(A). N Take cusp forms f ⇡D, i = 1, 2, 3 . We want to bound the global period integral i 2 i (3.1) I( f ,⇢(a([ ])) f , f ) = f (x) f (xa([ ])) f (x)dx. 1 N 2 3 1 2 N 3 ZX Let’s first specify our choices of local components for fi’s. 16 (i) At the places when all three representations are unramified, we will just choose local components to be spherical; (ii) For places in S , we will always pick new forms for all local components; (iii) For the remaining places, we will pick proper new forms or old forms to guarantee that the local integral I0 for some >0. In particular the local components of f and f can be v 1 2 chosen from a finite set of test vectors. (iv) fi’s are globally normalized with respect to (2.1), and locally normalized with respect to (2.30).

Remark 3.4. Note the level of ⇡3 is controlled by the levels of ⇡1 and ⇡2 for places outside S . (iii) was essentially proven in Lemma 6.3 and Lemma 6.4 of [24]. We briefly describe why it’s true here. If we fix the level of ⇡3, the parametrization of all possible representations with fixed 0 central character is compact. Theorem 2.4 guarantees that the local integral Iv is not zero with a 0 proper choice of test vectors. So Iv is bounded away from zero in an open neighborhood of the parametrization by continuity, and (iii) is true because of compactness. Now we can state our result on the upper bound of global period integral:

Proposition 3.5. Let ⇡i, i = 1, 2, 3 be three unitary automorphic cuspidal representations with ⇡1 and ⇡ fixed. Let f ⇡ , i = 1, 2, 3 be cusp forms with local components specified as above. Then 2 i 2 i (3.2) I( f ,⇢(a([ ])) f , f ) = f (x) f (xa([ ])) f (x)dx N . 1 N 2 3 1 2 N 3 ⌧ ZX (↵ 1/2)(2↵ 1/2) Here can be taken to be any positive number less than 4↵ 3 , where ↵ is a bound towards (↵ 1/2)(2↵ 1/2) 1 Ramanujan Conjecture. In particular when we take ↵ = 7/64, 4↵ 3 > 24 . Proof. We will basically generalize the proof in [22]. First we specify a signed measure on G(A) we are going to use. We will take = a µ , where µ is the measure associated to n th Hecke n n n n operator as defined in Section 2.3. We will choose the sequence of complex numbers an as follows: Let b be a fixed small positive real numberP to be chosen. For every finite place v, let l be the maximal prime ideal there. Let T be the set of prime ideals l such that Nm(l) [Nb, 2Nb] and 2 ⇡i’s are locally unramified. In particular by the choice of local component (i), fi’s are spherical at these places. As f1 and f2 have fixed conductor, and primes involved in the conductor of ⇡3 are asymptotically less than N✏ for any ✏>0 as N , T will essentially contain all such primes !1 with norm in [Nb, 2Nb]. More specifically by the distribution of primes, we have b ✏ b+✏ N T N . ⌧| |⌧ For z C we put sign(z) = z/ z for z , 0 and sign(0) = 1. Put 2 | | 2 sign(ˇn), if n T or n = l , l T; (3.3) an = 2 2 80, otherwise. <> Here ˇ is the eigenvalue of the Hecke operatorµ ˇ acting on f , as the local component of f at this n :> n 3 3 place is spherical. Then by the definition above, one can easily verify the following inequalities, which we will make use of later: b ✏ (3.4) anˇn ✏,F N . | n | X 1/2+✏ 2b+✏ (3.5) Nm(n) an ✏ N . n | |⌧ X 17 nm 2↵ 1/2 (4↵+1)b (3.6) (Nm( )) a a N d2 | n|| m|⌧ n,m d (n,m) X X| The first equality follows from Lemma 2.6. The second and the third inequalities are more direct to check. ↵ in the last inequality is a bound towards Ramanujan conjecture, and we need the fact that one can take ↵<1/4. Now for the measure defined as above, we have f ˇ = f , where 3 ⇤ 3

(3.7) = anˇn. n X Let (x) = f (x) f (xa([ ])) C1(X). Then 1 2 N 2 (3.8) I = (x)( f ˇ )(x)dx = ( )(x) f (x)dx ( 2dx)1/2 3 ⇤ ⇤ 3  | ⇤ | ZX ZX ZX 1/2 = ( (⇢(g) )(⇢(g0) )d(g)d(g0)dx) X g,g G(A) Z Z 02 1/2 = ( f1(xg) f2(xa([ ])g) f1(xg0) f2(xa([ ])g0)d(g)d(g0)dx) X g,g G(A) N N Z Z 02 1/2 = ( f1(xg) f2(xga([ ])) f1(xg0) f2(xg0a([ ]))d(g)d(g0)dx) . X g,g G(A) N N Z Z 02 Note that the support of commmutes with a([ ]) according to our choice of . So the last N equality above is valid. Now we want to change the order of the integral, separate the constant part and use Proposition 2.10 to bound the di↵erence. In particular, let hi(x) = fi(xg) fi(xg0), i = 1, 2, so the center acts trivially on hi(x). So we have

(3.9) h1(x)h2(xa( ))dx ( ) h1(x)(x)dx h2(x)(x)dx | X N 2 N X X | Z X=1 Z Z = < h ,⇢(a( ))h > < h ,⇢(a( )) h > | 1 N 2 P 1 N P 2 | = < h h ,⇢(a( ))(h h ) > | 1 P 1 N 2 P 2 ↵ 1/2+✏ N h 2 h 2 ⌧ || 1||L || 2||L ↵ 1/2+✏ N ⌧ The implicit constant depends on the compact open subgroups that stabilize f1 and f2 at places in 2 S , and is thus bounded. In the last inequality we have used h 2 f , which is finite and || i||L || i||L4 bounded because fi’s are normalized cusp forms chosen from a finite fixed collection for i = 1, 2. Combining (3.8) and (3.9), we have (3.10) 2 ↵ 1/2+✏ 2 1 1 I N + <⇢(g g0) f1, f1 >< ⇢(g g0) f2, f2 > d (g)d (g0), | | ⌧ || || 2 g,g0 | ⌦ ⌦ | | | | | X=1 Z where = a µ is the total variation measure associated to , = (X) is the total | | n | n| n || || | | variation of . P 1 1 Note that if we consider <⇢(g g0) f , f >< ⇢(g g0) f , f > as a function of g or g0, | 1 1 ⌦ 2 2 ⌦ | the center acts on it trivially as the central characters are unitary. So Hecke operators µn act on it 18 nicely. In particular, define (2) = . Then we can rewrite the above result as | |⇤| | 2 ↵ 1/2+✏ 2 (2) (3.11) I N + <⇢(g) f1, f1 >< ⇢(g) f2, f2 > d (g). | | ⌧ || || 2 g | ⌦ ⌦ | X=1 Z According to Lemma 2.5, we have the following formula for spherical local components on which the center acts trivially:

(2) (3.12) = a a µ 2 . | n|| m| nmd n,n d (n,m) X X| According to Lemma 2.8, one can easily prove that for spherical functions:

2↵ 1/2+✏ (3.13) <⇢(g) f1, f1 >< ⇢(g) f2, f2 > dµn(g) ✏ Nm(n) . g G(A) | ⌦ ⌦ | ⌧ Z 2 Moreover, for fixed g Supp(µ ), the inner product <⇢(g) f , f >is nonvanishing only if 2 n 1 1 ⌦ is unramified whenever f1 is unramified and the place does not divide n. The number of such ✏ ✏ quadratic characters is O✏(Nm(n) N ), where the implicit constant is allowed to depend on the base field F. Thus

2↵ 1/2+✏ ✏ (3.14) <⇢(g) f1, f1 >< ⇢(g) f2, f2 > dµn(g) ✏ Nm(n) N . 2 g G(A) | ⌦ ⌦ | ⌧ X=1 Z 2 One can also check that 1/2+✏ (3.15) ✏ Nm(n) an . || || ⌧ n | | X Now combine formulae (3.12), (3.14) and (3.15) into (3.11), we have 1/2+✏ 2 ↵ 1/2+✏ nm 2↵ 1/2+✏ 1/2 (( n Nm(n) an ) N + n,m d (n,m)(Nm( 2 )) an am ) (3.16) I N✏ | | | d | || | . | |⌧ a ˇ P | Pn nPn| According to the inequalities (3.4), (3.5) and (3.6), weP have 4b+↵ 1/2 (4↵+1)b 1/2 (N + N ) (3.17) I N✏ . | |⌧ Nb ↵ 1/2 Now choose b = 4↵ 3 > 0 as we can pick ↵<1/4. Then the above inequality becomes (↵ 1/2)(2↵ 1/2) +✏ (3.18) I N 4↵ 3 . | |⌧ (↵ 1/2)(2↵ 1/2) Again 4↵ 3 < 0. When we pick ↵ = 7/64, (↵ 1/2)(2↵ 1/2) 225 1 (3.19) = < . 4↵ 3 5248 24 ⇤

Remark 3.6. (1) The roles of f1 and f2 are interchangeable. One can also, for example, assume = with and relatively prime, and get a similar inequality N N1N2 N1 N2 (3.20) f (xa([ ])) f (xa([ ])) f (x)dx N . 1 N1 2 N2 3 ⌧ ZX 19 (2) It’s also possible to refine the proof and allow the conductors N1, N2 of f1 f2 to change at the same time, under somewhat restrictive condition that essentially Na, Na N for a large 1 2 | enough integer a (a = 9 should be enough). Such condition is used to control the implied constant in, for example, (3.9). One would also need this condition to control the L4 norm 2 using the normalized L norm and a proper bound of L1 norm in level aspect. Then one can also get a hybrid type subconvexity bound of triple product L-function in level aspect under the same condition. 4. Local integral for the triple product L-function In this section, we shall compute the local integral for the triple product L function at finite places. As we will work purely locally, let’s suppress subscript v in this section. Let ⇡ , i = 1, 2, 3 be three local irreducible unitary representations of GL . Let f ⇡ be the i 2 i 2 i normalized new forms for places v S according to our choice in the last section. Let 2 (g) =<⇡(g) f , f > for i = 1, 3 (g) =<⇡(ga ([ ])) f ,⇡ (a ([ ])) f >, i i i i 2 2 v N 2 2 v N 2 where a ([ ]) is as in Definition 3.3. v N We will compute in this section the following integral

(4.1) I ( f ,⇡ (a ([ ])) f , f ) = (g) (g) (g)dg. v 1 2 v N 2 3 1 2 3 F GLZ (F) ⇤\ 2 We will assume that c 2 max c , c , 1 . The di↵erence between this assumption and the condi- 3 { 1 2 } tion for the set of places S is the case c1 = c2 = 0 and c3 = 1. But this case was already considered in [24]. In general the exact value of the matrix coecient is very dicult to write out explicitly, and so is the local integral (4.1). But with the assumption c 2 max c , c , 1 , the computations 3 { 1 2 } turn out to be very nice and simple. We will consider all possible local irreducible unitary representations which can be classiﬁed into the following three types:

Type 1. ⇡ is either supercuspidal, or of form ⇡(µ1,µ2) where µi is ramified of level ki > 0 for i = 1, 2; Type 2. ⇡ is unramified or special unramified; Type 3. ⇡ is of form ⇡(µ1,µ2) where µ1 is unramified and µ2 is ramified of level k.

Note we don’t have to consider the case when µ1 is ramified and µ2 is unramified, as ⇡(µ1,µ2) 1/2 1/2 ⇡(µ ,µ ). Also when ⇡ is of form ( , ) where is ramified, we can pick the new form 2 1 |·| |·| similarly as in the second case of Type 1. So this case won’t be considered as a di↵erent case.

Theorem 4.1. Let ⇡1, ⇡2, ⇡3 be three local irreducible unitary representations of GL2, with levels satisfying c 2 max c , c , 1 . Let f ⇡ be normalized local new forms. Then the local integral 3 { 1 2 } i 2 i (1 A)(1 B) ,⇡ , = = , (4.2) Iv( f1 2(av([ ])) f2 f3) 1(g) 2(g) 3(g)dg c 1 N (q + 1)q 3 F GLZ (F) ⇤\ 2 where 1 1 $ 10 A =1( ) and B =2( c 1 ). 01 $ 3 1 ! ! More specifically we have the following tables of values of A and B for all three types of irreducible unitary representations ⇡1 Type 1 unramified of form ⇡(1,2) special unramified Type 3 1 1 1 2 1 1 A q 1 q+1 ( + + 1 q ) q 0 2 1 20 ⇡2 Type 1 unramified of form ⇡(⌘1,⌘2) special unramified Type 3 1 1 ⌘1 ⌘2 1 1 B q 1 q+1 ( ⌘ + ⌘ + 1 q ) q 0 2 1

c3 4.1. General strategy. As all the matrix coecients will be right K1($ ) invariant, it’s natural to break the integral on F⇤ GL (F) into integrals on the sets of the form \ 2 am 10 K ($c3 ) 01$i 1 1 ! ! am 10 for 0 i c . So one would like to know the values of on matrices of the form .   3 i 01$i 1 As we have assumed that c 2 max c , c , 1 , ⇡ will always be of Type 1. In particular! ⇡ is! 3 { 1 2 } 3 3 either supercuspidal, or of form ⇡(µ1,µ2) where µ1 and µ2 are ramiﬁed of same level c3/2. This is because we have assumed that the product of central characters is always trivial.

Lemma 4.2. Let ⇡3 be a supercuspidal representation or ⇡(µ1,µ2) where µ1 µ2 are both of level c3/2. am 10 (1) As a function in a, ( ) contains level 0 components only when i = c or 3 01$i 1 3 ! ! i = c3 1,v(a) = 0 and v(m) 1. Then we have the following special values and integral formula:

1, if v(m) 0 and i = c3; am 10 (4.3) ( ) = 1 , if v(m) = 1 and i = c ; 3 01$i 1 8 q 1 3 ! ! > 1 > q 1 , if v(m) 0 and i = c3 1. <> > When v(a) = 0,v(m) = 1 and i =>c3 1, :> am 10 1 (4.4) 3( c 1 )dm = . 01$ 3 1 q 1 v(mZ)= 1 ! ! am 10 (2) When i c /2, ( ) is supported at v(m) c /2 and consists of level 3 3 01$i 1 3  ! ! c3/2 components in a. am 10 (3) When 0 i < c /2, ( ) is supported at v(a) = 2i c and v(m) = i c .  3 3 01$i 1 3 3 As a function in a, it consists of! level c! i components. 3 Proof. When ⇡3 is supercuspidal, the above results follow directly from (actually is weaker than) Proposition 2.19. When ⇡3 is of the form ⇡(µ1,µ2) where µi are both of level c3/2, the claims basically follow from Lemma 2.12. By deﬁnition

am 10 (i) (4.5) ( ) = (m↵)W (a↵)W(c3)(↵)d⇤↵ 3 01$i 1 ! ! Z (i) = (m↵)W (a↵)d⇤↵.

v(↵Z)=0 21 am 10 To ﬁnd the level 0 component of ( ) in a is equivalent to ﬁnd the level 0 component 3 01$i 1 ! ! in W(i)(a↵) at v(↵) = 0, which only occurs when i = c or i = c 1 and v(a↵) = 0 from Lemma 3 3 2.12. Then we further need v(a) = 0 and v(m) 1 for the integral in ↵ to be nonzero. The special values and the special integral just follow from (ii) of Lemma 2.12. As an example, we will prove (4.4). When v(a) = 0 and i = c 1, 3

am 10 (c3 1) (4.6) 3( c 1 )dm = (m↵)W (a↵)dmd⇤↵ 01$ 3 1 v(mZ)= 1 ! ! v(↵Z)=0 v(mZ)= 1 (c 1) (c 1) 1 = W 3 (a↵)d⇤↵ = W 3 (↵)d⇤↵ = . q 1 v(↵Z)=0 v(↵Z)=0

Now to prove (2), suppose v(m) < c /2 in (4.5). Then (m↵) is of level c /2 + 1 in ↵. But 3 3 one can check explicitly from (ii) and (iii) of Lemma 2.12 that W(i) is of level c /2 in ↵. For  3 example, when i = c3/2,

c ( 3 ) W 2 (a↵)(4.7)

c3 c3 2 2 1 $ $ 1/2 v(a↵) = C µ1( c3 )µ2( a↵u) c3 ( a↵u)q du. + $ 2 | ↵ + $ 2 | c 1 u a u(1 u ) c Z 3 v(u) 3 ,u<$ 2 ( 1+$O )  2 F

c3 $ 2 As functions in u, µ1( c3 )µ2( a↵u) is of level c3/2 on the domain, (a↵u) is of level 1+u$ 2  v(a↵u). Then we need v(a↵u) c /2 for the integral in u ever to be nonzero. As a result, the 3 level of W(i)(a↵) in ↵ (and also in a) is c /2. Then (4.5) has to be zero as it’s the integral of  3 product of level c /2 + 1 components with level c /2 components. One can also see from this 3  3 argument that the level of in a is c /2. 3  3

(3) follows from (i) of Lemma 2.12: when i < c3/2, (4.8) i c c c (i) 1 $ i 3 i v(a↵)/2 i i 3 i 2i 3 v(a↵) W (a↵) = C µ ( )µ (a↵$ (1 $ 2 u))q (a↵$ (1 $ 2 u))q 2 du. 1 u 2 u ZO 2 ⇤F

$i c3 i 2 i i As functions in u, µ1( u ) is of level c3/2, µ2(a↵$ (1 $ u)) is of level i < c3/2, (a↵$ (1 c 3 i (i)  $ 2 u)) is of level 2i c /2 v(a↵). As v(↵) = 0 in (4.5), W (a↵) is not zero only when 3 v(a) = 2i c3. We will assume this for the remaining discussions. c c i 3 i i 3 i As functions in ↵, µ (a↵$ (1 $ 2 u)) is of level c /2, and (a↵$ (1 $ 2 u)) is now of 2 3 level i v(a↵) = c i > c /2. Then W(i)(a↵) as a function in ↵ is of level c i. Thus for the 3 3 3 integral in (4.5) to be vanishing, we require v(m) = i c3. am 10 In this argument, one can easily see that ( ) as a function in a for i < c /2 is of 3 01$i 1 3 ! ! level c3 i. ⇤ 22 Now we can explain the strategy to prove Theorem 4.1. As we mentioned earlier, we will add up the integrals on the double cosets of the form

am 10 K ($c3 ) 01$i 1 1 ! ! for 0 i c . We will show that the nonzero contributions will only come from i = c and   3 3 i = c 1, where we know special values or integrals for . In particular, we will prove the 3 3 following two claims about 1 and 2 for various types of representations: am 10 Claim 1. (1) When i c3/2, 1( ) is of level 0 in a for ﬁxed valuations. In 01$i 1 particular we have the following special! values:!

am 10 1, if v(a) = 0 and v(m) 0 ( ) = 1 01$i 1 A, if v(a) = 0 and v(m) = 1. ! ! 8 <> > am 10 (2) When i < c /2, v(a) = 2i c and v:(m) = i c , ( ) as a function in a is 3 3 3 1 01$i 1 of level c < c i. ! !  1 3 am 10 Remark 4.3. This claim should be clear by intuition. When i c /2 c , ( ) = 3 1 1 01$i 1 ! ! am ( ). Its special value at v(a) = 0 and v(m) 0 is just a matter of normalization. 1 01 ! am 10 Claim 2. (1) For i c3/2, and v(m) c3/2, 2( ) is of level 0 as a function in 01$i 1 a and independent of m. When i = c , v(a) = 0 and v!( m) !c /2, 3 3 am (4.9) ( ) = 1. 2 01 ! When i = c 1, v(a) = 0 and v(m) c /2, 3 3 am 10 (4.10) 2( c 1 ) = B. 01$ 3 1 ! ! am 10 (2) For i < c /2, v(a) = 2i c and v(m) = i c , ( ) is of level c < c i 3 3 3 2 01$i 1  2 3 as a function in a. ! !

Remark 4.4. Again the special value in the case i = c , v(a) = 0 and v(m) c /2 is just a matter 3 3 of normalization.

Now suppose that these two claims are always true for all three types of representations. When i < c 1, the level in a of and is strictly less than that of . So indeed the only nonzero 3 1 2 3 contribution to the ﬁnal integral will come from i = c and i = c 1. One can then have the 3 3 following table of values: 23 v(a) is always 0 1 2 3 i = c , v(m) 0 1 1 1 3 i = c , v(m) = 1 A 1 1 3 q 1 i = c 1, v(m) 1 B 1 3 q 1 0 i = c 1, v(m) = A B satisfying (4.4) 3 1 Then by Lemma 2.2, one can easily compute that

(4.11) 1(g)2(g)3(g)dg F GLZ (F) ⇤\ 2 1 1 q 1 1 1 = [1 + (q 1)A( )] + [B( ) + AB ] (q + 1)qc3 1 q 1 (q + 1)qc3 1 q 1 q 1 (1 A)(1 B) = . c 1 (q + 1)q 3 So the theorem will be proved if we can verify Claim 1 and Claim 2, and also ﬁgure out A and B for various types of representations. We will do this in the remaining of this section. First let’s give the formulae for 1 and 2 more explicitly in terms of Whittaker functionals. Let Wi be the corresponding Whittaker functionals for the normalized new forms fi, and ↵ 0 10 W( j)(↵) = W ( ) i i 01$ j 1 ! ! as in Section 2.5. is right K ($c1 ) invariant, and thus automatically K ($c3 ) invariant. When i c , 1 1 1 1

am 10 (c1) (c1) (4.12) ( ) = (m↵)W (a↵)W (↵)d⇤↵. 1 01$i 1 1 1 ! ! Z↵

When i < c1,

am 10 (i) (c1) (4.13) ( ) = (m↵)W (a↵)W (↵)d⇤↵. 1 01$i 1 1 1 ! ! Z↵

Now for 2, by deﬁnition , (4.14) c +c c +c c c c +c $ 3 2 0 $ 3 2 0 $ 3 2 0 $ 3 2 0 (g) =<⇡(g ) f ,⇡ ( ) f >=<⇡( g ) f , f >. 2 2 012 2 012 2 01 012 2 ! ! ! ! It is again right K ($c3 ) invariant as 1 c c c +c $ 3 2 0 $ 3 2 0 K ($c3 ) K ($c2 ), 011 01⇢ 1 ! ! and f is K ($c2 ) invariant. 2 1 Now if i c c , 3 2 c c c +c c c $ 3 2 0 am 10$ 3 2 0 am$ 3 2 10 i = i c +c , 0101$ 1 0101$ 3 2 1 ! ! ! 24 ! ! ! and

am 10 c c (i c3+c2) (c2) (4.15) ( ) = (m$ 3 2 ↵)W (a↵)W (↵)d⇤↵. 2 01$i 1 2 2 ! ! Z If i < c c , 3 2 c c c +c $ 3 2 0 am 10$ 3 2 0 0101$i 1 01 ! ! ! ! 2i+2c3 2c2 i+c3 c2 2i+2c3 2c2 c3 c2 i+c3 c2 i c +c a$ a($ $ ) + m$ 101 1 + $ = $ 3 2 , 011101 ! ! ! and (4.16) am 10 ( ) 2 01$i 1 ! ! i c +c i+c c 2i+2c 2c c c (0) 2i+2c 2c (c2) =w ($ 3 2 ) ((a($ 3 2 $ 3 2 ) + m$ 3 2 )↵)W (a$ 3 2 ↵)W (↵)d⇤↵. ⇡2 2 2 Z We can actually reduce many parts of Claim 1 and Claim 2 to the following simple lemma:

Lemma 4.5. Let ⇡ be a unitary local representation of GL2. Let W be a Whittaker functional ↵ 0 associated to a new form in ⇡. Then W( ) is of level 0 in ↵ and supported on v(↵) 0. 01 ! Proof. The claim follows directly from Lemma 2.12, Lemma 2.13 and Corollary 2.18 for Type 1 and Type 3. It’s also well know for unramified representations. For special unramified representa- 1/2 1/2 tions ⇡ (µ ,µ ) where µ is unramified, one can see, for example, [9]. There I proved |·| |·| that the Whittaker functional associated to a new form satisfies the following formula:

v(↵) ↵ 0 µ(↵)q , if v(↵) 0; W( ) = 01 80, otherwise. ! > <> ⇤ :> Now we shall prove Claim 1 and Claim 2 except part (2) of Claim 1 and explicit computations for A and B. We start with part (1) of Claim 1. Let i c , then 1

am 10 (c1) (c1) (4.17) ( ) = (m↵)W (a↵)W (↵)d⇤↵. 1 01$i 1 1 1 ! ! Z (c1) (c1) Both of W1 (a↵) and W1 (↵) are of level 0 in ↵ because of the lemma above. Then the result should be of level 0 in a. One can make a change of variable to see that A is well-deﬁned. c c Now we consider part (1) of Claim 2. If v(m) c /2, then v(m$ 3 2 ↵) 0 for v(↵) 0. So 3 when i c c , 3 2

am 10 c c (i c3+c2) (c2) (4.18) ( ) = (m$ 3 2 ↵)W (a↵)W (↵)d⇤↵ 2 01$i 1 2 2 ! ! v(↵Z) 0 (i c +c ) (c ) 3 2 2 = W2 (a↵)W2 (↵)d⇤↵. v(↵Z) 0 25 (i c +c ) 3 2 This is to ﬁnd level 0 components of W2 , and should be of level 0 in a. It’s also clearly independent of m. That’s why B is well-deﬁned. Similarly when c /2 i < c c , 3  3 2 (4.19) am 10 ( ) 2 01$i 1 ! ! i c +c i+c c 2i+2c 2c c c (0) 2i+2c 2c (c2) =w ($ 3 2 ) ((a($ 3 2 $ 3 2 ) + m$ 3 2 )↵)W (a$ 3 2 ↵)W (↵)d⇤↵ ⇡2 2 2 v(↵Z) 0

i c +c i+c c 2i+2c 2c (0) 2i+2c 2c (c2) =w ($ 3 2 ) (a($ 3 2 $ 3 2 )↵)W (a$ 3 2 ↵)W (↵)d⇤↵. ⇡2 2 2 v(↵Z) 0 i+c c 2i+2c 2c (0) 2i+2c 2c Again this integral is to ﬁnd level 0 components of (a($ 3 2 $ 3 2 )↵)W (a$ 3 2 ↵) 2 and should be of level 0 in a and independent of m. One can also prove (2) of Claim 2 without referring to any speciﬁc type of representation. When i < c /2, v(a) = 2i c and v(m) = i c , we know 3 3 3 am 10 i c +c c c i+c c 2i+2c 2c (4.20) ( ) = w ($ 3 2 ) (m$ 3 2 ↵) (($ 3 2 $ 3 2 )a↵) 2 01$i 1 ⇡2 ! ! v(↵Z) 0 (0) 2i+2c3 2c2 (c2) W2 ($ a↵)W (↵)d⇤↵.

(c ) c3 c2 By the previous lemma, W 2 (↵) is of level 0 in ↵, and v(↵) 0 in the integral. So v(m$ ↵) c3 c2 i c3 + c3 c2 = i c2 c2, which means (m$ ↵) is of level c2 in ↵. Now note that i+c c 2i+2c 2c (0) 2i+2c 2c  if (($ 3 2 $ 3 2 )a↵)W ($ 3 2 a↵) has a component of level j in ↵, then this 2 component is also of level j in a. As only those components of level c2 will be detected by the am 10  integral, ( ) can only be of level c in a. Another way to argue this is just to do 2 01$i 1  2 a change of variable ! for the! integral. As a result, we will only need to verify part (2) of Claim 1 and compute A and B explicitly for various types of unitary representations.

4.2. Type 1 occuring. First let’s consider supercuspidal representations. When ⇡1 is supercuspidal, part (2) of Claim 1 follows directly from Proposition 2.19 for k = 0 there and one can also 1 easily see that A = q 1 . c +c $ 3 2 0 When ⇡2 is supercuspidal, take k = c3 c2 in Proposition 2.19. Note ⇡2( )11,0 = 01 ! 11,c c , and thus 3 2 2(g) =<⇡2(g)11,c c , 11,c c >. 3 2 3 2 1 So again we can apply Proposition 2.19 and get B = q 1 . Now suppose ⇡1 is of form ⇡(1,2), where 1 and 2 are both ramiﬁed. For any m with v(m) = 1, 1 m 1 (4.21) ( ) = (m↵)d⇤↵ = . 1 01 q 1 ! v(↵Z)=0 26 This is the value of A. When i < c1, the proof of (2) of Claim 1 is actually similar to the proof of Lemma 4.2. We will leave this to the readers. Now let ⇡2 be of form ⇡(1,2), where 1 and 2 are both ramiﬁed. By formula (4.18), we have the following for v(a) = 0, i = c 1 and v(m) c /2: 3 3

am 10 (c2 1) 1 (4.22) ( ) = W (↵)d⇤↵ = . 2 01$i 1 2 q 1 ! ! v(↵Z)=0 In the last equality we have used Lemma 2.12. This gives the value B. 4.3. Type 2 occuring. In this subsection we will consider unramified and special unramified representations. We first recall the existing work of matrix coecients for these representations. For unramified representations, just recall Lemma 2.7. $n 0 01 For special unramified representations, let = , and ! = . n 01 10 ! ! 1/2 1/2 Lemma 4.6. Let ⇡ = ( , ) be a special unramified unitary representation of GL2. |·| |·| It has a normalized K ($) invariant new form. The associated matrix coecient for this new 1 form is bi-K ($) invariant and can be given in the following table for double K ($) cosets: 1 1 g 1 ! n !n n! !n! 1 n n n 1 n n 1 n n n (g) 1 q q q q q In this table n 1. This result is due to [24]. Now we consider unramified representations. Part (2) of Claim 1 is actually automatic in this case as is K invariant. Let’s figure out A and B values using Lemma 2.7. Let be the matrix 1 coecient as defined in Lemma 2.7. For v(m) = 1, 1 m 10m 0 1/m 1 = . 01 1/m 1 01/m 10 ! ! ! ! So we have

1 m m 0 1 ( ) =( ) = w ( )(4.23) 1 01 1 01/m ⇡1 2 ! ! 1 2 1 2 1 1 q (1 2q ) (2 1q ) = ( 1 2 ) 1 + q 1 1 2 1 2 1 1 2 1 = ( + + 1 q ). q + 1 2 1 This is the value A. On the other hand,

c c 1 $ 3 0 10$ 3 0 10 1 $ c 1 = 1 = ! !. 01$ 3 1 01 $ 1 01 ! ! ! ! ! 1 ⌘1 ⌘2 1 So we can similarly show that B = ( + + 1 q ) if ⇡ ⇡(⌘ ,⌘ ). q+1 ⌘2 ⌘1 2 1 2 1/2 1/2 Now let ⇡ be a special unramiﬁed representation of form ( , ), and let be the 1 |·| |·| matrix coecient as given in Lemma 4.6. When v(m) = 1, 1 m 10 1/m 0 10 = ! . 01 1/m 1 0 m 1/m 1 ! !27 ! ! So 1/m 0 1 1 2 1 1 A =(! ) = w (! ) = ( q ) = q . 0 m ⇡1 2 2 ! To check (2) of Claim 1, we just need to show that when i = 0 and v(m) = v(a) = c3, am10 ( ) is at most level 1. If v(a + m) > v(m), we have 1 0111 ! ! am10 a/m 0 1 m/a 10 = ! + , 0111 0 m 01 a m 1 ! ! ! ! m ! so

am10 a/m 0 c (4.24) ( ) =(! ) = w 3 (! ). 1 0111 0 m ⇡1 c3 ! ! ! If v(a + m) = v(m), we have am10 10 10 a+m 1 = ! ! m , 0111 1 1 0 m 0 a ! ! a+m ! ! a+m ! so

am10 10 c (4.25) ( ) =(! !) = w 3 (! !). 1 0111 0 m ⇡1 c3 ! ! ! am10 Put together, one can conclude that ( ) is at most level 1 in a when v(a) = v(m) = 1 0111 c . ! ! 3 Now we compute the value B for special unramiﬁed representations. Since

c 1 c +1 $ 3 0 10$ 3 0 10 (4.26) c 1 = 01$ 3 1 0111 ! ! ! ! 11 0111 = , 01 1001 ! ! ! we have

c3 1 c3+1 $ 0 10$ 0 1 (4.27) B =2( c 1 ) =(!) = q . 01$ 3 1 01 ! ! !

4.4. Type 3 occuring. In this subsection, we consider the representations of form ⇡(1,2), where 1 is unramiﬁed and 2 is of level k. The results will basically follow from Lemma 2.13. Let’s ﬁrst check part (2) of Claim 1. By (4.13), when i < k,

am 10 (i) (4.28) ( ) = (m↵)W (a↵)W(k)(↵)d⇤↵, 1 01$i 1 ! ! v(↵Z) 0 Here 1 v(↵)+k 2 v(↵) k q 1 ($) 2( m) ( m)dm, if v(↵) 0, k (4.29) W( )(↵) = v(m)= k 8 R >0, otherwise, <> > 28 :> and

(i) i i k i i k i 1 v(a↵) k+i (4.30) W (a↵) = ($) (a↵$ (1 $ u)) (a↵$ (1 $ u))q 2 du. 1 2 u ZOF 2 They are not normalized, but it turns out that this is enough. i k i i k i For fixed v(u) 0, (a↵$ (1 $ u)) is of level i v(u) in u, (a↵$ (1 $ u)) is additive 2  of level 2i k v(a↵) v(u) in u. For (4.30) to be nonzero, we need 2i k v(a↵) v(u) i v(u),  that is v(a↵) i k. This is because if 2i k v(a↵) v(u) > i v(u), then the integral will be automatically zero for fixed v(u) < i, and (4.31) i k i i k i i i k i (a↵$ (1 $ u)) (a↵$ (1 $ u))du = (a↵$ ) (a↵$ (1 $ u))du = 0, 2 2 v(uZ) i v(uZ) i k 2i as v(a↵$ ) < i. i k i i k i Then as functions in a, (a↵$ (1 $ u)) is of level k in a, and (a↵$ (1 $ u)) is of 2 level i v(a↵) k in a. In particular, W(i)(a↵) if of level k in a. So (2) of Claim 1 is verified for   this case. Now for v(m) = 1,

1 m (k) (4.32) ( ) = (m↵)W (↵)W(k)(↵)d⇤↵ 1 01 ! v(↵Z) 0 2k 2 v(↵) = q ( m) ( m)dm (m↵)q d⇤↵. | 2 | v(mZ)= k v(↵Z) 0

Here we have used the fact that for a representation of Type 3 to be unitary, both 1 and 2 have to v(↵) be unitary. Up to a nonzero constant, (m↵)q d⇤↵ is just v(↵) 0 R (m↵)d↵,

v(↵Z) 0 which is zero when v(m) = 1. So A = 0. Now let ⇡2 be of Type 3. By (4.15),

10 (k 1) (k) (4.33) 2( k 1 ) = W (↵)W (↵)d⇤↵. $ 1 ! v(↵Z) 0 Here

(k 1) k 1 k+1 k+1 1 v(↵) 1 (4.34) W (↵) = ($) (↵$ (1 $u)) (↵$ (1 $u))q 2 du. 1 2 u ZOF 2 k+1 k+1 As functions in ↵, 2(↵$ (1 $u)) is of level k, but (↵$ (1 $u)) is of level k 1 v(↵) (k 1)  k 1. Then the integral in (4.33) has to be zero as W (↵) doesn’t have any level 0 components. So B = 0. 29 5. Conclusion Using Theorem 4.1, we get the following lower bound for local integrals: Proposition 5.1. 0 1 ✏ (5.1) I ( f ,⇡ (a ([ ])) f , f ) N v 1,v 2,v v N 2,v 3,v v S Y2 Proof. The case when ⇡1,v and ⇡2,v are unramiﬁed and ⇡3,v is special unramiﬁed representation was considered in [24]. The remaining situations are covered by Theorem 4.1, as (1 A)(1 B) (1 A)(1 B) 1 = = . (5.2) Iv c 1 1 c (q + 1)q 3 (1 + q ) q 3 1 1 The factor qc3 here will contribute to the main part N . The other factors, including the nor- ✏ malizing L-factors, are uniformly bounded and can be absorbed into N when taking the product. This is to say if we have, for example, local inequalities

0 1 Iv C , qc3 then we are safe to take a product and claim that

0 1 ✏ I N . v v S Y2 1 This is because C will be finally strictly greater than qc3✏ , when either q + or c3 + . 1 !1 1 !2 1 So in particular we don’t have to worry about factors like 1 and = 1 q . We check 1+q ⇣v(2) here that A and B should be bounded away from 1. This is clear from Theorem 4.1 for Types 1 and 3 and also special unramified representations. For unramified representations, we have

1 1 2 1 1 2 q + q q + q ( + ) 1 A = | 2 1 | | 2 | | 1 | . | | q + 1 q + 1 This is clearly bounded below if the representation is tempered. When it’s not tempered, we need to use a bound ↵ towards Ramanujan Conjecture. So 1 2↵ 2↵ q + q (q + q ) 1 A | | q + 1 for ↵<1/4, and is clearly bounded below. ⇤ Corollary 5.2. For v S where S is as in Definition 3.1, we always have 2 ✏v(⇧v, 1/2) = 1. Proof. The claim just follows from Prasad’s work and that the local integrals in Theorem 4.1 are nonzero. ⇤ Now we can easily prove the main theorem: Proof of Theorem 1.4. By Ichino’s formula, ⇣2(2)L(⇧, 1/2) (5.3) I( f ,⇢(a[ ]) f , f ) 2 = F I0( f ,⇡ (a ([ ])) f , f ). | 1 N 2 3 | ⇧, , v 1,v 2,v v N 2,v 3,v 8L( Ad 1) v 30 Y According to [12], (5.4) L(⇧, Ad, 1) C(⇧)✏, ⌧ where the implicit constant depends continuously on the Langlands parameter of the infinity component. In particular when ⇡1 ⇡2 are fixed and ⇡3 has bounded infinity component, (5.5) L(⇧, Ad, 1) N✏. ⌧ By Proposition 3.5 and Remark 3.2,

(↵ 1/2)(2↵ 1/2) 2 2 +✏ (5.6) I( f ,⇢(a[ ]) f , f ) N 4↵ 3 . | 1 N 2 3 | ⌧ By Proposition 5.1 and Remark 3.4,

0 1 ✏ (5.7) I ( f ,⇡ (a ([ ])) f , f ) N . v 1,v 2,v v N 2,v 3,v Yv 7 Then one can prove the theorem by combining (5.3), (5.5), (5.6) and (5.7) and take ↵ = 64 . ⇤

Appendix A. Bound of global matrix coefficient In the appendix we will prove Proposition 2.10, which we record here again. Let D be a global 2 quaternion algebra. Let ⇢ denote the right regular representation of D⇤(A) on L (ZAD⇤(F) D⇤(A)). 2 \ Let F , F L (ZAD⇤(F) D⇤(A)) be two rapidly decreasing and K ﬁnite automorphic forms which 1 2 2 \ don’t have 1-dim components in their spectrum decompositions. Implicitly the center acts on Fi trivially. Let S be a ﬁnite set of non-archimedean places. We assume that D is locally the matrix algebra at the places in S . Let KS = v S Kv and 2

Ki,QS = Ki,v Ki = Ki,v, v S v ﬁnite Y2 Y where K stabilizes the local component of F at v. Let = $ev for e 0, and N = Nm( ). i,v i N v v v N Deﬁne the matrix e $ v 0 Q a([ ]) = , N 01 Yv ! which can be naturally thought of as an element of D⇤(A). Proposition A.1. With the setting as above, we have (A.1) 1/2 1/2 ↵ 1/2+✏ F (g)⇢(a([ ]))F (g)dg F [K : K ] [K : K ] N F 2 F 2 . | 1 N 2 |⌧✏, S 1,S S 2,S || 1||L || 2||L ZAD (FZ) D (A) ⇤ \ ⇤ The case when D is a division algebra is easier to prove. This is basically because there is 2 no continuous spectrum for L (ZAD⇤(F) D⇤(A)). First of all, for an irreducible unitary cuspidal \ representation, the global unitary pairing is equal to the product of local pairings. In the case when the automorphic forms are from a single cuspidal representation, one just need to take a product of local bounds in Lemma 2.8 according to Remark 2.9. In general we consider the spectrum decomposition for Fi:

Fi = < Fi, f > f, ⇡ f (⇡) 2B X X31 where (⇡) is an orthonormal basis under the unitary pairing for a cuspidal automorphic represen- B tation ⇡. If Fi is invariant under Ki, then its cuspidal component in ⇡:

< Fi, f > f f (⇡) X2B is also invariant under Ki. This is true because of Plancherel Theorem. As a result, one can apply the argument for the previous case for each such component, and then use Cauchy-Schwarz inequality. When D is the matrix algebra, one can argue similarly if F1, F2 have only cuspidal spectrums. But in general, they can have continuous spectrums. By intuition the continuous spectrums shouldn’t mess things up because they are related to Eisenstein series deﬁned by unitary characters, and they look like tempered representations locally. To argue more strictly, let’s ﬁrst recall some results.

2 A.1. The spectrums of L (ZAGL2(F) GL2(A)) and the Plancherel formula. For a more de- \ tailed reference of this subsection, see Section 2.2 in [16]. Let denote the set of pairs (M,), where M is a F Levi subgroup of a F parabolic subgroup X (containing a maximally F split torus T) of G, and is an irreducible subrepresentation of the space of functions on M(F) M(A) such that any f is square-integrable in the sense that \ 2 (A.2) f 2 =< f, f > = f 2 < . || || | | 1 Z M(FZ) M(A) M \ For G = GL2 there are two cases. When M is the whole group GL2, then is just a cuspidal representation. When M is the torus T, then is actually a unitary character on the torus. We can equip with a measure in the following way: we write X (A.3) = , X XM GM indexed by levis containing T. We require that for any continuous assignment of to f in 2XM the underlying space of ,

(A.4) f d 2 = f 2 d. | | || || Z M(FZ) M(A) Z Z M \ This uniquely speciﬁes a measure d on , and so also on . XM X (M,) is said to be equivalent to (M0,0) if there exists ! in the normalizer of T with Ad(!)M = M0 and Ad(!) = 0. There is a natural quotient measure on / . X ⇠ G(A) For = (M,) , we donte by () the unitary induced representation Ind , where P is 2X I P(A) any parabolic subgroup containing M. One can deﬁne a unitary pairing on () by I

(A.5) < f1, f2 >Eis= < f1(k), f2(k) > dk, ZK where K is equipped with Haar probability measure. When M = T and is just a unitary character of T, this pairing is just

(A.6) < f1, f2 >Eis= f1(k) f2(k)dk, ZK 32 which is directly a product of local integrals. With this pairing, one can talk about orthonormal basis for (). We will denote such an orthonormal basis by (). I B For any element ' (), one can deﬁne the corresponding Eisenstein series by just averaging 2I over P(F) G(F) and analytic continuation. We will denote the corresponding Eisenstein series by \ E,'. When M = G, E,' = '. For rapidly decreasing functions on ZAG(F) G(A), we have the following formulae \

(A.7) F = < F, E,' > E,'d, ' () Z/ 2B 2X ⇠ X

(A.8) < F1, F2 >= < F1, E,' > < F2, E,' >d. ' () Z/ 2B 2X ⇠ X A.2. Proof continued. Now we shall ﬁnish the proof of Proposition 2.10. As we’ve already proved the proposition for the cuspidal part and we can use Cauchy-Schwartz inequality to piece together, we assume from now on that F1 and F2 have only continuous spectrums. By (A.8),

(A.9) < F ,⇢(a([ ]))F > = < F , E > <⇢(a([ ]))F , E >d 1 N 2 1 ,' N 2 ,' ' () Z T 2B 2X X = < F , E > < F ,⇢(a([ ]) 1)E >d. 1 ,' 2 N ,' ' () Z T 2B 2X X Note that 1 ⇢(a([ ]) )E,' = E,⇢(a([ ]) 1)', N N 1 1 where ⇢(a([ ]) )' is still in () and K ﬁnite. In particular, we can decompose ⇢(a([ ]) )' N I N using the orthonormal basis (): B 1 1 (A.10) ⇢(a([ ]) )' = <⇢(a([ ]) )','0 > '0. N N Eis ' () 0X2B Note this is a purely local argument, and the sum on the right hand side is just a ﬁnite sum if we pick the basis properly. Correspondingly, 1 1 (A.11) ⇢(a([ ]) )E = <⇢(a([ ]) )','0 > E . N ,' N Eis ,'0 ' () 0X2B Now the part associated to in (A.9) becomes (A.12) < F , E > < F ,⇢(a([ ]) 1)E > 1 ,' 2 N ,' ' () X2B 1 = < F1, E,' >< ⇢(a([ ]) )','0 >Eis < F2, E,'0> ',' N X0 1 =< < F1, E,' >⇢(a([ ]) )', < F2, E,'0 >'0 >Eis ' N ' X X0

=< < F1, E,' >',⇢(a([ ])) < F2, E,'0 >'0 >Eis . ' N ' X X0 33 For each < Fi, E,' >'in the expression above we have the following lemma: ' () 2B P Lemma A.2. If Fi is Ki invariant, then for any cuspidal datum associated to T, (A.13) < Fi, E,' >' ' () X2B is also K invariant. i Proof. First of all (A.13) is independent of the choices of the basis. In particular one can pick an orthonormal basis for ()Ki ﬁrst, then extend it to an orthonormal basis for (). To prove the I I lemma, it is enough to show for this basis that if ' () ( ()Ki ), then < F , E >= 0. 2B BI i ,' By deﬁnition and the standard unfolding technique,

(A.14) < Fi, E,' > = Fi(g)E,'(g)dg

g ZAGL Z(F) GL (A) 2 2 \ 2

= Fi(g)'(g)dg

g ZA B(FZ) GL (A) 2 \ 2

= Fi(g) '(gk)dkdg.

1 g ZA B(F)ZGL2(A)/Ki k g ZA BZ(F)g K K 2 \ 2 \ i\ i Note that if < Fi, E,' > is ever going to be nonzero, both F and ' have to be invariant under 1 1 ZA K . If a g ZAB(F)g K , then there exists b ZAB(F) gK g such that \ i 2 \ i 2 \ i (A.15) ga = bg. 1 1 It’s easily seen that ZAB(F) gK g = ZA gK g = ZA K , thus \ i \ i \ i (A.16) '(gak) = '(bgk) = '(gk). So '(gk)dk will be a nontrivial multiple of '(gk)dk. They are simultaneously 1 k Ki k g ZA B(F)g Ki Ki zero2R or nonzero. 2 R \ \ Then we’d like to see for every g, whether the following integral is zero or not:

(A.17) '(gk)dk = 'v(gvkv)dkv. v ZKi YKZi,v Now ﬁx g. For every v, consider the double coset decomposition

GL2(Fv) = BaiKi,v, aai where a is the set of double coset representatives. Locally we can write g = b a k0 , where { i} v v g(v) v a a . Note for almost all places, K = K and g K , so b and a will be trivial there. g(v) 2{ i} i,v v v 2 v v g(v) Then (A.17) is zero if and only if

(A.18) 'v(ag(v)kv)dkv v YKZi,v is zero. 34 But for every ﬁxed g this integral is the same (up to a nonzero constant) as the pairing < , > · · Eis in () between ' and another element '0 () whose local component at v is only supported on I 2KI Ba K . This element '0 is clearly in () i , so by the choice of ', g(v) i,v I (A.19) <'0,'>Eis= 0. So (A.17) is zero and < F , E >= 0 for ' () ( ()Ki ). ⇤ i ,' 2B BI Recall the pairing < , > is directly a product of local pairings, and for the local pairing, · · Eis we can use Lemma 2.8 to bound the local matrix coecient. Note the local components of () I are always tempered. By taking a product and using the result that < Fi, E,' >'is Ki ' () 2B invariant, we can get P (A.20) < F , E > < F ,⇢(a([ ]) 1)E > 1 ,' 2 N ,' ' () X2B 1/2 1/2 1/2+✏ ✏,F [KS : K1,S ] [KS : K2,S ] N < Fi, E,' >' Eis < F2, E,' >' Eis ⌧ || ' || || ' || X X 1/2 1/2 ↵ 1/2+✏ 2 1/2 2 1/2 [KS : K1,S ] [KS : K2,S ] N ( < F1, E,' > ) ( < F2, E,' > ) ,  ' | | ' | | X X for any bound ↵ towards Ramanujan Conjecture. Finially when we do the integral in cuspidal datum , just apply Cauchy-Schwartz inequality.

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