GAUTAM CHINTA AND ADRIAN DIACONU

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1 DETERMINATION OF A GL 3 CUSPFORM BY TWISTS OF CENTRAL L-VALUES GAUTAM CHINTA AND ADRIAN DIACONU Abstract. Let π be a self-contragredient cusidal automorhic reresentations of GL 3 A Q ). We show that if the symmetric square L-function of π has a ole at s = 1, then π is determined by central values of quadratic twists of its L-function. That is, if π is another cusidal automorhic reresentations of GL 3 A Q ) for which L 1, π χ) = L 1, π χ) for sufficiently many quadratic characters χ, then π π. 1. Introduction Let π be a self-contragredient cusidal automorhic reresentation of GL 3 A Q ) and χ a Dirichlet character. Then the twisted L-function Ls, π χ) initially defined for Rs) > 1, is known to have an analytic continuation to C and to satisfy a certain functional equation relating the values at s to those at 1 s. The main result of this aer is that knowledge of these values at the central oint s = 1/ for twists by quadratic Dirichlet characters,at least when the symmetric square L-function Ls, π, sym ) has a ole at s = 1, is enough to determine π. Precisely, we rove Theorem 1.1. Let π and π be two self-contragredient cusidal automorhic reresentations of GL 3 A Q ). Suose that Ls, π, sym ) has a ole at s = 1. Fix an integer M and let X be the set of all quadratic Dirichlet characters of conductor relatively rime to M. If there exists a nonzero constant κ such that 1.1) L 1, π χ) = κl 1, π χ) for all χ X, then π π. Remarks 1) The L-functions in 1.1) are the full automorhic L functions, including the archimedean comonent. However, as will be clear from the roof of the theorem in section 5, if S is any finite set of laces of Q ossibly including the infinite lace), the conclusion of the theorem is still true if 1.1) is relaced by L S 1, π χ) = κls 1, π χ) for all χ X. Here L S s, π) = v S Ls, π v ). Later, we will also use the notation L S s, π) = v S Ls, π v ). Date: Setember 6,

2 GAUTAM CHINTA AND ADRIAN DIACONU ) The condition Ls, π, sym ) has a ole at s = 1 is satisfied when π is the Gelbart-Jacquet lift [4] of an automorhic reresentation on GL A Q ) with trivial central character. In fact, the converse is true as well. By the work of Ginzburg-Rallis-Soudry see e.g. [5]), for an irreducible cusidal automorhic reresentation π of GL 3 with the artial symmetric square L-function L S s, π, sym ) having a ole at s = 1, there exists an irreducible cusidal automorhic reresentation σ of S = SL, which lifts functorially to π. We thank Dihua Jiang and David Ginzburg for clarifying this oint for us. This fact will make the sieving argument of section 3 somewhat easier. For holomorhic newforms of congruence subgrous of SL Z), the analogous result was roved by Luo and Ramakrishnan [1]. Their idea is to consider the twisted averages of the form L 1, f, χ d)χ d r) d<x and show that asymtotics for these exressions as X involve the Hecke eigenvalues of f. We rove our result by a similar method. However, the averaging rocess for a GL 3 cusform is more delicate, and we use the method of double Dirichlet series, rather than the method of the aroximate functional equation used in [1]. The double Dirichlet aroach on GL 3 was first carried out by Bum- Friedberg-Hoffstein [] and Diaconu-Goldfeld-Hoffstein [3]. We rely heavily on the results of these aers. As the base field is Q, it is likely that the method of the aroximate functional equation will work here as well. Our reason for using the multile Dirichlet series aroach is that it rovides a considerably simler framework and makes the analysis quite easy. The extension to GL 3 A K ) for K an arbitrary number field, however, remains elusive from the oint of view of both aroaches. One goal of this aer is to illustrate the source of this difficulty in our method. The roblem arises from the need to aly a Lindelöf-on-average bound Lemma 3.) to carry out the sieving rocess of section 3. We establish this bound by aealing to a character sum estimate of Heath-Brown [6], which is valid only over Q. There are two ossible ways to solve this roblem: first, establish Lemma 3. over an arbitrary base field, or second, rove a uniqueness result for the finite Euler roducts P ψ1) 1/) of section, which would obviate the need to sieve altogether. The second method would be referable, but the first is of great interest in its own right. A recent illustration of the utility of the method of multile Dirichlet series in working over number fields is given by J. Li [11] in his thesis, where the original result of Luo and Ramakrishnan is extended to forms on GL A K ), for K an arbitrary number field. Such a result does not aear to be accessible by the method of the aroximate functional equation and large sieve. In section, we review some of the results of Bum-Friedberg-Hoffstein [] and construct the relevant double Dirichlet series. In section 3 we indicate how to sieve the double Dirichlet series of the revious section in order to obtain a series summed over only square-free discriminants d. In the final two sections we rove the main theorem. The authors thank S. Friedberg and J. Hoffstein for their encouragement and comments on an earlier version of this aer. We also thank D. Bum, D. Ginzburg, J. Hundley, D. Jiang and Y. Tian for their advice. Finally, we are grateful to the

3 DETERMINATION OF A GL 3 CUSPFORM BY TWISTS OF CENTRAL L-VALUES 3 referee and editor for several comments imroving the overall exosition of the aer.. Analytic Continuation of a Double Dirichlet Series Let π be a cusidal automorhic reresentation of GL 3 A Q ) of conductor N. Let S be a finite set of laces of Q including and the archimedean lace, such that π is unramified outside of S. We write S as the disjoint union S = S fin { }. Let M = S fin. Let R = Z/4MZ) Z/Z. Characters in the dual grou ˆR will be used to sieve out congruence classes. We note that characters in ˆR are recisely the quadratic characters of Z/4MZ). For the uroses of constructing L-functions below, we identify a character in ˆR with the rimitive quadratic Dirichlet character which induces it. Thus, given a divisor l of 4M, there is exactly one character in ˆR of conductor l if 8 does not divide l, and exactly two characters in ˆR of conductor l if 8 does divide l. We recall that L S and L S were defined in the remark following the statement of Theorem 1.1. Taking out the archimedean comonent from the L-function of π, we have the L-series L s, π) = m 1 c m m s = 1 α ) 1 s 1 β ) 1 s 1 γ ) 1 s, the Euler roduct being taken over all rimes of Q. For χ a Dirichlet character of conductor D relatively rime to M the twisted L-series L s, π χ) = c m χm) m s m 1 = 1 α ) 1 χ) s 1 β ) 1 χ) s 1 γ ) 1 χ) s, has a functional equation given by L s, π χ) = ɛ π τχ) 3 χ π D)χN)D 3 N) 1 s L 1 s, π χ) L 1 s, π χ). L s, π χ) Here χ π is the central character of π, τχ) is the normalized Gauss sum associated to χ and ɛ π = ɛ1/, π) is the central value of the ɛ-factor of π. From now on we will assume that π is self-contragedient with trivial central character and that χ is quadratic. In this case, τχ) is trivial. In [], a double Dirichlet series is constructed out of quadratic twists of the L-function of π. To describe this recisely we need some more notation. Let χ d denote the quadratic Dirichlet character associated to the extension Q d) of Q. For ψ 1, ψ in ˆR, it is shown that there exist finite Euler roducts P ψ1) s) such that the double Dirichlet series.1) Z M s, w, π; ψ, ψ 1 ) = d,m)=1 L S s, π χ d0 ψ 1 ) d w ψ d 0 )P ψ1) s)

4 4 GAUTAM CHINTA AND ADRIAN DIACONU has a meromorhic continuation to C. The sum is taken over ositive integers d = d 0 d 1, with d 0 squarefree. In fact Theorem.1. Let Ps, w) = ww 1)3s + w 5/)3s + w 3/). Then Ps, w)z M s, w, π; ψ, ψ 1 ) has an analytic continuation to an entire function of order 1 on C. This series has a olar line at w = 1 if and only if ψ = 1. In this case, for Rs) > 1/, the residue at w = 1 is comuted in [].) Res Z M s, w, π; 1, 1) = 1 1/) L S s, π, sym )ζ S 6s 1). w=1 M Moreover, the series satisfies certain functional equations as s 1 s and w 1 w, []. These are reviewed in the following section. For our alication we need to consider slightly different sums. For a rime number r relatively rime to M, let χ r denote the quadratic character with conductor r defined by χ r ) = r ). Let K be the set of all ositive integers d such that ψd) = 1 for all ψ ˆR. For d, M) = 1 we have the orthogonality relation 1 { 1 if d K.3) ˆR ψd) = 0 otherwise. ψ ˆR We let δ K be the characteristic function of the subset K, and define Zs, w, π; χ r δ K, ψ 1 ) := d K L s, π χ d0 ψ 1 ) d w χ r d)p ψ1) s). Proosition.. The double Dirichlet series Zs, w, π; χ r δ K, ψ 1 ) has a meromorhic continuation to Rs) > /5 and w C. In this region, the roduct is analytic. Ps, w)zs, w, π; χ r δ K, ψ 1 ) Proof. We exress Zs, w, π; χ r δ K, ψ 1 ) as a linear combination of the functions Z M s, w, π; ψ, ψ 1 ) defined above. Then the stated meromorhic continuation will follow from the known roerties of the Z M s, w, π; ψ, ψ 1 ). Note that for d K and S, we have χ d ) = 1. Hence, for Rs), Rw) sufficiently large, Zs, w, π; χ r δ K, ψ 1 ) = 1 ˆR L S fin s, π ψ 1 ) ψ ˆR d,m)=1 L S s, π χ d0 ψ 1 ) d w χ r d)ψ d)p ψ1) s), where we have used the orthogonality relation.3). Removing the r th -term from the Euler roduct of L S s, π χ d ψ 1 ) and letting S r = S {r}, we write the inner sum over d as d,rm)=1 L Sr s, π χ d0 ψ 1 ) d w k 0 c r kχ d0 ψ 1 )r k ) r ks χ r d)ψ d)p ψ1) s) = ψ 1 r)l 1,r s) Z Mr s, w, π; ψ, ψ 1 ) + L,r s) Z Mr s, w, π; ψ χ r, ψ 1 ),

5 DETERMINATION OF A GL 3 CUSPFORM BY TWISTS OF CENTRAL L-VALUES 5 say, where L 1,r s) = k 0 c r k+1 r k+1)s, L,rs) = k 0 c r k r ks. Thus Zs, w, π; χ r δ K, ψ 1 ) =.4) 1 ˆR L S fin s, π ψ 1 ) ψ ˆR ψ 1 r)l 1,r s) Z Mr s, w, π; ψ, ψ 1 )+ L,r s)z Mr s, w, π; ψ χ r, ψ 1 )). To comlete the roof of the Proosition, it remains to establish the analytic continuation of L i,r s), i = 1, ) to Rs) > /5. This is done in the Lemma below. Lemma.3. The series L i,r s), i = 1, ) converge absolutely for Rs) > /5. Moreover, we have the exlicit reresentations and L 1,r s) = L,r s) = c r + r s r s 1 α rr s )1 β r r s )1 γ r r s ) 1 + c r r s 1 α rr s )1 β r r s )1 γ r r s ). Proof. The absolute convergence of both series follows from the bound of Luo, Rudnick and Sarnak [13]: c n ɛ n 5 +ɛ, for any ɛ > 0. To evaluate the sum L 1,r s), we begin by writing the Fourier coefficient c r k in terms of the Satake arameters α = α r, β = β r, γ = γ r : α k+ β k+ γ k+ α β γ c r k = α β γ. α β γ We exand the determinants and evaluate the sum L i,r s) as a linear combination of geometric series. After some algebraic simlifications using the relations c r = α + β + γ = αβ + αγ + βγ we arrive at the desired result. Remark The roducts in the denominator can be similarly evaluated to give rational exressions of c r :.5) 1 α rr s )1 β r r s )1 γ r r s ) = 1 c r c r r s + c r c r r 4s 1 r 6s.

6 6 GAUTAM CHINTA AND ADRIAN DIACONU 3. Sieving the double Dirichlet series In this section we show that the imerfect double Dirichlet series without weighting olynomials P ψ1) s) has a meromorhic continuation to suitably large domain. Proosition 3.1. The series Z s, w, π; χ r δ K, 1) = d 0 K,d 0 sq.free d 0>0 L s, π χ d0 ) d w χ r d 0 ) 0 has a meromorhic continuation to a tube domain in C containing the oint s, w) = 1/, 1). More recisely, the roduct Ps, w)zs, w, π; χ r δ K, 1) is analytic in the union of the two regions {Rs) > 1 119, Rw) > 14 } and {0 Rs) 1 191, Rw) > 6 Rs) + 5 }. The roosition is roved by a sieving argument similar to that used in [3]. We assume familiarity with the methods of [3] and merely indicate in this section where modifications are needed. What makes the argument more difficult in our situation is the lack of a sufficiently owerful Lindelöf-on-average result for the mean square of twisted central values of the L-function of a GL 3 cusform. The best that one can resently do is Lemma 3.. Let π be a self-adjoint cusidal automorhic reresentation of GL r A Q ). For all ɛ > 0, we have the estimate L 1, π χ d ) ɛ x r +ɛ. d <x This Lemma follows readily from the following character sum estimate of Heath- Brown s [6]. Lemma 3.3. Let N, Q be ositive integers, and let a 1, a,..., a N be arbitrary comlex numbers. Then, for any ɛ > 0, a n χ d n) ɛ QN) ɛ Q + N) a n1 a n. d Q n N n 1,n N n 1 n = It is for this reason that we are forced to assume that the base field is Q. It would be of interest to establish an analogue of the Lemma over an arbitrary base field. With Lemma 3. in hand, what is needed for the sieving is the modified version of Proosition 4.1 of [3] given below. Proosition 3.4. Let w = ν + it. Let ɛ > 0, ɛ 1 4 ν. Let ψ 1, ψ ˆR. We will denote the conductor of ψ i ˆR by l i. The function Z M 1/, w, π; ψ, ψ 1 ) is an analytic function of w, excet for ossible oles at w = 3 4 and w = 1. If l 1, l ) = 1,, 4 or 8 and t > 1, then it satisfies the uer bounds Z M 1, ν + it, π; ψ, ψ 1 ) ɛ M ɛ,

7 DETERMINATION OF A GL 3 CUSPFORM BY TWISTS OF CENTRAL L-VALUES 7 for ɛ < ν, and 3.1) Z M 1, 1 4 ɛ + it, π; ψ, ψ 1 ) ɛ, π, t M 5+vɛ) ψ 3 ˆR d 0 sq. free d 0,M)=1 The function vɛ) is comutable and satisfies vɛ) 0 as ɛ 0. L 1, π χ d 0 ψ 3 ) d 5 4 +ɛ 0 l Proof. The fact that this function is analytic in w, excet for ossible oles at w = 3 4 and w = 1, is roved in [] and [3]. To rove the first estimate, fix x > 1. Then, d<x d,m)=1 L S, 1 π χ d0 ψ 1 ) d ν+it ψ d 0 )P ψ1) 1 ) ɛ M ɛ d 0<x, d 0 sq.free d 0,M)=1 ɛ M ɛ L 1, π χ d0 ψ 1 ) d ν 0 d 0<x, d 0 sq.free d 0,M)=1 d 1 < x d 0 d 1,M)=1 L 1, π χ d0 ψ 1 ) d ν 0 P ψ1) 1 ) d ν 1 d 1=1 d 1,M)=1 P ψ1) 1 ) d ν 1 where the last inner sum is absolutely convergent for ν > 5 4, and it is 1 indeendent of d 0 and M. It follows that 3.) Z M, 1 ν + it, π; ψ, ψ 1 ) ɛ, ν M ɛ ν > 4). 5 d 0 sq.free d 0,M)=1 L 1, π χ d0 ψ 1 ) d ν 0 The absolute convergence of the series in the right hand side, for ν > 5/4, can be easily justified by alying the Cauchy-Schwarz inequality and the estimate in Lemma 3.. To justify 3.1) we define two involutions on C : α : s, w) 1 s, 3s + w 3 ) and β : s, w) s + w 1, 1 w). If Z M s, w, π) denotes the column vector whose entries are Z M s, w, π; ψ, ψ 1 ) with ψ 1, ψ ˆR, then by Proositions 4. and 4.3 in [3], there exist matrices Ψ M s) and Φ M w) such that Z M s, w, π) = Ψ M s) Z M αs, w), π) ZM s, w, π) = Φ M w) Z M βs, w), π). Alying the transformation βαβαβ, one obtains the functional equation: where Z M s, w, π) = Ms, w) Z M s, 5 3s w, π), Ms, w) := Φ M w)ψ M s + w 1 )Φ M 3s + w )Ψ M s + w 1)Φ M 3s + w 3 ). We shall need to estimate the entries of the matrix M 1, 1 4 +it). To do this, we recall the exlicit descrition of, for instance, the β functional equation see 4.18),

8 8 GAUTAM CHINTA AND ADRIAN DIACONU in [3]). We continue to let l i denote the conductor of ψ i ˆR. If the conductor l of ψ is odd, we have 3.3) 1 +w ) Z M s, w, π; ψ, ψ 1 ) M/l ) = 1 l 1 w l 3,l 4 M/l ) µl 3 )ψ l 3 l 4 )l w 3 l 1+w 4 a=±1 L 1 w, χ a ψ ) L w, χ a ψ ) Z M βs, w), π; ψ, ψ 1 ψ 3 ψ 4 ) + az M βs, w), π; ψ, χ 1 ψ 1 ψ 3 ψ 4 )). Here χ 1 1, and χ 1 is the character defined by ) 4 { 1) m 1 + sgnm) 1 if m 1 mod ) χ 1 m) = = m 0 if m 0 mod ). When l is even, we have a similar exression. In fact, just the behavior at the finite lace changes. Using Stirling s formula, we obtain the estimate 3.4) 1 +Rw) ) Z M s, w, π; ψ, ψ 1 ) M/l ) w l 1 Rw) l 3,l 4 M/l ) l Rw) 3 l 1+Rw) 4 Z M βs, w), π; ψ, ψ 1 ψ 3 ψ 4 ). A similar estimate corresonding to the α functional equation can also be established: 3.5) Z M s, w, π; ψ, ψ 1 ) s where we have set M/l 1) l 3 3Rs) 1 l γ M/l 1) l β M/l 1) 1 α +s) 1 β +s) 1 γ +s) l α M/l 1) γ lγ l Rs) γ β l β l 1+Rs) β α lα l Rs) α l α M/l 1) l γ M/l 1) l β M/l 1) α l α l 1+Rs) α β lβ l Rs) β γ l γ l 1+Rs) γ Z M αs, w), π; ψ ψ α ψ β ψ γ ψ α ψ βψ γ, ψ 1 ) 3.6) α lα = l α α, β lβ = l β β, γ lγ = l γ γ, and similarly for α l α, β l β, γ l γ. The functional equations are more cleanly exressed in matrix notation. We will see that the corresonding matrices reresenting the right hand sides of the estimates 3.4) and 3.5) decomose as tensor roducts over the rimes dividing M. To this end, write 4M = M 0 M 1, where M 0 is a ower of and M 1 is odd and square-free. For the roof of Proosition 3.4, we need to bound Z M in terms of M. Because of the tensor roduct structure we will exhibit, we will see that the bounds

9 DETERMINATION OF A GL 3 CUSPFORM BY TWISTS OF CENTRAL L-VALUES 9 we obtain are multilicative in M. For this reason, we may ignore a finite number of rimes e.g. the rime ) and we may assume M 1 is rime. Under these simlifications, if M 1 = is rime, we fix Z s, w, π) = Z s, w, π; 1, 1) Z s, w, π; 1, ψ) Z s, w, π; ψ, 1) Z s, w, π; ψ, ψ) where ψ = χ is the character associated to the quadratic extension Q ) of Q, where = 1) 1. We now define two matrices which will be used to build u the matrices which reresent the right hand sides of the estimates 3.4) and 3.5) for the β and α functional equations, resectively. Let 1+ 1 w + 1+w 1 +w w Φ w + 1+w 1+ 1 w) = 1 +w w w w Ψ s) =, us) 0 vs) s 0 0 vs) 0 us) s, where us) and vs) are given by us) = 1 α +s )1 β +s )1 γ +s ), vs) = 1 α +s )1 β +s )1 γ +s ) with and = [ 1 + α 1 + β 1 + γ 1 ) +s + α + β + γ ) 1 + α + β + γ ) 3+s + α 1 + β 1 + γ 1 ) s + α 1 + β 1 + γ 1 ) + α + β + γ ) 1 s + 3] = [ α + β + γ ) 1+s + 3+3s + α + β + γ ) s + α + β + γ ) α 1 + β 1 + γ 1 ) +s + α + β + γ ) α 1 + β 1 + γ 1 ) 1 s + α 1 + β 1 + γ 1 ) 3+s + α 1 + β 1 + γ 1 ) s + 3s]. We digress a moment to exlain the aarent asymmetry in considering only the quadratic extensions Q ). The characters corresonding to the extensions Q ) will aear after we tensor with the matrices corresonding to the rime. These matrices act on the 16-dimensional vector whose comonents are Z s, w, π; ψ 1, ψ ), with ψ 1 and ψ each being one of the 4 rimitive quadratic characters of conductor a ower of. Fortunately, we will not need to write down these matrices as we are only interested in bounds for Z M in terms of M, as M grows large. Therefore, as remarked above, we can ignore finitely many small rimes.)

10 10 GAUTAM CHINTA AND ADRIAN DIACONU For M 1 odd and square-free, one can comute uer bounds for the matrices Ψ M1 s) and Φ M1 w) as follows. Let be a rime divisor of M 1, and let V be a comlex vector sace sanned by a basis {e l, l 1 ) : l 1, l }. We consider two additional comlex vector saces V α resectively, and let and V β {e α l, l 1 ) : l 1, l } {e β l, l 1 ) : l 1, l }, Ψ : V V α Φ : V V β sanned by be the linear mas corresonding to Ψ s) and Φ w). For instance, Φ is described by e l, l 1 ) 1 q +w ) 1 1 l w l3 w l4 1+w e β l, l 1 l 3 l 4 ). q /l ) q 1 l 3,l 4 /l ) Here, we made the convention that e β a, n b) = e β a, b). Let V M1 := M 1 V ; V α M 1 := M 1 V α ; V β M 1 := M 1 V β. Then, Ψ M 1 = M 1 Ψ and Φ M 1 = M 1 Φ. Now, for M 1 =, we have and where Φ M1 1 4) Φ M 1 1 4) Ψ M1 1 4) Ψ M 1 1 4) Φ M1 1) Φ M 1 1) c c c := α 1 + β 1 + γ 1.,, as, Recall that by the remark ) made after the statement of Theorem 1.1, the cusidal automorhic reresentation π is a Gelbart-Jacquet lift. It follows that c 9 by the bound of Kim and Shahidi in [9]. In fact, the weaker exonent 5/8 obtained by Bum, Duke, Hoffstein and Iwaniec in [1] would suffice for our uroses. It

11 DETERMINATION OF A GL 3 CUSPFORM BY TWISTS OF CENTRAL L-VALUES 11 follows easily that M 1, 1 4) To rove 3.1) consider the first three entries of Z s, w, π). Recall that in the statement of the roosition, we have the condition l 1, l ) = 1,, 4 or 8. For examle, we have that Z 1, 1 4 ɛ + it, π; 1, 1) vɛ) 5 Z 1, ɛ it, π; 1, 1) Z 1, ɛ it, π; 1, ψ) + 5 Z 1, ɛ it, π; ψ, 1) ). The factor vɛ) is a linear function of ɛ coming from reeated alications of the functional equations. For examle, when w = 1/4 ɛ and M 1 = is rime, the estimate 3.4) would roduce a ɛ. For general M, we can use the tensor roduct decomosition of Ψ M and Φ M obtaining Z M 1, 1 4 ɛ + it, π; ψ, ψ 1 ) M vɛ) ψ 3,ψ 4 ˆR l 3, l 4)=1 Now, the estimate 3.1) follows immediately from 3.). d 0 K,d 0 sq.free d 0>0 M 5 l Z M 1, ɛ it, π; ψ 4, ψ 3 ). To conclude the section, we now rove Proosition 3.1. We have Z L s, π χ d0 ) s, w, π; χ r δ K, 1) = d w χ r d 0 ). 0 As in the roof of Proosition., we can exress Z s, w, π; χ r δ K, 1) as a finite linear combination of the series ZMrs, L S s, π χ d0 ) w, π; ψ, 1) = d w ψ d 0 ) d 0,Mr)=1 d 0 sq.free Since r will be fixed for the rest of the section, we relabel Mr as M. We now indicate how to analytically continue ZM s, w, π; ψ, 1) to a region containing the oint 1/, 1). First, we write ZM s, w, π; ψ, 1) = µq)z M s, w, π; ψ, 1; q), q,m )=1 where the sum is over square-free q and Z M s, w, π; ψ, 1; q) := d 0d 1,M )=1 d 1 0 q) L S s, π χ d0 ) d w ψ d 0 )P d0,d 1 s).

12 1 GAUTAM CHINTA AND ADRIAN DIACONU In [] an exlicit descrition of the weighting olynomials P d0,d 1 s) is given. We need an estimate in the q asect for Z M, 1 w, π; ψ, 1; q) in a stri 1 4 ɛ < Rw) < ɛ with a small ɛ > 0. From the comutations of [], we know that P d0,d 1 s) = P ls) d0, l d 1 and, for rime, we have the following bound indeendent of d 0 P d0, ls) ) ls = 1 + Rw) c ) + c O + 1 Rs) 1 l=0 where c ) are the coefficients of Ls, π, sym ). Therefore for Rw) > 5/4 and Rs) 1/, we have Z M s, w, π; ψ, 1; q) q Rw) c ) q + c q ) q 1 Rs) + 1). To obtain an estimate for Rw) < 1/4 and Rs) 1/, we write 3.7) Z M s, w, π; ψ, 1; q) = l q where Z l) M s, w, π; ψ, 1) := d 0,M )=1 d 1,M l)=1 The oint is that one can decomose Z l) M Z M s, w, π; ψ, ψ 1 ). Proosition 3.5. We have 3.8) Z l) M s, w, π; ψ, 1) 1 α 1 l 3 l 1 l w 3 l s ) µl)z l) M s, w, π; ψ, 1), L S s, π χ d0 ) d w ψ d 0 )P d0,d 1 s). as a linear combination of the functions 1 β ) ) ) 1 α s 1 β s 1 γ s l 3 s ) 1 γ s m 1,m,m 3 l/l 3) ) = χ l3 m 1 m m 3 )α m1 β m γ m3 m 1 m m 3 ) s [Z M ls, w, π; ψ χ m1m m 3, χ l3 ) + Z M ls, w, π; ψ χ m1m m 3, χ l3 ) + χ 1 m 1 m m 3 )Z M ls, w, π; ψ χ m1m m 3, χ l3 ) χ 1 m 1 m m 3 )Z M ls, w, π; ψ χ m1m m 3, χ l3 )]. We recall that α m, β m and γ m were defined in 3.6). The roof is similar to that of Proosition 4.14 of [3] and will be omitted. Alying Proosition 3.4, it follows that, for Rw) < 1/4, Z l) M 1, w, π; ψ, 1) M q) 1 4 +ɛ ψ 3 ˆR q d 0 sq. free L 1, π χ d 0 ψ 3 ) d 1 Rw) 0 l 1 4 3, where ˆR q is the dual of R q = Z/4M qz) Z/Z and l 3 is the conductor of ψ 3. Clearly, the same estimate holds for Z M 1, w, π; ψ, 1; q). We now use the Phragmen-Lindelöf rincile on the holomorhic function Ps, w)z M 1, w, π; ψ, 1; q).

13 DETERMINATION OF A GL 3 CUSPFORM BY TWISTS OF CENTRAL L-VALUES 13 Since Proosition 3.5 and 3.7) allow us to exress Z M 1, w, π; ψ, 1; q) as a finite linear combination of the Z M s, the fact that the above roduct is holomorhic and of order 1 follows from Theorem.1. Therefore we may aly Phragmen-Lindelöf and obtain the estimate 3.9) Z M 1, w, π; ψ, 1; q) c ) q + c q + 1) q Rw)+vɛ) ψ 3 ˆR q d 0 sq. free for 1 4 ɛ < Rw) < ɛ. Here vɛ) 0 as ɛ 0. Using the above estimate, it follows that ZM, 1 w, π; ψ, 1) = µq)z M, 1 w, π; ψ, 1; q) q,m )=1 is absolutely bounded by the sum of the three series where, for instance, Σ 1 = q sq. free q,m )=1 Σ 1 + Σ + Σ 3, c ) q q Rw)+vɛ) ψ 3 ˆR q d 0 sq. free L 1, π χ d 0 ψ 3 ) d 5 4 +ɛ 0 l 1 4 3, L 1, π χ ) d 0 ψ 3. d 5 4 +ɛ 0 l Each of these sums may be bounded in the same way. As Σ 1 is the most difficult to bound, we rovide details in this case only. We decomose l 3 = l 1 l with l 1 4M and l q. If we write q = l n, then Σ 1 l 1 4M l sq. free l,m )=1 d 0 sq. free L 1, π χ d 0 χ l1 χ l ) c ) l d 5 4 +ɛ 0 l l 31 6 Rw) 89 4 vɛ) n 1 c ) n n Rw) 4 vɛ). Fix any ν 0 > Note that 31 6 ν > 1. Since by the work of Kim [8] and Kim-Shahidi [10], sym π) is automorhic recall that π is a Gelbart-Jacquet lift), it follows that, for Rw) > ν 0 and ɛ sufficiently small, the innermost sum is absolutely convergent. Therefore, Σ 1 l 1 4M l sq. free l,m )=1 d 0 sq. free L 1, π χ ) d 0 χ l1 χ l c ) l d 5 4 +ɛ 0 l l 31 6 Rw) 89 4 vɛ) Now, write d 0 = dd 1 and l = dl 0 with d 1, l 0 ) = 1. Introducing the Euler factors corresonding to the rimes dividing d, and then summing over d, we are reduced to estimating L 1, π χ ) d 1 χ l1 χ l0 ) c l 1 4M l 0,d 1)=1 l 0,M )=1 d 5 4 +ɛ 1 l l Rw) 89 4 vɛ) l 0..

14 14 GAUTAM CHINTA AND ADRIAN DIACONU Let ν w := min{ ɛ, 31 6 sufficiently small. Then, for fixed l 1, we have L 1, π χ ) d 1 χ l1 χ l0 ) c l 0 d 1 l 0 ) νw l 0,d 1)=1 l 0,M )=1 Rw) 89 4 vɛ)} Rw) > ν 0). Note that ν w > 5 4 for ɛ m sq. free a m L 1, π χ mχ l1 ) m νw, where a m = l 0 m c) l 0. To see that this final series is absolutely convergent, it suffices by the Cauchy-Schwartz inequality to establish that the two series a m m 1+δ and m sq. free m sq. free L 1, π χ mχ l1 ) m 3/+δ are absolutely convergent for any δ > 0. The convergence of the first follows from roerties of the Rankin-Selberg L-function of sym π) with itself, and the convergence of the second from Lemma 3.. This establishes the analytic continuation of Z s, w, π; χ r δ K, 1) to 3.10) {Rs) 1 119, Rw) > 14 }. To comlete the roof of Proosition 3.1, we note that, because of the functional equation, we have the bound Ls, π χ d0 ) ɛ d 0 3/+ɛ for Rs) = 0. Therefore, Z s, w, π; χ r δ K, 1) is holomorhic in the tube domain 3.11) {Rs) > 0, Rw) > 5 }. Alying Hartogs theorem [7] and taking the convex closure of the two tube domains 3.10) and 3.11) comletes the roof of Proosition Comuting the residue The series Z s, w, π; χ r δ K, 1) has olar lines at w = 1 and at w+3s = 5/ which it inherits from Zs, w, π; χ r δ K, 1). Excet for these two olar lines, Z s, w, π; χ r δ K, 1) is holomorhic in a neighborhood of the oint s, w) = 1/, 1). The residue of Z s, w, π; χ r δ K, 1) at w = 1 may be comuted from the known residue.) of the Z M s, w, π; χ r δ K, 1) and the exlicit sieving rocedure given in [3]. However, with the meromorhic continuation of Z s, w, π; χ r δ K, 1) already established in Proosition 3.1, we shall comute the residue more directly. We begin with the identity Z s, w, π; χ r δ K, 1) = L Sfin s, π) n,m)=1 d 0 K d 0 sq.free L S s, π χ d0 ) d w χ r d 0 ). 0 Interchanging the order of summation, we rewrite the sum as c n χ d n) χ r d) n s d w. d K d sq.free

15 DETERMINATION OF A GL 3 CUSPFORM BY TWISTS OF CENTRAL L-VALUES 15 Set n = n 0 n 1n where n 0 squarefree and n 1 = n 0. We concentrate on the inner sum: χ d n) χ r d) χ d n 0 ) χ r d) d w = d w d K, sq.free d K,d sq.free d,n )=1 = d K,d sq.free d,n )=1 d K, sq.free d,n )=1 χ n0 d)χ r d) d w by Quadratic Recirocity, as d 14). We again use the orthogonality.3) to conclude χ n0 d)χ r d) d w = 1 ˆR L Mn w, χ n0 ψχ r ) ψ ˆR where L bw, χ) = d sq.free d,b)=1 χd) d w. The residue of these L-functions is easily comuted: Lemma 4.1. Let χ be a rimitive quadratic Dirichlet character of conductor n and let b > 0. Then ζw)l b w, χ) can be meromorhically continued to Rw) > 0. It is analytic in this region unless n = 1 when it has exactly one simle ole at w = 1 with residue Res w=1 ζw)l bw, χ) = ) b Since n 0 and r are relatively rime to M, the L-function L Mn w, χ n0 ψχ r ) will have a ole at w = 1 iff ψ = 1 and n = r k+1 n, r, n ) = 1. Therefore, using Lemma 4.1, Res ζw) w=1 = = = n,m)=1 n,mr)=1 k=0 [ k=0 c r k+1 r k+1)s c n n s L Mn w, χ n0 χ r ) c r k+1 n r k+1 n )s ] n,mr)= ) L1,r s) r 1 r + L,rs) M Mn c n n s ) 1 n ) ) 1 M ) 1 n,m)=1 c n n s n ) 1 We refer the reader to Lemma.3 for the exlicit evaluations of L 1,r and L,r. Putting everything together, when r is a rime, r, M) = 1, Res w=1 Z s, w, π; χ r δ K, 1) = R 1 s; π) R r s; π),

16 16 GAUTAM CHINTA AND ADRIAN DIACONU where and R 1 s; π) = L Sfin s, π) 1 ˆR M 4.1) R r s; π) = For r = 1 a similar argument yields ) 1 c n n s n,m)= ) L1r s) r 1 r + L,rs). Res w=1 Z s, w, π; δ K, 1) = R 1 s; π). n ) 1, Next we note that R 1 s; π) is well-aroximated by the symmetric square L- function for Rs) 1/. Proosition 4.. We have R 1 s; π) = L S s, π, sym )B M s; π) where B M s; π) is an absolutely convergent Euler roduct for Rs) > 9/0. Proof. The Dirichlet series in the definition of R 1 s; π) has the Euler roduct reresentation c n n s ) 1 = Q s), n,m)=1 n,m)=1 where Q s) = 1 + c k + 1 ks k 1 = c k ks k 0 k 1 c k ks = L, s) + O s+1/5 ɛ) ). On the other hand the symmetric square L-function has an Euler roduct L S s, π, sym ) = T s), where T s) = 1 α s Thus the quotient L, s) T s) ) 1 = 1 β s ) 1 1 γ s 1 α β s,m)=1 ) 1 ) 1 1 β ) 1 γ s 1 γ ) 1 α s 1 + c ) s 1 α ) β s 1 β ) γ s 1 γ ) α s = 1 + O 4/5 4s ). For the final bound we have again used the fact that, for π self-contragredient, c = α + β + γ = α β + α γ + β γ.

17 DETERMINATION OF A GL 3 CUSPFORM BY TWISTS OF CENTRAL L-VALUES 17 Now, an infinite roduct ) 1 + O 4/5 4s ) will converge absolutely rovided 4 5 4Rs) < 1, i.e. rovided Rs) > 9/0. Thus B M s; π) =,M)=1 L, s) T s) converges absolutely in this range. To roceed further we now use the assumtion that L S s, π, sym ) has a simle ole at s = 1/. By virtue of the revious roosition, R 1 s, π) has a simle ole at s = 1 as well. We set C M π) = lim s 1/)R 1s; π). s 1/ As noted at the start of this section, the double Dirichlet series Z s, w, π; χ r δ K, 1) is analytic in a neighborhood of the oint s, w) = 1/, 1) excet for the two olar lines w = 1 and w + 3s = 5/. Hence near the oint 1/, 1) we have the exansion 4.) Z s, w, π; χ r δ K, 1) = + A 0 w 1)s 1/) + A 1s) w 1 A 0 w + 3s 5/)s 1/) + A 1s) + Hs, w), w + 3s 5/) where A 0 = lim s 1/ lim w 1 s 1/)w 1)Z s, w, π; χ r δ K, 1) A 0 = lim s 1/ lim s 1/)w + 3s w 5/ 3s 5/)Z s, w, π; χ r δ K, 1), and A 1 s), A 1s) and Hs, w) are analytic near 1/, 1). Fix w > 1 and let s 1/ in 4.). The limit on the left hand side exists, therefore we conclude that A 0 = A 0. Hence at s = 1/, 4.3) Z 1/, w, π; χ r δ K, 1) = 3A 0 w 1) + B 1 w 1 + Iw), for Iw) an analytic function in a neighborhood of w = 1. In conclusion { CM π) if r = 1 4.4) A 0 = A 0 r; π) = C M π)r r 1/; π) otherwise. We note that Lemma.3 and 4.1) ensure that R r s; π) makes sense for Rs) > 9/0.

18 18 GAUTAM CHINTA AND ADRIAN DIACONU 5. Proof of Theorem 1.1 We are finally in a osition to rove our main result. We assume π 1 and π are two self-contragredient cusidal automorhic reresentations of GL 3 A Q ) of trivial character and levels N 1, N resectively. We suose further that Ls, π 1, sym ) has a simle ole at s = 1. Choose the finite set S to contain, the archimedean lace, and all the laces of bad ramification of π 1 and π. We assume there exists a nonzero constant κ such that L1/, π 1 χ d ) = κl1/, π χ d ) for all ositive squarefree integers d K. Let Z s, w, π 1 ; χ r δ K, 1) and Z s, w, π ; χ r δ K, 1) be the associated double Dirichlet series. Letting r = 1, we see w 1) Z 1/, w, π 1 ; δ K, 1) = 3C M π 1 ) + Ow 1). by 4.3) and 4.4). On the other hand, w 1) Z 1/, w, π 1 ; δ K, 1) = κw 1) Z 1/, w, π ; δ K, 1) = 3κC M π )+Ow 1), whence Similarly, if r is rime and r, M) = 1, C M π 1 ) = κc M π ). 3C M π 1 )R r 1/; π 1 ) + Ow 1) = w 1) Z 1/, w, π 1 ; χ r δ K, 1) as w 1. Equivalently = κw 1) Z 1/, w, π ; χ r δ K, 1) = 3κC M π )R r 1/; π ) + Ow 1), R r 1/; π 1 ) = R r 1/; π ). Lemma 5.1. There exists a rational function h r t) such that h r c r π 1 )) = R r 1/; π 1 ). Moreover, the function h r t) is monotone for r sufficiently large and t < r 1 ɛ, for any ɛ > 0. Proof. Combining Lemma.3 with Eq. 4.1) and Eq..5), we get R r 1/; π 1 ) = r + 1)r 3/ rc r π 1 ) + 1 c r π 1 ) r r ) + c r π 1 )r 3 + r r) + r 4 + r 3 1). Let h r t) be the function defined by Note that h r t) := r + 1)r 3/ rt + 1 t r r ) + tr 3 + r r) + r 4 + r 3 1). h rt) = r5/ r 1 ) 1 + r + r + r 3 + t + r t ) 1 + r 3 + r 4 r t + r t + r 3 t + r t r t ). Fix ɛ > 0. If r is sufficiently large in terms of ɛ, the derivative is strictly ositive for t < r 1 ɛ. Thus h r t) is monotone for t in this range. To comlete the roof of Theorem 1.1, we use Lemma 5.1 to conclude that if R r 1/; π 1 ) = R r 1/; π ), then c r π 1 ) = c r π ) for all r sufficiently large. The strong multilicity one theorem now imlies that π 1 π.

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