DETERMINATION OF GL(3) CUSP FORMS BY CENTRAL VALUES OF GL(3) GL(2) L-FUNCTIONS, LEVEL ASPECT

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1 DETERMIATIO OF GL(3 CUSP FORMS BY CETRAL ALUES OF GL(3 GL( L-FUCTIOS, LEEL ASPECT SHEG-CHI LIU Abstract. Let f be a self-dual Hecke-Maass cusp form for GL(3. We show that f is uniquely determined by central values of GL( twists of its L-function. More precisely, if g is another self-dual GL(3 Hecke-Maass cusp form such that L (, f h = L (, g h for all h S0(q, for infinitely many primes q, then f = g.. Introduction In their paper [LR], Luo and Ramakrishnan proved that a holomorphic cusp form is uniquely determinated by the central values of twisted L-functions. More precisely, let f(z Sk new ( and g(z Snew k ( be cuspidal normalized newforms of level and respectively. They proved that if L(/, f χ d = cl(/, g χ d for all quadratic characters χ d with fundamental discriminant d and some constant c, then f = g. Instead of twists by quadratic characters, Luo [Lu] considered twists by normalized newforms h(z Sl new (p for all primes p and proved the same result. Later Ganguly, Hoffstein and Sengupta [GHS] considered varying the twists h(z in the weight aspect and obtained the same result. It is interesting to consider the analogous question for GL(n cusp forms. For a GL(3 self-dual cusp form, such a result is true proved by Chinta and Diaconu [CD] with twists by quadratic characters and proved by the author [Liu] with twists by GL( cusp forms in the weight aspect. In this paper we will prove the same result with twists by GL( cusp forms in the level aspect (Theorem. Let Sk (q denote the set of primitive cuspidal newforms on Γ 0(q with trivial nebentypus and weight k. Theorem. Let f and g be self-dual Hecke-Maass forms for SL(3, Z of types (ν, ν and (µ, µ respectively. Let c be a constant. If ( ( L, f h = cl, g h for all h S 0(q, for infinitely many primes q, then f = g. Key words and phrases. central values of L-functions, multiplicity one theorem.

2 SHEG-CHI LIU As in [Liu], Theorem follows from the asymptotic formulas for the first moments of twisted L-functions at the critical points (Theorem 3 and the multiplicity one theorem. It is known that the symmetric square lift (Gelbart-Jauet lift of a GL( Hecke cusp form f is a self-dual GL(3 cusp form. The following theorem is a consequence of our Theorem and a result of Ramakrishnan [R]. Theorem. Let f and g be Hecke-Maass cusp forms for SL(, Z. Let c be a constant. If ( ( L, sym f h = cl, sym g h for all h S 0(q, for infinitely many primes q, then f = g.. Preliminaries.. GL( holomorphic cusp forms. For k and q integers, k and χ a character modulo q, let S k (q, χ denote the space of weight k holomorphic cusp forms with level q and nebentypus χ. Let H k (q, χ be a normalized Heck basis of S k (q, χ. For each h H k (q, χ, it has the Fourier expansion h(z = n= λ h (nn k e(nz where e(z = e πiz and λ h ( =. The following Petersson trace formula is well-known and can be found in [IK]. Proposition (Petersson trace formula. ω h λ h(mλ h (n = δ m,n + πi k h H k (q,χ c 0 (mod q c>0 ( S χ (m, n; c 4π mn J k, c c where ω h = (4πk Γ(k h, δ m,n equals if m = n and 0 otherwise, S χ (m, n; c is the Kloosterman sum with character χ defined below, and J k is the J-Bessel function. The Kloosterman sum is defined as S χ (m, n; c = and for χ =, A. Weil proved that where τ(c is the divisor function. aā (mod c ( ma + nā χ(ae, c S(m, n; c (m, n, c c τ(c,

3 DETERMIATIO OF GL(3 CUSP FORMS 3 Let Sk (q, χ denote the set of primitive newforms. In the case when χ is primitive or k < and χ is trivial, we have H k (q, χ = Sk (q, χ. For h Sk (q with q squarefree and k <, the Hecke L-function of h is defined by λ h (n L(s, h = = ( λ n s h (pp s + χ(pp s. n= p It is entire and satisfies the functional equation ( ( s + k Λ(s, h = q s/ π s s + k+ Γ Γ L(s, h = ε h Λ( s, h where ε h = i k µ(qλ h (qq / = ± and µ is the Möbius function. Proposition. Let q be a prime. For any m, n, we have ω h λ h(mλ h (n = δ m,n π h S 0 (q ε h = h S 0 (q ε h = +q / (δ m,nq π c= c= ( S(m, n; 4π mn ( S(m, nq; 4π mnq. Proof. Since ε h = q / λ h (q, we have ω h λ h(mλ h (n = ( + ε h ω h λ h(mλ h (n h S0 (q = h S 0 (q ω h λ h(mλ h (n + q / ω h h S0 (q λ h(mλ h (qλ h (n. Because h is primitive and q is the level, λ h (qλ h (n = λ h (nq for all n. Hence the result follows from the Petersson trace formula... Automorphic forms on GL(3. Here we use a classical setting for GL(3 Maass forms. One can see the details and notation in Goldfeld s book [Go]. Let f be a normalized self-dual Hecke-Maass form for SL(3, Z of type (ν, ν and let A(m, n be its (m, n-th Fourier coefficient. ote that A(, = and A(n, m = A(m, n. Moreover we have (see [Go, Remark..8] (. A(m, n f. m n

4 4 SHEG-CHI LIU By Cauchy s inequality and (., one can deduce that (. A(m, n f m. n The Godement-Jauet L-function associated to f is defined by L(s, f = n= A(, n n s = p ( A(, pp s + A(p, p s p 3s. It is entire and satisfies the functional equation γ ν (sl(s, f = γ ν ( sl( s, f where Set ( γ ν (s = π 3s s + 3ν ( s Γ Γ Γ α = 3ν +, β = 0, γ = 3ν. ( s + 3ν. Suppose k = 0, and ψ is a smooth compactly supported function on (0,. Define ψ(s := 0 ψ(xx s dx x. For σ > max{ Rα, Rβ, Rγ}, define ψ k (x := +s+k+α (π 3 s Γ( Γ( +s+k+β Γ( +s+k+γ x πi Rs=σ Γ( s+k α Γ( s+k β Γ( s+k γ ψ( sds, (.3 and (.4 Ψ + (x = ( ψ π 3/ 0 (x + i ψ (x Ψ (x = ( ψ π 3/ 0 (x i ψ (x. The following oronoi formula on GL(3 was first proved by Miller and Schmid [MS] and a simpler proof was given by Goldfeld and Li [GL].

5 n DETERMIATIO OF GL(3 CUSP FORMS 5 Proposition 3 ([MS], [GL]. Let ψ(x Cc (0,. Let d, d, c Z with c 0, (c, d =, and dd (mod c. Then we have ( nd A(m, ne ψ(n c = c n cm n >0 +c n cm n >0 ( A(n, n S(md, n ; mcn n n Ψ + n n c 3 m ( A(n, n S(md, n ; mcn n n Ψ n n c 3 m. 3. The twisted first moment of GL(3 GL( L-functions Let h S 0(q with q prime and let f be a self-dual Hecke-Maass form of type (ν, ν. The Rankin-Selberg L-function of f and h is defined by L(s, f h = m n It is entire and satisfies the functional equation where ε f h = ε h = ± and ( s + 9 γ(s = π 3s Γ λ h (na(m, n (m n s. Λ(s, f h = ε f h Λ( s, f h Λ(s, f h = q 3 s γ(sl(s, f h, α ( s + Γ α ( s + 9 Γ Γ β ( s + β ( s + 9 Γ γ Γ ( s + γ Remarks.. Luo-Rudnick-Sarnak [LRS] proved that Rα, Rβ, Rγ. 0. The above functional equation can be obtained by using the template in [Go, p.35]. Theorem 3. Let f be a self-dual Hecke-Maass form for SL(3, Z, normalized such that A(, =. Then for any large prime q, we have h S 0 (q ε h = ω h L (, f h = L(, f + O(q 8 +ε.

6 6 SHEG-CHI LIU and for any fixed prime p, we have h S 0 (q ε h = ( ( A(, p ω h L, f h λ h (p = L(, f + O(q p / p 3/ 8 +ε. The implied constants depend on ε, f and p. ote that L(, f 0 (see [JS]. We will prove Theorem 3 in Section 4. The proof is based on a general framework which uses the approximate functional equation and the Petersson trace formula. The main term comes from the diagonal term and the others contribute to error terms. To estimate the offdiagonal terms, we will need the oronoi formula on GL(3. As explained in [Li], L(, f h 0 by Lapid s theorem [La]. For q large enough such that the twisted first moment in Theorem 3 is nonvanishing, we have the following corollary. Corollary 4. Let f be a fixed self-dual Hecke-Maass form for SL(3, Z. For each prime q large enough, there exists h S0(q such that L(, f h > Proof of Theorem Approximate functional equation. Let G(u = e u. We have the following approximate functional equation (see [IK, Theorem 5.3], (4. ( L, f h = λ h (na(m, n (m n / m n ( m n q 3/, where (4. (y = y u G(u γ( + u du πi (3 γ( u. Moreover the derivatives of (y satisfy (see [IK, Proposition 5.4] (4.3 y a (a (y ( + y A. Here the implied constant depends on a and A. We will only show the second asymptotic formula in Theorem 3 since the proof of the first one is similar. By the approximate functional equation (4.,

7 DETERMIATIO OF GL(3 CUSP FORMS 7 Proposition, and since δ p,nq = 0 for all n if q > p, we have ( ω h L, f h λ h (p h S 0 (q ε h = = h S 0 (q ε h = = m ω h λ h (na(m, n (m n / m n A(m, p (m p / π m n πq / m := D E E. ( m p q 3/ A(m, n (m n / n 4.. Diagonal terms. Lemma. D = A(m, n (m n / Proof. Using the relation A(m, n = ( m n q 3/ c= ( m n q 3/ ( m n q 3/ λ h (p ( S(p, n; 4π pn c= ( A(, p L(, f + O(q 3 p / p 3/. d (m,n ( m ( µ(da d, A, n, d ( S(p, nq; 4π pnq one can get D = A(, p ( A(m, m (m p p ( A(m, m / q 3/ (m p 3 n / q 3/ m m A(, p ( A(m, m u p = G(u γ( + u du p / πi (3 m q 3/ γ( u m ( A(m, m p 3 u G(u γ( + u du p 3/ πi m q 3/ γ( u = A(, p p / p 3/ (3 m πi πi (3 (3 ( p L( + u, f q 3/ ( p 3 L( + u, f q 3/ u G(u γ( + u du γ( u u G(u γ( + u du γ( u.

8 8 SHEG-CHI LIU We shift the contour to ( and only pick up a simple pole at u = 0 (see Remarks. The residue at u = 0 is ( A(, p L(, f p / p 3/ and the integral over the line Ru = is q 3/ Off-diagonal terms. We will prove the off-diagonal terms, E and E, only contribute the error terms. By taking smooth dyadic subdivisions (see [IM], it suffices to estimate E, := m and n E, := q / m A(m, n (m n / w n ( m n A(m, n (m n / w ( m n ( m n q 3/ c= ( m n q 3/ ( S(p, n; 4π pn c= ( S(p, nq; 4π pnq where w(ξ is essentially a fixed smooth function with support contained in [, ] and q 3/+ε. Lemma. Proof. For m n, we have x = 4π pn E, q /+ε. = 4π p(m n / m Hence x < for q sufficient large. Recall that (4.4 4π p( / J k (x min(, x k. q /4+ε. Using Weil s bound for Kloosterman sums, (4.4, (4.3 and (., we have E, ( A(m, n m (m n / w n ( n ( /+ε / 9 m n c q 9/+ε A(m, n (m n9/ n / m n q 9/+ε 6 q /+ε.

9 DETERMIATIO OF GL(3 CUSP FORMS 9 Lemma 3. E, q /8+ε. Proof. Let R = q / m c> 8π p q 3/34 m n A(m, n (m n / w ( m n ( S(p, nq; 4π pnq ( m n q 3/ and R = q / m c 8π p q 3/34 m n A(m, n (m n / w ( m n ( S(p, nq; 4π pnq ( m n. q 3/ Then E, = R + R. We will show R q /8+ε and R is negligible in the next two sections Estimate of R. For c > 8π p q 3/34 m and m n, we have x = 4π pnq = 4π p(m n / / m q /7. Hence x < for q sufficient large. Using Weil s bound for Kloosterman sums, (4.4, (4.3 and (., we have R q ( / A(m, n m (m n / w n ( nq 9 ( /+ε m n q 9/+ε m n q 9/+ε / A(m, n (m n / n9/ m n q 5/4+ε 3/4+ε q /8+ε. c> 8π p q 3/34 m c> 8π p q 3/34 m c 9/+ε A(m, n n 9/ ( / q 3/34 m 7/+ε

10 0 SHEG-CHI LIU 4.5. Estimate of R. To estimate R, we will use the oronoi formula on GL(3. After opening the Kloosterman sums in R, we will need to estimate the following sum over n, H := n A(m, n w n / ( m n ( m n q 3/ ( 4π pnq e Applying the GL(3 oronoi formula (Proposition 3 with ( ( ( m ψ(y = y / y m y 4π pqy w J q 3/ 9, we have H = n A(m, ne = c n cm n >0 +c n cm n >0 ( na ψ(n c ( A(n, n S(ma, n, mcn n n Ψ + n n c 3 m ( A(n, n S(ma, n, mcn n n Ψ n n c 3 m ( na. c where Ψ + (x and Ψ (x are defined in (.3 and (.4. As explained in [Li], ψ (x has similar asymptotic behavior to ψ 0 (x, so we only consider ψ 0 (x. Since c 8π p, q 3/34 m n n c 3 m m q39/34 / q 7/68 ε. Hence we can apply [Li, Lemma.] for x = n n c 3 m K ψ 0 (x = π 4 xi ψ(y (4.5 ( +O 0 j= (q 7/68 ε K+ 3 which gives c j cos(6πx /3 y /3 + d j sin(6πx /3 y /3 dy (π 3 xy j/3 where c j and d j are some constants. In particular c = 0 and d = 3π. We only deal with the first term since the other terms can be analyzed in a similar way. So we have to estimate the integral where x 0 ψ(y sin(6πx/3 y /3 (π 3 xy /3 dy = x /3 ( m r(y = y 5/6 y w ( m y q 3/ 0 r(y sin(a(ydy ( 4π pqy

11 DETERMIATIO OF GL(3 CUSP FORMS and a(y = 6πx /3 y /3. ( /6 Using (4.3 and x l J (l k, we have r m (y. Moreover we have a (y q3/34 m, so 7/6 a (y[r (y] q 3/34 /3 m 5/3 q 3/34 /6 q 9/68 ε. By integration by parts, one shows that H is negligible and so is R. 5. Proofs of Theorems and. The proofs of Theorems and are the same as in [Liu]. For completeness, we include them here. Proof of Theorem. Let A(m, n and B(m, n be the (m, n-th Fourier coefficients of f and g respectively. Then the assumption L (, f h = cl (, g h and Theorem 3 imply L(, f = cl(, g and ( A(, p ( B(, p L(, f = cl(, g p / p 3/ p / p 3/ for all primes p. Hence A(, p = B(, p for all primes p. So f = g by the multiplicity one theorem [JS]. Proof of Theorem. Recall that (see [MS] for example L(s, sym f = λ ( f (nn s where λ ( f (n = λ f (m. ml =n Hence λ ( f (p = λ f(p. Let F and G be the Gelbart-Jauet lifts of f and g respectively. Then F and G are GL(3 self-dual Maass cusp forms which satisfy L(s, F = L(s, sym f and L(s, G = L(s, sym g. Moreover their Fourier coefficients n= A F (, n = λ ( f (n, A G(, n = λ ( g (n. As in the proof of Theorem, our assumption gives us A F (, p = A G (, p for all prime p and hence λ f (p = λ g (p for all prime p. Using the Hecke relation λ f (p = λ f (p +, we have λ f (p = λ g (p. Then we use (Ramakrishnan[R], Corollary 4..3 to conclude f = g.

12 SHEG-CHI LIU Acknowledgement. The author would like to thank Professor Matt Young for his valuable comments and encouragement. References [CD] G. Chinta and A. Diaconu, Determination of a GL 3 cuspform by twists of central L-values, IMR 005 (005, [GHS] S. Ganguly, J. Hoffstein, and J. Sengupta, Determining modular forms on SL (Z by central values of convolution L-functions, Math. Ann. 345 (009, [GL] D. Goldfeld and X. Li, oronoi formulas on GL(n, Int. Math. Res. ot. 006, Art. ID 8695, 5pp. [Go] D. Goldfeld, Automorphic forms and L-functions for the group GL(n, R. With an appendix by Kevin A. Broughan. Cambridge Studies in Advanced Mathematics, 99. Cambridge University Press, Cambridge, 006. [IK] H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, 53. American Mathematical Society, Providence, RI, 004. [IM] H. Iwaniec and P. Michel, The second moment of the symmetric square L-function, Ann. Acad. Sci. Fenn. Math. 6 (00, [JS] H. Jauet and J.A. Shalika, A non-vanishing theorem for zeta functions of GL n, Invent. Math. 38 (976/77, no., -6. [JS] H. Jauet and J.A. Shalika, On Euler products and the classification of automorphic representations I, Amer. J. Math. 03 (98, no. 3, [La] E. Lapid, On the nonnegativity of Rankin-Selberg L-function at the center of symmetry, I.M.R.. (003, no., [Li] X. Li, Bounds for GL(3 GL( L-functions and GL(3 L-functions, Ann. of Math. 73 (0, [Liu] S.-C. Liu, Determination of GL(3 cusp forms by central values of GL(3 GL( L-functions, Int. Math. Res. ot. 00, Art. ID rnq0, 7pp. [Lu] W. Luo, Special L-values of Rankin-Selberg convolutions, Math. Ann. 34 (999, no. 3, [LR] W. Luo and D. Ramakrishnan, Determination of modular forms by twists of critical L-values, Invent. math. 30 (997, [LRS] W. Luo, Z. Rudnick and P. Sarnak, On the generalized Ramanujan conjecture for GL(n, Automorphic forms, automorphic representation, and arithmetic (Fort Worth, TX, 996, 30-30, Proc. Sympos. Pure Math., 66, Part, Amer. Math. Soc., Providence, RI, 999. [MS] S.D. Miller and W. Schmid, Automorphic distribution, L-functions, and oronoi summation for GL(3, Ann. of Math. ( 64, (006, no., [MS] S.D. Miller and W. Schmid, Summation formula, from Poisson and oronoi to the presentoncommutative harmonic analysis, , Progr. Math., 0, Birkhuser Boston, Boston, MA, 004. [R] D. Ramakrishnan, Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(, Ann. of Math. ( 5, (000, 45-. Department of Mathematics, Texas A&M University, College Station, TX , U.S.A. address: scliu@math.tamu.edu