To G.Shimura with admiration

On the Generalized Ramanujan Conjecture for GL(n)

Wenzhi Luo, Zeév Rudnick, and Peter Sarnak ^{1}

1 Introduction

In this paper we describe a modification to the method that we introduced in [11] for giving bounds towards the generalized Ramanujan conjectures. The modification allows us to treat the case of a general number field. The main new difficulty in dealing with a number field is the presence of an infinite group of units. The method in [11] and the earlier related method of [5] made use of Dirichlet characters of varying conductors; in general the unit group limits the construction of the analogous ray class characters. The modification mentioned above allows us to exploit in a crucial way the positivity of the coefficients of the Rankin-Selberg L-function. This, together with Rohrlich’s construction [14] of special moduli which have many ray class characters, enable us to overcome this difficulty.

Let F be a number field of degree [F : Q] = d with ring of integers O = O_{F}, unit group U = U_{F}, and D = D_{F} is the discriminant of F. For n ≥ 2 let π be an cuspidal automorphic form on GL_{n}(A_{F}) with unitary central character and contragredient representation π̃. The standard L-function L(s, π) associated with π is of the form (for Re(s) > 1 )

1Research partially supported by grants from the NSF and the Bi-National Science Foundation.

L(s, π) = ∏_{v} L(s, π_{v}).

The product is over all the places of F and for each such v, L(s, π_{v}) is the local Euler factor. At an unramified finite place of π, this local factor is given by

L(s, π_{v}) = ∏_{i=1}^{n}(1 − α_{i,π}(v)(Nv)^{−s})^{−1},

for suitable complex numbers α_{i,π}(v), i = 1, ⋯, n, and where N(v) is the absolute norm of v viewed as a prime ideal of F. At an unramified archimedean place v for π, the local factor is given in the form

L(s, π_{v}) = ∏_{i=1}^{n} Γ_{v}(s − μ_{j,π}(v)),

where

Γ_{v}(s) = π^{−s/2}Γ(s/2), if v is real,

and

Γ_{v}(s) = (2π)^{−s}Γ(s), if v is complex.

In this notation the generalized Ramanujan conjecture for π asserts that for such unramified v,

|α_{j,π}(v)| = 1, if v is finite;
Re(μ_{j,π}(v)) = 0, if v is archimedean.

Jacquet and Shalika [8], using the genericity of π and local methods, showed that

|α_{j,π}(v)| < N(v)^{1/2}, if v is finite;  (1)
|Re(μ_{j,π}(v))| < 1/2, if v is archimedean.  (2)

For v finite, as was first observed by Serre [16], one can do better than the above bound by using the Rankin-Selberg L-function (assuming its analytic properties) in conjunction with Landau’s lemma [10]. This leads to

|α_{j,π}(v)| ≤ N(v)^{1/2−1/(dn^{2}+1)}

To describe our main results let S be any finite set of places of F. For B an integral ideal of F let X_{B} denote the set of all wide ray class characters χ (see [20] for definitions) of conductor B such that χ(v) = 1, for any v ∈ S. Let X_{B}^{∗} be the subset of X_{B} consisting of primitive characters. Set X = ∪_{B}X_{B} and X^{∗} = ∪_{B}X_{B}^{∗}. For χ ∈ X_{B}^{∗}, the partial Rankin-Selberg L-function L_{S}(s, (π ⊗ χ) × π̃) is defined to be

L_{S}(s, (π ⊗ χ) × π̃) = ∏_{v∉S} L(s, (π_{v} ⊗ χ_{v}) × π̃_{v}).

At an unramified place v of π the local factor is given by

L(s, (π_{v} ⊗ χ_{v}) × π̃_{v}) = ∏_{i,j=1}^{n}(1 − χ(v)α_{i,π}(v)ᾱ_{j,π}(v)(Nv)^{−s})^{−1}, if v is finite; (3)
= ∏_{i,j=1}^{n} Γ_{v}(s − μ_{i,π}(v) − μ̄_{j,π}(v)), if v is archimedean. (4)

With these notations our main result is : Theorem 1. Given n, π, S as the above and any β > 1 − \frac{2}{n^{2}+1}, there are infinitely many χ ∈ X^{∗} such that

L_{S}(β, (π ⊗ χ) × π̃) ≠ 0.

Combining Theorem 1 with the known analytic properties of the Rankin-Selberg L-functions, we establish Theorem 2. Given n, π as the above and let v be a place of F at which π is unramified. Then for any j, 1 ≤ j ≤ n,

(a) | log_{N(v)} |α_{j,π}(v)|| ≤ \frac{1}{2} − \frac{1}{n^{2} + 1}, if v is finite;
(b) |Re(μ_{j,π}(v)| ≤ \frac{1}{2} − \frac{1}{n^{2} + 1}, if v is archimedean.

For n ≥ 3, part (b) is the first general result going beyond the bound (2). Part (a) has the feature of being uniform (i.e. independent of the degree d).

In the case n = 2, we can do better than the bound in Theorem 2 by combining the n = 3 case of Theorem 2 together with the Gelbart-Jacquet lift [6] from GL_{2}(A_{F}) to GL_{3}(A_{F}) (which is in turn based on Shimura’s symmetric square theory [19]) to obtain Theorem 3. For π an unitary cuspidal automorphic representation of GL_{2}(A_{F}), unramified at v and j = 1, 2, we have

(a) | log_{N(v)} |α_{j,π}(v)|| ≤ 1/5, if v is finite;
(b) |Re(μ_{j,π}(v))| ≤ 1/5, if v is archimedean.

Remark 1. The bound in part (a) of Theorem 3 is not new. It coincides essentially with the best known bound at the finite places obtained by Shahidi [18] ( note that he also showed that the equality cannot occur in (a)) by a very different method. Part (b) gives a uniform generalization of our result [11] over Q to an arbitrary number field F. Remark 2. In the special case that F = Q or F is an imaginary quadratic field (so that the group of units U is finite), one can apply the method of [11] to the symmetric square L-function on GL_{n}(A_{F}) (note that the method in [11] does not use the positivity of the coefficients of the Rankin-Selberg L-functions). This leads to a slight improvement of Theorem 2 and hence in Theorem 3 for such F ’s. We will not pursue this here.

It is expected that the Ramanujan conjecture on GL_{n}(A_{F}) will play a fundamental role in understanding more general groups. In a few cases where this connection has been established we can apply Theorems 2 and 3. We exhibit a couple of examples: (A). The Jacquet-Langlands correspondence together with Theorem 3 implies the following. Let X = Γ\H^{2} be a Shimura curve, by which we mean that Γ is a lattice in SL_{2}(R) coming from a congruence subgroup of a quaternion algebra D defined over a totally real number field K (if v_{1}, ⋯, v_{l} are the archimedean places of K, then we require that v_{1} ⊗ D ≅ SL_{2}(R) while v_{j} ⊗ D ≅ Hamilton quaternion algebra for j = 2, ⋯, l ). Then

λ_{1}(X) ≥ \frac{21}{100},  (5)

where λ_{1} is the first positive eigenvalue of the Laplacian on the hyperbolic surface X. Knowing the full Ramanujan conjectures would allows us to replace 0.21 by 0.25. (B). Using the method of Ramanujan duals (see [2], [3]) together with Theorem 3 we obtain the following result. Let Γ be a congruence subgroup of an orthogonal group G defined by a quadratic form over totally real number field. Assume that the form is of signature (n, 1) at one such real place and is definite at all the other places. Then Γ acts discontinuously on hyperbolic n-space H^{n} and X = Γ\H^{n} is a hyperbolic n-manifold (i.e. it has constant sectional curvature ≡ −1). For n ≥ 3

λ_{1}(X) ≥ \frac{3n}{5} − \frac{24}{25},  (6)

where again λ_{1} is the first positive eigenvalue of the Laplacian. Previously it was known that (see [3])

λ_{1}(X) ≥ \frac{n}{2} − \frac{3}{4},  (7)

while the Ramanujan conjecture for these X’s asserts that

λ_{1}(X) ≥ n − 2.  (8)

Again (7) would follows from the Ramanujan conjecture for GL_{2}(A_{F}).

2 Proofs

We begin by showing that Theorem 3 follows from Theorem 2 (applied with n = 3), using the Gelbart-Jacquet lift [6]. Indeed if π is an unitary cuspidal automorphic representation on GL_{2}(A_{F}), let Π be its lift to GL_{3}(A_{F}). Those π’s for which Π is not cuspidal satisfy the full Ramanujan conjectures, and so we can assume that Π is a unitary cuspidal automorphic representation on GL_{3}(A_{F}). Its local Euler factor L(s, Π_{w}) at an unramified w of π is given in terms of the local parameters of π at w:

L(s, Π_{w}) = (1 − α_{1,π}^{2}(w)(Nw)^{−s})^{−1}(1 − α_{1,π}(w)α_{2,π}(w)(Nw)^{−s})^{−1}(1 − α_{2,π}^{2}(w)(Nw)^{−s})^{−1};

if w is finite,

L(s, Π_{w}) = Γ_{w}(s − 2μ_{1,π}(w))Γ_{w}(s − μ_{1,π}(w)μ_{2,π}(w))Γ_{w}(s − 2μ_{2,π}(w)), if w is archimedean.

Inspecting the local parameters and applying Theorem 2 to Π yields Theorem 3. Remark 3. Theorem 3 can also be made to apply to other π, such as odd Maass forms for the modular group, which are ramified at infinity. The point is that their lifts to GL(3) are still unramified at infinity and so we may apply Theorem 2.

Next we show that Theorem 2 follows from Theorem 1. The global Rankin-Selberg L-function L(s, (π ⊗ χ) × π̃) is known to be holomorphic on the whole complex plane except for simple poles at s = 1 and s = 0 if χ ≡ 1. This is a consequence of the works of Jacquet, Shalika, Piatetski-Shapiro, Shahidi, Moeglin and Waldspurger [7],[8],[9],[12],[17]. Now

L(s, (π ⊗ χ) × π̃) = L^{S}(s, (π ⊗ χ) × π̃) · L_{S}(s, (π ⊗ χ) × π̃),  (9)

where

L^{S}(s, (π ⊗ χ) × π̃) = ∏_{v∈S} L(s, (π_{v} ⊗ χ_{v}) × π̃_{v}).  (10)

Let v be a place of F at which π is unramified. Choose S = {v}. For χ ∈ X^{∗}, χ(v) = 1 so that (see (3),(4))

L(s, (π_{v} ⊗ χ_{v}) × π̃_{v}) = L(s, π_{v} × π̃_{v}).

It follows that any pole of L(s, π_{v} × π̃_{v}) must be a zero of L_{S}(s, (π ⊗ χ) × π̃). Again from (3) and (4) we see that if β is chosen so that for some j, |α_{j,π}(v)|^{2} = N(v)^{β} if v is finite, and so that 2Re(μ_{j,π}(v)) = β if v is archimedean, then for any χ ∈ X^{∗}, we have

L_{S}(β, (π ⊗ χ) × π̃) = 0.  (11)

According Theorem 1, we must therefore have

β ≤ 1 − \frac{2}{n^{2} + 1}.

That is, for any j = 1, ⋯, n,

|α_{j,π}(v)| ≤ N(v)^{\frac{1}{2} − \frac{1}{n^{2}+1}} , if v is finite;
Re(μ_{j,π}(v)) ≤ \frac{1}{2} − \frac{1}{n^{2} + 1}, if v is archimedean.

The unitarity of π_{v} ensures that

{α_{j,π}^{−1}(v)}_{j=1}^{n} = {ᾱ_{j,π}(v)}_{j=1}^{n};

and

{−μ_{j,π}(v)}_{j=1}^{n} = {μ̄_{j,π}(v)}_{j=1}^{n}.

Hence Theorem 2 is established.

It remains to prove Theorem 1. For any place v and s ∈ C, L(s, (π_{v}⊗χ_{v})×π̃_{v}) ≠ 0 [8]. Hence if we can establish Theorem 1 for S, then it holds for any S′ ⊂ S. We may therefore assume that S contains all the archimedean places of F as well as the places at which π is ramified and those at which F ramifies. This assumption about S will be made from now on.

We need to know that there are many Q’s for which |X_{Q}| is large (that is, it should be almost of the order of N(Q) ). This is accomplished using Rohrlich’s construction [14]. For ϵ > 0 (fixed but arbitrarily small) he shows using a variant of the Bombieri-Vinogradov theorem due to Murty-Murty [13], that there are infinitely many integral ideals Q of F such that (i). The integer q = N(Q) is a product of distinct primes, and (Q, S) = 1. (ii). Any element in (O/Q)^{∗} has order at most q^{ϵ}.

As Rohrlich shows, these imply that the group C(Q) of ray classes (mod Q) has order ≫ q^{1−ϵ}. In fact from (ii) and the fact that for any ideal B, B^{h} is principal (here h = |C(1)| is the class number of F), we see that the order of any subgroup of C(Q) generated by a fixed number r of elements, is ≪ q^{rϵ}. Choosing these r elements to be the finite primes v of S, we conclude that for the above sequence of Q’s

|X_{Q}| ≫ q^{1−rϵ}.  (12)

As mentioned above, the L-function L(s, π × π̃) is holomorphic except for simple poles at s = 1 and s = 0. Also it has non-negative coefficients in its series expansion (see [15]):

∑_{A} a(A)(NA)^{−s},  (13)

with a(A) ≥ 0. From this one has

∑_{N(A)≤X} a(A) ∼ C_{π}X, C_{π} ≠ 0.  (14)

For χ ∈ X_{B}^{∗}, the L-function L(s, (π ⊗ χ) × π̃) satisfies the following functional equation[11]:

L(s, (π ⊗ χ) × π̃)
= χ(f(π × π̃))W (χ)^{n^{2}}W (π × π̃)N^{1/2−s}L(1 − s, (π̃ ⊗ χ̄) × π),

where N = D^{n^{2}}N(f(π × π̃))N(B)^{n^{2}}, W (π × π̃) is the root number of π × π̃ (it is non-zero), and W (χ) is the normalized Gauss sum corresponding to χ, and f(π × π̃) is the conductor of π × π̃.

By our choice of S, χ ≡ 1 at the places of ramification of π×π̃, so that χ(f(π×π̃)) = 1. Denote by L^{S}(s, π × π̃) the product

L^{S}(s, π × π̃) = ∏_{v∈S} L(s, π_{v} × π̃_{v}).

Then since χ(v) = 1 for v ∈ S, we have

L(s, (π ⊗ χ) × π̃) = L^{S}(s, π × π̃) · L_{S}(s, (π ⊗ χ) × π̃).  (15)

Denote by L_{S,χ}(s) the partial L-function L_{S}(s, (π ⊗ χ) × π̃) and by G(s) the quotient

G(s) = \frac{L^{S}(1 − s, π × π̃)}{L^{S}(s, π × π̃)}.

The local bounds of [8] which are applicable to any place v, assert that L(s, π_{v}×π̃_{v}) is analytic in Re(s) > β_{v}, where β_{v} < 1 (and depends on π). S being finite, it follows that L^{S}(s, π×π̃) is analytic in Re(s) > β_{0} for some β_{0} = β_{0}(π) < 1. Since L^{S}(s, π×π̃) is nowhere zero, we have that

G(1 − s) is analytic in Re(s) > β_{0},  (16)

for some β_{0} < 1.

The functional equation for L_{S,χ}(s) when χ ∈ X_{B}^{∗} becomes

L_{S,χ}(s) = W (χ)^{n^{2}}W (π × π̃)N^{1/2−s}G(s)L_{S,χ̄}(1 − s).  (17)

The series representation for L_{S,χ}(s) is

∑_{(A,S)=1} a(A)χ(A)(NA)^{−s}.  (18)

We are ready to prove Theorem 1. For β ≥ 1, L(β, (π ⊗ χ) × π̃) ≠ 0 (see [17]), so we can assume 1 − \frac{2}{n^{2}+1} < β < 1. The idea is to show that for N(Q) = q large enough (as in (i), (ii) above)

| ∑_{B|Q,B≠O} ∑_{χ∈X_{B}^{∗}} L_{S,χ}(β)| ≫ q^{1−ϵ}.  (19)

This immediately implies Theorem 1.

Let f ∈ C_{c}^{∞}(0, ∞) with supp(f) = [A, B], 0 < A < B, such that f(y) ≥ 0 and ∫_{0}^{∞} \frac{f(y)}{y}dy = 1. Define

k(s) = ∫_{0}^{∞} f(y)y^{s−1}dy,  (20)

and

f_{1}(y) = \frac{1}{2πi} ∫_{Re(s)=2} k(s)y^{−s} \frac{ds}{s},  (21)

for y > 0. Then, k(s) is entire and for s = σ + it, |σ| ≤ a, we infer, by partial integration several times, that

k(s) ≪_{a,m} \frac{1}{(|t| + 1)^{m}} , for any positive integer m.  (22)

It is easy to see, by means of Mellin inversion formula, that

f_{1}(y) = ∫_{y}^{∞} \frac{f(x)}{x}dx.  (23)

Thus, we have: 0 ≤ f_{1}(y) ≤ 1; f_{1}(y) = 1, if 0 < y ≤ A; f_{1}(y) = 0, if y ≥ B. Next we set f_{2}(y) to be

f_{2}(y) = \frac{1}{2πi} ∫_{Re(s)=2} k(−s)G(−s + β)y^{−s} \frac{ds}{s},  (24)

for y > 0. Then, as in our previous paper [11] and using (16), we have (i). f_{2}(y) ≪_{m} y^{−m} for y ≥ 1 and any positive integer m. (ii). f_{2}(y) ≪_{ϵ} 1 + y^{1−β_{0}−β−ϵ} for 0 < y ≤ 1 and any ϵ > 0.

The proofs are the same except in (ii) the integrand might have infinitely many poles on the line Re(s) = max(0, β + β_{0} − 1), coming from ∏_{v∈S} L(1 + s − β, π_{v} × π̃_{v}). However it causes no problem in view of the estimate of k(s) in (22).

The functional equation (17) leads to the following approximate functional equation: For χ ∈ X_{B}^{∗}, B ≠ O, 0 < β < 1, and any Y > 1,

L_{S,χ}(β) = ∑_{(A,S)=1} \frac{a(A)χ(A)}{(NA)^{β}}f_{1} (\frac{N(A)}{Y}) + W (π × π̃)(D^{n^{2}}N(f)N(B)^{n^{2}})^{\frac{1}{2} −β}

× ∑_{(A,S)=1} \frac{a(A)χ̄(A)}{(NA)^{1−β}}W (χ)^{n^{2}}f_{2} (\frac{N(A)Y}{D^{n^{2}}N(f)N(B)^{n^{2}}} )

= ∑_{(1)} + ∑_{(2)}, say.

Since the derivation of the above formula is almost identical to the Lemma 3.2 in [11], we omit the proof.

Now for any ideal A,

∑_{B|Q} ∑_{χ∈X_{B}^{∗}} χ(A) ≥ 0.

Indeed, write Q = Q_{1}Q_{2}, where

Q_{1} = ∏_{P^{a}||Q,P|(A,Q)} P^{a}.

Then (Q_{2}, Q_{1}A) = 1 and we have

∑_{B|Q} ∑_{χ∈X_{B}^{∗}} χ(A) = ∑_{B|Q_{2}} ∑_{χ∈X_{B}^{∗}} χ(A) = ∑_{χ∈X_{Q_{2}}} χ(A) ≥ 0,  (25)

since X_{Q_{2}} is a subgroup of (C(Q_{2}))^{Λ}.

We see that the contribution from A = O and the positivity imply that

∑_{B|Q} ∑_{χ∈X_{B}^{∗}} ∑_{(1)} ≥ ∑_{B|Q} ∑_{χ∈X_{B}^{∗}} 1 = |X_{Q}| > q^{1−rϵ},  (26)

by (12). Moreover the contribution from B = O to (26) is estimated by

∑_{χ∈X_{O}} ∑_{(1)} = ∑_{χ∈X_{O}} ∑_{(A,S)=1} \frac{a(A)χ(A)}{(NA)^{β}}f_{1} (\frac{NA}{Y})

≪ ∑_{i≥0, 2^{i}≪Y} ∑_{2^{i}≤N(A)<2^{i+1}} \frac{a(A)}{(NA)^{β}} f_{1} (\frac{NA}{Y})
≪ ∑_{i≥0, 2^{i}≪Y} (2^{i})^{1−β} ≪ Y^{1−β+ϵ},

using (14).

Hence we deduce that for q large,

∑_{B|Q,B≠O} ∑_{χ∈X_{B}^{∗}} ∑_{(1)} > q^{1−rϵ} + O(Y^{1−β+ϵ}).  (27)

Next we treat ∑_{(2)}. For a fractional ideal M = A^{−1}, and B|Q, B ≠ O, we have, as in (66), (67) of [14] and also (32) and (33) in [1]( which appeals to Deligne’s bound [4] on Kloosterman sums), that

∑_{χ∈X_{B}^{∗}} χ(M)W (χ)^{n^{2}} ≪ q^{− \frac{1}{2} +ϵ} ∑_{k (mod q)} | ∑_{χ∈X_{B}^{∗}} χ(kT )|,

where T is some fractional ideal relatively prime to B. But we have

∑_{k (mod q)} | ∑_{χ∈X_{B}^{∗}} χ(kT )| ≤ | < {P, v_{P} ∈ S, v_{P} < ∞} > |

·|image of U in (O/Q)^{∗}| · ∑_{c∈C_{1}(B)} | ∑_{χ∈X_{B}^{∗}} χ(c)| ≪ q^{ϵ} ∑_{c∈C_{1}(B)} | ∑_{χ∈X_{B}^{∗}} χ(c)|,

where C_{1}(B) = C(B)/< {P, v_{P} ∈ S, v_{P} < ∞} >. Now from the relation (χ^{0} stands for the trivial character )

∑_{A|B}( ∑_{χ∈X_{A}^{∗}} χχ^{0}) = ∑_{χ∈X_{B}} χ,

and Möbius inversion, we have

∑_{χ∈X_{B}^{∗}} χχ^{0} = ∑_{A|B}( ∑_{χ∈X_{B/A}} χ)μ(A),

where μ(·) is the Möbius function defined on ideals. Thus,

∑_{c∈C_{1}(B)} | ∑_{χ∈X_{B}^{∗}} χ(c)| ≤ ∑_{A|B} ∑_{c∈C_{1}(B)} | ∑_{χ∈X_{B/A}} χ(c)|
≪ ∑_{A|B} |X_{B/A}|\frac{|X_{B}|}{|X_{B/A}|} ≪ d(B)|X_{B}|.

Hence, we have

∑_{χ∈X_{B}^{∗}} χ(M)W (χ)^{n^{2}} ≪ q^{− \frac{1}{2} +ϵ}N(B).  (28)

We infer from this, using the estimates for f_{2}(y), (14) and (28), that

∑_{B|Q} ∑_{χ∈X_{B}^{∗}} ∑_{(2)} ≪ (q^{n^{2}})^{\frac{1}{2} −β}q^{\frac{1}{2} +ϵ} max_{B|Q,B≠O} ( ∑_{(A,S)=1} \frac{a(A)}{(NA)^{1−β}} |f_{2} (\frac{N(A)Y}{D^{n^{2}}N(f)N(B)^{n^{2}}} )|)
≪ (q^{n^{2}})^{\frac{1}{2} −β}q^{\frac{1}{2} +ϵ}q^{n^{2}β}Y^{−β} ≪ q^{\frac{n^{2}+1}{2} +ϵ}Y^{−β}.

Choosing Y = q^{\frac{n^{2}+1}{2}} , we conclude that

| ∑_{B|Q,B≠O} ∑_{χ∈X_{B}^{∗}} L_{S,χ}(β)| > q^{1−rϵ} + O(Y^{1−β+ϵ} + q^{\frac{n^{2}+1}{2} +ϵ}Y^{−β}) ≫ q^{1−rϵ},

provided ϵ is sufficiently small, q is sufficiently large and β > 1− \frac{2}{n^{2}+1}. This completes our proof of (19) and hence of Theorem 1.

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Wenzhi Luo: Department of Mathematics, Princeton University, Princeton, New Jersey 08544, USA. Zeév Rudnick: Raymond and Beverly Sackler Faculty of Exact Sciences, School of Mathematical Sciences, Tel Aviv University, Ramat Aviv 69978 Tel Aviv, Israel. Peter Sarnak: Department of Mathematics, Princeton University, Princeton, New Jersey 08544, USA.