contents the averages we consider to the fourier-whittaker coe cients of certain genuine automorphic...

CLASS GROUP TWISTS OF GLn-AUTOMORPHIC

L-FUNCTIONS

JEANINE VAN ORDER

Abstract. Fix n ≥ 2 an integer, and let F be a totally real number field. We

derive nonvanishing estimates for the finite parts of the L-functions of cuspidalGLn(AF )-automorphic representations twisted by class group characters of

quadratic extensions K/F , evaluated at arguments 12− 1

n2+1< <(s) < 1 in the

critical strip. Assuming the generalized Ramanujan conjecture at infinity, we

obtain estimates for all arguments in the critical strip. The estimates we obtain

for higher dimensions are completely new, even for the rational number field.Specializing to the central point s = 1/2 gives various arithmetical applications

such as to Deligne’s rationality conjectures for central critical values of certain

families of Rankin-Selberg L-functions for GLn(AF )×GLn−1(AF ).

Contents

1. Introduction 12. Gamma factors 73. Approximate functional equations 84. Average values 105. Fourier-Whittaker expansions 126. Relation to metaplectic Fourier-Whittaker coefficients 146.1. Metaplectic forms 146.2. Integral presentations 156.3. Bounds via spectral decompositions 19References 24

1. Introduction

Fix n ≥ 2 an integer, and let F be a totally real number field with adele ringAF . We estimate the finite parts of L-functions of cuspidal GLn(AF )-automorphicrepresentations twisted by class group characters of quadratic extensions K of F ,evaluated at arguments 1

2 −1

n2+1 < <(s) < 1 in the critical strip 0 < <(s) < 1.

In fact, if the GLn(AF )-representation is tempered at each of the real places of F ,as predicted by the generalized Ramanujan conjecture, then we obtain nonvanish-ing estimates for any argument in the critical strip. The estimates we obtain indimensions n ≥ 3 are completely new, even for the rational number field F = Q.The key input here is to use Cogdell’s theory of Eulerian integral presentations forautomorphic L-functions on GLn(AF )×GL1(AF ) to relate the off-diagonal termsin the averages we consider to the Fourier-Whittaker coefficients of certain genuineautomorphic forms on the metaplectic cover of GL2(AF ). This reduces us to the

1

2 JEANINE VAN ORDER

better-understood setting of GL2(AF ), where spectral decompositions of automor-phic forms on the metaplectic cover of GL2(AF ) can be used to derive suitablenonvanishing estimates in the style of [24], [23], and [2].

The estimates we obtain have various arithmetical applications when specializedto the central value s = 1/2. For example, we obtain a first special case of the non-vanishing hypothesis of Grobner-Harris-Lin [11, Hypothesis 0.6, Conjecture 3.7] onaverage. In this way, our result has applications to Deligne’s rationality conjecturesfor central values of certain families of GLn(AF ) × GLn−1(AF ) Rankin-SelbergL-functions, namely when the GLn−1(AF )-automorphic representation is inducedfrom a collection of finite order anticyclotomic characters of a totally imaginaryquadratic extension K of F . It seems likely that the techniques used here could bedeveloped to deal with more general cases of this hypothesis for ring class charac-ters, however we focus on the simplest case of class group characters here to simplifythe exposition. Our estimates also apply to study the analytic properties of thestandard L-function of Π, as developed in the forthcoming sequel work [9].

To describe our results, let F be any number field. Fix Π = ⊗Πv an irreduciblecuspidal GLn(AF )-automorphic representation of conductor f(Π) ⊂ OF and uni-tary central character ω = ⊗vωv of A×F /F

×. We consider the standard L-function

Λ(s, π) = L(s,Π∞)L(s,Π) =∏v≤∞

L(s,Πv)

of Π, whose local components at places v of F where Πv is unramified take the form

L(s,Πv) =

∏nj=1 (1− αj,vNv−s)

−1for v nonarchimedean∏n

j=1 ΓR(s− µj,v) for v | ∞ real∏nj=1 ΓC(s− µj,v) for v | ∞ complex.

Here, the complex numbers αj,v and µj,v correspond to the Satake parameters of Πv,and we use the shorthand notations ΓR(s) = π−

s2 Γ(s2

)and ΓC(s) = 2(2π)−sΓ(s).

This L-function Λ(s, π) is holomorphic except for simple poles at s = 0 and s = 1when Π is the trivial representation. It also satisfies the functional equation

L(s,Π) = ε(s,Π)L(1− s, Π),

where Π denotes the corresponding contragredient representation, and the ε-factorε(s,Π) is given explicitly by the formula

ε(s,Π) = (DnFNf(Π))

12−sW (Π).

Here, DF denotes the absolute discriminant of F , Nf(Π) the absolute norm ofthe conductor f(Π), and W (Π) ∈ S1 the root number. Note that the generalizedRamanujan conjecture predicts that for each place v of F for which Πv is unramified,

|αj,v| = 1 for each 1 ≤ j ≤ n if v is nonarchimedean

|<(µj,v)| = 0 for each 1 ≤ j ≤ n if v is archimedean.

We know thanks to Luo-Rudnick-Sarnak [16, Theorem 2] that we have the followingapproximations to this conjecture (for places v such that Πv is unramified):

| logN(v) |αj,v|| ≤1

2− 1

n2 + 1for each 1 ≤ j ≤ n if v is nonarchimedean

|<(µj,v)| ≤1

2− 1

n2 + 1for each 1 ≤ j ≤ n if v is archimedean.

CLASS GROUP TWISTS OF GLn-AUTOMORPHIC L-FUNCTIONS 3

Let us now assume that F is a totally real number field. Let us also assume thatthe archimedean component Π∞ of Π is spherical unless the generalized Ramanujanconjecture is known or taken for granted (cf. [17], [16]). Fix a totally imaginaryquadratic extension K of F , writing D = DK/F to denote the relative discriminant,and η = ηK/F the corresponding idele class character of F . Let χ denote a characterof the ideal class group of K, and π(χ) the automorphic representation of GL2(AF )it induces. We shall consider the GLn(AF )×GL2(AF ) Rankin-Selberg L-function,defined first for <(s)� 1 by an Euler product of the form

Λ(s,Π× π(χ)) = L(s,Π∞ × π(χ)∞)L(s,Π× π(χ)) =∏v≤∞

L(s,Πv × π(χ)v).

This completed function Λ(s,Π × π(χ)) is entire, and satisfies the following func-tional equation ([3, Proposition 4.1]): Assuming that the relative discriminant D iscoprime to the conductor f(Π), we have for χ any class group character of K that

L(s,Π× π(χ)) = ε(s,Π× π(χ))L(1− s, Π× π(χ)),(1)

where the epsilon factor ε(s,Π× π(χ)) is given by the formula

ε(s,Π× π(χ)) = (DnKNf(ΠK))

12−s ε(1/2,Π× π(χ)),

and the root number ε(1/2,Π× π(χ)) ∈ S1 by the formula

ε(1/2,Π× π(χ)) = ηω(−f(Π))W (Π).

Here we write ΠK to denote the basechange of Π to GLn(AK), as well as use theequality of L-functions Λ(s,Π×π(χ)) = L(s,ΠK ⊗χ) (as explained below). Noticethat the root number ε(1/2,Π×π(χ)) does not depend on the choice of class groupcharacter χ. We are therefore justified in dropping the χ from the notation, andwriting W (ΠK) = ε(1/2,Π×π(χ)) (for each χ) to denote this root number. Let usalso consider the quotient of archimedean factors defined by

F (s) =L(1− s, Π∞ × π(χ)∞)

L(s,Π∞ × π(χ)∞).(2)

As explained below (Lemma 2.1), this factor also does not depend on the choice ofclass group character χ, and so we are justified in dropping the χ from the notation.Notice too that F (δ) 6= 0 for any δ ∈ C in the interval 1

2 −1

n2+1 < <(δ) < 1 thanks

to the theorem of Luo-Rudnick-Sarnak [16] (see also [17]). Moreover, if we know orassume the generalized Ramanujan conjecture, then F (δ) 6= 0 for any δ ∈ C in thecritical strip 0 < <(δ) < 1.

Let us now fix an argument δ ∈ C in the critical strip 0 < <(δ) < 1. In the event

that δ = 1/2 and Π is self-contragredient (Π ∼= Π), we shall assume additionally thatthe root number W (ΠK) is not −1. That is, we shall exclude this situation whichcorresponds to forced vanishing of central values by the functional equation (1).Writing Cl(OK) to denote the ideal class group of K, and Cl(OK)∨ its charactergroup, we then estimate the corresponding average of values

XD(Π, δ) =1

# Cl(OK)

∑χ∈Cl(OK)∨

L(δ,Π× π(χ)).

To describe the estimate we obtain for this average, we must first introduce thesymmetric square L-function Λ(s,Sym2 Π) =

∏v≤∞ L(s,Sym2 Πv) (see [21], [20]).

4 JEANINE VAN ORDER

Note that at a finite place v of F where Πv is unramified, we have that

L(s,Sym2 Πv) =∏

1≤i≤j≤n

(1− αi,vαj,vNv−s

)−1.

Let us write cΠ to denote the L-function coefficients of Π, so that the finite partL(s,Π) of the standard L-function Λ(s,Π) = L(s,Π∞)L(s,Π) has the Dirichletseries expansion L(s,Π) =

∑n⊂OF cΠ(n)Nn−s for <(s) � 1. We can then write

the Dirichlet series expansion of L(s,Sym2 Π) for <(s)� 1 as

L(s,Sym2 Π) = L(2s, ω)∑

n⊂OF

cΠ(n2)

Nns=∑

m⊂OF

ω(m)

Nm2s

∑n⊂OF

cΠ(n2)

Nns.

Note that by Shahidi [20, Theorem 5.1] and more generally [21], we know thatL(1,Sym2 Π) does not vanish. The same is true for any argument L(s0,Sym2 Π)with <(s0) > 0 (in the range of absolute convergence). We refer to [4], [10], and[7] for my on the analytic properties satisfied by these L-functions. Here, we shallconsider the partial L-series defined by the following Dirichlet series expansions:Writing 1 = 1F to denote the principal class of F , i.e. so that 1 is simply the ringof integers OF when F has class number one, let us consider

L1(s,Sym2 Π) = L1(2s, ω)∑n⊂1

cΠ(n2)

Nns=∑m⊂1

ω(m)

Nm2s

∑n⊂1

cΠ(n2)

Nns.

Note that this Dirichlet series does not generally admit an Euler product. Weobtain the following estimate for the average XD(Π, δ), given in terms of thesepartial symmetric square L-values L1(s,Sym2 Π) (and the partial values L1(2s, ω)).

Theorem 1.1. Let Π be a cuspidal GLn(AF )-automorphic representation of levelf(Π) ⊂ OF and central character ω. Let K/F be a quadratic extension of relativediscriminant D ⊂ OF prime to f(Π), and write η to denote the corresponding ideleclass character of F . Let δ ∈ C be any argument in the critical strip 1 < <(δ) < 1.Assume that Π is not dihedral, in other words not induced from a Hecke characterof a quadratic extension of the basefield F . If δ = 1/2 and Π is self-contragredient,then let us assume additionally that we are not in the situation where the rootnumber W (ΠK) = η(−f(Π)) is −1. Writing DK to denote the absolute discriminant

of K, and Y = (DnKNf(ΠK))

12 the square root of the conductor of each L-function

in the average XD(Π, δ) we derive the following estimate: For any choice of ε > 0,

XD(Π, δ) =2L(2δ, ηω) · L1(4(1− δ), ω)

L1(4δ, ω)· L1(2δ Sym2 Π)

+ ωη(−f(Π))W (Π)Y 1−2δ · F (δ) · 2L(2− 2δ, ηω)L1(2− 2δ, Sym2 Π)

+OΠ,ε

(D

54 +n

2 (σ0− θ02 −1−<(δ)+ε)K

).

Here, we write 0 ≤ σ0 ≤ 1/4 to denote the best known approximation towards thegeneralized Lindelof hypothesis for GL2(AF )-automorphic forms in the level aspect(with σ0 = 0 conjectured), and 0 ≤ θ0 ≤ 1/2 the best known approximation towardsthe generalized Ramanujan/Petersson conjecture for GL2(AF )-automorphic forms(with θ0 = 0 conjectured). Taking θ0 = 7/64 via the theorem of Blomer-Brumley


[1], we can then take σ0 = 103/512 via Blomer-Harcos [2, Corollary 1] to derive

XD(Π, δ) =2L(2δ, ηω) · L1(4(1− δ), ω)

L1(4δ, ω)· L1(2δ Sym2 Π)

+ ωη(−f(Π))W (Π)Y 1−2δ · F (δ) · 2L(2− 2δ, ηω)L1(2− 2δ, Sym2 Π)

+OΠ,ε

(D

54 +n

2 ( 75512−1−<(δ)+ε)

K

).

Observe that for the central point δ = 1/2, we obtain from this latter result that

XD(Π, 1/2) =2L(1, ηω)L1(2, ω)

L1(2, ω)· L1(1 Sym2 Π)

+ ωη(−f(Π))W (Π) · F (1/2) · 2L(1, ηω)L1(1,Sym2 Π)

+OΠ,ε

(D

54 +n

2 (− 693512 +ε)

K

).

Since 5·1284·128 = 640

512 <693512 , and since F (1/2) 6= 0, it is easy to deduce that the average

XD(Π, 1/2) does not vanish for all sufficiently large ND. Consequently, as long as

we are not in the degenerate situation with Π ∼= Π and W (ΠK) = −1, we derive thefollowing result: If ND is sufficiently large, then we can find a class group characterχ of the corresponding quadratic field for which the central values L(1/2,Π×π(χ))does not vanish. In fact, we can show the following more general result.

Corollary 1.2. Let δ ∈ C be any argument in the interval 12 −

1n2+1 ≤ <(δ) < 1,

or indeed any argument in the critical stip 0 < <(δ) < 1 granted the generalizedRamanujan conjecture is assumed or known. Then, XD(Π, δ) 6= 0 for ND � 1.As a consequence, assuming as we do that Π is not self-contragredient with rootnumber W (ΠK) = η(−f(Π)) = −1 in the case of δ = 1/2, we derive the followingresult: If ND is sufficiently large, then there exists a class group character χ of thecorresponding quadratic field for which the value L(δ,Π× π(χ)) does not vanish.

Proof. Suppose first that <(δ) ≥ 1/2. In this case, the first residue term

2L(2δ, ηω) · L1(4(1− δ), ω)

L1(4δ, ω)· L1(2δ Sym2 Π)

is seen easily to be nonvanishing, in particular as the symmetric square L-value(which does not vanish for s = 1) is in the range of absolute convergence. It is alsoeasy to see that the error term in our estimate for XD(Π, δ) is bounded above by

OΠ,ε

(D

54−

n·6931024

K

),

this being the error term for δ = 1/2 (taking θ0 = 7/64 and σ0 = 103/512). Theclaim is then easy to deduce from the corresponding estimate. Suppose now that12 −

1n2+1 ≤ <(δ) < 1/2. In this case, the second (twisted) residue term

ωη(−f(Π))W (Π)Y 1−2δ · 2F (δ) · L(2− 2δ, ηω)L1(2− 2δ, Sym2 Π)

is seen easily to be nonvanishing. To prove the claim, it is then easy to verify that

Y 1−2δ = (DnKNf(ΠK))

12 (1−2δ) �Π D

54 +n

2 (σ0− θ02 −1−<(δ)+ε)K

for any δ ∈ C in the region <(δ) < 2n3

(n− 5

4 −n2

(σ0 − θ0

2

))< 1

2 . �

6 JEANINE VAN ORDER

This latter result is completely new in higher dimensions n ≥ 3, even for therational number field F = Q. It has several antecedents in the literature, not onlythe theorem of Luo-Rudnick-Sarnak [16], but also that of Barthel-Ramakrishan [3]dealing with regions in the critical strip closer to the edge <(s) = 1. In the caseof n = 2, there is the theorem of Rohrlich [19] which deals with central valuesof the completed L-function Λ(s,ΠK ⊗ χ) for χ a Hecke character with proscribedramification. There is also the estimate of Templier [22] for central derivative values

corresponding to the degenerate case of Π ∼= Π with W (ΠK) = −1 described above,as well as the estimates implicit in the shifted convolution estimates of Templier-Tsimerman [23] (see also [24] for the setting of totally real basefields). We remarkthat this setting of central derivative values, together with the more general caseof ring class Hecke characters with proscribed ramification, should be accessible bythese techniques. We also remark that in this level of generality, apart from somespecial and relatively well-understood settings coming from constructions of p-adicL-functions, the only other result for central values s = 1/2 appears to be thecriterion of Ginzburg-Jiang-Rallis [5, Main Theorem] for certain Rankin-SelbergL-functions on GLn(AF )×GLm(AF ) in terms of period integrals.

The main novelty in this work is our usage of Cogdell’s theory of Eulerian integralpresentations for automorphic L-functions for GLn(AF )×GL1(AF ) (see e.g. [6]),and more specifically the construction of a certain projection operator P = Pn2from the space of cuspidal GLn(AF )-automorphic forms to the space of cuspidalGL2(AF )-automorphic forms. In particular, we make use of the Fourier-Whittakerexpansion of such a projected GLn(AF )-automorphic cusp form (Proposition 5.1)to reduce the task of bounding our off-diagonal sums to a setting which is equivalent,up to a manageable scaling factor, to the better-understood setting of GL2(AF ).This utility of this construction does not appear to have been exploited fully inthe literature, although it plays an important role in the derivation of the Voronoisummation formulae for GLn(AF ) of Ichino-Templier [12] (see especially [12, §4.1]).Roughly speaking, starting with a cuspidal GLn(AF )-automorphic form ϕ, Cogdellconstructs a cuspidal automorphic form Pϕ on some mirabolic subgroup P (AF ) ofGL2(AF ) whose Fourier-Whittaker expansion is given by a sum over γ ∈ F× thethe L-function coefficients cΠ(γ) of Π. More generally, given ξ = ⊗ξv any Heckecharacter of F , the L-function L(s,Π⊗ ξ) on GLn(AF )×GL1(AF ) has an integralpresentation as a Mellin transform (shifted by s = 1/2) of Pϕ twisted by ξ ,

∫A×F /F

×Pϕ((

y1

))ξ(y)|y|s− 1

2 dy.(3)

Using this construction, together with the surjectivity of the archimedean Kirillovmap (see Proposition 6.1, [12, Lemma 5.1]), we can choose a decomposable vectorϕ ∈ VΠ in such a way that the off-diagonal sum in the average XD(Π, δ) can beviewed as a finite sum of Fourier-Whittaker coefficients of genuine automorphicforms on the metaplectic cover of GL2(AF ) (see Corollary 6.3). The argumentthen proceeds in the usual way via spectral decompositions to derive bounds in thestyle of [23], [24], and [2]. We refer to Theorem 6.4 and Proposition 6.5 for details.

Finally, let us note that these estimates can be used to determine more aboutthe analytic properties of the standard L-function Λ(s,Π). However, thanks tothe appearance of the potentially vanishing factor F (δ) in the second residue term,


the arguments are not direct and require substantial innovation. Such issues areexplored in the forthcoming work [9].

Acknowledgements. It is a pleasure to thank Dorian Goldfeld for many so helpfuldiscussions and clarifications; the special case of n = 2 is part of a joint collaborativework with him. It is also a pleasure to thank Peter Sarnak for helpful discussionsand encouragement, as well as Philippe Michel for helpful comments, and MichaelHarris for suggesting this type of averaging problem some time ago for its potentialapplications to Deligne’s conjectures.

2. Gamma factors

Let us keep all of the notations and setup described above. We first note that thecompleted Rankin-Selberg L-function Λ(s,Π×π(χ)) is equivalent to the L-functionassociated to Π and χ by quadratic basechange. That is, we have an equivalence ofL-functions Λ(s,Π × π(χ)) = Λ(s,ΠK ⊗ χ), where ΠK denotes the basechange ofΠ to GLn(AK). This latter GLn(AK)×GL1(AK) L-function is also analytic, and

satisfies a functional equation Λ(s,ΠK ⊗χ) = ε(s,ΠK ⊗χ)Λ(1− s, ΠK ⊗χ). Usingthis equivalence, we deduce the following description of local archimedean factors:

Lemma 2.1. If an idele class character χ = ⊗χw of a number field K is aclass group character, then the archimedean component χ∞ is trivial, and henceL(s,ΠK,∞ ⊗ χ∞) = L(s,ΠK,∞). As a consequence, for a quadratic extension ofnumber fields K/F , we have for any class group character χ of K that

L(s,Π∞ × π(χ)∞) = L(s,ΠK,∞) = L(s,Π∞)L(s,Π∞ ⊗ η∞).

In particular, the archimedean component L(s,Π∞×π(χ)∞) of Λ(s,Π×π(χ)) doesnot depend on the choice of class group character χ of K.

Proof. Recall that we have Λ(s,Π× π(χ)) = Λ(s,ΠK ⊗ χ), where

Λ(s,ΠK ⊗ χ) = L(s,ΠK,∞ ⊗ χ∞)L(s,ΠK ⊗ χ) =∏w≤∞

L(s,ΠK,w ⊗ χw)

denotes the GLn(AK)×GL1(AK) automorphic L-function obtained from twistingΠK by the idele class character χ, with Euler product taken over all places w of K.Notice too that we have the Artin decomposition

Λ(s,ΠK) = Λ(s,Π)Λ(s,Π⊗ η) = L(s,Π∞)L(s,Π∞ ⊗ η∞)L(s,Π)L(s,Π⊗ η)(4)

for the standard L-function Λ(s,ΠK) = L(s,ΠK,∞)L(s,ΠK) of ΠK .Now, it is a classical result (see e.g. [18, Corollary (6.10)]) that the archimedean

component χ∞ of any class group character χ is trivial, in which case we derive(via (4)) the immediate simplification

L(s,ΠK,∞ ⊗ χ∞) = L(s,ΠK,∞) = L(s,Π∞)L(s,Π∞ ⊗ η∞).

�

Recall that we consider the quotient of archimedean factors F (s) defined in (2)above. Let us make the following observation about F (s) for future reference:

Corollary 2.2. The function F (s) has no poles in the region 0 < <(s) < 12 + 1

n2+1 .

8 JEANINE VAN ORDER

Proof. Note that by Lemma 2.1, we have the equivalent expressions

F (s) =L(1− s, ΠK,∞)

L(s,ΠK,∞)=L(1− s, Π∞)L(1− s, Π∞ ⊗ η∞)

L(s,Π∞)L(Π∞ ⊗ η∞).

Writing (µK,j,w)w|∞ to denote the Satake parameters of the basechange componentΠK,∞, we know thanks to Luo-Rudnick-Sarnak [16, Theorem 2] that

|<(µj,v)| ≤1

2− 1

n2 + 1for each 1 ≤ j ≤ n and each archimedean place v | ∞ of F

|<(µK,j,v)| ≤1

2− 1

n2 + 1for each 1 ≤ j ≤ n and each archimedean place w | ∞ of K .

The claim is then easy to deduce from the definition L(s,ΠK,∞) =∏w|∞ L(s,ΠK,w).

�

3. Approximate functional equations

Keep the setup described above, with χ = ⊗χw a class group character of thequadratic extension K/F (viewed as idele class character of K). Notice that thefunctional equation (1) can be written in the equivalent form

L(s,Π× π(χ)) = ε(1/2,Π× π(χ)) (DnKNf(ΠK))

12−s F (s)L(1− s, Π× π(χ)),(5)

where F (s) is the quotient of archimedean factors defined in (2), and the Dirichletseries L(s,Π× π(χ)) corresponds to the Euler product over the finite places of F .Here, we use the description of Lemma 2.1 above, which in particular shows thatthis quotient does not depend on the choice of class group character χ. Fix anargument δ ∈ C in the critical strip 0 < <(δ) < 1. Let us also note here that theDirichlet series expansion of L(s,Π× π(χ)) for <(s) > 1 is given by

L(s,Π× π(χ)) =∑

m⊂OF

ωη(m)

Nm2s

∑n⊂OF

cΠ(n)cχ(n)

Nns,(6)

where each sum is taken over nonzero integral ideals of F , cΠ denotes the L-functioncoefficient of Π (so that L(s,Π) =

∑n⊂OF cΠ(n)Nn−s for <(s)� 1), and

cχ(n) =∑

A∈Cl(OK)

χ(A)rA(n)

the coefficient of the Hecke L-function L(s, χ). To be clear, this latter sum is takenover the classes A of the ideal class group Cl(OK) of K, and each rA(n) counts thenumber of ideals in the class A whose image under the relative norm homomorphismNK/F : K −→ F equals n.

Choose a smooth and compactly supported test function f ∈ C∞c (R>0), and let

k0(s) =∫∞

0f(y)ys dyy denote its Mellin transform. Assume that k0(0) = 1. We can

and do suppose that the Mellin transform k0(s) vanishes at any possible poles ofF (s) in the region 1

2 + 1n2+1 < <(s) < 1 (see Corollary 2.2). Let us then define k(s)

to be the function of s ∈ C defined by

k(s) = k0(s)L1(4(1− s− δ), ω).


Writing F (s) again to denote the quotient of archimedean factors defined in (2),we consider the following functions defined on a real variable y ∈ R>0:

V1 (y) =

∫<(s)=2

k(s)

sy−s

ds

2πi

and

V2 (y) = V2,δ (y) =

∫<(s)=2

k(−s)F (−s+ δ)

sy−s

ds

2πi.

Lemma 3.1. We have the following expression for L(s,Π× π(χ)) at any δ ∈ C:

L(δ,Π× π(χ)) =∑

m⊂OF

ωη(m)

Nm2δ

∑n⊂OF

cΠ(n)cχ(n)

NnδV1

(N(m2n)

Y

)

+ ηω(−f(Π))W (Π)Y 1−2δ∑

m⊂OF

ωη(m)

Nm2(1−δ)

∑n⊂OF

cΠ(n)cχ(n)

Nn1−δ V2

(N(m2n)

Y

).

Proof. The argument is standard (see e.g. [17, Lemma 3.2]). Consider the integral∫<(s)=2

k(s)

sL(s+ δ,Π× π(χ))Y s

ds

2πi.

Shifting the range of integration to <(s) = −2, we cross a simple pole at s = 0 ofresidue equal to L(δ,Π× π(χ)). The remaining integral∫

<(s)=−2

k(s)


ds

2πi

is then evaluated using the functional equation (5) to obtain the expression

ε(1/2,Π× π(χ))

∫<(s)=−2

k(s)

sY 1−2(s+δ)F (s+ δ)L(1− s+ δ, Π× π(χ))Y s

ds

2πi

= ε(1/2,Π× π(χ))Y 1−2δ

∫<(s)=−2

k(s)F (s+ δ)

sL(δ − s, Π× π(χ))Y −s

ds

2πi

= ε(1/2,Π× π(χ))Y 1−2δ

∫<(s)=2

k(−s)F (−s+ δ)

sL(δ + s, Π× π(χ))Y s

ds

2πi.

Hence, we have shown that

L(δ,Π× π(χ)) =

∫<(s)=2

k(s)


ds

2πi

+ ε(1/2,Π× π(χ))Y 1−2δ

∫<(s)=2

k(−s)F (−s+ δ)

sL(δ + s, Π× π(χ))Y s

ds

2πi.

Expanding out the absolutely convergent Dirichlet series (according to (6) above)then gives the stated formula for the value L(δ,Π× π(χ)). �

Lemma 3.2. The cutoff functions V1 and V2 = V2,δ decay as follows.

(i) We have that that Vj(y) = OC(y−C) for any C > 0 as y →∞ (j = 1, 2).(ii) We have that V1(y) = OB(yB) for any B ≥ 1 as y →∞.

(iii) We have that V2(y)� 1 +Oε(y1−<(δ)−δ0+ε) as y → 0, where

δ0 = maxj<(µj) ≤

1

2− 1

n2 + 1.

10 JEANINE VAN ORDER

Proof. The proof is also a standard contour argument; see [17, Lemma 3.1]. �

4. Average values

Keep the setup above, fixing δ ∈ C in the critical strip 0 < <(δ) < 1, and

writing Y = (DnKNf(ΠK))

12 to denote the square root of the conductor of each

L-function L(s,Π × π(χ)) = L(s,ΠK ⊗ χ) in the average XD(Π, δ). In the eventthat δ = 1/2 and Π is self-contragredient, let us assume additionally that the rootnumber W (ΠK) = η(−f(Π)) is not −1, so as to exclude the situation of forcedvanishing coming from the functional equation (1). Using Lemma 3.1 togetherwith the orthogonality of characters on Cl(OK), it is easy to see by inspection that

XD(Π, δ) =∑

m⊂OF

ωη(m)

Nm2δ

∑n⊂OF

cΠ(n)r(n)

NnδV1

(N(m2n)

Y

)

+ ηω(−f(Π))W (Π)Y 1−2δ∑

m⊂OF

ωη(m)

Nm2(1−δ)

∑n⊂OF

cΠ(n)r(n)

Nn1−δ V2

(N(m2n)

Y

),

where r(n) denotes the number of principal ideals in the class group Cl(OK) whoseimage under the relative norm homomorphism NK/F : K −→ F equals n ⊂ OF .

Let us now fix an OF -basis [1,√d] of OK . We can then parametrize r(n) as pairs

of F -integers a and b for which a2 − b2d = n, up to the action of units of OK :

r(n) = #{a, b ∈ OF : a2 − b2d = n

}/O×K .(7)

Expanding out in terms of such a parametrization gives us the more explicit formula

XD(Π, δ) =∑

m⊂OF

ωη(m)

Nm2δ

∑(a,b)∈O2

F/O×K

a2−b2d 6=0

cΠ(a2 − b2d)

N(a2 − b2d)δV1

(N(m2(a2 − b2d))

Y

)

+ ηω(−f(Π))W (Π)Y 1−2δ∑

m⊂OF

ωη(m)

Nm2(1−δ)

∑(a,b)∈O2

F/O×K

a2−b2d 6=0

cΠ(a2 − b2d)

N(a2 − b2d)1−δ V2

(N(m2(a2 − b2d))

Y

).

Lemma 4.1. Let Y = (DnKNf(ΠK))

12 . The contribution from b = 0 terms in

XD(Π, δ) is estimated for any choice of constant C > 0 as

2L(2δ, ηω) · L1(4(1− δ), ω)

L1(4δ, ω)· L1(2δ, Sym2 Π)

+ ωη(−f(Π))W (Π)Y 1−2δ · F (δ) · 2L(2− 2δ, ηω)L1(2− 2δ, Sym2 Π) +OC,Π(Y −C).

Proof. We estimate the contribution of the sums∑m⊂OF

ωη(m)

Nm2δ

∑a∈OFa 6=0

cΠ(a2)

Na2δV1

(N(m2a2)

Y

)

+ ηω(−f(Π))W (Π)Y 1−2δ∑

m⊂OF

ωη(m)

Nm2(1−δ)

∑a∈OFa 6=0

cΠ(a2)

Na2(1−δ)V2

(N(m2a2)

Y

).


Opening up the definition of the cutoff function V1 in the first sum, we have that

∑m⊂OF

ωη(m)

Nm2δ

∑a∈OFa 6=0

cΠ(a2)

Na2δV1

(N(m2a2)

Y

)

=∑

m⊂OF

ωη(m)

Nm2δ

∑a∈OFa6=0

cΠ(a2)

Na2δ

∫<(s)=2

k0(s)L1(4(1− s− δ), ω)

s

(N(m2a2)

Y

)−sds

2πi

=

∫<(s)=2

k0(s)L1(4(1− s− δ), ω)

sL(2δ + 2s, ωη)

L1(2δ + 2s,Sym2 Π)

L1(4δ + 4s, ω)Y s

ds

2πi,

which after shifting the contour to <(s) = −2 (crossing a simple pole at s = 0)gives us the equivalent expression

L(2δ, ωη) · L1(4(1− δ), ω)

L1(4δ, ω)· L1(2δ, Sym2 Π)

+

∫<(s)=−2

k0(s)L1(4(1− s− δ), ω)

sL(2δ + 2s, ωη)

L1(2δ + 2s,Sym2 Π)

L1(4δ + 4s, ω)Y s

ds

2πi.

Shifting the contour leftward, this latter integral is bounded using the well-knownanalytic properties of L(s, ηω) and L(s,Sym2 Π) with Stirling’s approximation the-orem as OC,Π(Y −C). Opening up the second cutoff function V2 = V2,δ gives us

∑m⊂OF

ωη(m)

Nm2(1−δ)

∑a∈OFa6=0

cΠ(a2)

Na2(1−δ)V2

(N(m2a2)

Y

)

=∑

m⊂OF

ωη(m)

Nm2(1−δ)

∑a∈OFa6=0

cΠ(a2)

Na2(1−δ)

∫<(s)=2

k(−s)F (−s+ δ)

s

(N(m2a2)

Y

)−sds

2πi

=

∫<(s)=2

k(−s)F (−s+ δ)

sL(2(1− δ) + 2s)

L1(2(1− δ) + 2s,Sym2 Π)

L1(4(1− δ) + 4s, ω)

ds

2πi

=

∫<(s)=2

k0(−s)F (−s+ δ)

sL(2(1− δ) + 2s)L1(2(1− δ) + 2s,Sym2 Π)

ds

2πi,

which after shifting the contour to <(s) = −2 gives us the equivalent expression

F (δ) · L(2(1− δ), ωη) · L1(2(1− δ),Sym2 Π)

+

∫<(s)=−2

k0(−s)F (−s+ δ)

sL(2(1− δ) + 2s)L1(2(1− δ) + 2s,Sym2 Π)

ds

2πi.

The latter integral is bounded in the same way as for the first sum as OC,Π(Y −C)for any choice of constant C > 0. Taking C � 1− 2<(δ) then shows the claim. �


Corollary 4.2. Suppose that the quadratic extension K/F is totally imaginary.Then, to estimate XD(Π, δ) it will suffice to estimate the truncated sum defined by

X†D(Π, δ) =∑

m⊂OF

ωη(m)

Nm2δ

∑a∈OF

∑b∈OF ,b 6=0

1≤Nb2≤ YN(m2d)

cΠ(a2 − b2d)

(a2 − b2d)δV1

(N(m2(a2 − b2d))

Y

)

+ ηω(−f(Π))W (Π)Y 1−2δ∑

m⊂OF

ωη(m)

Nm2(1−δ)

∑a∈OF

∑b∈OF ,b 6=0

1≤Nb2≤ YN(m2d)

cΠ(a2 − b2d)

N(a2 − b2d)1−δ V2

(N(m2(a2 − b2d))

Y

).

Proof. The result is easy to see from Lemma 4.1, using the rapid decay of thecutoff functions V1 and V2 in the region defined by N(m2(a2 − b2d)) ≥ Y . Here,we have used that K/F is totally imaginary to rule out the possible existence offundamental units, and hence to decompose the (a, b)-sum in this way. �

Our strategy to bound this truncated sum X†D(Π, δ) is twofold. First, we useCogdell’s projection operator P = Pn2 (see e.g. [6, §2]) to reduce to computationswith Fourier-Whittaker expansions of GL2(AF )-automorphic forms. This allows

us to approximate X†D(Π, δ) in terms of Fourier-Whittaker coefficients of certainautomorphic forms on the metaplectic cover of GL2(AF ) corresponding to Hilbertmodular forms of half-integral weight when K/F is totally imaginary. We thendecompose these metaplectic forms spectrally to derive an estimate (Theorem 6.4)

to bound the truncated sum X†D(Π, δ) (Proposition 6.5). In the general case oneK/F , we consider a simpler argument to derive an integral presentation for theentire average XD(Π, δ).

5. Fourier-Whittaker expansions

Let ϕ be any cuspidal GLn(AF )-automorphic form. We now consider the imagePϕ of ϕ under Cogdell’s projection operator to a certain mirabolic subgroup ofGL2(AF ) (see e.g. [6, §2.2.1]), and more precisely its Fourier-Whittaker expansion.

Let U ⊂ GLn denote the unipotent subgroup of upper-triangular matrices, and

U ′ = {u = (ui,j) ∈ U : u1,2 = 0} .

Let ψ′ denote a fixed nontrivial additive character on U(AF ), as well as its inducedcharacter on U ′(AF ). We shall assume this ψ′ is compatible with our fixed choiceof standard additive character ψ of AF /F . We consider the mirabolic subgroupP ⊂ GL2 determined by the stabilizer of this character ψ′ on U ′,

P =

{(∗ ∗0 1

)}⊂ GL2 −→ GL2×GL1× · · · ×GL1 ⊂ GLn .

Using Cogdell’s theory of Eulerian integrals (for GLn×GL1), one can constructfrom a cuspidal form ϕ on GLn(AF ) a cuspidal form Pϕ on the mirabolic subgroupP (AF ) ⊂ GL2(AF ) via the rule on p ∈ P (AF ) given by

Pϕ(p) = |det(p)|−(n−2)/2

∫U ′(F )\U ′(AF )

ϕ

(u

(p

1n−2

))ψ′(u)du.(8)


Proposition 5.1 (Cogdell). The projection Pϕ of a cuspidal GLn(AF )-automorphicform ϕ defined in (8) above determines a cuspidal automorphic form on the sub-group P (AF ) ⊂ GL2(AF ), and has the Fourier-Whittaker expansion

Pϕ (p) = |det(p)|−(n−2)/2∑γ∈F×

Wϕ

((γ

1n−1

)(p

1n−2

)).

In particular, taking p =

(y x

1

)for an idele y ∈ A×F and an adele x ∈ AF ,

Pϕ((

y x1

))= |y|−(n−2)/2

∑γ∈F×

Wϕ

((γ

1n−1

)(y

1n−1

))ψ(γx),

(9)

where ψ denotes the standard additive character on AF /F .

Proof. The first two assertions are well-known from Cogdell’s theory, see [6, §2.2.1](taking m = 1) for instance. Note as well that the calculation of [6, Lemma §2.2.1]works for any p ∈ P (AF ), though a specialization to p as in our third assertionwith x = 1 is made in the last step of the proof (cf. also [12, Lemma 2.3, § 4.1]).To be clear, taking for granted that Pϕ determines a cuspidal automorphic form onP (AF ) ⊂ GL2(AF ), and writing ϕ′(p) = |det(p)|(n−2)/2Pϕ(p) to lighten notations,we know that ϕ′ has a Fourier-Whittaker expansion of the form

ϕ′(p) =∑γ∈F×

Wϕ′

(p

(γ

1

)).(10)

Here, Wϕ′ denotes the Whittaker integral defined over the maximal unipotent sub-group N2 ⊂ GL2 by

Wϕ′(p) =

∫N2(F )\N2(AF )

ϕ′(np)ψ′(n)dn.

To derive the stated expansion for Pϕ, we open up the integral in the definition ofϕ′(p) = |det(p)|(n−2)/2Pϕ(p) in this latter expression to obtain

Wϕ′(p) =

∫N2(F )\N2(AF )

ϕ′(n′p)ψ′(n′)dn′

=

∫N2(F )\N2(AF )

∫U ′(F )\U ′(AF )

ϕ

(u

(n′p

1n−2

))ψ′(u)duψ′(n′)dn′.

As explained in [6, § 2.2.1, Lemma] (where U ′ is denoted by Yn,1), it is not hardto see that the maximal unipotent subgroup U = Nn ⊂ GLn factorizes into thesemidirect productN2nU ′. Hence, writing any element u ∈ U as a product u = n′u′

with n′ ∈ N2 and u′ ∈ U ′, and using the factorization ψ′(u) = ψ′(n′)ψ′(u′), wecan we express the latter double integral as an integral over the maximal unipotentsubgroup U :

Wϕ′(p) =

∫U(F )\U(AF )

ϕ

(u

(p

1n−2

))ψ′(u)du = Wϕ

((p

1n−2

)).

That is, we express the coefficients Wϕ′ in terms of the GLn(AF ) Fourier-Whittakercoefficients Wϕ. Substituting this back into (10) then gives the second stated


Fourier-Whittaker expansion. To derive the third expansion, we specialize to

p =

(y x

1

)=

(1 x

1

)(y

1

).

It is then easy to deduce that we have the more explicit expansion

Wϕ′

((1 x

1

)(y

1

))=∑γ∈F×

Wϕ′

((γ

1

)(y

1

))ψ(γx).

�

6. Relation to metaplectic Fourier-Whittaker coefficients

We now explain how to express the truncated sum X†D(Π, δ) defined in Corollary4.2 above in terms of Fourier-Whittaker coefficients of certain metaplectic forms.Here, we shall make use of the Fourier-Whittaker expansion (9) of the image underCogdell’s projection operator P = Pn2 of a cuspidal form ϕ on GLn, which (naturally)describes the usual expansion when n = 2. We shall also make use of Gelbart’sdescription of automorphic forms on the metaplectic cover of GL2(AF ) (see [8]).

Using such a description of X†D(Π, δ), we can then use spectral decompositionsof metaplectic forms to bound this sum in terms of the best existing bounds forFourier coefficients of GL2(AF )-automorphic forms, as well as those for the Fouriercoefficients of automorphic forms on the metaplectic cover of GL2(AF ). In fact, wederive a rather general shifted convolution bound in Theorem 6.4 below, and then

apply this to X†D(Π, δ) (Corollary 6.5) to derive our main theorem. These boundsappear to be completely new for dimensions n ≥ 3.

Notations and conventions. Let us first establish the following additional notationsand conventions for this section (in the style of [24], [2]). Throughout, F is atotally real number field of degree d = [F : Q], ring of integers OF , ring of adelesAF , absolute discriminant DF , and different dF . We trust that there should be noconfusion between the degree d = [F : Q] and the fixed OF -basis [1,

√d] of the ring

of integers of K, but nevertheless take this opportunity to point out the distinction.We embed F as a Q-algebra into F∞ =

∏v|∞ Fv = Rd via the real embeddings of

F . Hence, F×∞ = (Rd)× describes the archimedean part of A×F . Finally, we takeψ = ⊗ψv : AF −→ S1 to be the unique additive character which is trivial on F ,agrees with the function x 7→ exp (2πi(x1 + . . .+ xd)) on F∞, and for each realplace v is trivial on the the local inverse different d−1

F,v but nontrivial on v−1d−1F,v.

6.1. Metaplectic forms. Let us begin with a few words about automorphic formson the metaplectic cover of GL2(AF ), especially concerning spectral decompositionsand Fourier-Whittaker expansions. The main reference is [8], though some relevantdescription of the Fourier-Whittaker expansions we consider is given in [24] (cf. [14]).Let G denote the metaplectic cover of GL2, as constructed via cocycles in [8, §2].The group of adelic points G(AF ) is thus a central extension of GL2(AF ) by thesquare roots of unity Z2 = {±1}, and fits into the short exact sequence

0 −→ Z2 −→ G(AF ) −→ GL2(AF ) −→ 0.

This sequence splits over GL2(F ), as well as the unipotent radical of the standardBorel subgroup of GL2(F ). An automorphic form on G(AF ) is said be to genuineif it transforms nontrivially under the action of Z2, in which case it corresponds to


a Hilbert modular form of half-integral weight. A form which transforms triviallyunder Z2 is said to be non-genuine, and corresponds to a lift of a Hilbert modularform of integral weight (see [8, Proposition 3.1, and p. 57]). As shown in [8], thespace L2(GL2(F )\G(AF )) decomposes into the direct sum of a discrete spectrumL2

disc(GL2(F )\G(AF )) and a continuous spectrum L2cont(GL2(F )\G(AF )) spanned

by the analytic continuations of Eisenstein series:

L2(GL2(F )\G(AF )) = L2disc(GL2(F )\G(AF ))⊕ L2

cont(GL2(F )\G(AF )).

The space L2disc(GL2(F )\G(AF )) decomposes into a subspace of residual forms

L2res(GL2(F )\G(AF )) which occur as residues of metaplectic Eisenstein series, with

orthogonal complement given by the space of cuspidal forms L2cusp(GL2(F )\G(AF ))

(defined by the usual vanishing condition, [8, p. 53]):

L2disc(GL2(F )\G(AF )) = L2

res(GL2(F )\G(AF ))⊕ L2cusp(GL2(F )\G(AF )).

We shall be most concerned with the metaplectic theta series θ = θQ associatedto the quadratic form Q(x) = x2. This theta series has the following Fourier-Whittaker expansion (see [24, §2] and [8, Proposition 2.35]). Given y ∈ A×F anyidele with idele norm |y|, and x ∈ AF any adele, we have that

θ

((y x

1

))= |y| 14

∑γ∈F

ψ(Q(γ)(x+ iy)).

6.2. Integral presentations. We now derive integral presentations for the sum

X†D(Π, δ) defined in Corollary 4.2 above. Let us first recall the following key result,which comes from the surjectivity of the archimedean Kirillov map.

Let ϕ ∈ VΠ be a vector as above, hence viewed as a cuspidal form on GLn(AF ).Note that we shall always take this vector to be a pure tensor ϕ = ⊗vϕv. To sucha vector, we have a corresponding Whittaker function defined on g ∈ GLn(AF ) by

Wϕ (g) =

∫Nn(AF )/Nn(F )

ϕ(ng)ψ(−n)dn.

Here again, Nn ⊂ GLn denotes the maximal unipotent subgroup, and we also writeψ to denote the extension of our fixed additive character of AF /F to this subgroupin the usual way (see e.g. [6, §1]). Let us now consider the archimedean componentof this function defined on y∞ ∈ F×∞ by

Wϕ(y∞) := Wϕ

((y∞

1n−1

)).

Proposition 6.1. Let W be any function in the space L2(F×∞). There exists a(pure) vector ϕ ∈ VΠ whose corresponding archimedean Whittaker function Wϕ(y∞)satisfies the relation Wϕ(y∞) = W (y∞).

Proof. The result is shown in [13, (3.8)]. See also [12, Lemma 5.1]. �

We now apply this result as follows. Given a pure tensor ϕ ∈ VΠ, let us write thecorresponding decomposition of the global Whittaker function Wϕ into archimedeanand nonarchimedean components as follows. Given an an idele y = yfy∞ ∈ A×Fwith yf ∈ A×F,f factoring through the finite ideles A×F,f and y∞ ∈ F×∞, we write

Wϕ

((y

1n−1

))= ρϕ(yf )Wϕ(y∞) = ρϕ(yf )Wϕ

((y∞

1n−1

)).


Proposition 6.2. Fix a nonzero F -integer q, and let us use the same symbol q todenote its image under the diagonal embedding q = (q, q, . . .) ∈ A×F . Let Y � Nqbe any positive real number, and let Y∞ = (Y∞,j)

dj=1 ∈ F×∞ be any vector/element

whose (idelic) norm |Y∞| = NY∞ equals Y . Given any function W ∈ L2(F×∞),there exists a vector ϕ ∈ VΠ for which we have the integral presentation

Y14−

(n−2)2

∫AF /F

Pϕ · θ((

1Y∞

x

1

))ψ(−qx)dx =

∑γ∈F

γ2+q 6=0

cΠ(γ2 + q)

(γ2 + q)12

W

(γ2 + q

Y∞

).

Proof. Let us first consider the simpler the case of n = 2. Given φ ∈ VΠ any newvector, we have for x ∈ AF and y = yfy∞ ∈ A×F the Fourier-Whittaker expansion

φ

((y x

1

))= φ

((1 x

1

)(y

1

))=∑γ∈F×

Wφ

((γy

1

))ψ(γx).

Writing θ again to denote the metaplectic theta series associated to the quadraticform Q(γ) = γ2, we also have the (metaplectic) Fourier-Whittaker expansion

θ

((y x

1

))= |y| 14

∑γ∈F

ψ(−Q(γx))ψ(iyQ(γ)) = |y| 14∑γ∈F

ψ(−γ2x)ψ(iyγ2).

Given any nonzero F -integer q, we then compute∫AF /F

φθ

((y x

1

))ψ(−qx)dx

=

∫AF /F

φ

((y x

1

))θ

((y x

1

))ψ(−qx)dx

= |y| 14∫AF /F

∑γ1∈F×

Wφ

((γ1y

1

))ψ(γ1x)

∑γ2∈F

ψ(iyγ22)ψ(−γ2

2x)γ(−qx)dx

= |y| 14∑

γ1∈F×Wφ

((γ1y

1

)) ∑γ2∈F

ψ(iyγ22)

∫AF /F

ψ(γ1x− γ22x− qx)dx

= |y| 14∑γ∈F

γ2+q 6=0

Wφ

(((γ2 + q)y

1

))ψ(γ2iy),

where in the last step we use the orthogonality of additive characters on AF /Fto evaluate the integral. Decomposing the Whittaker integral into its archimedeanand nonarchimedean components, we then obtain the identity∫

AF /F

φθ

((y x

1

))ψ(−qx)dx = |y| 14

∑γ∈F

γ2+q 6=0

ρφ((γ2 + q)yf )Wφ((γ2 + q)y∞)ψ(iγ2y).

which after specializing to y = y∞ ∈ F×∞ (with yf = 1) gives us the simpler relation

∫AF /F

φθ

((y∞ x

1

))ψ(−qx)dx = |y∞|

14

∑γ∈F

γ2+q 6=0

ρφ(γ2 + q)Wφ((γ2 + q)y∞)ψ(iγ2y∞).

(11)

Fix Y � Nq any real number, and let Y∞ = (Y∞,j)dj=1 ∈ F×∞ be any element/vector

of norm Y . Fix any smooth function W ∈ L2(F×∞). By Proposition 6.1 above, we


can choose φ in such a way that for any y = (yj)dj=1 ∈ F×∞ we have the relation

Wφ(y∞) = W (y∞)ψ(−iy∞)ψ

(iq

Y∞

).

Making such a choice, it is then easy to see that (11) gives us the identity

Y14

∫AF /F

φθ

((1Y∞

x

1

))=

∑γ∈F

γ2+q 6=0

ρφ(γ2 + q)W

(γ2 + q

Y∞

)

=∑γ∈F

γ2+q 6=0

cΠ(γ2 + q)

(γ2 + q)12

W

(γ2 + q

Y∞

).

Let us now consider the general case of n ≥ 2. Fix a new vector ϕ ∈ VΠ, andconsider the cuspidal form on GL2(AF ) defined by Pϕ. Using the Fourier-Whittakerexpansion (9) described in Proposition 5.1 above, we compute∫

AF /F

Pϕ · θ((

y x1

))ψ(−qx)dx

=

∫AF /F

Pϕ((

y x1

))θ

((y x

1

))ψ(−qx)dx

=

∫AF /F

|y|−(n−2)

2

∑γ1∈F×

Wϕ

((γ1y

1n−1

))ψ(γ1x)|y| 14

∑γ2∈F

ψ(iyγ22)ψ(−γ2

2x)γ(−qx)dx

= |y| 14−(n−2)

2

∑γ1∈F×

Wϕ

((γ1y

1n−1

)) ∑γ2∈F

ψ(iyγ22)

∫AF /F

ψ(γ1x− γ22x− qx)dx

= |y| 14−(n−2)

2

∑γ∈F

γ2+q 6=0

Wϕ

(((γ2 + q)y

1n−1

))ψ(γ2iy)

= |y| 14−(n−2)

2

∑γ∈F

γ2+q 6=0

ρϕ((γ2 + q)yf )Wϕ

(((γ2 + q)y∞

1n−1

))ψ(γ2iy)

which after specialization to y = y∞ (with yf = 1) is the same as

|y∞|14−

(n−2)2

∑γ∈F

γ2+q 6=0

ρϕ(γ2 + q)Wϕ

(((γ2 + q)y∞

1n−1

))ψ(γ2iy∞).

Let us again take W ∈ L2(F×∞) to be any smooth function, with Y∞ ∈ F∞ andelement/vector of norm equal to Y . Choosing ϕ ∈WΠ in such a way that

Wϕ(y∞) = W (y∞)ψ(−iy∞)ψ

(iq

Y∞

),

we then derive the desired identity

Y14−

(n−2)2

∫AF /F

Pϕθ((

1Y∞

x

1

))ψ(−qx)dx =

∑γ∈F

γ2+q 6=0

ρPϕ(γ2 + q)W

(γ2 + q

Y∞

)

=∑γ∈F

γ2+q 6=0

cΠ(γ2 + q)

(γ2 + q)12

W

(γ2 + q

Y∞

).


Here, in the second equality, we use Cogdell’s integral presentation (see [6, §2.2.2])to view the L-function coefficients cΠ in terms of the Fourier-Whittaker coefficientsof Pϕ via a Mellin transform of Pϕ shifted by 1/2, as in (3) above. �

We obtain from this the following integral presentation for the truncated sum

X†D(Π, δ) defined in Corollary 4.2 above, which recall is

X†D(Π, δ) =∑

m⊂OF

ωη(m)

Nm2δ

∑a∈OF

∑b∈OF ,b 6=0

1≤Nb2≤ YN(m2d)

cΠ(a2 − b2d)

(a2 − b2d)δV1

(N(m2(a2 − b2d))

Y

)

+ ηω(−f(Π))W (Π)Y 1−2δ∑

m⊂OF

ωη(m)

Nm2(1−δ)

∑a∈OF

∑b∈OF ,b 6=0

1≤Nb2≤ YN(m2d)

cΠ(a2 − b2d)

N(a2 − b2d)1−δ V2

(N(m2(a2 − b2d))

Y

).

Corollary 6.3. Put Y = (DnKNf(ΠK))

12 , and let Y∞ ∈ F×∞ be any element/vector

of norm Y . Let us for each F -integer b in the region 1 ≤ Nb ≤ Y12 /(NmNd

12 )

choose vectors ϕ1,b ∈ VΠ and ϕ2,b ∈ VΠ in such a way that for any y∞ ∈ F×∞,

Wϕ1,b(y∞) = Ny

12−δ∞ V1 (Ny∞)ψ(−iy∞)ψ

(i · b

2d

Y∞

)and

Wϕ2,b(y∞) = Ny

δ− 12∞ V2 (Ny∞)ψ(−iy∞)ψ

(i · b

2d

Y∞

).

Then, writing m to denote both an ideal of OF and its corresponding representativein AF , we have the integral presentation

X†D(Π, δ)

= Y74−δ−

n2

∑

m⊂OF

ηω(m)

Nm72−n

∑b∈OF

1≤N(bm)≤( YNd )

12

∫AF /F

Pϕ1,b · θ((

m2

Y∞x

1

))ψ(b2dx)dx

+ηω(−f(Π))W (Π)∑

m⊂OF

ηω(m)

Nm72−n

∑b∈OF

1≤N(bm)≤( YNd )

12

∫AF /F

Pϕ2,b · θ((

m2

Y∞x

1

))ψ(b2dx)dx

.

Proof. Fix an F -integer b in the region 1 ≤ Nb ≤ Y12 /(NmNd

12 ). It is easy to see

from Proposition 6.2 above that we have the identities∫AF /F

Pϕ1,b · θ((

m2

Y∞x

1

))ψ(b2dx)dx

=

(Nm2

Y

) 14−

(n−2)2(Nm2

Y

) 12−δ ∑

a∈OF

cΠ(a2 − b2d)

(a2 − b2d)δV1

(N(m2(a2 − b2d))

Y

)


and∫AF /F

Pϕ2,b · θ((

m2

Y∞x

1

))ψ(b2dx)dx

=

(Nm2

Y

) 14−

(n−2)2(Nm2

Y

)δ− 14 ∑a∈OF

cΠ(a2 − b2d)

(a2 − b2d)1−δ V2

(N(m2(a2 − b2d))

Y

),

or equivalently∑a∈OF

cΠ(a2 − b2d)

(a2 − b2d)δV1

(N(m2(a2 − b2d))

Y

)

=

(Nm2

Y

) 34−δ ∫

AF /F

Pϕ2,b · θ((

m2

Y∞x

1

))ψ(b2dx)dx

and ∑a∈OF

cΠ(a2 − b2d)

(a2 − b2d)1−δ V2

(N(m2(a2 − b2d))

Y

)

=

(Nm2

Y

)δ− 14−

(n−2)2∫AF /F

Pϕ2,b · θ((

m2

Y∞x

1

))ψ(b2dx)dx

Substituting these two expressions into the formula for X†D(Π, δ) then gives

Y34−δ−

(n−2)2

∑

m⊂OF

ηω(m)

Nm32−(n−2)

∑b∈OF

1≤N(bm)≤( YNd )

12

∫AF /F

Pϕ1,b · θ((

m2

Y∞x

1

))ψ(b2dx)dx

+ηω(−f(Π))W (Π)∑

m⊂OF

ηω(m)

Nm32−(n−2)

∑b∈OF

1≤N(bm)≤( YNd )

12

∫AF /F

Pϕ2,b · θ((

m2

Y∞x

1

))ψ(b2dx)dx

.

�

6.3. Bounds via spectral decompositions. We now derive bounds for the sumsappearing in Proposition 6.2 and Corollary 6.3 by using spectral decompositions ofautomorphic forms on the metaplectic cover G(AF ) of GL2(AF ).

Let us begin with the following result, noting that while the statement appears tobe completely new (cf. also [24, Theorem 1.2]), the proof is reduced to the relativelywell-understood case of GL2(AF ) thanks to Proposition 6.2. Hence, the followingresult follows from the proof of [24, Theorem 1.1], which generalizes the shiftedconvolutions estimates of Templier-Tsimerman [23, Theorem 1] to the setting oftotally real number fields considered in Blomer-Harcos [2].

Theorem 6.4. Let Π = ⊗vΠv be a cuspidal GLn(AF )-automorphic representationwith L-function coefficients cΠ, so that the standard L-function of π has finite partcorresponding to the Dirichlet series L(s,Π) =

∑m⊂OF cΠ(m)Nm−s for <(s)� 1.

Let 0 ≤ θ0 ≤ 1/2 denote the best known approximations towards the generalizedRamanujan conjecture for GL2(AF )-automorphic forms (with θ0 = 0 conjectured),


and 0 ≤ σ0 ≤ 1/4 the best known approximation towards the generalized Lindelofhypothesis for GL2(AF )-automorphic forms in the level aspect (with σ0 conjec-tured). Fix q any nonzero F -integer, and let us also use the same symbol to denoteits image under the diagonal embedding q 7→ (q, q, . . .) ∈ A×F . Let W be any smooth

function in L2(F×∞), subject to the decay condition W (i) � 1 for all i ≥ 0. GivenY∞ ∈ F×∞ any vector with NY∞ � Nq, we have for any choice of ε > 0 thefollowing estimate:∑γ∈F×

cΠ(γ2 + q)

N(γ2 + q)12

W

(γ2 + q

Y∞

)

= MΠ,qNY14−

(n−2)2∞ · I(W ) +OΠ,ε

NY14−

(n−2)2∞ Nqσ0− 1

2

(Nq

NY∞

) 12−

θ02 −ε

.

Here, I(W ) denotes some linear functional in the chosen weight function W , andMΠ,q is a constant depending (only) on Π and q which vanishes unless Π is dihedral.

Proof. See the proofs of [24, Theorems 1.1, 1.2], which carry over to this settingthanks to the integral presentation of the sum given in Proposition 6.2 above. Weoutline the proof for the convenience of the reader, giving references for details foreach point in the argument. We start with the result of Proposition 6.2 above,which shows that there exists a new vector ϕ ∈ VΠ for which

∑γ∈F×

cΠ(γ2 + q)

N(γ2 + q)12

W

(γ2 + q

Y∞

)= NY

14−

(n−2)2∞

∫AF /F

Pϕ · θ((

1Y∞

x

1

))ψ(−qx)dx.

(12)

Let us now consider the genuine automorphic form Φ on the metaplectic coverG(AF ) of GL2(AF ) defined by Pϕ · θ(= θPϕ). We know (by [8]) that any genuineform Φ on G(AF ) decomposes into a linear combination of:

• An orthonormal basis of cuspidal forms {φj}j• An orthogonal basis of residual forms generated by theta series {θj}j• A basis {Ej}j of the continuous spectrum spanned by analytic continuations

of metaplectic Eisenstein series.

Let us decompose the metaplectic form Φ in such a way, writing

Φ =∑

τ∈RES

cτθτ +∑

τ∈NRES

cτfτ(13)

to denote the corresponding components of the residual spectrum and the non-residual spectrum (including Eisenstein series) respectively. One can show that Φis bounded in the Sobolev norm topology (see [24, Lemma 2.7], [23, Lemma 6.4]).This justifies inserting the sum (13) into the integral appearing in (12) above, givingrise to the corresponding contributions

IRES = IRES(W ) = NY14−

(n−2)2∞

∫AF /F

∑τ∈RES

cτθτ

((1Y∞

x

1

))ψ(−qx)dx

and

INRES = INRES(W ) = NY14−

(n−2)2∞

∫AF /F

∑τ∈RES

cτfτ

((1Y∞

x

1

))ψ(−qx)dx.


The argument of [24, Proposition 2.8] (cf. [23, §6.8]) shows that

IRES = MΠ,qNY14−

(n−2)2∞ · I(W ).

Note that we use a different choice of scaling for the integral in the discussion here.Using the bounds for Whittaker functions established in [23, §7] (see also [24, §2]),the argument of [24, Proposition 2.9] (cf. [23, §6.7]) shows that for any ε > 0,

INRES �Π,ε NY14−

(n−2)2∞ Nqσ0− 1

2

(Nq

NY∞

) 12−

θ02 −ε

.

Note again that we use a different choice of scaling for the integral here. The statedestimate follows from putting together these two bounds. �

Using this, we derive the following bounds for the truncated sum X†D(Π, δ):

Proposition 6.5. Assume that the cuspidal GLn(AF )-automorphic representationΠ is non-dihedral, and also that the quadratic extension K/F is totally imaginary.

We have the following bounds for the truncated sum X†D(Π, δ) defined in Corollary4.2 above, as described in terms of metaplectic Fourier-Whittaker coefficients inCorollary 6.3 above: For any choice of ε > 0,

X†D(Π, δ)�Π,ε D54 +n

2 (σ0− θ02 −1−<(δ)+ε)K .

Here again, 0 ≤ σ0 ≤ 1/4 denotes the best known approximation towards thegeneralized Lindelof hypothesis for GL2(AF )-automorphic forms in the level aspect,and 0 ≤ θ0 ≤ 1/2 the best known approximation towards the generalized Ramanu-jan conjecture for GL2(AF )-automorphic forms. Hence, taking θ0 = 7/64 via thetheorem of Blomer-Brumley [1], we can then take σ0 = 103/512 via the theorem ofBlomer-Harcos [2, Corollary 1] to obtain the more explicit bound

X†D(Π, δ)�Π,ε D54 +n

2 ( 75512−1−<(δ)+ε)

K .

Proof. Let us again take Y = (DnKNf(ΠK))

12 . Putting together the description of

X†D(Π, δ) given in Corollary 6.3 with the estimate of Theorem 6.4, and separating

out the scaling factor of Y14−

(n−2)2 from the start of the argument, we find that


X†D(Π, δ)

�Π,ε Y34−

(n−2)2 −<(δ)

∑

m⊂OF

1

Nm32−(n−2)

∑b∈OF ,b 6=0

1≤NmNb≤( YNd )

12

N(b2d)σ0− 12

(N(b2d)

Y

) 12−

θ02 −ε

= Y14 +

θ02 −

(n−2)2 −<(δ)+ε

∑

m⊂OF

1

Nm32−(n−2)

∑b∈OF ,b 6=0

1≤NmNb≤( YNd )

12

N(b2d)σ0− θ02 +ε

= Y14 +

θ02 −

(n−2)2 −<(δ)+εNdσ0− θ02 −ε

∑

m⊂OF

1

Nm32−(n−2)

∑b∈OF ,b 6=0

1≤NmNb≤( YNd )

12

Nb2(σ0− θ02 +ε)

.

Using that Y = (DnKNf(ΠK))

12 and that Nd = O(DK), this bound simplifies to

X†D(Π, δ)�Π,ε Dn2 ( 1

4−(n−2)

2 −<(δ))+σ0− θ02 +ε

K

∑

m⊂OF

1

Nm32−(n−2)

∑b∈OF ,b 6=0

1≤NmNb≤(YDK

) 12

Nb2(σ0− θ02 +ε)

= Dn2 ( 5

4−n2−<(δ))+σ0− θ02 +ε

K

∑

m⊂OF

1

Nm32−(n−2)

∑b∈OF ,b 6=0

1≤NmNb≤(YDK

) 12

Nb2(σ0− θ02 +ε)

.

To estimate the inner double sum, we first argue that the number of ideals m ⊂ OFand nonzero F -integers b ∈ OF subject to the constraint NmNb ≤ (Y/DK)

12 can be

estimated via the number of lattice points under the hyperbola defined by xy = r

for r = (Y/DK)12 . A classical estimate describes the number of such points as

r log(r) + (2γ − 1)r = Oε(r1+ε), where γ denotes the Euler-Mascheroni constant.

Using such an estimate together with the trivial bounds, we argue that

∑m⊂OF

1

Nm32−(n−2)

∑b∈OF ,b 6=0

1≤NmNb≤(YDK

) 12

Nb2(σ0− θ02 +ε)

�ε

(Y

DK

) 12 +ε(

Y

DK

) 12 ((n−2)− 3

2 )( Y

DK

)σ0− θ02 +ε

=

(Y

DK

) 12 + 1

2 (n− 72 )+σ0− θ02 +2ε

.


Using that YDK

= Dn2−1

K Nf(ΠK)12 = OΠ

(D

n−22

K

), this latter bound simplifies to

�Π,ε

(D

n−22

K

) 12 + 1

2 (n− 72 )+σ0− θ02 +ε

= Dn−22 (n2−

54 +σ0− θ02 +ε)

K = D54 +n2

4 −5n8 −

n2 +

(n−2)2 (σ0− θ02 +ε)

K .

Putting this together with the previous estimate then gives the stated bound. �

Hence, we derive the proof of Theorem 1.1 for the case of K/F totally imaginaryby putting together the results of Lemma 4.1 and Proposition 6.5. Now, to dealwith the general case on the quadratic extension K/F , we reduce to the CM caseas follows. Let E be a totally imaginary quadratic extension of F , and let us fixan OF -basis [1, ϑ] of the principal class in the ring of integers OE . We deduce fromthe arguments above (proving Theorem 1.1 in this case and retaining all notations)that for some constant(s) κj = κj(δ) (j = 1, 2), we have the upper bounds

Y34−δ(

n−22 )

∑b∈OFb 6=0

∫AF /F

Pϕ1,bθ

((1Y∞

x

1

))ψ(−b2ϑ2x)dx� Y κ1

and

Y34−δ−(n−2

2 )∑b∈OFb6=0

∫AF /F

Pϕ2,bθ

((1Y∞

x

1

))ψ(−b2ϑ2x)dx� Y κ2 .

Let us for each F -integer b 6= 0 choose vectors ϕ?1,b ∈ VΠ such that for all y∞ ∈ F×∞,the corresponding Whittaker integrals are specified by

Wϕ?1,b(y∞) = Ny

12−δ∞ V (Ny∞)ψ(iy∞)ψ

(−i · b

2ϑ2

Y∞

).

Similarly, let us for each b 6= 0 choose vectors ϕ?b,2 ∈ VΠ specified by

Wϕ?2,b(y∞) = Ny

δ− 12∞ V (Ny∞)ψ(iy∞)ψ

(−i · b

2ϑ2

Y∞

).

It is easy to see that for each index j = 1, 2, we have the upper bounds

Wϕ?j,b(y∞)�Wϕj,b(y∞),

where the vectors ϕj,b denote those chosen explicitly according to Corollary 6.3above. Since these vectors vary over the same metaplectic representation and itscontragredient, we then argue that it is not hard to establish the upper bounds

Y34−δ(

n−22 )

∑b∈OFb 6=0

∫AF /F

Pϕ?j,bθ((

1Y∞

x

1

))ψ(−b2ϑ2x)dx

� Y34−δ−(n−2

2 )∑b∈OFb 6=0

∫AF /F

Pϕj,bθ((

1Y∞

x

1

))ψ(b2ϑ2x)dx� Y κj

for each index j = 1, 2. On the other hand, we argue that these latter sums makeup the off-diagonal terms in the averages X−D(Π, δ) of the non-CM quadratic fieldcorresponding to E/F (where XD(Π, δ) denotes the average corresponding to E/F ).The estimate of Theorem 1.1 is then easy to deduce, using Lemma 4.1 to estimatethe corresponding b = 0 contributions in the same way.


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contents the averages we consider to the fourier-whittaker coe cients of certain genuine automorphic...

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