INTEGRAL PRESENTATIONS OF THE SHIFTED CONVOLUTION PROBLEM AND

SUBCONVEXITY ESTIMATES FOR GLn-AUTOMORPHIC L-FUNCTIONS

JEANINE VAN ORDER

Abstract. Fix n ≥ 2 an integer, and let F be a totally real number field. We reduce the shifted convo-

lution problem for L-function coefficients of GLn(AF )-automorphic forms to the better-understood setting

of GL2(AF ) via new integral presentations derived from Fourier-Whittaker expansions of projected cuspforms. As one application of this reduction, we derive the following uniform asymptotic subconvexity bound

for GLn(AF )-automorphic L-functions twisted by Hecke characters. Let Π be an irreducible cuspidal au-tomorphic representation of GLn(AF ), and let χ be a Hecke character of F of conductor q. Let θ0 be the

best known approximation to the generalized Ramanujan-Petersson conjecture for GL2(AF )-automorphic

forms; hence θ0 = 0 is conjectured, and θ0 = 7/64 is admissible by the theorem of Blomer and Brumley.Writing L(s,Π⊗χ) to denote the finite part of the standard L-function of Π⊗χ, normalized to have central

value at s = 1/2, we show that for any ε > 0 we have the upper bound

L(1/2,Π ⊗ χ) �Π,χ∞,ε Nqn4

( 12

+θ0)+ 516− θ0

2 ,

and even L(1/2,Π ⊗ χ) �Π,χ∞,ε Nqn4

( 12

+θ0)+ε if n ≥ 4. Here, the implied constants depend only on therepresentation Π, the archimedean component χ∞ of χ, and the totally real field F . This estimate appears

to the the first of its kind in higher dimensions.

Contents

1. Introduction 11.1. Analytic continuation of Dirichlet series 31.2. Application to the subconvexity problem in higher dimensions 31.3. Idea of proof 42. Fourier-Whittaker expansions 82.1. Projection operators and their expansions 82.2. Relation to L-function coefficients 92.3. Extensions to GL2(AF ) 102.4. Relations of Fourier-Whittaker coefficients 133. Integral presentations 173.1. Quadratic progressions via metaplectic theta series 183.2. Quadratic progressions via binary theta series 193.3. Shifts of arbitrary positive definite quadratic forms 193.4. Linear progressions of two cuspidal forms 204. Bounds for the shifted convolution problem 214.1. Integral presentations revisited 224.2. Whittaker functions and their decay properties 274.3. Sobolev norms 314.4. Spectral decompositions 325. Remarks on analytic continuation of shifts by quadratic forms 506. Estimates for twisted GLn(AF )-automorphic L-functions 536.1. Esquisse 546.2. Reductions via approximate functional equations 556.3. Amplified second moments 58References 64

1

1. Introduction

Let F be a totally real number field of degree d = [F : Q] and adele ring AF . Let A×F denote the ideles,with F∞ = F ⊗R ≈ Rd the archimedean component, and | · | the idele norm. Fix an integer n ≥ 2. LetΠ = ⊗vΠv be an irreducible cuspidal automorphic representation of GLn(AF ) of unitary central characterω. Let Λ(s,Π) = L(s,Π∞)L(s,Π) denote the standard L-function of Π, where for <(s) > 1 we write theDirichlet series expansion of the Euler product over finite places v of F (identified with primes v ⊂ OF ) as

L(s,Π) =∏v⊂OFv prime

L(s,Πv) =∑

m⊂OF

cΠ(m)Nm−s.

We show in this work essentially how to reduce the shifted convolution problem for sums of the L-functioncoefficients cΠ to the better-understood setting of dimension n = 2, so that various arguments such as thoseof Blomer-Harcos [6], Templier-Tsimerman [33] and others for GL2(AF ) apply to derive completely newestimates in dimensions n ≥ 3. The key insight is that such shifted convolution sums can be realizedas Fourier-Whittaker expansions of certain L2-automorphic forms of trivial central character on GL2(AF )and its two-fold metaplectic cover G(AF ). More precisely, we derive integral presentations for sums ofthe coefficients cΠ along progressions defined by linear shifts of arbitrary positive definite quadratic forms(Propositions 3.4 and 4.2 (A)), as well as for linear shifts with the L-function coefficients of an auxiliarycuspidal GLm(AF )-automorphic representation for any dimension m ≥ 2 (Propositions 3.5 and 4.2 (B)). Toderive such presentations, we use the classical projection operator Pn1 sending cuspidal automorphic formson GLn(AF ) to cuspidal automorphic forms on the mirabolic subgroup P2(AF ) ⊂ GL2(AF ), together withCogdell’s theory of Eulerian integral presentations for L-functions on GLn(AF )×GL1(AF ), and the abilityto choose a pure tensor ϕ = ⊗vϕv ∈ VΠ whose nonarchimedean local components ϕv are each essential Whit-taker vectors (as we can at ramified places thanks to Matringe [28]). That is, we use purely representationtheoretic methods to describe the shifted convolution problem for GLn(AF ) in this unconventional way. TheFourier-Whittaker expansions of such projected pure tensors can be related via a shifted Mellin transformto the Dirichlet series L(s,Π), as we explain for Theorem 2.2 and Corollary 2.3 below. Hence, the sumswe wish to consider can be related to the Fourier-Whittaker coefficients of certain L2-automorphic formson the mirabolic subgroup P2(AF ) of GL2(AF ) . Using strong approximation and Iwasawa decompositionto lift to a well-defined L2-automorphic form of trivial weight and central character on GL2(AF ) (in §2.3below), relating Fourier-Whittaker coefficients (see Propositions 2.8, 2.9, 2.10, and 4.2 especially), we canthen reduce the shifted convolution problem for GLn(AF ) to the less mysterious setting of dimension n = 2.That is, decomposing a certain lifted L2-automorphic form on GL2(AF ) or G(AF ) which represents theshifted convolution problem spectrally, we can justify the convergence of the corresponding spectral coef-ficients by either a standard argument with Sobolev norms for the case of dimension n = 2 (Lemma 4.8),or else via a direct estimate in terms of the inner product with some Poincare series (Proposition 4.9) inthe style of [31, §2] for the case of dimensions n ≥ 3. This in effect allows us to derive bounds for theshifted convolution problem for GLn(AF ) in terms of existing estimates for Whittaker functions and genericbounds for Fourier-Whittaker coefficients of automorphic forms on GL2(AF ) and G(AF ) as predicted bythe generalized Ramanujan conjecture and the generalized Lindelof hypothesis respectively. In this way, wederive the following uniform estimates for the shifted convolution problem for coefficients of automorphicL-function coefficients in all dimensions n ≥ 2, which are completely new in dimensions n ≥ 3. Let us writeF×∞,+

∼= Rd>0 to denote the totally positive archimedean adeles.

Theorem 1.1. Fix n ≥ 2 an integer. Let Π = ⊗vΠv be a cuspidal GLn(AF )-automorphic representationof unitary central character ω and L-function coefficients cΠ. Let α be any nonzero F -integer, which wealso identify with its image under the diagonal embedding α → (α, α, · · · ) ∈ A×F . Let Y∞ ∈ F×∞ be anyarchimedean idele of idele norm |Y∞| > |α|. Writing 0 ≤ θ0 ≤ 1

2 to denote the best uniform approximationtowards the generalized Ramanujan conjecture for GL2(AF )-automorphic forms (with θ0 = 0 conjectured),and 0 ≤ δ0 ≤ 1

4 that towards the generalized Lindelof hypothesis for GL2(AF )-automorphic forms in the levelaspect (with δ0 = 0 conjectured), we derive the following uniform estimates

(A) Let f(a1, . . . , ak) be an arbitrary positive definite F -rational quadratic form in k ≥ 1 many variables,and p(a1, · · · , ak) a spherical polynomial for f (possibly trivial). Let W be any smooth function

2

of y∞ ∈ F×∞,+ ∼= Rd>0 satisfying satisfying the moderate decay condition W (|y∞|) = O(|y∞|κ) for

|y∞| → 0 for some 0 < κ < 1, which decays rapidly for |y∞| → ∞, and which is is square integrableor else is compactly supported. Let us also assume that W (i) � 1 for all i ≥ 1. Then for any choiceof ε > 0, we have the estimate

∑a=(a1,...,ak)∈OkF

p(a) · cΠ(f(a) + α)

|f(a) + α| 12W

(f(a) + α

Y∞

)= OΠ,α,f,p,W (1) +OΠ,ε

(|Y∞|

2−k4 +

θ02 +ε|α|δ0+

θ02 −ε

).

Here, the leading term OΠ,α,f,p,W (1) = MΠ,αI(W ) is given by a constant MΠ,α ≥ 0 times a linearfunctional I(W ) in W ; this constant MΠ,α vanishes unless the F -integer α is totally positive andthe Dirichlet series corresponding to the shifted convolution sum (as described below) has a pole. Inthe setting where f(x) = x2, this constant term vanishes unless the symmetric square L-functionL(s,Sym2 Π) has a pole at s = 1, equivalently unless Π is orthogonal (and hence self-dual). Finally,the exponent δ0 can be replaced with θ0 when the number of variables k ≥ 2 is even.

(B) Let m ≥ 2 be an integer, and π = ⊗πv any cuspidal GLm(AF )-automorphic representation withL-function coefficients cπ. Let Wj be smooth functions of y∞ ∈ F×∞,+

∼= Rd>0 for j = 1, 2, each

satisfying the moderate growth condition Wj(|y∞|) = O(|y∞|κj ) as |y∞| → 0 for some 0 < κj < 1for each of j =, 2, which decay rapidly for |y∞| → ∞, and which are are square integrable or else is

compactly supported. Let us also assume that W(i)j � 1 for all i ≥ 1. Then for any choice of ε > 0,

we have the estimate

∑γ1,γ2∈F×γ1−γ2=α

cΠ(γ1)cπ(γ2)

|γ1γ2|12

W1

(γ1

Y∞

)W2

(γ2

Y∞

)

= OΠ,π,W1,W2,α(1) +OΠ,π,ε

(|Y∞|

12 +

θ02 +ε |α|θ0+ε

).

Here, the constant term vanishes unless the Rankin-Selberg L-function L(s,Π× π) has a pole.

Notice in particular that the exponents 0 ≤ θ0 ≤ 12 and 0 ≤ δ0 ≤ 1

4 describe the best existing approxi-mations towards the generalized Ramanujan conjecture and the generalized Lindelof hypothesis respectivelyfor GL2(AF )-automorphic forms, rather than the corresponding exponents for GLn(AF )-automorphic forms(!). Hence, we can take θ0 = 7/64 thanks to the theorem of Blomer-Brumley [5], and δ0 = 103/512 thanks toBlomer-Harcos [6, Corollary 1]. Let us also remark that while we follow the works of Blomer-Harcos [6] andTemplier-Tsimerman [33] (cf. also [34]) closely in places to derive these bounds, the work is closer in spiritto those of Bernstein-Reznikov [2], [3], [4] and Krotz-Stanton [24] using analytic continuation of automor-phic representations and other representation theoretic methods. Although we do not explore links to theseseminal works here, we do give the following immediate applications to Dirichlet series and subconvexityestimates for automorphic L-functions on GLn(AF )×GL1(AF ) via more standard analytic arguments.

1.1. Analytic continuation of Dirichlet series. In the distinct setting where the F -integer α is takento be zero for the sums appearing in Theorem 1.1 (1), so that there is no shift, we also explain in Corollary5.3 below how to derive the analytic continuation of the Dirichlet series

D(s,Π, f, p) :=∑

a1,··· ,ak∈OFf(a1,...,ak)6=0

p(a1, . . . , ak)cΠ(f(a1, . . . , ak))

|f(a1, . . . , ak)|s,(1)

which is defined a priori only for s ∈ C with <(s) sufficiently large. Here, we also use integral presentationsderived from the projection operator Pn1 , together with a semi-classical unfolding argument, but omit a fullydetailed account for simplicity. Let us remark however, as in the setting of [6, Theorem 3, Remark 14],that Selberg [32] posed the question of showing the analytic continuation of such Dirichlet series associatedto shifted convolution sums. It would be interesting to give a more precise account of this variation of the

3

Rankin-Selberg theory, making clear the analytic properties (such as existence or location of poles) from theGL2(AF ) and metaplectic Eisenstein series for the cases of k even and k odd respectively.

1.2. Application to the subconvexity problem in higher dimensions. Using a variation of Theorem1.1 (B) to estimate an amplified second moment in the style of Blomer-Harcos [6, §3.3] (cf. [36]) and Cogdell[15], as well as unpublished work of Cogdell-Piatetski-Shapiro-Sarnak [11], we can also derive the followingestimate for central values of the L-functions of cuspidal GLn(AF )-automorphic representations twisted byidele class character os F . To be more precise, we derive the following subconvexity estimate, essentially asan application of Theorem 1.1 (B):

Theorem 1.2 (Theorem 6.6, Corollary 6.7). Let Π = ⊗Πv be an irreducible cuspidal GLn(AF )-automorphicrepresentation of unitary central character ω, and let χ be a Hecke character of F of conductor q ⊂ OF . Wehave for any choice of parameters L = Nqu with 0 < u < 1 and ε > 0 the estimate

|L(1/2,Π⊗ χ)|2 �Π,χ∞,ε Nq1+εL−1 + Nqn2 ( 1

2 +θ0)+εL52 +θ0+ε.

For instance, taking u = 1/4− θ0/2 gives us the estimate

L(1/2,Π⊗ χ)�Π,χ∞,ε Nq38 +

θ04 + Nq

n4 ( 1

2 +θ0)+(5−8θ0)

16 +ε.

Taking u = (1− 6θ0)/(14− 4θ0) when n = 3 gives us the estimate

L(1/2,Π⊗ χ)�Π,χ∞,ε Nq13+2θ0

2(14−4θ0)+ε

+ Nq34 ( 1

2 +θ0)+(5−28θ0−12θ20)

4(14−4θ0) ,

and taking u = 0 when n ≥ 4 gives us the estimate

L(1/2,Π⊗ χ)�Π,χ∞,ε Nqn4 ( 1

2 +θ0)+ε.

Note that in all of these estimates, 0 ≤ θ0 ≤ 1/2 denotes the best uniform approximation to the generalizedRamanujan conjecture for all GL2(AF )-automorphic forms at the finite places, and in particular that onecannot simply assume θ0 = 0 if the generalized Ramanujan conjecture is known for the given cuspidal auto-morphic representation Π = ⊗vΠv of GLn(A) (e.g. for important spectral cases where Π is cohomological).We give a high-level sketch of the proof of this latter application via amplified second moments following[6] in §6.1 below. As we explain in the next paragraph, the key step in our approach here is to reduce theproblem to one of estimating the Fourier-Whittaker coefficients of some L2-automorphic form on GL2(AF )via the derivation of suitable integral presentations. One this key reduction step is achieved, the claimedestimates can be derived by a relatively straightforward technical generalization of the main theorems ofBlomer-Harcos [6, Theorems 1 via Theorem 3]. The bound we derive in this way appears to be completelynew in higher dimensions, with some scope to improve the exponent. This approach also appears to leadto some subtler questions about cuspidal automorphic forms on the mirabolic subgroup P2(AF ) and theirinclusion in L2(GL2(F )\GL2(AF ),1). That is, certain extensions of L2-automorphic forms on the mirabolicsubgroup P2(AF ) to L2-automorphic forms on GL2(AF ) play a crucial role in our reduction, as does thesubsequent study of relations between Fourier-Whittaker coefficients and compact restictions with respectto a chosen fundamental domain for the action of GL2(OF ) on GL2(F∞). Our findings here are by no meansexhaustive, and there appears to be scope to develop a better theory of the arithmetic of such mirabolicautomorphic forms. As well, it seems that our discussion of the analytic continuation could be developedto derive some triple-product integral presentations of the sums we consider. This in turn suggests that thetechniques of Venktesh [35] and Michel-Venkatesh [29] could perhaps also be extended to this higher-ranksetting using variations of the ideas developed here.

1.3. Idea of proof. The proofs of Theorems 1.1 and 6.6 reduce the corresponding problems to derivingbounds for Fourier-Whittaker coefficients of certain continuous and compactly supported L2-automorphicforms on GL2(AF ) or its two-fold metaplectic cover G(AF ). Once such a reduction is established, it remainsto show that the coefficients in the corresponding spectral expansion are bounded. If the form is smoothor sufficiently smooth that its Sobolev norm converges, then a relatively standard argument (Lemma 4.8)can be used to bound these spectral coefficients in terms of the spectral parameters of the correspondingbasis forms. In the main case we consider, where the form is continuous and compactly supported – butnot in general smooth – we introduce an auxiliary Poincare series Pφ to derive bounds in terms of the

4

inner product with Pφ (Proposition 4.9). Although the bounds we derive in this more general setting carrysome dependence on the chosen real parameter Y = |Y∞| > |α|, we can in fact proceed after this point todecompose the form spectrally to derive the stated bounds in terms of classical Whittaker functions andFourier-Whittaker coefficients of automorphic forms on GL2(AF ) and G(AF ). Let us now describe thesequence of steps required to make such a reduction to GL2(AF ), written in such a way that an expert couldreconstruct the proof of the main results without much trouble from the existing theory for GL2(AF ). Here,we shall assume we are in the generic case of dimension n ≥ 2, where we make modifications to both theprojected form Pn1ϕ and the second form (e.g. the metaplectic theta series) we multiply with to derive theintegral presentation for the corresponding shifted convolution problem.

(1). The first observation is that the shifted convolution sums of Theorem 1.1 (A) and (B) can be encodedin the Fourier-Whittaker coefficients at α ∈ OF of certain L2-automorphic forms on the mirabolic subgroup

P2(AF ) =

{(y x0 1

): y ∈ A×F , x ∈ AF

}⊂ GL2(AF )

or its metaplectic cover via the classical projection operator Pn1ϕ taking L2-automorphic forms on GLn(AF )to the L2-automorphic forms on P2(AF ). This operator Pn1 could be thought of as a kind of partial Whittakertransform. To be more precise, writing Yn,1 to denote the unipotent radical of the standard parabolicsubgroup corresponding to the partition (2, 1, . . . 1) of n, so that the standard unipotent subgroup Nn ⊂ GLnof upper triangular matrices decomposes into a semi-direct product Nn ∼= N2 nYn,1, this Pn1 takes a cuspidalautomorphic form ϕ on GLn(AF ) to the function on p ∈ P2(AF ) defined by

Pn1ϕ(p) = |det(p)|−(n−22 )∫Yn,1(F )\Yn,1(AF )

ϕ

(y

(p

1n−2

))ψ−1(y)dy.

Here, ψ denotes the standard additive character on AF , extended in the usual way to Nn. Now, we canand do replace the cuspidal automorphic form ϕ with a pure tensor ϕ = ⊗vϕv ∈ VΠ in the correspondingrepresentation space VΠ of Π = (Π, VΠ). That the shifted convolution sums of Theorem 1.1 (A) and (B)can be expressed in terms of the Fourier-Whittaker coefficients of certain L2-automorphic forms on P2(AF )or its two-fold metaplectic cover by this operator is not obvious. However, it is relatively simple to deducefrom the following powerful and arguably deep results applied the choice of pure tensor ϕ = ⊗vϕv ∈ VΠ:

(i). Using the surjectivity of the archimedean local Kirilov map, we can and do choose ϕ∞ = ⊗v|∞ϕv insuch a way that the corresponding Whittaker function Wϕ(y∞) on y∞ ∈ F∞ is given by the chosen weightfunction W (y∞), or rather Wj(y∞) (j = 1, 2) for the case Theorem 1.1 (B).

(ii). Using the theory and existence of essential Whittaker vectors (completed recently by Matringe [28]),we can and do choose each of the nonarchimedean local vectors ϕv ∈ VΠv to be an essential Whittaker vector.Essentially, this allows us to relate the local L-factor L(s,Πv) to a shifted Mellin transform of ϕv.

(iii). Using Cogdell’s theory of Eulerian integral presentations for the L-function Λ(s,Π) = L(s,Π∞)L(s,Π),or more generally for Λ(s,Π⊗ ξ) = L(s,Π⊗ ξ∞)L(s,Π⊗ ξ) with ξ = ⊗vξv a Hecke character of F , we canthen deduce from the corresponding integral presentation for the finite part

L(s,Π⊗ ξ) =∑

m⊂OF

cΠ(m)ξ(m)Nm−s (first for <(s)� 1)

as

L(s,Π⊗ ξ) =∏v<∞

L(s,Πv) =

∫A×F,f

Wϕ

((hf

1n−1

))ξ(hf )|hf |s−(n−1

2 )dhf

that the finite coefficients in the Fourier-Whittaker expansion of the projected pure tensor Pn1ϕ are given byshifts of the L-function coefficients cΠ. In particular, the projected form Pn1ϕ for ϕ chosen according to (ii)and (iii) has the Fourier-Whittaker expansion for any idele y = yfy∞ ∈ A×F

∼= A×F,f ×F×∞ and adele x ∈ AF

5

(Corollary 2.3):

Pn1ϕ((

y x1

))= |y|−(n−2

2 )∑γ∈F×

cΠ(γyf )

|γyf |n−12

Wϕ(γy∞)ψ(γx).

It is easy to deduce integral presentations from this latter expansion (see Propositions 3.4 and 3.5). Thatis, writing θf,p to denote the (metaplectic) theta series associated to the quadratic form f(a) = f(a1, · · · ak),

with θf,p = T−1θf,p its image under the Hecke operator at infinity sending x → −x ∈ AF , the shiftedconvolution sum of Theorem 1.1 (A) can be realized as a Fourier-Whittaker coefficient∫

AF /F

Pn1ϕ · θf,p((

1Y∞

x

1

))ψ(−αx)dx

=

∫I∼=[0,1]d⊂F∞

Pn1ϕ · θf,p((

1Y∞

x∞1

))ψ(−αx∞)dx∞.

Similarly, the shifted convolution sum of Theorem 1.1 (B) can be realized as a Fourier-Whittaker coefficient∫AF /F

Pn1ϕ · Pm1 ϕ′((

1Y∞

x

1

))ψ(−αx)dx

=

∫I∼=[0,1]d⊂F∞

Pn1ϕ · Pm1 ϕ′((

1Y∞

x∞1

))ψ(−αx∞)dx∞

for some similarly chosen pure tensor ϕ′ = ⊗vϕ′v ∈ Vπ in the representation space corresponding to thecuspidal automorphic representation π = (π, Vπ) on GLm(AF ).

(2). The second observation (which applies in all dimensions n ≥ 2) is that we may use a combination of thestrong approximation theorem and the Iwasawa decomposition of GL2(F∞) to extend the L2-automorphicform Pn1ϕ on P2(AF ) to an L2-automorphic form on GL2(AF ) with trivial central character and trivial rightaction by the maximal compact subgroup K. We note that this procedure works for an arbitrary automorphicform on GL2(AF ) or its two-fold metaplectic cover G(AF ), and is not unlike taking the classical descent ofsuch a form. To be more precise, we show in Theorem 2.5 how a combination of the strong approximationtheorem with the Iwasawa decomposition allows us to derive a certain unique factorization of elementsg ∈ GL2(AF ), which we then use to define an extension

Pn1ϕ ∈ L2(GL2(F )\GL2(AF ),1)K

to GL2(AF ) in Definition 2.6 and Proposition 2.7. This extension is analogous to the classical passagefrom Hilbert Maass forms on the d-fold upper-half plane Hd ∼= P2(F∞) to automorphic forms on GL2(AF ),and requires that we fix a fundamental domain for the action of GL2(OF ) on GL2(F∞). This choice offundamental domain corresponds in a natural way to choosing a collection of fundamental domains for theaction of SL2(Z) on H indexed by the real places of F . Moreover, it imposes some constraints on thearchimedean idele y∞ = (y∞,j)

dj=1 ∈ F×∞ and adele x∞ = (x∞,j)

dj=1 ∈ F∞ coordinates of the mirabolic

matrices P2(F∞) ∼= Hd, and moreover the extended function Pn1ϕ is not smooth along the boundary of thechosen fundamental domain. However, we explain in §2 and §4 how to surmount these issues.

(3). The third step of the reduction is to use the various automorphy properties of both the mirabolic cusp

form Pn1ϕ and its extension Pn1ϕ to GL2(AF ) to show (in Lemma 2.8, Proposition 2.9, and Proposition 4.2)that for any Y∞ ∈ F×∞ in our chosen fundamental domain, we can relate the unipotent integrals∫

I∼=[0,1]d⊂F∞Pn1ϕ · θf,p

((Y∞ x∞

1

))ψ(−αx∞)dx∞

=

∫I∼=[0,1]d⊂F∞

Pn1ϕ · θf,p((

Y∞ x∞1

))ψ(−αx∞)dx∞

6

and ∫I∼=[0,1]d⊂F∞

Pn1ϕ · Pm1 ϕ′((

Y∞ x∞1

))ψ(−αx∞)dx∞

=

∫I∼=[0,1]d⊂F∞

Pn1ϕ · Pm1 ϕ((

Y∞ x∞1

))ψ(−αx∞)dx∞.

The arguments for this step are elementary, but delicate. For instance, we require the automorphy of themirabolic form Pn1ϕ, and remark that this L2-automorphic form on P2(AF ) could not simply be replaced bya random function (e.g. a formal power series having values of the Mobius function as coefficients). Here, weneed to assume in addition that Wϕ(y∞) = Wϕ(−y∞). We also need to fix a smooth partition of unity anddyadic decomposition to derive a more delicate integral presentation for the shifted convolution problem, asexplained in Proposition 4.2 below.

(4). The penultimate step of the reduction, given in Theorem 2.10, Proposition 4.2, and Proposition 4.9, isto argue that the spectral coefficients of the function defined by

Φ =

{Pn1ϕ · θf,p for Theorem 1.1 (A)

Pn1ϕ · Pm1 ϕ′ for Theorem 1.1 (B)

are bounded. The theta series θf,p in this latter definition is really some sort of descent defined in a similar

way as the lift Pn1ϕ via Iwasawa decompostion of the metaplectic theta series θf,p introduced above. We referto Propositions 4.1 and 4.2 below for more details. After this point, there are at least two ways to proceed.One is to take the inner product with some suitably chosen Poincare series Pφ whose underlying smoothfunction φ ∈ C∞(N2(AF )Z2(AF )\GL2(AF );ψ) is compactly supported, and moreover supported only onsome domain J = J(Y∞)× I ⊂ P2(F∞) ∼= Hd where the function Φ is smooth. As we show in Proposition4.9 below, some relatively simple calculations with unfolding and orthogonality then allow us to reduce tothe standard argument for n = 2 described in Lemma 4.8 to derive bounds for the coefficients. As explainedin a subsequent remark, we could also consider the convolution Φ ? K with a smoothing kernel K to derivethe same bounds for the spectral coefficients of Φ.1

(5). Taking for granted the convergence of spectral coefficients, we can then decompose Φ on G(AF ) orGL2(AF ) spectrally to derive the claimed bounds. Again, this accounts for how all of the exponentsappearing in the bounds we derive come from the best exponents in approximations towards the gener-alized Ramanujan conjecture for GL2(AF )-automorphic forms and the generalized Lindelof hypothesis forGL2(AF )-automorphic forms in the level aspect.

(6). The application to subconvexity estimates is derived via a direct generalization of [6, Theorem 3] givenin Corollary 6.5 and Theorem 6.6 below via the shifted convolution sums estimate of Theorem 1.1 (B). Wegive a sketch of the method in § 6.1 via amplified second moments (following [6, §3]), but note that it isa well-known and standard application of bounds for the corresponding shifted convolution problem. In anutshell, a minor technical variation of Propositions 3.5 and 4.2 (B) above allows us to derive a suitableintegral presentation for the corresponding off-diagonal term as a non-constant Fourier-Whittaker coefficientof some continuous and compactly supported L2-automorphic form on GL2(AF ). Decomposing this formspectrally according to the discussion above then allows us to extend the main estimate of [6, Theorem 3]to this setting to derive the stated subconvexity bounds.

Outline of the paper. We first review Fourier-Whittaker expansions in §2, leading to a description of theexpansion of Pn1ϕ for ϕ ∈ VΠ a carefully chosen pure tensor, and give the key arguments with extensionsto GL2(AF ) and relations of the Fourier-Whittaker coefficients. We then explain how to derive integralpresentations via mirabolic coefficients in the §3, starting with the special case of the metaplectic thetaseries associated to the quadratic form q(x) = x2 (Proposition 3.2), followed by the classical theta seriesassociated to a positive definite binary quadratic form (Proposition 3.3), the general case of a theta seriesassociated to a positive definite quadratic form (Proposition 3.4), and then the setting of linear shifts oftwo forms (Proposition 3.5). We then prove Theorem 1.1 in §4, starting with some discussion of dyadic

1I am grateful to Peter Sarnak and to Akshay Venkatesh for suggesting these respective approaches to me.7

decompositions and modified integral presentations (leading to Proposition 4.2) as required to deal with ourfixed choice of fundamental domain for the lifting of the mirabolic form. This part of the paper also containskey argument concerning spectral decompositions and uniform bounds for coefficients (namely Lemma 4.8and Proposition 4.9). The remaining bounds for Whittaker functions and Fourier-Whittaker expansionsare then relatively standard, but nevertheless given in detail. After showing these main bounds for theshifted convolution problem , we explain in §5 how to derive analytic continuations for the correspondingDirichlet series. Finally in §6, we derive subconvexity bounds via amplified second moments in the style ofBlomer-Harcos [6]. As we explain, this latter application is derived from a variation of Theorem 1.1 (B).

Notations. We write the set of real embeddings of F as (τj)dj=1, and embed F as a Q-algebra into F∞ = Rd

via these embeddings. We also write F×∞,+ = Rd>0 to denote the set of totally positive elements of F∞, and

F diag∞,+ = {(x, . . . , x) : x ∈ R>0}. We decompose the idele group A×F into its corresponding nonarchimedean

and archimedean components as AF = A×F,f × F×∞, so that A×F,f denotes the finite ideles. We write | · | todenote the idele norm, which on idele representatives of integral ideals m ⊂ OF coincides with the absolutenorm Nm = [OF : mOF ]. We also write ψ = ⊗ψv to denote the standard additive character of AF /F .Hence, ψ : AF → F is the unique continuous additive character on AF which is trivial on F , agrees withx 7→ exp(2πi(x1 + · · ·+xd)) on F∞, and at each nonarchimedean completion Fp is trivial on the local inverse

different d−1F,p but nontrivial on p−1d−1

F,p. In general, we use many of the same notations and conventions as in

Blomer-Harcos [6], but use F instead of K to denote the totally real field. However, for each nonarchimedeanplace v of F where Πv is ramified, we choose a measure on F×v in such a way that the local zeta integralsattached to our chosen essential Whittaker vectors are characterized in terms of the local Euler factor L(s,Πv)for the special case of m = 1 corresponding to a twist by the trivial idele class character as in Matringe [28,Corollary 3.3] (see Theorem 2.2).

Acknowledgements. I should like to thank Valentin Blomer, Jim Cogdell, Paul Garrett, Andre Reznikov,Peter Sarnak, Werner Muller, Akshay Venkatesh, and Jean-Loup Waldspurger for various helpful exchanges.I am especially grateful to Valentin Blomer, Dorian Goldfeld, Kevin Kwan, Werner Muller, Waldspurger,and anonymous referees/correspondents for careful readings of this work – including pointing out a gap inan earlier version – and also for comments which have lead to vast improvements.

2. Fourier-Whittaker expansions

Let Π = ⊗vΠv be an irreducible cuspidal automorphic representation of GLn(AF ) of unitary centralcharacter ω = ⊗vωv. Let ϕ = ⊗ϕv ∈ VΠ be a pure tensor in the space of smooth vectors VΠ, chosen so thateach of the nonarchimedean local vectors ϕv is an essential Whittaker vector, as we can thanks to Matringe[28, Theorem 1.3]. Let us also fix a nontrivial additive character ψ = ⊗vψv on AF /F , which we extend inthe usual way to the corresponding quotient Nn(F )\Nn(A) of the maximal unipotent subgroup Nn ⊂ GLn(see e.g. [13], [14]). Again, we shall take this ψ to be the standard additive character throughout all of thediscussion that follows. Recall that for g ∈ GLn(AF ), we have the usual Fourier-Whittaker expansion

ϕ(g) =∑

γ∈Nn−1(F )\GLn−1(F )

Wϕ

((γ

1

)g

),

where

Wϕ(g) = Wϕ,ψ(g) =

∫Nn(F )\Nn(AF )

ϕ(ng)ψ−1(n)dn.

2.1. Projection operators and their expansions. We now review the construction and properties of theprojection operator Pn1 , in particular as it relates to Cogdell’s theory of Eulerian integrals for automorphicL-functions on GLn×GL1. We refer to [13, Lecture 5] and [14, §2.2.1] for details.

Let Yn,1 ⊂ GLn denote the unipotent radical of the standard parabolic subgroup attached to the partition(2, 1, . . . , 1) of n. Note that we have the semi-direct product decomposition Nn = N2 n Yn,1. Note as wellthat our fixed additive character ψ extends to the quotient Yn,1(F )\Yn,1(AF ) via the inclusion Yn,1 ⊂ Nn,

8

and also that Yn,1 is normalized by GL2 ⊂ GLn. Let P2 ⊂ GL2 denote the mirabolic subgroup determinedby the stabilizer in GL2 of ψ,

P2 =

{(∗ ∗0 1

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

Fix a pure tensor ϕ = ⊗vϕv ∈ VΠ as above. We shall also view ϕ as its corresponding cuspidal automorphicform on GLn(AF ). The projection operator Pn1 from the space of cuspidal automorphic forms on GLn(AF )to the space of cuspidal automorphic forms on P2(AF ) ⊂ GL2(AF ) is defined for any p ∈ P2(AF ) by thepartial Whittaker integral

Pn1ϕ(p) = |det(p)|−(n−22 )∫Yn,1(F )\Yn,1(AF )

ϕ

(y

(p

1n−2

))ψ−1(y)dy,(2)

where 1m for a positive integer m denotes the m×m identity matrix. Note that the integral in (2) is takenover a compact domain, and hence converges absolutely. This projection has the following basic properties.

Proposition 2.1. Given a cuspidal automorphic form ϕ on GLn(AF ), the projection Pn1ϕ defined by theintegral (2) is a cuspidal automorphic form on P2(AF ) having the Fourier-Whittaker expansion

Pn1ϕ(p) = |det(p)|−(n−22 )

∑γ∈F×

Wϕ

((γ

1n−1

)(p

1n−2

)).

In particular, for x ∈ AF a generic adele and y ∈ AF a generic idele, we have that

Pn1ϕ((

y x1

))= |y|−(n−2

2 )∑γ∈F×

Wϕ

((yγ

1n−1

))ψ(γx).

Proof. See [13, Lemma 5.2] or [14, §2.2.1] for the first two statements. The third is an easy consequence ofspecialization. To be clear, writing ϕ′ to denote the normalized function defined on p ∈ P2(AF ) by

ϕ′(p) = |det(p)|n−22 Pn1ϕ(p),

we specialize the expansion of the second statement to p =

(y x

1

). It is then easy to check that

Pn1ϕ((

y x1

))=∑γ∈F×

Wϕ′

((γ

1

)(1 x

1

)(y

1

))

=∑γ∈F×

Wϕ′

((1 γx

1

)(γ

1

)(y

1

))

=∑γ∈F×

Wϕ′

((yγ

1

))ψ(γx).

�

2.2. Relation to L-function coefficients. Let us retain all of the setup described above. Given an ideleclass character ξ = ⊗vξv of F , we now consider the shifted Mellin transform

I(s, ϕ, ξ) =

∫A×F /F

×Pn1ϕ

((h

1

))ξ(h)|h|s− 1

2 dh,

defined first for s ∈ C with <(s) > 1. As explained in Cogdell [13, Lecture 5] or [14, §2.2] (with m = 1), wecan open up the Fourier-Whittaker expansion of Pn1ϕ in this integral to derive the integral presentation

I(s, ϕ, ξ) =

∫A×F /F

×

∑γ∈F×

Wϕ

((γh

1n−1

))ξ(h)|h|s−

12−(n−2

2 )dh

=

∫A×F

Wϕ

((h

1n−1

))ξ(h)|h|s−(n−1

2 )dh.

9

Since we choose the pure tensor ϕ = ⊗vϕv ∈ VΠ in such a way that each nonarchimedean local componentϕv is an essential Whittaker vector (thanks to Matringe [28, Corollary 3.3]), we deduce from Cogdell’s theoryof Eulerian integrals for GLn×GL1 that we have the following exact integral presentation for the finite partL(s,Π⊗ ξ) of the standard L-function Λ(s,Π⊗ ξ) = L(s,Π∞ ⊗ ξ∞)L(s,Π⊗ ξ) of Π⊗ ξ.

Theorem 2.2. Let ϕ = ⊗vϕv ∈ VΠ be a pure tensor whose local nonarchimedean components ϕv are eachessential Whittaker vectors. Let ξ = ⊗vξv be an idele class character of F , and consider the standard L-function Λ(s,Π⊗ξ) = L(s,Π∞⊗χ∞)L(s,Π⊗ξ) of Π⊗ξ. Then, the finite part L(s,Π⊗ξ) of this L-functionthen has the following integral presentation:

L(s,Π⊗ ξ) =∏v<∞

∫F×v

Wϕv

((hv

1n−1

))ξv(hv)|hv|

s−(n−12 )

v dhv

=

∫A×F,f

Wϕ

((hf

1n−1

))ξ(hf )|hf |s−(n−1

2 )dhf .

Here, for each nonarchimedean place v of F not dividing the conductor of Π or ξ, we choose a measure onF×v according to Matringe [28, Corollary 3.3]. As well, for an idele h = (hv)v ∈ A×F write the corresponding

decomposition into nonarchimedean and archimedean components as h = hfh∞ for hf ∈ A×F,f and h∞ ∈ F×∞.In particular, taking ξ to be the trivial character, we have the following relation of the specialized coefficients

ρϕ(hf ) := Wϕ

((hf

1n−1

))and Wϕ(h∞) := Wϕ

((h∞

1n−1

)),(3)

to the coefficients in the Dirichlet series of L(s,Π) (first for <(s) > 1): Fixing a finite idele representativeof each nonzero integral integral ideal m ⊂ OF , and using the same symbol to denote this so that m ∈ A×F,f ,

L(s,Π) :=∑

m⊂OF

cΠ(m)

Nms=∑

m⊂OF

ρϕ(m)

Nms−(n−12 )

.(4)

Proof. The first claim is deduced from Cogdell’s theory of Eulerian integrals (see e.g. [13, Lecture 9]) withthe theorem of Matringe [28, Corollary 3.3] to describe the local zeta integrals at each nonarchimedean placev for which Πv or ξv is ramified. This latter result gives the identification

L(s,Πv ⊗ ξv) =

∫F×v

Wϕv

((hv

1n−1

))ξv(hv)|hv|

s−(n−12 )

v dhv

for each nonarchimedean place v of F where Πv or ξv is ramified, the unramified case being well-understood(see e.g. [13, Lecture 7]). The second claim follows after specialization to the trivial character, comparingMellin coefficients. �

Using (4), we can now relate the Fourier-Whittaker coefficients of Pn1ϕ to the L-function coefficients cΠ:

Corollary 2.3. Let ϕ = ⊗vϕv ∈ VΠ be a pure tensor whose nonarchimedean local components are eachessential Whittaker vectors. Given a generic adele x ∈ AF and a generic idele y ∈ A×F , the projected cuspform Pn1ϕ has the Fourier-Whittaker expansion

Pn1ϕ((

y x1

))= |y|−(n−2

2 )∑γ∈F×

cΠ(γyf )

|γyf |n−12

Wϕ (γy∞)ψ(γx).

Proof. The expansion follows from Proposition 2.1, and decomposing Whittaker coefficients as in (3), andthen using the relation to L-function coefficients implied by (4). �

2.3. Extensions to GL2(AF ). Recall that a function φ : GL2(AF ) −→ C is said to be an L2-automorphicform on GL2(AF ) of a given unitary central character ξ = ⊗vξv : A×F → S1 if

• It is measurable and∫Z2(AF ) GL2(F )\GL2(AF )

|φ(g)|2dg <∞.

• It is left GL2(F )-invariant, so φ(γg) = φ(g) for all γ ∈ GL2(F ) and g ∈ GL2(AF ).

10

• The centre Z2(AF ) ∼= A×F of GL2(AF ) acts via ξ, so φ(zg) = ξ(z)φ(g) for all z ∈ Z2(AF ) andg ∈ GL2(AF ).

We write L2 (GL2(F )\GL2(AF ), ξ) to denote the corresponding Hilbert space of such functions. Note thatGL2(AF ) acts naturally on this space by right translation, giving it the structure of a unitary representation.

Let

K =∏v≤∞

Kv = O2(F∞)∏v<∞

GL2(OFv )

denote the maximal compact subgroup of GL2(AF ). Let us also remark that the discussion that followsapplies to any dimension n ≥ 2, although we shall focus on the generic higher dimensional setting n ≥ 3where the projection operator Pn1 is not trivial (and hence why such a discussion is needed). Keeping withthe setup described above, we now show how to extend an L2-automorphic form Pn1ϕ on the mirabolicsubgroup P2(AF ) ⊂ GL2(AF ) to an L2-automorphic form on GL2(AF ) of trivial central character whichis also right K-invariant. To be more precise, we argue as follows that we can construct an extensionof Pn1ϕ to L2(GL2(F )\GL2(AF ),1) via the strong approximation theorem for A×F together with the Iwa-sawa decomposition for GL2(F∞), analogous to the classical setting of Hilbert-Maass cusp forms on thed-fold upper-half plane. Let us write ι : GL2(F ) → GL2(AF ) to denote the diagonal embedding, andν : GL2(F∞) → GL2(AF ) the embedding sending a matrix x to the idele with archimedean component xand all other components trivial. We shall sometimes drop these notations when the context is clear. Recallthat the strong approximation theorem for A×F gives us the identification

A×F /F×∞F

×O×F = A×F /F×∞F

×∏v<∞

O×Fv ∼= C(OF ),

where C(OF ) denotes the ideal class group of F . Hence, fixing a set of idele representatives ∆ of C(OF ),the strong approximation theorem for A×F implies that

D := F×∞∏v<∞

O×Fv∐ζ∈∆

ζ

is a fundamental domain for the idele class group A×F /F×, and so we derive the disjoint union decomposition

A×F =∐α∈F×

αD.(5)

Given a representative ζ ∈ ∆, let us write hζ =

(ζ

ζ

)to denote the corresponding diagonal matrix,

A×F∼= Z2(AF ), ζ 7→ hζ :=

(ζ

ζ

).

We have the following well-known application of this theorem to GL2(AF ) via the determinant map.

Proposition 2.4 (Strong approximation). We have that GL2(AF ) =∐ζ∈∆ GL2(F ) GL2(F∞)Khζ .

Proof. Cf. [16, Appendix 3] and [19, Proposition 4.4.2] or [10, §3.3.1]. To derive the version we state here,observe that we can view GL2(F ) GL2(F∞)K ⊂ GL2(AF ) as a subgroup, and that the homomorphism

GL2(AF )det−−→ A×F → C(OF )

factors to given an identification

GL2(AF )/GL2(F ) GL2(F∞)K ∼= C(OF ).

The stated identification is then easy to deduce. �

We can now use some version of the Iwasawa decomposition for GL2(F∞) to deduce the following useful“unique factorization” result for elements of GL2(AF ) (cf. [19, Theorem 4.4.4]).

11

Theorem 2.5 (Unique decomposition). Fixing a fundamental domain for GL2(OF )\GL2(F∞), we obtainfrom the strong approximation decomposition GL2(AF ) ∼=

∐ζ∈∆ GL2(F )P2(F∞)hζZ2(F∞)K described above

that each element g ∈ GL2(AF ) can be expressed uniquely as a product of matrices of the form

g =∐ζ∈∆

γ ·(y∞ x∞

1

)·(r∞

r∞

)·(ζ

ζ

)· k(6)

or more precisely

g =∐ζ∈∆

ι(γ) · ν((

y∞ x∞1

))· ν((

r∞r∞

))· ι((

ζζ

))· k,

where γ ∈ GL2(F ) is a rational element, (y∞ x∞

1

)∈ P2(F∞)

is a mirabolic element with archimedean coordinates,(r∞

r∞

)∈ Z2(F∞)

is a central element with archimedean idele coordinates, and k ∈ K is an element of the maximal compactsubgroup. Here, after fixing a fundamental domain, the archimedean coordinates in this presentation (6)are constrained as follows so that the (6) presentation is unique: For each index 1 ≤ j ≤ d, the adelex∞ = (x∞,j)

dj=1 and the ideles r∞ = (r∞,j)

dj=1 and y∞ = (y∞,j)

dj=1 satisfy the simultaneous constraints

r∞,j > 0, y∞,j > 0, 0 ≤ x∞,j ≤ 1/2, x2∞,j + y2

∞,j ≥ 1.

Proof. Cf. [19, Theorem 4.4.4], where it is explained how to derive a similar (classical) result via the Iwasawadecomposition for GL2(F∞) in the strong approximation decomposition described above. To be more precise,let us first recall the Iwasawa decompostion for GL2(F∞), as described in [19, Proposition 4.1.1] (for instance).Hence, let us fix a fundamental domain for the action of GL2(OF ) on GL2(F∞), as described in more detailbelow. Fixing such a choice of fundamental domain, a minor variation of the standard argument given for[19, Proposition 4.1.1] allows us to deduce that each matrix g∞ ∈ GL2(F∞) can be expressed uniquely as

g∞ =

(1 x∞

1

)·(y∞

1

)· k ·

(r∞

r∞

)(7)

with k ∈ O2(F∞), x∞ ∈ F∞, and y∞, r∞ ∈ F×∞ constrained as in the statement. To be more precise here,we consider for each real place of F the identification

H := {z = x+ iy ∈ C : x ∈ R, y ∈ R>0} ∼= GL2(R)/O2(R) ·R×,

so that GL2(Z) ⊂ GL2(R) acts on H via left matrix multiplication. We then use the natural identification ofmatrices of the from appearing in (7) above with x∞+iy∞ ∈ Hd to form a fundamental domain following theclassical example of SL2(Z)\H for each component of the adele coordinate x∞ = (x∞,j)

dj=1 ∈ F∞ ∼= Rd and

the idele coordinate y∞ = (y∞,j)dj=1 ∈ F×∞ ∼= (Rd)×. That is, for each component j corresponding to a real

place of F , we consider the standard fundamental domain D for SL2(Z) acting on the complex upper-halfplane z = x+ iy ∈ H given by

D =

{z ∈ H : −1

2≤ <(z) ≤ 1

2, zz ≥ 1

}.

Following a similar argument to what is given in [19, Theorem 4.4.4], we use the elementary matrix identity(−1

1

)(y x

1

)(−1

1

)=

(y −x

1

).

for each x ∈ R and y ∈ R× to deduce that a fundamental domain for GL2(Z)\GL2(R) is half of afundamental domain for SL2(R)\H in each adele coordinate x = x∞,j . The constraints stated above, whichcorrespond to choosing D with 0 ≤ x∞,j ≤ 1/2 for each 1 ≤ j ≤ d, are then easy to deduce. In particular,this justifies that the x∞ ∈ F∞ and y∞, r∞ ∈ F×∞ are constrained as stated.

12

Now, we can use this unique decomposition of g∞ ∈ GL2(F∞) according (7) in the strong approximationtheorem GL2(AF ) =

∐ζ∈∆ GL2(F )P2(F∞)KhζZ2(F∞) to deduce that each element g ∈ GL2(AF ) can be

expressed uniquely as

g =∐ζ∈∆

γ ·(y∞ x∞

1

)·(r∞

r∞

)·(ζ

ζ

)· k

or more precisely

g =∐ζ∈∆

ι(γ) · ν((

y∞ x∞1

))· ν((

r∞r∞

))·(ζ

ζ

)· k

with γ ∈ GL2(F ) and k ∈ K = O2(F∞)∏v<∞Kv. This proves the stated claim. �

We can now define an extension Pn1ϕ of the mirabolic form Pn1ϕ to GL2(AF ) via Theorem 2.5 as follows:

Definition 2.6. Let ϕ ∈ VΠ be a pure tensor for the irreducible cuspidal GLn(AF )-automorphic represen-

tation Π, and let Pn1ϕ denote its projection to the mirabolic subgroup P2(AF ) ⊂ GL2(AF ). We take Pn1ϕ tobe the function defined on g ∈ GL2(AF ) decomposed uniquely as in (6) above by the rule

Pn1ϕ(g) := Pn1ϕ((

y∞ x∞1

)).

Remark Note that this definition does not depend on the choice of fundamental domain for GL2(OF )\GL2(F∞).

We now check that this extension Pn1ϕ in fact determines and L2-automorphic form on GL2(AF ) of trivialcentral character which is also right K-invariant.

Proposition 2.7. The function Pn1ϕ(g) = Pn1ϕ((

y∞ x∞1

))from Definition 2.6 above determines an

L2-automorphic form of trivial central character on GL2(AF ) which is also K-finite (but not Z-finite),

Pn1ϕ ∈ L2 (GL2(F )\GL2(AF ),1)K↪→ L2 (GL2(F )\GL2(AF ),1) .

Proof. Cf. [19, Proposition 4.8.4]. Let us write φ = Pn1ϕ to lighten notation. We first argue that φ ismeasurable and convergent in the L2-norm as a consequence of the corresponding properties satisfied by thepure tensor ϕ ∈ VΠ on GLn(AF ). To be more precise, writing || · || to denote the standard matrix norm ong ∈ GLn(AF ), we know that there exist constants C,B > 0 such that

ϕ(g) =∑

γ∈Nn−1(F )\GLn−1(F )

Wϕ

((γ

1

)g

)< C · ||g||B .

Comparing Fourier-Whittaker expansions, it is easy to deduce that at a similar bound holds for the projection:

Pn1ϕ(p) = |det(p)|−(n−22 )

∑γ∈F×

Wϕ

((γ

1n−1

)(p

1n−2

))< C · || diag(p,1n−2)||B .

We now check the required invariance properties. It is easy to see from Definition 2.6 that φ(γ · g) = φ(γ)for all γ ∈ GL2(AF ) and g ∈ GL2(AF ). To check invariance under the action of Z2(AF ), we argue followingthe discussion of strong approximation (5) above that given z ∈ Z2(AF ), we can find α ∈ F×, w∞ ∈ F×∞and k ∈ K such that z can be expressed as

z = ι

((α

α

))· ν((

w∞w∞

))· k ·

∐ζ∈∆

hζ =∐ζ∈∆

hζ · ι((

αα

))· ν((

w∞w∞

))· k.

It is then easy to check from Definition 2.6 that z ∈ Z2(AF ) acts trivially (with g · z in the right form):

φ(z · g) = φ(g · z) = φ

g · ∐ζ∈∆

hζ · ι((

αα

))· ν((

w∞w∞

))· k

= φ(g).

13

Finally, observe that the extension φ(g) is trivially K-finite, since

φ(g · k) = Pn1ϕ((

y∞ x∞1

))= φ(g)

for any g ∈ GL2(AF ) and k ∈ K as a consequence of Definition 2.6. Hence, it is trivial to deduce that theset {φ(gk) : k ∈ K} of all right translates of φ generates a finite dimensional vector space. �

2.4. Relations of Fourier-Whittaker coefficients. We now consider how to relate the Fourier-Whittakercoefficients of a mirabolic cusp form Pn1ϕ to those of its extension Pn1ϕ to GL2(AF ) according to Theorem2.5, Definition 2.6, and Proposition 2.7 above. We also consider localizations of the latter functions to certaincompact subdomains of the chosen fundamental domain of Theorem 2.5 for our subsequent arguments. Letus start with the following basic general observation.

Lemma 2.8. Let φ ∈ L2(GL2(F )\GL2(AF ),1) be any L2-automorphic form on GL2(AF ) of trivial centralcharacter 1 which is also right K-invariant. Then for any idele y ∈ A×F , we have the identification

φ

((y

1

))= φ

((y−1

1

)).(8)

Proof. Let us suppose more generally for illustration that φ ∈ L2(GL2(F )\GL2(AF ), ω) is any L2-automorphicform on GL2(AF ) of central character ω which is right K-invariant. A direct computation shows that(

11

)(y

1

)(1

1

)=

(1

y

).(9)

Since we can view the long Weyl element (1

1

)as an element of both the rational subgroup GL2(F ) and the maximal compact subgroup K, it follows fromthe left GL2(F )-invariance and right K-invariance of φ that we have the idenfications

φ

((1

1

)(y

1

)(1

1

))= φ

((y

1

))= φ

((1

y

)).

That is, the first identity follows from the invariance of φ, while the second follows from the calculation (9)applied the the expression on the left. Factoring out by a central element on the right, we then deduce that

φ

((y

1

))= ω(y)−1φ

((y−1

1

)).

This implies the stated identity (8) for the special case of trivial central character ω = 1. �

Remark Notice that for any archimedean idele y∞ ∈ F×∞, Lemma 2.8 implies we have the identification

Pn1ϕ((

y−1∞

1

))= Pn1ϕ

((y∞

1

)).

Thus if |y∞| > 1 is contained in the fundamental domain of Theorem 2.5, we can consider the decomposition(1

1

)(y−1∞

1

)(1

1

)=

(1

y−1∞

)=

(y−1∞

y−1∞

)(y∞

1

)(10)

to deduce that the diagonal matrix on the left hand side of (10) is put into the correct form for the uniquedecomposition of Theorem 2.5 by the expression on the right hand side. Hence by Definition 2.6, we have

Pn1ϕ((

y−1∞

1

))= Pn1ϕ

((y∞

1

))= Pn1

((y∞

1

)).(11)

14

Let us now consider relations between the Fourier-Whittaker coefficients of a given mirabolic form Pn1ϕand its extension Pn1ϕ as defined above. In particular, keeping the idele coordinate y∞ ∈ F×∞ inside of thechosen fundamental domain of Theorem 2.5, we can show that there is a matching of each of the Fourier-Whittaker coefficients, provided that the pure tensor ϕ = ⊗vϕv ∈ VΠ on GLn(AF ) has the property that itslocal archimedean Whittaker function is even, i.e. so that

Wϕ

((y∞

1n−1

))= Wϕ

((−y∞

1n−1

))(12)

as functions of y∞ ∈ F×∞. To justify this claim, we shall first describe the Fourier-Whittaker coefficientsequivalently in semi-classical terms, as integrals over the compact domains I ∼= [0, 1]d ⊂ F∞ ∼= Rd against thecorresponding archimedean additive character ψ∞(x∞) = exp(2πiTr(x∞)) = exp(2πi(x∞,1 + · · · + x∞,d)).To be more precise, recall that given an L2-automorphic form φ ∈ L2(GL2(F )\GL2(AF ), ω) evaluated ong ∈ GL2(AF ) and α ∈ OF an F -integer, the coefficient in the Fourier-Whittaker expansion of φ(g) alongthe unipotent subgroup of upper triangular matrices N2 ⊂ GL2 at α is given by the unipotent integral∫

AF /F

φ

((1 x

1

)g

)ψ(−αx)dx.

This integral can be described equivalently as one over the domain I ∼= [0, 1]d ⊂ F∞ ∼= Rd, using that thisforms a full orthogonal set for the additive characters ψ(αx∞) = ψ∞(αx∞) = exp(−2πiTr(αx∞)),∫

AF /F

φ

((1 x

1

)g

)ψ(−αx)dx =

∫I∼=[0,1]d⊂F∞

φ

((1 x∞

1

)g

)ψ(αx∞).(13)

Note that the identity (13) reflects the well-known equivalence of the Fourier-Whittaker expansion of aGL2(AF )-automorphic form with that of its underlying GL2(F∞)-automorphic form, or that of its underlyingMaass form on the d-fold upper-half plane, and that we shall often take such identifications for granted.

Proposition 2.9. Let ϕ = ⊗vϕv ∈ VΠ be any pure tensor in the representation space of the cuspidalautomorphic representation Π = ⊗vΠv of GLn(AF ). Assume that the archimedean local Whittaker coefficientWϕ satisfies condition (12), so that Wϕ(y∞) = Wϕ(−y∞) as functions of y∞ ∈ F×∞. Let Pn1ϕ denote the

image of ϕ under the projection operator Pn1 . Let Pn1ϕ ∈ L2(GL2(F )\GL2(AF ),1)K denote the extensionof Pn1ϕ to an L2-automorphic form on GL2(AF ) via Theorem 2.5, Definition 2.6, and Proposition 2.7. Lety∞ ∈ F×∞ be any archimedean idele with |y∞| > 1 contained in our fixed fundamental domain of Theorem 2.5.Then for any F -integer α ∈ OF , we have the corresponding identification of Fourier-Whittaker coefficients∫

I∼=[0,1]dPn1ϕ

((y∞ x∞

1

))ψ(−αx∞)dx∞ =

∫I∼=[0,1]d

Pn1ϕ((

y∞ x∞1

))ψ(−αx∞)dx∞.

In other words, the Fourier-Whittaker coefficients of such an L2-automorphic form Pn1ϕ evaluated on diagonal

elements diag(y∞, 1) ∈ P2(F∞) match those of its extension Pn1ϕ to GL2(AF ) evaluated on diagonal elementsdiag(y∞, 1) ∈ GL2(F∞) ∼= P2(F∞)Z2(F∞)O2(F∞).

Proof. Consider ∫I∼=[0,1]d

Pn1ϕ((

y∞ x∞1

))ψ(−αx∞)dx∞.

Let us split the integral into intervals I1 ∼= [0, 1/2]d ⊂ F∞ and I2 ∼= (1/2, 1]d ⊂ F∞ as∫I1∼=[0,1/2]d

Pn1ϕ((

y∞ x∞1

))ψ(−αx∞)dx∞ +

∫I2∼=(1/2,1]d

Pn1ϕ((

y∞ x∞1

))ψ(−αx∞)dx∞.(14)

Since the coordinates in the first integral of the latter expression are well-defined with respect to the chosenfundamental domain of Theorem 2.5, it is easy to see from Definition 2.6 that∫

I1∼=[0,1/2]dPn1ϕ

((y∞ x∞

1

))ψ(−αx∞)dx∞ =

∫I1∼=[0,1/2]d

Pn1ϕ((

y∞ x∞1

))ψ(−αx∞)dx∞.

15

To identify the second integral over I2 in this way, let us first fix any adele x∞ = (x∞,j)dj=1 ∈ I2 ∼= (1/2, 1]d.

Observe that since Pn1ϕ is left invariant by

(1 −1

1

)∈ GL2(F ), we have the identification

Pn1ϕ((

y∞ x∞1

))= Pn1ϕ

((y∞ x∞ − 1

1

)).

On the other hand, since the GL2(AF )-automorphic form Pn1ϕ is both left invariant by

(−1

1

)∈ GL2(F )

and right invariant by

(−1

1

)∈ K, it is easy to deduce that

Pn1ϕ((

y∞ x∞ − 11

))= Pn1ϕ

((−1

1

)(y∞ x∞ − 1

1

)(−1

1

))= Pn1ϕ

((y∞ 1− x∞

1

)),

where the latter function is well-defined with respect to the chosen fundamental domain of Theorem 2.5.That is, we then have by Definition 2.6 the identification

Pn1ϕ((

y∞ 1− x∞1

))= Pn1ϕ

((y∞ 1− x∞

1

)).

Now, since the mirabolic form Pn1ϕ is left invariant by

(1 −1

1

)∈ P2(F ), we find that

Pn1ϕ((

y∞ 1− x∞1

))= Pn1ϕ

((y∞ −x∞

1

)).

Since the mirabolic form is also left invariant by

(−1

1

)∈ P2(F ), we then find that

Pn1ϕ((

y∞ −x∞1

))= Pn1ϕ

((−y∞ x∞

1

)).

Now, it is easy to deduce from the condition (12) that we have the identification

Pn1ϕ((−y∞ x∞

1

))= Pn1ϕ

((y∞ x∞

1

)).

Indeed, opening up the Fourier-Whittaker expansion of the function on the left hand side, we check that

Pn1ϕ((−y∞ x∞

1

))= | − y−1

∞ |−(n−22 )

∑γ∈F×

Wϕ

((−γy∞

1n−1

))ψ(γx∞)

= |y−1∞ |−(n−2

2 )∑γ∈F×

Wϕ

((γy∞

1n−1

))ψ(γx∞)

= Pn1ϕ((

y∞ x∞1

)).

16

Hence, we have shown that∫I2∼=(1/2,1]d

Pn1ϕ((

y∞ x∞1

))ψ∞(−αx∞)dx∞

=

∫I2∼=(1/2,1]d

Pn1ϕ((

y∞ x∞ − 11

))ψ∞(−αx∞)dx∞

=

∫I2∼=(1/2,1]d

Pn1ϕ((

y∞ 1− x∞1

))ψ∞(−αx∞)dx∞

=

∫I2∼=(1/2,1]d

Pn1ϕ((

y∞ 1− x∞1

))ψ∞(−αx∞)dx∞

=

∫I2∼=(1/2,1]d

Pn1ϕ((

y∞ −x∞1

))ψ∞(−αx∞)dx∞

=

∫I2∼=(1/2,1]d

Pn1ϕ((−y∞ x∞

1

))ψ∞(−αx∞)dx∞

=

∫I2∼=(1/2,1]d

Pn1ϕ((

y∞ x∞1

))ψ∞(−αx∞)dx∞.

The claim follows after inserting this identification for the second integral in (14). �

Remark Observe that the argument given above requires the automorphy of the mirabolic cusp form Pn1ϕ,and so would not simply work for a random function defined on P2(F∞) ∼= Hd extended to an L2-automorphicform in the space L2(GL2(F )\GL2(AF ),1)K by Theorem 2.5, Definition 2.6, and Proposition 2.7.

Note that we also obtain from this identification of coefficients the following useful result.

Corollary 2.10. Fix y∞ ∈ F×∞ an idele in the chosen fundamental domain of Theorem 2.5. Then, thecorresponding function of the adele coordinate x∞ ∈ I ∼= [0, 1]d ⊂ F∞

Pn1ϕ((

y∞ x∞1

))is continuous.

Proof. The claim is a direct consequence of the algorithmic calculation of Proposition 2.9, which shows thatfor any fixed adele x∞ ∈ I2 ∼= (1/2, 1]d ∈ F∞, we have the series of identifications

(15)

Pn1

((y∞ x∞

1

))= Pn1ϕ

((y∞ x∞ − 1

1

))= Pn1ϕ

((y∞ 1− x∞

1

))= Pn1ϕ

((y∞ 1− x∞

1

))= Pn1ϕ

((y∞ −x∞

1

))= Pn1ϕ

((y∞ x∞

1

)).

Since it is clear from the definition that the same identification (15) holds for any x∞ ∈ I1 ∼= [0, 1/2]d ∈ F∞,we deduce the claim. That is, we verify that the corresponding function on y∞ ∈ F with the adele coordinatex∞ ∈ F∞ varying freely in the interval I ∼= [0, 1]d ⊂ F∞ is continuous. This is because we have verified thatit matches the underlying mirabolic function Pn1ϕ in this region, the mirabolic function being continuous inx∞ ∈ I ∼= [0, 1]d ⊂ F∞. �

17

3. Integral presentations

We now show how to use the expansion of Corollary 2.3 to derive the following integral expansions forthe shifted convolution problem for GLn(AF ). In this section, we shall work entirely with the miraboliccusp form Pn1ϕ coming from a pure tensor ϕ = ⊗vϕv ∈ VΠ on GLn(AF ) with proscribed local vectors ϕvas described above, but defer considering relations to the extended form Pn1ϕ following Proposition 2.9 untillater. The key property we shall use here is the surjectivity of the archimedean local Kirillov map, whichcan be viewed as a vector space isomorphism VΠ

∼= L2(F×∞), φ → Wφ. To be more precise, we shall use thefollowing key result due in this generality to Jacquet and Shalika.

Proposition 3.1. Let W be any smooth and compactly supported function on F×∞, or more generally anysmooth, summable function on F×∞ of moderate decay near zero which decays rapidly at infinity. There existsa smooth vector ϕ ∈ V∞Π whose corresponding archimedean local Whittaker coefficient Wϕ(y∞) satisfies

Wϕ(y∞) := Wϕ

((y∞

1n−1

))= W (y∞),

i.e. as functions of y∞ ∈ F×∞.

Proof. See Jacquet-Shalika [23, (3.8)]; cf. also [22, Lemma 5.1] and [6, § 2.5]. �

Remark Note that for applications, one can relax the conditions on the chosen function W for Proposition3.1 via a standard dyadic subdivision argument which we present in the subsequent section.

Notations. We identify any F -rational number α ∈ F× with its image in A×F under the diagonal embedding

α 7→ (α, α, . . .) ∈ A×F , and write and y = yfy∞ ∈ A×F to denote any idele with nonarchimedean component

yf ∈ A×F,f and archimedean component y∞ = (y∞,j)dj=1 ∈ F×∞ ∼= (R×)d. The notation αy∞ then refers to

the product αyfy∞ with yf = (1, 1, . . .) ∈ A×F,f trivial, so that αy∞ = (α, α, . . .)αy∞ has nonarchimedean

component (α, α, . . .) ∈ A×F,f and archimedean component αy∞ = (αy∞,j)dj=1 ∈ F×∞. We shall use this

notation without comment throughout the rest of the work. Let us remark however that the idele norm|αy∞| is then given by the archimedean norm on the component (αy∞) = (αy∞,j)

dj=1 ∈ F×∞ ∼= (R×)d.

3.1. Quadratic progressions via metaplectic theta series. We take for granted some basic backgroundabout automorphic forms on the metaplectic cover G of GL2 following Gelbart [17], as well as the relevantdiscussions in [33] and [34]. Let us first consider metaplectic theta series θq corresponding to the F -rationalquadratic form q(x) = x2. In fact, we shall consider only the corresponding partial theta series associatedto the principal class 1 in the ideal class group C(OF ) of OF , as this is all we require for our applications.Viewed as an automorphic form on G(AF ), this partial theta series has the following Fourier-Whittakerexpansion: Taking x∞ ∈ F∞ with y∞ ∈ F×∞ with norm |y∞|, we have the expansion

θq

((y∞ x∞

1

))= θq,1

(((y∞ x∞

1

), 1

))= |y∞|

14

∑a∈OF

ψ(q(a)(x∞ + iy∞)).

Proposition 3.2. Let W be any smooth, summable function of y∞ ∈ F×∞ of moderate decay near zero andrapid decay at infinity. Fix an F -integer α, and let us use the same symbol α to denote its image under thediagonal embedding α = (α, α, . . .) ∈ A×F if α 6= 0. Let Y∞ ∈ F×∞ be any archimedean idele of idele norm|Y∞| > |α|. Let ϕ = ⊗vϕ ∈ VΠ be any pure tensor whose nonarchimedean local components are essentialWhittaker vectors, and whose archimedean local component has corresponding local Whittaker function2

Wϕ(y∞) = |y∞|n−22 ψ(−iy∞)ψ

(i · αY∞

)W (y∞) .

2Note that the rapidly decaying function W can be chosen with some flexibility after making a standard dyadic subdivision

argument, restricting the chosen Whittaker function to dyadic intervals of the form [2k, 2k+1], then checking that the sum overintegers k converges. Although we do not require such an argument here (taking W to be sufficiently rapidly decaying), such a

reduction is crucial in many applications.

18

given as a function of y∞ ∈ F×∞. Then, we have the integral presentation

|Y∞|14

∫I∼=[0,1]d⊂F∞

Pn1ϕ · θq((

1Y∞

x∞1

))ψ(−αx∞)dx∞

=∑a∈OF

q(a)+α6=0

cΠ(q(a) + α)

|q(a) + α| 12W

(q(a) + α

Y∞

).

Proof. We open up Fourier-Whittaker expansions using Corollary 2.3 and the discussion above, switch theorder of summation, and then use the orthogonality of additive characters on the compact abelian groupI ∼= [0, 1]d ∼= (R/Z)d ⊂ F∞ to compute∫I∼=[0,1]d⊂F∞

Pn1ϕ · θq((

y∞ x∞1

))ψ(−αx∞)dx∞

=

∫I∼=[0,1]d⊂F∞

Pn1ϕ((

y∞ x∞1

))θq

((y∞ x∞

1

))ψ(−αx∞)dx∞

=

∫I∼=[0,1]d⊂F∞

|y∞|−(n−2)

2

∑γ∈F×

Wϕ

((γy∞

1n−1

))ψ(γx∞)|y∞|

14

∑a∈OF

ψ(iy∞q(a))ψ(−q(a)x∞)ψ(−αx∞)dx∞

= |y∞|14−

(n−2)2

∑γ∈F×

Wϕ

((γy∞

1n−1

)) ∑a∈OF

ψ(iyq(a))

∫I∼=[0,1]d⊂F∞

ψ(γx∞ − q(a)x∞ − αx∞)dx∞

= |y∞|14−

(n−2)2

∑a∈OF

q(a)+α6=0

Wϕ

(((q(a) + α)y∞

1n−1

))ψ(q(a)iy∞)

= |y∞|14−

(n−2)2

∑a∈OF

q(a)+α6=0

ρϕ((q(a) + α))Wϕ ((q(a) + α))ψ(q(a)iy∞).

The claimed relation then follows after specialization to y∞ = 1/Y∞ ∈ F×∞, after using the relation toL-function coefficients described in (4) above, and the explicit choice of ϕ. �

3.2. Quadratic progressions via binary theta series. Let us now consider the theta series associatedto a positive definite F -rational binary quadratic form Q(a, b) having w = wQ many automorphs. We shallonly the consider the theta series associated to the principal class 1 of such binary quadratic forms of agiven discriminant (or equivalently of the ideal class group of the corresponding CM extension of F ). LetθQ = θQ,1 denote the corresponding theta series, viewed as a GL2(AF )-automorphic form. Taking x∞ ∈ F∞and y∞ ∈ F×∞, this theta series has the expansion

θQ

((y∞ x∞

1

))= |y∞|

12

1

w

∑a,b∈OF

ψ(Q(a, b)(x∞ + iy∞)).

Keeping the same conventions from the discussion above, we derive the following

Proposition 3.3. Let W be any smooth, summable function of y∞ ∈ F×∞ of moderate decay near zeroand rapid decay at infinity. Let α be any F -integer, as well as its image under the diagonal embeddingα → (α, α, . . .) ∈ A×F if α 6= 0. Let Y∞ ∈ F×∞ be any archimedean idele of idele norm |Y∞| > |α|. Letϕ = ⊗vϕv ∈ VΠ be a smooth vector whose nonarchimedean local components are essential Whittaker vectors,and whose archimedean local Whittaker function is given as a function of y∞ ∈ F×∞ by

Wϕ(y∞) = |y∞|n−22 ψ (−iy∞)ψ

(i · αY∞

)W (y∞)

19

for any archimedean idele y∞ ∈ F×∞. Then, we have the integral presentation

|Y∞|12

∫I∼=[0,1]d⊂F∞

Pn1ϕθQ((

y∞ x∞1

))ψ(−αx∞)dx∞

=1

w

∑a,b∈OF

q(a,b)+α6=0

cΠ(Q(a, b) + α)

|Q(a, b) + α| 12W

(Q(a, b) + α

Y∞

).

Proof. The proof is derived in the same was as for Proposition 3.2 via the expansions of Pn1ϕ and θQ, usingthe orthogonality of additive characters on the compact abelian group I ∼= [0, 1]d ⊂ F∞. �

3.3. Shifts of arbitrary positive definite quadratic forms. Let us now record the following naturalgeneralization of Propositions 3.3 and 3.2 to arbitrary positive definite quadratic forms, which does not seemto appear in literature on the shifted convolution problem, and which might be of independent interest.

Let f = f(a1, . . . , ak) be an F -rational positive definite quadratic form in k ≥ 1 many variables. Letp = (a1, . . . , ak) be a homogeneous polynomial on Rn which is harmonic with respect to f . Hence ∆fp = 0,where ∆f is the unique homogeneous differential operator of order two which is invariant under the orthogonalgroup O(f) of f . Let θf,p denote the corresponding theta series, which determines an automorphic form on

GL2(AF ) if k is even, and a genuine automorphic form on the metaplectic cover G(AF ) if k is odd. Again,we shall really only consider the corresponding partial theta series associated to the principal class in C(OF )here, and not mention it again. Taking x∞ ∈ F∞ again to be a generic archimedean adele and y = y∞ ∈ F×∞a generic archimedean idele, this (partial, metaplectic) theta series θf,p has the Fourier series expansion

θf,p

((y∞ x∞

1

))= θf,p

(((y∞ x∞

1

), 1

))= |y∞|

k4

∑a1,...,ak∈OF

p(a1, . . . , ak)ψ(f(a1, . . . , ak)(x∞ + iy∞)).

Proposition 3.4. Let W be any smooth, summable function of y∞ ∈ F×∞ of moderate decay near zero andrapid decay at infinity. Fix an F -integer α, and let us also write α to denote the imagine under the diagonalembedding α→ (α, α, · · · ) ∈ A×F if α 6= 0. Let Y∞ ∈ F×∞ be any archimedean idele of idele norm |Y∞| > |α|.Let ϕ = ⊗vϕv ∈ VΠ be a pure tensor whose nonarchimedean local components are essential Whittaker vectors,and whose archimedean component has the corresponding Whittaker coefficient

Wϕ(y∞) = |y∞|n−22 ψ(−iy∞)ψ

(i · αY∞

)W (y∞)

for any y∞ ∈ F×∞. Writing a = (a1, · · · , ak) to denote a k-tuple of F -integers, we have that

|Y∞|k4

∫I∼=[0,1]d⊂F∞

Pn1ϕ · θf,p((

1Y∞

x∞1

))ψ(−αx∞)dx∞

=∑

a=(a1,...,ak)∈OkF

p(a)cΠ(f(a) + α)

|f(a) + α| 12W

(f(a) + α

Y∞

).

20

Proof. Let us again lighten notation by writing a = (a1, . . . , ak) to denote a k-tuple of F -integers aj ∈ OF .Again, we open up Fourier-Whittaker expansions and switch the order of summation to compute∫

I∼=[0,1]d⊂F∞Pn1ϕ · θf,p

((y∞ x∞

1

))ψ(−αx∞)dx∞

= |y∞|k4−(n−2

2 )∑γ∈F×

Wϕ

((γy∞

1n−1

)) ∑a=(a1,...,ak)∈OkF

p(a)ψ(iy∞f(a))

×∫I∼=[0,1]d⊂F∞

ψ(γx∞ − f(a)x∞ − αx∞)dx∞

= |y∞|k4−(n−2

2 )∑

a=(a1,...,ak)∈OkF

f(a)+α 6=0

Wϕ

((y∞(f(a) + α)

1n−1

))p(a)ψ(iy∞f(a))

= |y∞|k4−(n−2

2 )∑

a=(a1,...,ak)∈OkF

f(a)+α 6=0

ρϕ (f(a) + α)Wϕ((f(a) + α)y∞)p(a)ψ(iy∞f(a))

= |y∞|k4−(n−2

2 )∑

a=(a1,...,ak)∈OkF

f(a)+α 6=0

cΠ(f(a) + α)

|f(a) + α|n−12

Wϕ((f(a) + α)y∞)p(a)ψ(iy∞f(a)).

Specializing to y∞ = 1/Y∞ ∈ F×∞ as above then gives∫I∼=[0,1]d⊂F∞

Pn1ϕ · θf,p((

1Y∞

x∞1

))ψ(−αx∞)dx∞

=

∣∣∣∣ 1

Y∞

∣∣∣∣ k4−(n−22 ) ∑

a=(a1,...,ak)∈OkF

f(a)+α 6=0

cΠ(f(a) + α)

|f(a) + α|n−12

Wϕ

(f(a) + α

Y∞

)ψ

(i · f(a)

Y∞

)p(a).

Choosing ϕ = ⊗vϕv ∈ VΠ as above then gives the stated integral presentation. �

3.4. Linear progressions of two cuspidal forms. Let us now fix an integer m ≥ 2, together with acuspidal automorphic representation π = ⊗vπv of GLm(AF ).

Proposition 3.5. Let Wj for j = 1, 2 be any smooth, summable function of y∞ ∈ F×∞ of moderate decaynear zero and rapid decay at infinity. Fix a nonzero F -integer α, and let us also write α to denote theimage under the diagonal embedding α → (α, α, . . .) ∈ A×F . Fix an archimedean idele Y∞ ∈ F×∞ of idelenorm |Y∞| > |α|. Let ϕ = ⊗vϕv ∈ VΠ and ϕ′ = ⊗vϕ′v ∈ Vπ be pure tensors whose nonarchimedean localcomponents are essential Whittaker vectors, and whose respective archimedean local Whittaker coefficientsare specified as functions of y∞ ∈ F×∞ by

Wϕ(y∞) = |y∞|n−22 W1 (y∞)

and

Wϕ′(y∞) = |y∞|m−2

2 W2 (y∞) .

Then, we have the integral presentation∫I∼=[0,1]d⊂F∞

Pn1ϕ · Pm1 ϕ′((

1Y∞

x∞1

))ψ(−αx∞)dx∞

=∑

γ1,γ2∈F×γ1−γ2=α

cΠ(γ1)cπ(γ2)

|γ1γ2|12

W1

(γ1

Y∞

)W2

(γ2

Y∞

).

21

Proof. Opening up Fourier-Whittaker expansions, switching the order of summation, and evaluating viaorthogonality of additive characters again, we find that∫

I∼=[0,1]d⊂F∞Pn1ϕPm1 ϕ′

((y∞ x∞

1

))ψ(−αx∞)dx∞

= |y∞|−(n−22 )−(m−2

2 )∑

γ1,γ2∈F×γ1−γ2=α

cΠ(γ1)cπ(γ2)

|γ1|n−12 |γ2|

m−12

Wϕ(γ1y∞)Wϕ′(γ2y∞).

Specializing to y∞ = 1/Y∞ as above then gives the identity∫I∼=[0,1]d⊂F∞

Pn1ϕPm1 ϕ′((

1Y∞

x∞1

))ψ(−αx∞)dx∞

=

∣∣∣∣ 1

Y∞

∣∣∣∣−(n−22 )−(m−2

2 ) ∑γ1,γ2∈F×γ1−γ2=α

cΠ(γ1)cπ(γ2)

|γ1|n−12 |γ2|

m−12

Wϕ

(γ1

Y∞

)Wϕ′

(γ2

Y∞

).

Choosing the pure tensors ϕ = ⊗vϕv ∈ VΠ and ϕ′ = ⊗vϕ′v ∈ Vπ as we do then gives the stated formula. �

4. Bounds for the shifted convolution problem

Fix a nonzero F -integer α ∈ OF and a totally positive archimedean idele Y∞ = (Y∞,j)dj=1 ∈ F×∞ with

idele norm |Y∞| > |α|. We can and do assume that each component Y∞jis contained within the chosen

fundamental domain F = (Fj)dj=1 ⊂ Hd ∼= P2(F∞) of Theorem 2.5 above. We now derive bounds for thesums of L-function coefficients appearing in the integral presentations of Propositions 3.2, 3.3, 3.4 and 3.5,or rather the shifted convolution sums appearing in the statement of the main theorem. Here, we shalluse spectral decompositions of automorphic forms on GL2(AF ) and its metaplectic cover G(AF ), building

on ideas from [6], [33], and [34], and also developing the setting described above with the lift Pn1ϕ of themirabolic form Pn1ϕ to GL2(AF ). We shall also treat the special case of dimension n = 2 separately fromthe generic case on all dimensions n ≥ 2, using Theorem 2.5 and Definition 2.6.

To begin, we first derive a variation of the integral presentations described above in terms of the lifted

form Pn1ϕ (Proposition 4.2). The constraints coming from the choice of fundamental domain above leadus to fix a smooth partition of unity and dyadic subdivision, which allows us to make a suitable choiceof archimedean local vectors in the pure tensor to derive integral presentations in terms of the lift. Afterreviewing some setup with Whittaker functions and Sobolev norms, we then argue that the L2-automorphicform(s) Φ we consider on GL2(A) or its two-fold metaplectic cover G(A) for the integral presentation ofthe corresponding shifted convolution problem have bounded spectral coefficients. This is automatic if theform Φ is smooth (see Lemma 4.8), as is the case for the standard setup of dimension n = 2. In the genericcase of n ≥ 2 however, we give an additional argument with inner products of suitable Poincare series(Proposition 4.9) or convolutions with smoothing kernels (described in the subsequent remark) to show thatthe spectral coefficients are bounded uniformly in a suitable sense. We then derive bounds in all casesafter decomposing the form Φ whose Fourier-Whittaker coefficient at α describes the corresponding shiftedconvolution problem spectrally. The arguments after this point are relatively standard, and given in termsof the best uniform approximations to the generalized Ramanujan conjecture or the generalized Lindelofhypothesis for GL2(AF )-automorphic forms in the level aspect, i.e. uniform bounds for all such forms – asopposed to the individual forms we consider.

4.1. Integral presentations revisited. Let us start by returning to the integral presentations derived inPropositions 3.4 and 3.5 for the generic case of dimension n ≥ 2, replacing the mirabolic cusp form Pn1ϕ by

its corresponding extended form Pn1ϕ ∈ L2(GL2(F )\GL2(A¯F

),1)K . In the setup of case (B) correspondingto linear shifts as in Proposition 3.5, we shall also make a similar modification, replacing Pm1 ϕ′ by its

corresponding localized extended form Pm1 ϕ′ ∈ L2(GL2(F )\GL2(AF ),1)K . Here, we shall first introduce asmooth partition of unity and corresponding dyadic decompositions to reduce to the setting of compactlysupported local weight functions. Although relatively standard, we spell out the details as we shall relyon this setup at some stages of the later argument. We shall also use the following observations about

22

the metaplectic group G(AF ) and L2-automorphic forms on it, leading to some convenient modifications tothe metaplectic theta series θf,p in the style of the liftings discussion of Theorem 2.5, Proposition 2.7, andProposition 2.9 which we shall use to derive suitable integral presentations for our shifted convolution sums.

Proposition 4.1. The following assertions about G(AF ) and automorphic functions on it are true.

(1) Each metaplectic element g ∈ G(AF ) can be expressed uniquely as g = (g, µ) with

g =∐ζ∈∆

γ ·(y∞ x∞

1

)·(r∞

r∞

)·(ζ

ζ

)· k ∈ GL2(AF )

decomposed uniquely according to Theorem 2.5 above (with the same conditions and constraints),and µ ∈ C2

∼= {±1} a square root of unity.

(2) Let φ ∈ L2(GL2(F )\G(AF ), ω) be any (genuine) L2-automorphic form on G(AF ) with some central

character ω. Then, the function φ defined on an element g = (g, µ) ∈ G(AF ) decomposed uniquelyaccording to (1) above by the rule

φ(g) = φ((g, µ)) := φ

(((y∞ x∞

1

), µ

))determines a (genuine) automorphic form on G(AF ) in the L2 sense having trivial central characterand trivial right action by the maximal compact subgroup K ⊂ GL2(AF ). Moreover, for y ∈ F×∞contained strictly within the chosen fundamental domain of Theorem 2.5, we have a matching ofFourier-Whittaker coefficients: For each F -integer α, we have that∫

I∼=[0,1]d⊂F∞φ

(((y∞ x∞

1

)), µ

)ψ(−αx∞)dx∞

=

∫I∼=[0,1]d⊂F∞

φ

(((y∞ x∞

1

)), µ

)ψ(−αx∞)dx∞.

Proof. The first claim (1) follows in a direct way from Theorem 2.5, i.e. applied to the matrix g ∈ GL2(AF )in a generic metaplectic element g = (g, µ) ∈ G(AF ). The second claim (2) can then be deduced by a minorvariation of the arguments given above for Propositions 2.7 and 2.9, i.e. if not a more direct argument. �

4.1.1. Choices of weight functions. Recall that fix smooth functions W and Wj (for j = 1, 2) as in thestatement of the main theorem. That is, we fix a smooth function W on y∞ ∈ F×∞,+ which decays as

W (y∞) =

{OC(|y∞|−C) for any C > 0 as |y∞| → ∞O(|y∞|κ) for some 0 < κ < 1 as |y∞| → 0,

and whose derivatives are bounded by W (i) � 1 for all i ≥ 1. Similarly, for each index j = 1, 2, we fixsmooth weight functions Wj on y∞ ∈ F×∞,+ which decay as

Wj(y∞) =

{OC(|y∞|−C) for any C > 0 as |y∞| → ∞O(|y∞|κj ) for some 0 < κj < 1 as |y∞| → 0,

and whose derivatives are bounded by W(i)j � 1 for all i ≥ 1. Here, we also assume that 0 < κ1 + κ2 < 1.

4.1.2. A smooth partition of unity and dyadic decompositions. Let us now introduce a smooth partition ofunity and corresponding dyadic decompositions. That is, a standard result in analysis shows that we canfix a smooth partition of unity as follows: There exists a smooth function U ∈ C∞c (R>0) supported on [1, 2]with bounded derivatives U (i) � 1 for each i ≥ 0 together with a collection {R} of ranges R ∈ R>0 suchthat for any y ∈ R>0, we have the partition of unity∑

{R}

U( yR

)= 1.

23

Moreover, there exists for each integer l ∈ Z a constant number (independent of l) of ranges R contained inthe dyadic interval [2l, 2l+1], and so we can assume without loss of generality that this takes the form∑

l∈Z

∑R∈{R}

R∩[2l,2l+1]6=∅

U( yR

)=∑l∈Z

∑R∈{R}

R∩[2l,2l+1]6=∅

U( yR

)= 1.(16)

Here, the notation means that we take a sum over all integers l ∈ Z, where for each corresponding dyadicinterval [2l, 2l+1] there is a single positive real number R ∈ [2l, 2l+1] ∈ R>0 which contributes to the partitionof unity. We shall sometimes omit the reference to each single range {R} in the subsequent discussion tolighten the notation, as in the second sum of the latter expression, taking for granted that this preliminarydiscussion makes the context clear. Now, given any generic sum∑

r≥1

A(r),

we can then consider the corresponding dyadic decomposition induced by (16)∑l∈Z

∑R∈{R}

R∩[2l,2l+1] 6=∅

∑r≥1

A(r)U( rR

)=∑l∈Z

A(r)∑R∈{R}

R∩[2l,2l+1]6=∅

U( rR

).

4.1.3. Reduction to local weight functions. Recall that for a fixed nonzero F -integer α ∈ OF , and for weightfunctions W and Wj as above (and keeping the setup described in the introduction), we seek to estimatethe shifted convolution sums ∑

a=(a1,...,ak)∈OkF

p(a)cΠ(f(a) + α)

|f(a) + α| 12W

(f(a) + α

Y∞

)and ∑

γ1,γ2∈F×γ1−γ2=α

cΠ(γ1)cπ(γ2)

|γ1γ2|12

W1

(γ1

Y∞

)W2

(γ2

Y∞

).

Using the partition of unity (16), each of these sums can be decomposed respectively as a sum over sums oflocal weight functions defined by∑

l∈Z

∑a=(a1,...,ak)∈OkF

p(a)cΠ(f(a) + α)

|f(a) + α| 12W

(f(a) + α

Y∞

) ∑R∈{R}

R∩[2l,2l+1]6=∅

U

(|f(a) + α|

R

)

and ∑l1,l2∈Z

∑γ1,γ2∈F×γ1−γ2=α

cΠ(γ1)cπ(γ2)

|γ1γ2|12

W1

(γ1

Y∞

) ∑R1∈{R1}

R1∩[2l1 ,2l1+1]6=∅

U

(|γ1|R1

)W2

(γ2

Y∞

) ∑R2∈{R2}

R2∩[2l2 ,2l2+1]6=∅

U

(|γ2|R2

).

Thus for each integer l ∈ Z, writing Y = |Y∞| as we do, and taking the local weight functions Wl and Wj,l

(for j = 1, 2) defined on y∞ ∈ F×∞,+ by

Wl (y∞) = W (y∞)∑R∈{R}

R∩[2l,2l+1]6=∅

U

(|y∞|YR

)(17)

and

Wl,j (y∞) = Wj(y∞)∑

Rj∈{Rj}

Rj∩[2lj ,2

lj+1]6=∅

U

(|y∞|YR

),(18)

24

we have the respective decompositions into local sums

(19)

∑a=(a1,...,ak)∈OkF

p(a)cΠ(f(a) + α)

|f(a) + α| 12W

(f(a) + α

Y∞

)

=∑l∈Z

∑a=(a1,...,ak)∈OkF

p(a)cΠ(f(a) + α)

|f(a) + α| 12Wl

(f(a) + α

Y∞

)and

(20)

∑γ1,γ2∈F×γ1−γ2=α

cΠ(γ1)cπ(γ2)

|γ1γ2|12

W1

(γ1

Y∞

)W2

(γ2

Y∞

)

=∑

l1,l2∈Z

∑γ1,γ2∈F×γ1−γ2=α

cΠ(γ1)cπ(γ2)

|γ1γ2|12

W1,l1

(γ1

Y∞

)W2,l2

(γ2

Y∞

).

4.1.4. Integral presentations for local weight functions. Recall that we want to derive bounds for the sums(19) and (20). Let us now observe that to derive upper bounds for these shifted convolution sums, it willsuffice by the rapid decay properties of the weight functions W and Wj (for j = 1, 2) to bound the truncatedsums in terms of the length Y = |Y∞| as∑

a=(a1,...,ak)∈OkF

|f(a)+α|≤Y

p(a)cΠ(f(a) + α)

|f(a) + α| 12W

(f(a) + α

Y∞

)(21)

and ∑γ1,γ2∈F×γ1−γ2=α

|γ1|,|γ2|≤Y

cΠ(γ1)cπ(γ2)

|γ1γ2|12

W1

(γ1

Y∞

)W2

(γ2

Y∞

).(22)

It is easy to deduce from the fact that the smooth function U used to define the partition of unity (16) issupported on [1, 2] that for a given range R ∈ [2l, 2l+1], the function defined on y∞ ∈ F×∞,+ by

U

(|y∞|YR

)is supported on y∞ ∈ F×∞,+ in the interval

2l

Y≤ |y∞| ≤

2l+2

Y.

Scaling out by Y in the definition, it is then easy to deduce that the truncated sums (21) and (22) canbe reparametrized respectively as finite sums over local sums indexed by integers 0 ≤ l ≤ log Y/ log 2 and0, l1, l2,≤ log Y/ log 2. In particular, we deduce that it will suffice to bound the finite sums over local sums∑

0≤l≤log Y

∑a=(a1,...,ak)∈Ok

F|f(a)+α|≤Y

p(a)cΠ(f(a) + α)

|f(a) + α| 12Wl

(f(a) + α

Y∞

)(23)

and ∑0≤l1,l2≤log Y

∑γ1,γ2∈F×γ1−γ2=α

|γ1|,|γ2|≤Y

cΠ(γ1)cπ(γ2)

|γ1γ2|12

W1,l1

(γ1

Y∞

)W2,l2

(γ2

Y∞

).(24)

We shall now take this fact for granted, and in each case work to derive bounds for each of the inner sumsindexed by an integer 0 ≤ l ≤ log Y or a pair of integers 0 ≤ l1, l2 ≤ log Y . First, let us explain how wecan derive integral presentations for any of the local sums corresponding to an integer l ∈ Z in the dyadicdecomposition (19) or a pair of integers (l1, l2) ∈ Z2 in the dyadic decomposition (20).

Taking the discussion above with Proposition 4.1 for granted, we can now derive the following integralpresentations for the shifted convolution problems we consider in terms of the Fourier-Whittaker coefficientsat a given nonzero F -integer α of certain L2-automorphic forms on GL(AF ) or its metaplectic cover G(AF ).

25

Proposition 4.2. Fix a nonzero F -integer α ∈ OF , and let Y∞ = (Y∞,j)dj=1 ∈ F×∞ be any totally positive

archimedean idele of norm Y = |Y∞| > |α| contained within our chosen fundamental domain of Theorem 2.5above. Let Π = ⊗vΠv be an irreducible cuspidal automorphic representation of GLn(AF ). Given an integer2 ≤ m ≤ n, we also consider a cuspidal automorphic representation π = ⊗vπv of GLm(AF ). Let us also fix0 ≤ l ≤ log Y any integer in the subdivision (23) above, together with any pair of integers 0 ≤ l1, l2 ≤ log Yin the subdivision (24). We have the following integral presentation for each of the corresponding inner sums.

(A) Let W ∈ L2(F×∞) be any smooth function of moderate decay near zero which decays rapidly at infinityor is compactly supported, and which decays near zero as described above. Fixing a smooth partition of unity(16) and corresponding dyadic decomposition (19), let Wl denote the local weight function (17) corresponding

to some given integer l ∈ Z in this decomposition. Let use then take ϕ(l) = ⊗vϕ(l)v ∈ VΠ to be a pure tensor

whose nonarchimedean local components are each essential Whittaker vectors, and whose archimedean localcomponent is chosen so that we have the identification of functions of y∞ ∈ F×∞

Wϕ(l)(y∞) := Wϕ(l)

((y∞

1n−1

))= |y∞|

n−22 ψ(−i|y∞|)ψ (iαY∞)Wl

(1

|y∞|

).

Note that Wϕ(l)(y∞) = Wϕ(l)(−y∞). Let θf,p be the theta series introducted above, with θf,p = T−1θf,p its

image under the Hecke operator corresponding to x 7→ −x ∈ AF , and θf,p extension or descent according toProposition 4.1 (ii). We have the integral presentation∫

I∼=[0,1]d⊂F∞Pn1ϕ(l) · θf,p

((Y∞ x∞

1

))ψ(−αx∞)dx∞

= |Y∞|k4

∑a=(a1,··· ,ak)∈OkF

p(a)cΠ(a+ α)

|f(a) + α| 12Wl

(∣∣∣∣f(a) + α

Y∞

∣∣∣∣) .(B) Let W1,W2 ∈ L2(F×∞) by any smooth functions of moderate decay near zero which decay rapidly atinfinity, or are compactly supported, and which decay near zero as described above. Fixing a smooth partitionof unity (16) and corresponding dyadic decomposition (20) as above, let Wlj for j = 1, 2 denote the localweight functions corresponding to some given pair of integers l1, l2 ∈ Z as defined in (18). Let us then take

ϕ(l1) = ⊗vϕ(l1)v ∈ VΠ be a pure tensor whose nonarchimedean local components are each essential Whittaker

vectors, and whose archimedean local component is chosen so that we have the identification of functions ofy∞ ∈ F×∞

Wϕ(l1)(y∞) := Wϕ(l1)

((y∞

1n−1

))= |y∞|

n−22 W1,l1

(1

|y∞|

).

Note that Wϕ(l1)(y∞) = Wϕ(l1)(−y∞). Let us also take ϕ′(l2) = ⊗vϕ′(l2) ∈ Vπ to be a pure tensor whosenonarchimedean local components are each essential Whittaker vectors, and whose archimedean local compo-nents are chosen so that as functions of y∞ ∈ F×∞

Wϕ′(l2)(y∞) := Wϕ′(l)

((y∞

1m−1

))= |y∞|

m−22 W2,l2

(1

|y∞|

).

Note again that Wϕ′(l2)(y∞) = Wϕ′(l2)(−y∞). Then, we have the integral presentation∫I∼=[0,1]d⊂F∞

Pn1ϕ(l1) · ˜Pm1 ϕ′(l2)

((Y∞ x∞

1

))ψ(−αx∞)dx∞

=∑

γ1,γ2∈F×γ1−γ2=α

cΠ(γ1)cπ(γ2)

|γ1γ2|12

W1,l1

(∣∣∣∣ γ1

Y∞

∣∣∣∣)W2,l2

(∣∣∣∣ γ2

Y∞

∣∣∣∣) .26

Proof. Let us start with (A), which we reduce to the calculation of Proposition 3.4 using Proposition 2.9.We split in the integral into two parts following the proof of Proposition 2.9, so that∫

I∼=[0,1]dPn1ϕ(l)

((Y∞ x∞

1

))θf,p

((Y∞ x∞

1

))ψ(−αx∞)dx∞

=

∫I∼=[0,1]d

Pn1ϕ(l)

((Y∞ x∞

1

))θf,p

((Y∞ x∞

1

))ψ(−αx∞)dx∞

=

∫I1∼=[0,1/2]d

Pn1ϕ(l)

((Y∞ x∞

1

))θf,p

((Y∞ x∞

1

))ψ(−αx∞)dx∞

+

∫I2∼=(1/2,1]d

Pn1ϕ(l)

((Y∞ x∞

1

))θf,p

((Y∞ x∞

1

))ψ(−αx∞)dx∞.

For any x∞ ∈ I1 ∼= [0, 1/2]d in the first integral, we can use Definition 2.6 to make the easy identification∫I1∼=[0,1/2]d

Pn1ϕ(l)

((Y∞ x∞

1

))θf,p

((Y∞ x∞

1

))ψ(−αx∞)dx∞

=

∫I1∼=[0,1/2]d

Pn1ϕ(l)

((Y∞ x∞

1

))θf,p

((Y∞ x∞

1

))ψ(−αx∞)dx∞.

For any x∞ ∈ I2 ∼= (1/2, 1]d in the second integral, we first argue as in Proposition 2.9 that we have

Pn1ϕ(l)

((Y∞ x∞

1

))= Pn1ϕ(l)

((Y∞ x∞

1

))= Pn1ϕ(l)

((Y∞ x∞ − 1

1

))= Pn1ϕ(l)

((Y∞ 1− x∞

1

)).

We then use Definition 2.6 again to make the identification

Pn1ϕ(l)

((Y∞ 1− x∞

1

))= Pn1ϕ(l)

((Y∞ 1− x∞

1

)),

whence we argue again following the proof of Proposition 2.9 that

Pn1ϕ(l)

((Y∞ 1− x∞

1

))= Pn1ϕ(l)

((Y∞ −x∞

1

))= Pn1ϕ(l)

((−Y∞ x∞

1

))= Pn1ϕ(l)

((Y∞ x∞

1

)),

so that

Pn1ϕ(l)

((Y∞ x∞

1

))= Pn1ϕ(l)

((Y∞ x∞

1

)).

The same argument works to make such a series of identifications for the modified form θf,p. Thus, we have∫I2∼=(1/2,1]d

Pn1ϕ(l)

((Y∞ x∞

1

))θf,p

((Y∞ x∞

1

))ψ(−αx∞)dx∞

=

∫I2∼=(1/2,1]d

Pn1ϕ(l)

((Y∞ x∞

1

))θf,p

((Y∞ x∞

1

))ψ(−αx∞)dx∞,

from which we derive the identification of unipotent integrals∫I∼=[0,1]d

Pn1ϕ(l)

((Y∞ x∞

1

))θf,p

((Y∞ x∞

1

))ψ(−αx∞)dx∞

=

∫I∼=[0,1]d

Pn1ϕ(l)

((Y∞ x∞

1

))θf,p

((Y∞ x∞

1

))ψ(−αx∞)dx∞.

27

We can then evaluate the latter integral by a minor variation of the proof of Proposition 3.4. That is,∫I∼=[0,1]d

Pn1ϕ(l) · θf,p((

Y∞ x∞1

))ψ(−αx∞)dx∞

=

∫I∼=[0,1]d

Pn1ϕ(l)

((Y∞ x∞

1

))θf,p

((Y∞ x∞

1

))ψ(−αx∞)dx∞

= |Y∞|k4−(n−2

2 )∑γ∈F×

Wϕ(l)

((γY∞

1n−1

)) ∑a=(a1,...,ak)∈OkF

p(a)ψ (i · f(a)Y∞)

×∫I∼=[0,1]d

ψ(x∞(γ − f(a)− α))dx∞

= |Y∞|k4−(n−2

2 )∑

a=(a1,...,ak)∈OkF

f(a)+α 6=0

p(a)ρϕ(l)(f(a) + α)Wϕ(l) ((f(a) + α)Y∞)ψ (i · f(a)Y∞)

= |Y∞|k4

∑a=(a1,...,ak)∈Ok

Ff(a)+α 6=0

p(a)cΠ(f(a) + α)

|f(a) + α| 12Wl

(f(a) + α

Y∞

).

To proof for (B) works in the same way, using (11) and the arguments of Proposition 2.9 to relate eachof the integrals to the ones computed already in Proposition 3.5 above. �

4.2. Whittaker functions and their decay properties. Let us first give a brief account of the normalizedWhittaker functions we shall consider for our estimates below, and particularly the uniform bounds theysatisfy, following [33, §7] and [6, §2]. Given complex numbers κ, ν ∈ C, let Wκ,ν denote the classicalWhittaker function as described in [33, §7.1] (for instance), whose Mellin transform for <(s) > 1

2 ± ν can bedescribed according to the calculation of [20, 7.621-11] as∫ ∞

0

e−y2Wκ,ν(y)ys

dy

y=

Γ(

12 + s+ ν

)Γ(

12 + s− ν

)Γ (1 + s− κ)

.(25)

Taking the inverse Mellin transform then gives us the more convenient integral presentation

e−y2Wκ,ν(y) =

∫<(s)=σ

Γ(

12 + s+ ν

)Γ(

12 + s− ν

)Γ (1 + s− κ)

y−sds

2πi(26)

for any sufficiently large real number σ > 0.

4.2.1. Estimates. The latter integral presentation (26) can be used to derive the following estimates for theWhittaker function Wκ,ν(y) as y → 0 (and also as y →∞) according to the arguments given in [33, §7]. Todescribe these in more detail, let us first record the following auxiliary result.

Proposition 4.3. The following assertions about the gamma function Γ(s) are true:

(i) For any real parameter σ ∈ R, the function t 7−→ Γ(σ + it) is a decreasing function of |t|.

(ii) (Stirling approximation). Fix ε > 0. For any s ∈ C with | arg(s)| > π − ε, we have that

Γ(s) =

(2π

s

) 12 (s

e

)s(1 +O

(1

|s|

)).

(iii) For any fixed δ > 0, the value Γ(t+ i|t|1+δ) is exponentially small as t→ ±∞.

28

(iv) Fix ε > 0 and A ≥ 1 any (large) integer. Given z ∈ C, write ||z|| = minn∈Z |z − n| to denotethe nearest integer. Then, we have the following uniform bounds for any a, b ∈ C and κ ∈ R with=(a) = =(b) and ||a+ κ||, ||b− κ|| < ε:

h−2A � |Γ (a+ κ) Γ (b− κ)||Γ (a+ {κ}) Γ (b− {κ})|

� h2A,

where h := |κ|+ |=(a)|+ 1, and {κ} ∈ [0, 1] denotes the fractional part of κ ∈ R.

Proof. See [33, Lemma 7.1] for (i), [33, Lemma 7.2] for (ii), [33, Lemma 7.3] for (iii), and [33, Lemma 7.4]for (iv). �

Proposition 4.4. We have the following uniform bounds for y ∈ R>0 as y → 0.

(i) If κ, r ∈ R, then for some constant A > 0 and any choice of ε > 0,

Wκ,ir(y)

Γ(

12 + ir + κ

) �ε (|κ|+ |r|+ 1)Ay

12−ε

(ii) If κ ∈ R and 0 < ν < 12 , then for some constant A > 0 and any choice of ε > 0,

Wκ,ν(y)

Γ(

12 + κ

) �ε (|κ|+ 1)Ay

12−ν−ε.

(iii) If κ, ν ∈ R with κ− ν − 12 ≥ 0 and ν > − 1

2 + ε, then for some A > 0 and any choice of ε > 0,

Wκ,ν(y)

Γ(

12 + κ− ν

)Γ(

12 + κ+ ν

) �ε (|κ|+ |ν|+ 1)Ay

12−ε.

Proof. See [33, Proposition 3.1], which is proven in [33, §7]. Let us give a sketch of the argument for futurereference. In each case, starting with the integral presentation (26), we shift the line of integration leftwardto <(s) = σ = 0, crossing poles from the gamma functions.

For (i), we estimate the normalized contour integral presentation

Wκ,ir(y)

Γ(

12 + κ+ ir

) =ey2

Γ(

12 + κ+ ir

) ∫<(s)=σ

Γ(

12 + s+ ir

)Γ(

12 + s− ir

)Γ (1 + s− κ)

y−sds

2πi.

Shifting the line of integration to <(s) = σ = 0, we cross poles at s = − 12 − ir and s = − 1

2 + ir. It is easy

to see that the pole at s = − 12 − ir contributes

ey2 · Γ (−2ir)

Γ(

12 + κ+ ir

)Γ(

12 + κ− ir

) · y 12−ir � (|κ|+ |r|+ 1)

By

12

while the pole at s = − 12 + ir contributes

ey2 · Γ (2ir)

Γ(

12 + κ+ ir

)Γ(

12 + κ− ir

) · y 12−ir � (|κ|+ |r|+ 1)

By

12

Here, we use Proposition 4.3 (iv) and 4.3 (ii) (Stirling approximation) to derive the stated bounds. Note

that in either case, we us the fact that for 0 < y < 1, we have 1 < ey2 < 2, and hence the exponential term

is absorbed into the larger quantity on the right. To estimate the remaining integral at <(s) = σ = 0, puts = it and ν = ir so that we reduce to considering the ratio of gamma factors

Γ(

12 + it+ ir

)Γ(

12 + it− ir

)Γ(

12 + ir + κ

)Γ(

12 + ir − κ

) .Using the exponential decay of the gamma function on vertical lines described in Proposition 4.3 (i), wemay modify the denominator so that the two gamma functions are estimated along the same imaginary partmax(|r|, |t|). We can then apply the estimate of Proposition 4.3 (iv) to replace the real parameter κ ∈ R by

29

its fractional part {κ}, and apply Stirling approximation to estimate the remaining integral, which is thenseen to have rapid decay as t→∞ with polynomial control in κ and r. This implies the stated bound.

Th argument for (ii) is similar: Starting with the integral presentation

e−y2Wκ,ν(y)

Γ(

12 + κ

) =1

Γ(

12 + κ

) ∫<(s)=σ

Γ(s+ 1

2 + ν)

Γ(s+ 1

2 − ν)

Γ (s+ 1− κ)y−s

ds

2πi,

we shift the line of integration to <(s) = σ = 0. The pole at s = − 12 − ν contributes a residue of

ey2 · Γ(−2ν)

Γ(

12 + κ

)Γ(

12 − ν − κ

) · y 12 +ν � (1 + |κ|)B · y 1

2 +ν

and the pole at s = − 12 + ν contributes a residue of

ey2 · Γ(2ν)

Γ(

12 + κ

)Γ(

12 + ν − κ

) · y 12−ν � (1 + |κ|)B · y 1

2−ν

To estimate the remaining integral over σ = 0, we use the same argument as given for (i) to estimate thecorresponding ratio of gamma factors

Γ(

12 + it+ ν

)Γ(

12 + it− ν

)Γ (1 + it− κ) Γ

(12 + κ

) .

The argument for (iii) is also similar: Starting with the integral presentation

e−y2Wκ,ν(y)

Γ(

12 + κ− ν

)Γ(

12 + κ+ ν

) =1

Γ(

12 + κ− ν

)Γ(

12 + κ+ ν

) ∫<(s)=σ

Γ(s+ 1

2 + ν)

Γ(s+ 1

2 − ν)

Γ (s+ 1− κ)y−s

ds

2πi,

we shift the line of integration to <(s) = σ = 0. Now, observe that since κ− ν − 12 ∈ Z, the function

s 7−→Γ(

12 + s− ν

)Γ (1 + s− κ)

has no pole. There is a residue as s = − 12 − ν which would contribute

ey2 · Γ(−2ν)

Γ(

12 + κ+ ν

)Γ(

12 − κ− ν

)2 · y 12 +ν ,

however we do not have to consider such a contribution as ν > − 12 . The remaining integral over <(s) = 0 is

bounded using a similar argument as for (i) and (ii). That is, it will do to bound the ratio

Γ(

12 + ν + it

)Γ(

12 − ν + it

)√|Γ(

12 + κ− ν

)Γ(

12 + κ+ ν

)| · Γ (1 + it− κ)

.

Observe that√|Γ(

12 + κ− ν

)Γ(

12 + κ− ν

)| ≥

√Γ(κ), from which we deduce that it will suffice to bound

Γ (it+ κ)

Γ (κ)·

Γ(

12 + ν + it

)Γ(

12 − ν + it

)|Γ (it+ κ) Γ (1 + it− κ) |

.

Here, we can apply Proposition 4.3 (i) to derive the bound Γ(it + κ)/Γ(κ) � 1. Moreover, we can applyProposition 4.3 (ii) and (iii) to deduce that this quantity Γ(it+ κ)/Γ(κ) decays exponentially as |t| > κ1+δ.Now, since ν ∈ R, we can apply Proposition 4.3 (iv) to the produce of gamma factors in the remainingnumerator. This allows us to replace the contribution of ν by its fractional part {ν}. Similarly, applyingProposition 4.3 (iv) to the product of gamma factors in the denominator allows us to replace the contributionof κ by its fractional part {κ}. We can then use Stirling approximation (Proposition 4.3 (ii)) to deduce thatthis remaining ratio is bounded uniformly by a polynomial in κ and ν, as required. �

Let us now record how a variation of this argument allows us to derive similar quantitative uniform boundsas y →∞. In fact, a variation of the same argument gives the same uniform, quantitative bounds as y →∞:

Corollary 4.5. We have the following uniform bounds for y ∈ R>0 as y →∞.

30

(i) If κ, r ∈ R, then for some constant A > 0 and any choice of ε > 0,

Wκ,ir(y)

Γ(

12 + ir + κ

) �ε (|κ|+ |r|+ 1)Ay

12 +ε

(ii) If κ ∈ R and 0 < ν < 12 , then for some constant A > 0 and any choice of ε > 0,

Wκ,ν(y)

Γ(

12 + κ

) �ε (|κ|+ 1)Ay

12 +ν+ε.

(iii) If κ, ν ∈ R with κ− ν − 12 ≥ 0 and ν > − 1

2 + ε, then for some A > 0 and any choice of ε > 0,

Wκ,ν(y)

Γ(

12 + κ− ν

)Γ(

12 + κ+ ν

) �ε (|κ|+ |ν|+ 1)Ay

12 +ε.

Proof. We proceed in a similar way as for Proposition 4.4, shifting contours to the right. That is, fixing thereal parameter y > 1, we assume the integral in (26) is defined for <(s) = σ > 0 sufficiently large, then shift

the contour to the right beyond the line determined by yσ = ey2 . We then have to consider residues coming

from the corresponding gamma factors at − 12 ± ν + [σ], where [σ] denotes the integer part of σ > 0, i.e. so

that σ = [σ] +{σ}. The remaining contours are seen easily to be negligible, in contrast the case of 0 < y < 1shown above, though the final estimates (reflecting the contributions of the residual terms) are the same.

For (i), we estimate the contribution of

Wκ,ir(y)

Γ(

12 + ir + κ

) =ey2

Γ(

12 + ir + κ

) ∫<(s)=σ

Γ(s+ 1

2 + ir)

Γ(s+ 1

2 − ir)

Γ (s+ 1− κ)y−s

ds

2πi.

Taking σ > 0 to be large enough so that yσ ≥ ey2 , we shift the contour far enough to the right that we cross

poles from each of the gamma factors in the numerator. The pole at s = − 12 − ir + [σ] contributes

ey2

yσ· Γ (−2ir + [σ])

Γ(

12 + ir + [σ] + κ

)Γ(

12 + ir + [σ]− κ

) � (|κ|+ |r|+ 1)By

12 .

where again we use Proposition 4.3 (iv) to replace [σ] by its fractional part {[σ]} = 0 followed by Proposition4.3 (ii) (Stirling approximation). Similarly, we see that the pole at s = − 1

2 + ir + [σ] contributes

ey2

yσΓ (2ir + [σ])

Γ(

12 + ir + [σ] + κ

)Γ(

12 + ir + [σ]− κ

) � (|κ|+ |r|+ 1)By

12 .

It is easy to see that the remaining integral is bounded above by a comparatively negligible quantity.For (ii), we argue in a similar way to estimate the contribution of

Wκ,ν(y)

Γ(

12 + κ

) =ey2

Γ(

12 + κ

) ∫<(s)=σ

Γ(s+ 1

2 + ν)

Γ(s+ 1

2 − ν)

Γ (s+ 1− κ)y−s

ds

2πi.

Shifting the contour to the right in the same way, the pole at s = − 12 − ν − [σ] contributes

ey2

yσ· Γ (−2ν + [σ])

Γ(

12 + κ

)Γ(

12 − ν + [σ]

) � (1 + |κ|)By 12 +ν ,

and the pole at s = − 12 + ν + [σ] contributes

ey2

yσ· Γ (2ν + [σ])

Γ(

12 + κ

)Γ(

12 − ν + [σ]

) � (1 + |κ|)By 12−ν .

Again, we argue that the remaining integral is bounded above by a comparatively negligible quantity.Finally for (iii), we also argue in a similar way to estimate the contribution of

Wκ,ν(y)

Γ(

12 + κ− ν

)Γ(

12 + κ+ ν

) =ey2

Γ(

12 + κ− ν

)Γ(

12 + κ+ ν

) ∫<(s)=σ

Γ(s+ 1

2 + ν)

Γ(s+ 1

2 − ν)

Γ (s+ 1− κ)y−s

ds

2πi.

31

Shifting the contour to the right in the same way, we pick up a pole at s = − 12 − ν + [σ] which contributes

ey2

yσ· Γ (−2ν + [σ])

Γ(

12 + κ+ ν

)Γ(

12 + κ− ν

)Γ(

12 − ν + [σ]− κ

) � (|κ|+ |ν|+ 1)By

12 +ν

and a pole at s = − 12 + ν + [σ] which contributes

ey2

yσ· Γ (2ν + [σ])

Γ(

12 + κ+ ν

)Γ(

12 + κ− ν

)Γ(

12 + ν + [σ]− κ

) � (|κ|+ |ν|+ 1)By

12−ν

Again, the remaining integral is seen to be bounded by a comparatively negligible quantity to derive thestated estimate. �

4.2.2. Remark on normalizations. Note that we could also consider the following normalized Whittakerfunctions here, in the style of Blomer-Harcos [6, §2.4]. To describe these normalized Whittaker functions, letus now take k ∈ Z any integer, and assume that the corresponding spectral parameter ν in constrained by

ν ∈

{(12 + Z

)∪ iR ∪ (− 1

2 ,12 ) if k ≡ 0 mod 2

Z ∪ iR if k ≡ 1 mod 2

For such parameters, we define the normalized Whittaker function on y ∈ R× by the formula

W ?k2 ,ν

(y) =isgn(y) k2Wsgn(y) k2 ,ν

(4π|y|)√Γ(

12 − ν + sgn(y)k2

)Γ(

12 + ν + sgn(y)k2

) .As explained in [6, §2.4] (cf. erratum), these normalized Whittaker functions are more natural to considerfor spectral arguments, as they can be shown according to [9, §4] to form an orthonormal basis of the spaceof square integrable functions on R× with respect to the Haar measure d×y = dy/|y|: For each choice ofparity l ∈ {0, 1}, we have the Hilbert space decomposition

L2(R×, d×y) =⊕k∈Z

k≡l mod 2

CW ?k2 ,ν, 〈W ?

k12 ,ν

,W ?k22 ,ν〉 =

{1 if k1 = k2

0 if k1 6= k2

.

It is easy to deduce that these normalized functions can be bounded uniformly as in Proposition 4.4 above.

4.2.3. Extension of estimates to totally real fields. Let us also note that since we work over a totally realnumber field F of degree d, the explicit functions appearing in the Fourier-Whittaker expansions describedabove can be realized as d-fold products of these classical Whittaker functions, indexed by d-tuples of weightsκ = (κj)

dj=1 and spectral parameters ν = (νj)

dj=1. Hence, we define the corresponding Whittaker functions

on y∞ = (yj)dj=1 ∈ F×∞ via the d-fold products

Wκ,ν(y∞) =

d∏j=1

Wκj ,νj (yj) and W ?k2 ,ν

(y∞) =

d∏j=1

W kj2 ,νj

(yj).(27)

Plainly, these functions can be estimated for |y∞| → 0,∞ via Proposition 4.4 and Corollary 4.5 above.

4.3. Sobolev norms. Let us now give some background on Sobolev norms for later arguments. Here, werefer to the general theory as described in [35, §2.9] and [29, §2.4], which apply to smooth or sufficientlysmooth L2-automorphic functions on GL2(AF ). Note that we shall later supply additional justification totreat the L2-automorphic functions we constructed from the automorphic functions Pn1ϕ on the mirabolicsubgroup P2 as considered above, which are not generally smooth.

The action of GL2(F∞) on the space of L2-automorphic forms L2(GL2(F )\GL2(AF ), ω) induces anaction of the Lie algebra gl2(F∞) of GL2(F∞) on this space. The corresponding action of the Lie subalgebrag = sl2(F∞) induces an action of its universal enveloping algebra U(g) on L2(GL2(F )\GL2(AF ), ω) viahigher-order differential operators. To be more explicit, writing ej for a given index 1 ≤ j ≤ d to denote

32

the standard basis vector with j-th entry equal to one and all others zero, the action of g is generated for1 ≤ j ≤ d by the linearly independent vectors

Hj =

(ej 00 ej

), Rj =

(0 ej0 0

), Lj =

(0 0ej 0

).

Writing k(ϑ) ∈ SO2(F∞) for ϑ = (ϑj)dj=1 ∈ (R/Z)d to denote a generic element, i.e. so that

k(ϑ) =

(cosϑ sinϑ− sinϑ cosϑ

)∈ SO2(F∞)

the differential operators corresponding to these basis elements can be described explicitly as

dHj = −2yj sin(2ϑj)∂xj + 2yj cos(2ϑj)∂yj + sin(2ϑj)∂ϑj

dRJ = yj cos(2ϑj)∂xj + yj sin(2ϑj)∂yj + sin2(ϑj)∂ϑj

dLj = yj cos(2ϑj)∂xj + yj sin(2ϑj)∂yj − cos2(ϑj)∂ϑj .

Note that this discussion extends directly to describe an action of the universal enveloping algebra U(g) onthe corresponding space of genuine L2-automorphic forms on the metaplectic cover L2(GL2(F )\G(AF ), ω),as there is an identification of the corresponding Lie subalgebras (cf. [33, §6.2], [29, §2.4]). Thus in eithercase, we can define Sobolev norms in a natural way as follows:

Definition 4.6. Given an integer B ≥ 1 and a smooth function φ ∈ L2(GL2(F )\GL2(AF ), ω) or moregenerally φ ∈ L2(GL2(F )\G(AF ), ω), we define the corresponding Sobolev norm ||φ||SB by the rule

||φ||SB =∑

ord(D)≤B

||Dφ||,

where the sum runs over all monomials D ∈ U(g) in Hj , Rj, and Lj of degrees less than or equal to B.The norm || · || on the right hand side of the definition denotes the norm coming from the inner product onL2(GL2(F )\GL2(AF ), ω) or L2(GL2(F )\G(AF ), ω) respectively.

Now, observe that if we use the surjectivity of the archimedean local Kirillov map to choose an L2-automorphic form φ ∈ L2(GL2(F )\GL2(AF ), ω) in such a way that the archimedean local component of thecorresponding Whittaker function Wφ(y∞) is smooth as a function of y∞ ∈ F×∞, then ||φ||SB is convergentfor each integer B ≥ 1 (cf. [33, §2.4, Proposition 2.1]). The same assertion is true after multiplication by ametaplectic theta series θ. That is, each norm || · ||SB can be extended to the corresponding Hilbert spaceL2(GL2(F )\G(AF ), ω) of L2-automorphic forms on the metaplectic cover G(AF ). It is then relatively easyto deduce that ||φθ||SB � 1 for any choice of integer B ≥ 1. Moreover, we have the following general result.

Proposition 4.7. Let φ be any smooth automorphic form on GL2(F )\GL2(AF ) or GL2(F )\G(AF ) with agiven central character. Then, we have for each integer B ≥ 0 the Sobolev norm bound ||φ||SB � 1, wherethe implied constant depends only on the choice of B.

Proof. See [33, Lemma 6.1] together with the more general discussion given in [29, §2.4] or [35, §2.9, 8]. Thatis, the result is standard for smooth cuspidal functions, and can be extended via height functions accordingto the argument of [33, Lemma 6.1] to treat the more general setting described in the statement. �

4.4. Spectral decompositions. We now begin the proof of Theorem 1.1. We first review some backgroundon spectral decompositions for the Hilbert spaces of L2-automorphic forms we consider. We then argue thatthe coefficients in the spectral expansion(s) of the form(s) Φ we consider are bounded uniformly in anycase; see Lemma 4.8, Proposition 4.9, and the subsequent remark. We then proceed to bound the Fourier-Whittaker coefficient at α of Φ via spectral decomposition to prove Theorem 1.1.

4.4.1. Spectral decompositions of (genuine and non-genuine) metaplectic forms. Let us first describe thespace of L2-automorphic forms on the metaplectic cover G(AF ) briefly. The two-fold cover G of the algebraicgroup GL2 can be defined via cocycles, as described in Gelbart [17]. Writing µ2 = {±1} to denote the squareroots of unity, the adelic points G(AF ) fit into the exact sequence

1 −→ µ2 −→ G(AF ) −→ GL2(AF ) −→ 1.

33

This sequence splits over the group of F -rational points GL2(F ), as well as the unipotent radical of itsstandard Borel subgroup. Recall that automorphic forms on G(AF ) which transform nontrivially underµ2 are said to be genuine. These genuine forms correspond to modular forms of half-integral weight, theprototype example being the metaplectic theta series θq described above, and more generally the theta seriesθf,p associated to a positive definite quadratic form f with an odd number of variables k. Note that thesetheta series can be described equivalently in terms of the Weil representation rf associated to f .

Let Φ be any genuine automorphic form in the space L2(GL2(F )\G(AF ), ω), where ω is some given ideleclass character of F . Hence, Φ transforms nontrivially under the group of square roots of unity µ2. Wecan decompose any L2-automorphic form Φ ∈ L2(GL2(F )\G(AF ), ω) spectrally as follows, according to thedescription given in Gelbart [17]. That is, the space of genuine forms L2(GL2(F )\G(AF ), ω) decomposesinto a Hilbert space direct sum of a discrete spectrum L2

disc(GL2(F )\G(AF ), ω) plus a continuous spectrum

L2cont(GL2(F )\G(AF ), ω) spanned by analytic continuations of metaplectic Eisenstein series. The discrete

spectrum decomposes further into subspaces of cuspidal and residual forms

L2disc(GL2(F )\G(AF ), ω) = L2

cusp(GL2(F )\G(AF ), ω)⊕ L2res(GL2(F )\G(AF ), ω).

Note that the residual forms (by definition) occur as residues of metaplectic Eisenstein series, and moreoverby [17, Theorem 6.1] all occur as translates of the metaplectic theta series associated to the quadratic formq(x) = x2. Let us fix a basis B for this space consisting of:

• An orthonormal basis {φi}i of cuspidal forms φi ∈ L2cusp(GL2(F )\G(AF ), ξ), whose respective

weights we denote by κi = (κi,j)dj=1, and whose respective spectral parameters we denote by

νi = (νi,j)dj=1.

• An orthonormal basis {ϑξ}ξ of residual forms ϑξ ∈ L2res(GL2(F )\G(AF ), ω) whose respective weights

we denote by κξ = (κξ,j)dj=1, and whose respective spectral parameters we denote by νξ = (νξ,j)

dj=1.

• An orthonormal basis {E$}$ of Eisenstein series whose analytic continuations span the continuousspectrum L2

cont(GL2(F )\G(AF ), ω). Here, we denote the respective weights by κ$ = (κ$,j)dj=1 and

the respective spectral parameters (which depend on a complex variable s ∈ C) by νs,$ = (νs,$,j)dj=1.

Here, let us clarify that we choose a basis of smooth vectors implicitly. That is, we fix an orthonormal basisB for the dense subspace of smooth forms in L2(GL2(F )\G(AF ), ω) and similarly for L2(GL2(F )\GL2(AF ), ω).We can then consider the decomposition of any smooth L2-automorphic form Φ ∈ L2

(GL2(F )\G(AF ), ω

)in

terms of such a basis B. To spell this out, writing 〈·, ·〉 to denote the inner product on L2(GL2(F )\G(AF ), ω),we have the spectral decomposition

Φ(g) =∑i

〈Φ, φi〉 · φi(g) +∑

ξ(ξ2=ω)

〈Φ, ϑξ〉 · ϑξ(g) +∑$

∫<(s)=1/2

〈Φ, E$(∗, s)〉 · E$(g, s)ds

2πi.(28)

Note that we can view (28) as an identification of functions on g = (g, ζ) ∈ G(AF ), although we shall only con-sider specializations to elements of the form (g, 1) ∈ G(AF ) with g ∈ GL2(AF ) for our subsequent arguments.In the event that we evaluate at such an element g = (g, 1) ∈ G(AF ) with g ∈ GL2(AF ), we shall simply writeg = g. Note as well that the spectral decomposition (28) is completely analogous to the well-known spectraldecomposition of L2(GL2(F )\GL2(AF ), ω) into subspaces of cuspidal forms L2

cusp(GL2(F )\GL2(AF ), ω),

residual forms L2res(GL2(F )\GL2(AF ), ω), and forms spanned by analytic continuations of Eisenstein series

L2cont(GL2(F )\GL2(AF ), ω), which we write here as

L2cusp(GL2(F )\GL2(AF ), ω)⊕ L2

res(GL2(F )\GL2(AF ), ω)⊕ L2cont(GL2(F )\GL2(AF ), ω).(29)

Essentially, we shall use this latter decomposition (29) to the treat the non-genuine cases of (A) and (B)considered above, i.e. for a quadratic form f with an even number of variables k ≥ 2 for (A) or any version of(B) in the setup with integral presentations described above (e.g. Proposition 4.2). The discussion of spectraldecompositions of non-genuine forms factoring through L2(GL2(F )\GL2(AF ), ω) is formally identical to the

34

case of genuine forms in L2(GL2(F )\G(AF ), ω) for all of our subsequent arguments. We shall therefore onlygive full details for the genuine metaplectic case, the non-genuine case being easy to deduce as a consequence.

4.4.2. Convergence properties of the spectral coefficients. Let us now consider convergence properties of thespectral coefficients appearing in the decomposition (28), as well as the analogous coefficients appearing inthe corresponding decomposition (29) with respect to a similar choice of orthonormal basis of non-genuineforms. Recall that if the form Φ ∈ L2(GL2(F )\G(AF ), ω) is smooth (even if it is not K-finite), then weknow by Proposition 4.7 that this decomposition is convergent in the Sobolev norm topology, i.e. that theSobolev norm ||Φ||SB converges for any integer B ≥ 1. We can then use the following well-known argumentto bound the coefficients in the corresponding spectral decomposition (28) of Φ.

Lemma 4.8. Suppose that the genuine form Φ ∈ L2(GL2(F )\G(AF ), ω) is smooth, and hence that it hasconvergent Sobolev norm, or more generally that Φ is sufficiently smooth so as to have convergence Sobolevnorm. Then, the spectral coefficients of Φ appearing in the decomposition (28) can be bounded as follows:For any choices for real numbers C ∈ R, we have the bounds

〈Φ, φi〉 �C

d∏j=1

|νi,j |C , 〈Φ, ϑξ〉 �C

d∏j=1

|νξ,j |C 〈Φ, E$(∗, s)〉 �C

d∏j=1

|ν$,s,j |C

for all φi, ϑξ, and E$(∗, s) in a fixed orthonormal basis B as described above. Here, the implied constantsdepend on the choice of form Φ, and in particular the representation Π.

Proof. See [33, §5-6], [6], or [29]. The idea is that for any arbitrary integer A ≥ 1, we can apply the A-folditerate ∆(A) of the generalized Laplacian operator ∆ to either side of (28) to derive the relation

∆(A)Φ =∑i

〈Φ, φi〉 ·d∏j=1

(1

4+ ν2

i,j

)A· φi +

∑ξ

〈Φ, ϑξ〉 ·D∏j=1

(1

4+ ν2

ξ,j

)A· ϑξ

+∑$

∫<(s)=1/2

〈Φ, E$(∗, s)〉 ·d∏j=1

(1

2+ ν2

$,s,j

)A· E$(∗, s) ds

2πi.

Note that this relation is also justified for non-K-finite forms Φ: In classical terms, such a form Φ can beviewed as a convergent infinite linear combination of some fixed K-finite basis form under iterates of Maassweight raising operators, and this can be used to determine the weights of the corresponding basis formswhich contribute to (28). Taking L2-norms || · || = || · ||L2 = 〈·, ·〉 then gives the corresponding relation

||∆(A)Φ|| =∑i

〈Φ, φi〉2 ·d∏j=1

(1

4+ ν2

i,j

)2A

· ||φi||+∑ξ

〈Φ, ϑξ〉2 ·d∏j=1

(1

4+ ν2

ξ,j

)2A

· ||ϑξ||

+∑$

∣∣∣∣∣∣∣∣∣∣∣∣∫<(s)=1/2

〈Φ, E$(∗, s)〉2 ·d∏j=1

(1

2+ ν2

$,j,s

)A· E$(∗, s) ds

2πi

∣∣∣∣∣∣∣∣∣∣∣∣ .

The key point here is that ||∆(A)Φ|| � ||Φ||SB(A) � 1 is bounded for any such choice of integer A ≥ 1,i.e. where the exponent B = B(A) ≥ 1 for the Sobolev norm is some integer depending only on the choice ofA. Applying Parseval’s formula to the contribution of the discrete spectrum, and using adjointness propertiesof the inner product for the contribution of the continuous spectrum, the claimed bounds for the spectralcoefficients 〈Φ, φi〉, 〈Φ, ϑξ〉, and 〈Φ, E$(∗, s)〉 are then easy to deduce. Here, we use the fact that the integerA ≥ 1 can be taken to be arbitrarily large. �

Let us now suppose more generally that Φ ∈ L2(GL2(F )\G(AF ), ω) is one of the forms appearing inthe integral presentations of Proposition 4.2 above. In particular, we now consider the more general settingwhere the L2-automorphic form Φ we consider is not generally smooth, specifically:

Φ =

Pn1ϕ(l) · θf,p corresponding to case (A) of Proposition 4.2

Pn1ϕ(l1) · ˜Pm1 ϕ′(l2) corresponding to case (B) of Proposition 4.2,(30)

35

so that

Φ ∈

{L2(GL2(F )\G(AF ),1)K corresponding to case (A) of Proposition 4.2

L2(GL2(F )\GL2(AF ),1)K corresponding to case (B) of Proposition 4.2.

In any case (lifting to the metaplectic cover G(AF ) if needed for (A)), we should like to consider the spectraldecomposition of Φ with respect to a fixed orthonormal basis, similar to the basis B described above. To beclear, the form Φ can be decomposed with respect to an orthonormal basis B′ of either of the correspondingHilbert spaces: Writing {fi}i to denote an orthonormal basis of the cuspidal subspace, {θξ}ξ an orthonormalbasis for the subspace of residual forms, and {E$(∗, s)}$ an orthonormal basis for the subspace of Eisensteinseries spanning the continuous spectrum, we have

Φ =∑i

〈Φ, fi〉 · fi +∑ξ

〈Φ, θξ〉 · ϑξ +∑$

∫<(s)=1/2

〈Φ, E$(∗, s)〉 · E$(∗, s) ds2πi

.(31)

Now, we are going to argue that this decomposition can be approximated by the corresponding linearcombination of the smooth basis B,

Φ ≈∑i

〈Φ, φi〉 · φi +∑ξ

〈Φ, ϑξ〉 · ϑξ +∑$

∫<(s)=1/2

〈Φ, E$(∗, s)〉 · E$(∗, s) ds2πi

,(32)

and that the coefficients in this latter expression can be bounded using a variation of the standard argumentgiven for Lemma 4.8 above, using inner products with Poincare series.

Proposition 4.9. The spectral coefficients of Φ ∈ L2(GL2(F )\G(AF ),1) defined in (30) appearing in thedecomposition (31) can be bounded as follows: We have for any choices for real numbers C ∈ R the bounds

〈Φ, fi〉 �C

d∏j=1

|νi′,j |C , 〈Φ, θξ〉 �C

d∏j=1

|νξ′,j |C 〈Φ, E$(∗, s)〉 �C

d∏j=1

|ν$,s,j′ |C .

Here, the indices i′, ξ′, and $′ on the right hand side corresponding to elements of the smooth basis B canbe chosen arbitrarily. That is, the bounds for the coefficients appearing in the spectral expansion (31) withrespect to the fixed orthonormal basis B′ are given in terms of the spectral parameters of the basis B of smoothforms described above, but without specification to a particular form within the respective cuspidal, residual,or continuous subspaces. The implied constants depend on Φ, and in particular the choices of archimedeanlocal vectors determining the weight functions, the representation Π, and the theta series θf,p correspondingto case (A) of Proposition 4.2 or the representation π corresponding to case (B) of Proposition 4.2. Theyalso depend on the choice of some auxiliary Poincare series.

Proof. We argue in the style of [31, §2], replacing the polynomial P = P (φ1, · · ·φk) with a Poincare seriesPφ ∈ L2(GL2(F )\G(A),1) corresponding to a suitably chosen smooth function φ. That is, we shall firstderive bounds for the linear combination appearing on the right hand side of (32) by taking the inner productwith some suitable-chosen Poincare series. We shall then argue that (32) gives a uniform approximation ofthe spectral decomposition (31), and in particular that the coefficients in (31) can be bounded similarly.

Recall that we write ψ = ⊗vψv to denote the standard additive character on AF /F . Consider the space

C∞ (N2(AF )Z2(AF )\GL2(AF );ψ)

of smooth functions φ : GL2(AF )→ C which are left invariant by the centre Z2(AF ), and which satisfy

φ(ng) = ψ(n)φ(g) for all n ∈ N2(AF ) and g ∈ GL2(AF ).

It is well-known (e.g. [11, §5]) that for a suitably chosen decomposable function φ = ⊗vφv in this space, thecorresponding Poincare series defined on g ∈ GL2(AF ) by the summation formula

Pφ(g) =∑

γ∈N2(F )Z2(F )\GL2(F )

φ(γg)

converges absolutely, and uniformly on compact subsets. Moreover, for

(33) g =∐ζ∈∆

γ ·(y∞ x∞

1

)·(r∞

r∞

)·(ζ

ζ

)· k ∈ GL2(AF )

36

decomposed uniquely according to Proposition 2.5 above, one can show that Pφ(g) � |y∞|−M+1 for somereal parameter M > 0 depending on the choice of φ (e.g. [11, Proposition 5.2]). Note that with respect tothe decomposition (33) as in Theorem 2.5, we can assume that

φ

((y∞

1

))= |y∞|−σ exp (−ε|y∞|)(34)

for some choices of real parameters σ, ε > 0 (see e.g. [11, §5]). Let us henceforth choose such a functionφ = ⊗vφv ∈ C∞(Z2(AF )Z2(AF )\GL2(AF );ψ) as in (34) with ε > σ. Granted the constant M > 0 issufficiently large, we can view the corresponding Poincare series Pφ as an L2-automorphic form in the space

L2(GL2(F )\GL2(AF ),1), or after lifting to the two-fold metaplectic cover G(AF ) in the natural way as anL2-automorphic form in the space L2(GL2(F )\G(AF ),1). Let us also assume that this function φ = ⊗vφvis chosen more precisely in the following way. Fix Y∞ ∈ F×∞ with |Y∞| � 1 as in the statement of the maintheorem, and let J(Y∞) be a compact neighbourhood around Y∞ which does not intersect the boundary ofthe chosen fundamental domain of Theorem 2.5. Put I ∼= [0, 1]d ⊂ F∞ as above, and let J = J(Y∞)× I bethe corresponding compact domain in P2(F∞) ∼= Hd. We then choose φ = ⊗vφv so that:

• The component φ∞ = ⊗v|∞φv is compactly supported on J = J(Y∞)× I ⊂ P2(F∞) ⊂ GL2(F∞).

• The function φ ∈ C∞c (Z2(AF )N2(AF )\GL2(AF );ψ) is right K-invariant.

Taking this choice of function φ = ⊗vφv ∈ C∞c (Z2(AF )Z2(AF )\GL2(AF );ψ) for granted, we now considerthe corresponding Poincare series Pφ. Taking for the granted the analytic properties described above, it iswell-known that we can decompose Pφ into a basis of smooth forms as described above (see e.g. [12, §I.1]).Hence, we decompose spectrally in terms of our fixed basis B described above as

Pφ =∑i

〈Pφ, φi〉 · φi +∑ξ

〈Pφ, ϑξ〉 · ϑξ +∑$

∫<(s)=1/2

〈Pφ, E$(∗, s)〉 · E$(∗, s) ds2πi

.(35)

Notice that via the argument of Lemma 4.8, we can deduce that the spectral coefficients 〈Pφ, φi〉, 〈Pφ, ϑξ〉,and 〈Pφ, E$(∗, s)〉 in this expansion are bounded in terms of the spectral parameters of the respective basisforms φi, ϑξ, and E$(∗, s). That is, for any choices for real numbers C ∈ R, we have the upper bounds

(36) 〈Pφ, φi〉 �C

d∏j=1

|νi,j |C , 〈Pφ, ϑξ〉 �C

d∏j=1

|νξ,j |C 〈Pφ, E$(∗, s)〉 �C

d∏j=1

|ν$,s|C

Notice as well that we can assume without loss of generality that each of these coefficients is nonvanishing,whence we have the strict lower bounds

|〈Pφ, φi〉| > 0, |〈Pφ, ϑξ〉| > 0, and |〈Pφ, E$(∗, s)〉|<(s)=1/2 > 0(37)

for each index i, ξ, and $ parametrizing our fixed basis B. Let us now consider the inner product of Pφagainst the form Φ defined in (30) above, i.e. where we lift to the metaplectic cover g = (g, 1) ∈ G(AF ) inany case, and consider inner product

〈Pφ,Φ〉 =

∫Z2(AF ) GL2(F )\GL2(AF )

Pφ(g)Φ(g)dg.

To be clear, we take

Φ(g) =

{Φ((g, 1)) for case (A) corresponding to Theorem 4.2 (i)

Φ(g) for case (B) corresponding to Theorem 4.2 (ii)

in this definition, which is the same for both the genuine and non-genuine cases (cf. [17, p. 56]). Observethat after decomposing Pφ spectrally according to (35) above, this inner product is the same as

〈Pφ,Φ〉 =∑i

〈Pφ, φi〉 · 〈φi,Φ〉+∑ξ

〈Pφ, ϑξ〉 · 〈ϑξ,Φ〉+∑$

∫<(s)=1/2

〈Pφ, E$(∗, s)〉 · 〈E$(∗, s),Φ〉 ds2πi

.(38)

37

In particular, it is simple to see by inspection of (38) with the lower bounds (37) of the spectral coefficientsappearing in (35) that each of the spectral coefficients 〈φi,Φ〉, 〈ϑξ,Φ〉, and 〈E$(∗, s),Φ〉 of Φ can be boundedabove by this inner product 〈Pφ,Φ〉. On the other hand, unfolding with the Poincare series Pφ, we find that

〈Pφ,Φ〉 =

∫Z2(AF ) GL2(F )\GL2(AF )

Pφ(g)Φ(g)dg

=

∫Z2(AF ) GL2(F )\GL2(AF )

∑γ∈Z2(F )N2(F )\GL2(F )

φ(γg)Φ(g)dg

=

∫Z2(AF )N2(F )\GL2(AF )

φ(g)Φ(g)dg.

Here, we alter the appearance of the complex conjugation for simplicity (as we may). Using the definition(30) with choices of vectors and Definition 2.6 (via Theorem 2.5) with the compact domain J = J(Y∞)× Idescribed above for φ, it is easy to see via unfolding with the unique decomposition of Theorem 2.5 that

〈Pφ,Φ〉 =

∫J(Y∞)

∫I∼=[0,1]d⊂F∞

φ · Pn1ϕ(l) · θf,p((

y∞ x∞1

))dx∞

dy∞y2∞

(39)

for the case corresponding to Proposition 4.2 (A), and by

〈Pφ,Φ〉 =

∫J(Y∞)

∫I∼=[0,1]d⊂F∞

φ · Pn1ϕ(l1) · ˜Pm1 ϕ′(l2)

((y∞ x∞

1

))dx∞

dy∞y2∞

(40)

for the case corresponding to Proposition 4.2 (B). Let us first consider the constant coefficients determinedby the inner unipotent integrals these expressions. Here, we assume (crucially) that the archimedean idelecoordinate y∞ is contained in our chosen fundamental domain, and hence that it is totally positive. Usingthe identification of Fourier-Whittaker coefficients from Proposition 2.9 for the extended mirabolic formsand theta series, we can open up Fourier-Whittaker expansions and evaluate via orthogonality of additivecharacters to calculate

KA(y∞) :=

∫I∼=[0,1]d⊂F∞

φ · Pn1ϕ(l) · θf,p((

y∞ x∞1

))dx∞

=

∫I∼=[0,1]d⊂F∞

φ

((y∞ x∞

1

))Pn1ϕ(l)

((y∞ x∞

1

))θf,p

((y∞ x∞

1

))dx∞

=

∫I∼=[0,1]d⊂F∞

φ

((y∞ x∞

1

))Pn1ϕ(l)

((y∞ x∞

1

))θf,p

((y∞ x∞

1

))dx∞

= φ

((y∞

1

))|y∞|−(n−2

2 )∑γ∈F×

Wϕ(l)

((γy∞

1n−1

)) ∑a=(a1,···ak)∈OkF

p(a)ψ (i · f(a)y∞)

×∫I∼=[0,1]d⊂F∞

ψ(x∞ + γx∞ − f(a)x∞)dx∞

= φ

((y∞

1

))|y∞|

k4−(n−2

2 )∑

a=(a1,··· ,ak)∈OkF

Wϕ(l)

(((f(a)− 1)y∞

1n−1

))p(a)ψ (i · f(a)y∞)

= φ

((y∞

1

))ψ (i · αY∞)ψ (i · y∞) |y∞|

k4

∑a=(a1,··· ,ak)∈OkF

p(a)CΠ(f(a)− 1)

|f(a)− 1| 12Wl

(|f(a)− 1||y∞|

)38

for the first unipotent integral, and

KB(y∞) :=

∫I∼=[0,1]d⊂F∞

φ · Pn1ϕ(l1) · ˜Pm1 ϕ′(l2)

((y∞ x∞

1

))dx∞

=

∫I∼=[0,1]d⊂F∞

φ

((y∞ x∞

1

))Pn1ϕ(l1)

((y∞ x∞

1

))˜Pm1 ϕ′(l2)

((y∞ x∞

1

))dx∞

=

∫I∼=[0,1]d⊂F∞

φ

((y∞ x∞

1

))Pn1ϕ(l1)

((y∞ x∞

1

))Pm1 ϕ′(l2)

((y∞ x∞

1

))dx∞

= φ

((y∞

1

))|y∞|−(n−2

2 )∑

γ1∈F×Wϕ(l1)

((γy∞

1n−1

))|y∞|−(m−2

2 )∑

γ2∈F×Wϕ′(l2)

((γ2y∞

1n−1

))×∫I∼=[0,1]⊂F∞

ψ(x∞ + γ1x∞ − γ2x∞)dx∞

= φ

((y∞

1

))|y∞|−(n−2

2 )−(m−22 )

∑γ1,γ2∈F×γ1−γ2=−1

Wϕ(l1)

((γ1y∞

1n−1

))Wϕ′(l2)

((γ2y∞

1n−1

))

= φ

((y∞

1

)) ∑γ1,γ2∈F×γ1−γ2=−1

cΠ(γ1)cπ(γ2)

|γ1γ2|12

W1,l1

(|γ1||y∞|

)W2,l2

(|γ2||y∞|

)

for the second. Note that we have used the explicit choice of pure tensors described in Proposition 4.2 toevaluate these constant coefficient kernels. We also use the choice of compact domain J(Y∞) to ensure thateach archimedean idele y∞ = (y∞,j)

dj=1 ∈ F×∞ in this calculation is totally positive. Now, recall we assume

that the chosen global weight functions W and Wj on y ∈ R>0 (for j = 1, 2) decay moderately as{W (y) = O (yκ) for some constant 0 < κ < 1 when 0 < y < 1

Wj(y) = O(yκj ) for some constants 0 < κj < 1 when 0 < y < 1 for either j = 1, 2

for y → 0, and decay rapidly as{W (y) = OB(y−B) for any choice of constant B > 0 when y > 1

Wj(y) = OB(y−B) for any choice of constant B > 0 when y > 1 for either j = 1, 2

as y →∞. Recall as well that for the respective fixed integers 0 ≤ l ≤ log Y and 0 ≤ lj ≤ log Y the dyadicsubdivisions coming from our fixed partition of unity (16) above, we consider the local weight functionsdefined (respectively) by (17) and (18). That is, as functions of y∞ ∈ F×∞, these are given by

Wl(y∞) = W (y∞)∑R∈{R}

R∩[2l,2l+1]6=∅

U

(|y∞|YR

)

and (for each of j = 1, 2) by

Wj,l(y∞) = Wj(y∞)∑R∈{R}

R∩[2l,2l+1]6=∅

U

(|y∞|YR

).

Here again, the summation notation for∑R∈[2l,2l+1] denotes the single range 2l ≤ R ≤ 2l+1 in our fixed

partition of unity (16), i.e. so that the sum consists of this one element. Now, it is easy to deduce that theselocal weight functions are bounded in a similar way as for the global weight functions, uniformly in the rangeparameter R ∈ [2l, 2l+1]. In particular, taking

CU := max1≤t≤2

U(t)(41)

39

to be the maximum obtained by the smooth test function U , which recall is only supported on the compactinterval [1, 2], we deduce that for any integer l ∈ Z, we have the uniform bounds

Wl(y∞) =

{O (|CU · y∞|κ) for some 0 < κ < 1 when 0 < y < 1

OB(|CU · y∞|−B

)for any choice of constant B > 0 when y > 0

(42)

and

Wj,l(y∞) =

{O (|CU · y∞|κj ) for some 0 < κj < 1 when 0 < y < 1

OB(|CU · y∞|−B

)for any choice of constant B > 0 when y > 0

(43)

for each index j = 1, 2. That is, these bounds (42) and (43) are uniform in the integer l ∈ Z parametrizingthe range R ∈ [2l, 2l+1] in the corresponding dyadic decomposition. We can then derive the following uniformbounds for the kernel functions KA(y∞) and KB(y∞) and their partial Mellin transforms in terms of thesebounds (42) and (43). Namely, using the choice φ of function as described in (34) above, we find

KA(Y∞)�Pφ,Π,f,p,W CU · Y −σ+ k4 +(1−κ)e−εY · e−2π|α+1|Y

and

KB(Y∞)�Pφ,Π,π,WjCU · Y −σ+(1−κ1−κ2) · e−εY

Observe that these bounds are uniform in the choice of dyadic range 2l ≤ R ≤ 2l+1. Consequently, takingthe compact domain J(Y∞) to be sufficiently small, and assuming as we do that 0 < κ < 1 and 0 < κj < 1,we can bound the corresponding partial Mellin transforms defining (39) and (40) respectively as

〈Pφ,Φ〉 =

∫J(Y∞)

KA(y∞)dy∞y2∞�φ,Π,f,p CU · Y −σ+ k

4 +(1−κ)−2 · e−εY · e−2π|α+1|Y � CU(44)

and

〈Pφ,Φ〉 =

∫J(Y∞)

KB(y∞)dy∞y2∞�φ,Π,π CU · Y −σ+(1−κ1−κ2)−2 · e−εY � CU(45)

Again, note that these bounds are uniform with respect to the choice of range R in the corresponding dyadicdecomposition, with CU = maxt∈[1,2] U(t). As well, we obtain identical upper bounds for the correspondinginner products 〈Pφ,Φ〉.

Now, to use these estimates (44) and (45) for the inner product 〈Pφ,Φ〉 to bound the spectral coefficients

of our chosen L2-automorphic form Φ defined above, we first consider arbitrary D-fold iterates ∆(D) of thesuitable Laplacian operator ∆ acting on the Poincaree series Pφ. Thus for any integer D ≥ 1, we apply the

D-fold iterate ∆(D) to each side of the decomposition (35) to derive the corresponding decomposition

(46)

∆(D)Pφ =∑i

〈Pφ, φi〉 ·d∏j=1

(1

4+ ν2

i,j

)D· φi +

∑ξ

〈Pφ, ϑξ〉 ·d∏j=1

(1

4+ νξ,j

)D· ϑξ

+∑$

∫<(s)=1/2

〈Pφ, E$(∗, s)〉 ·d∏j=1

(1

2+ ν2

$,s,j

)D· E$(∗, s) ds

2πi.

Let us now consider the inner product of ∆(D)Pφ with our chosen L2-automorphic form Φ,

〈∆(D)Pφ,Φ〉 =

∫Z2(AF ) GL2(F )\GL2(AF )

∆(D)Pφ(g)Φ(g)dg.

Observe that by using the decomposition (46) to describe the contribution of ∆(D) ·Φ in this expression, weobtain the corresponding spectral decomposition

(47)

〈∆(D)Pφ,Φ〉 =∑i

〈Pφ, φi〉 · 〈φi,Φ〉 ·d∏j=1

(1

4+ ν2

j,i

)D+∑ξ

〈Pφ, ϑξ〉 · 〈ϑξ,Φ〉 ·d∏j=1

(1

4+ ν2

ξ,j

)D

+∑$

∫<(s)=1/2

〈Pφ, E$(∗, s)〉 · 〈E$(∗, s),Φ〉 ·d∏j=1

(1

2+ ν2

$,s,j

)Dds

2πi.

40

Note that the decompositions (46) and (47) hold for any choice of integer D ≥ 1. On the other hand, observeas well that a simple variation of the unfolding calculations about allows us to derive the bounds(48)

〈∆(D)Pφ,Φ〉

=

∫J(Y∞)

∆(D)φ

((y∞

1

))ψ(iαY∞)|y∞|

k4

∑a=(a1,...,ak)∈OkF

cΠ(f(a)− 1)

|f(a)− 1| 12p(a)ψ(iy∞)Wl

(|f(a)− 1||y∞|

) dy∞y2∞

�∆(D)φ,Π,f,p CU · Y −σ+ k4 +(1+κ)−2 · e−εY · e−2π|α+1|Y

and

(49)

〈∆(D)Pφ,Φ〉

=

∫J(Y∞)

∆(D)φ

((y∞

1

)) ∑γ1,γ2∈F×γ1−γ2=−1

cΠ(γ1)cπ(γ2)

|γ1γ2|12

W1,l1

(|γ1||y∞|

)W2,l2

(|γ2||y∞|

) dy∞y2∞

�∆(D)φ,Π,π CU · Y −σ+(1+κ1+κ2)−2 · e−εY

Since the Poincare series Pφ is smooth and hence convergent in the Sobolev norm topology, we argue as inLemma 4.8 above, after taking L2 norms, that its spectral coefficients are bounded uniformly in terms ofthe spectral parameters of the basis forms as in (36). Putting these upper bounds together with the lowerbounds (37) and upper bounds for the inner products (48) and (49) then allows to extend the standardargument (i.e. as given in Lemma 4.8) to derive the following uniform bounds: For each of the forms φi, ϑξ,and E$(∗, s) in our fixed orthonormal basis B, we have for any choice of real parameter C ∈ R the bounds

〈Φ, φi〉 �C

d∏j=1

|νi′,j |

C

, 〈Φ, ϑξ〉 �C

d∏j=1

|νξ′,j |

C

, 〈Φ, E$(∗, s)〉|<(s)=1/2 �C

d∏j=1

|ν$′,s,j |<(s)=1/2

C

.

(50)

Again, the indices i′, ξ′, and $′ appearing on the right hand side, corresponding to elements of the smoothbasis B, can be chosen arbitrarily. Hence, the coefficients appearing on the right hand side of the expansion(32) are bounded according to the familiar setup described in Lemma 4.8. Now, to argue that this linearcombination gives a uniform approximation to the spectral decomposition (31), we first observe that thelinear combination defined by

ΦS :=∑i

〈Φ, φi〉 · φi +∑ξ

〈Φ, ϑξ〉 · ϑξ +∑$

∫<(s)=1/2

〈Φ, E$(∗, s)〉 · E$(∗, s) ds2πi

must in fact be the image of our chosen from Φ in the dense subspace of smooth forms. Indeed, it is easy tocompare inner products with the Poincare series Pφ (decomposing Pφ spectrally) to find that

〈Pφ,ΦS〉 = 〈Pφ,Φ〉.

We then argue that ΦS must be a uniform approximation of Φ in the dense subspace of smooth forms, inwhich case it is easy to deduce that the spectral coefficients of Φ as described in (31) can be bounded in termsof those of ΦS . Another way to see this is to expand the Poincare series Pφ with respect to the orthonormalbasis B′ described above, and then to compare spectral expansions. In this way, ordering the bases B′ and Bso that each cusp form φi, residual form ϑξ, and Eisenstein series E$(∗, s) in B′ has corresponding cusp formfi, residual form θξ, or Eisenstein series E$(∗, s) respectively in the smooth basis B, we derive the relations

(51)

Pφ =∑i

〈Pφ, fi〉 · fi +∑ξ

〈Pφ, θξ〉 · θξ +∑$

∫<(s)=1/2

〈Pφ, E$(∗, s)〉 · E$(∗, s) ds2πi

=∑i

〈Pφ, φi〉 · φi +∑ξ

〈Pφ, ϑξ〉 · ϑξ +∑$

∫<(s)=1/2

〈Pφ, E$(∗, s)〉 · E$(∗, s) ds2πi

41

and

(52)

〈Pφ,Φ〉 =∑i

〈Pφ, fi〉 · 〈fi,Φ〉+∑ξ

〈Pφ, θξ〉 · 〈θξ,Φ〉+∑$

∫<(s)=1/2

〈Pφ, E$(∗, s)〉 · 〈E$(∗, s),Φ〉 ds2πi

=∑i

〈Pφ, φi〉 · 〈φi,Φ〉+∑ξ

〈Pφ, ϑξ〉 · 〈ϑξ,Φ〉+∑$

∫<(s)=1/2

〈Pφ, E$(∗, s)〉 · 〈E$(∗, s),Φ〉 ds2πi

= 〈Pφ,ΦS〉.

Since both bases B and B′ are orthonormal, we derive from (51) and (36) the corresponding bounds

〈Pφ, fi〉 �C

d∏j=1

|νi,j |C , 〈Pφ, θξ〉 �C

d∏j=1

|νξ,j |C 〈Pφ, E$(∗, s)〉 �C

d∏j=1

|ν$,s|C

for any choice(s) of real parameter C ∈ R. We can then derive the stated bounds from (52) and (50). �

4.4.3. Convergence of spectral coefficients via convolution with smoothing kernels. Note that we could alsoconsider the convolution of the L2-automorphic form Φ defined in (30) with a smoothing kernel K, e.g. ofthe form described in [8, §3]. The corresponding convolution Φ ? K determines a smooth automorphic form,and hence it can be decomposed spectrally in terms of a basis of smooth forms B as described above. Inparticular, the standard Sobolev norms argument presented in Lemma 4.8 applies to bound the spectralcoefficients of this convolution Φ ? K uniformly in the spectral parameters of the forms in the basis B. Letus now explain how we could also proceed to argue in this way, albeit at the expense of complicating ourdiscussion of integral presentations in Proposition 4.2.

Let us now describe this in more detail, building on the setup of [8, §3]. Recall that we fix an identificationP2(F∞) ∼= Hd of P2(F∞) with the d-fold upper-half plane Hd. Let us write

z∞ = (z∞,j)dj=1 = x∞ + iy∞ = (x∞,j + iy∞,j)

dj=1 ∈ H

with x∞ = (x∞,j)dj=1 ∈ F∞ ∼= Rd and y∞ = (y∞,j)

dj=1 ∈ F×∞ ∼= (R×)d to denote a generic element, and also

dµ(z∞) =dx∞,1dy∞,1

y2∞,1

· · · dx∞,ddy∞,dy2∞,d

the corresponding measure on Hd ∼= P2(F∞). As well, we shall put

Φ(z∞) = Φ(x∞ + iy∞) := Φ

((y∞ x∞

1

)).

Let K(z∞, w∞) = K(x∞ + iy∞, u∞ + it∞) be a point pair invariant on Hd × Hd ∼= P2(F∞) × P2(F∞). Weconsider the convolution Φ ? K(z∞) defined by

Φ ? K(z∞) =

∫Hd

K(z∞, w∞)Φ(w∞)dµ(w∞)

=

∫(t∞ u∞0 1

)∈P2(F∞)

K(x∞ + iy∞, u∞ + it∞)Φ

((t∞ u∞

1

))dµ(u∞ + it∞).

Now, it is easy to see that this Φ ? K determines a smooth automorphic form on SL2(OF )\Hd. Indeed,smoothness is simple to check. To see the automorphy, observe that for any γ ∈ SL2(OF ), we have that

Φ ? K(γ · z∞) =

∫Hd

K(γ · z∞, w∞)Φ(w∞)dµ(w∞) =

∫Hd

K(z∞, γ−1 · w∞)Φ(γ−1 · w∞)dµ(w∞)

=

∫Hd

K(z∞, w∞)Φ(w∞)dµ(w∞) = Φ ? K(z∞).

Thus, we can view the convolution Φ ? K as a smooth automorphic form, and decompose it with respect toan orthonormal basis B as described above, with a simple variation of Lemma 4.8 showing that the spectralcoefficients are bounded uniformly in the spectral parameters of the basis forms.

Is it also easy to see that we can extract a given Fourier-Whittaker coefficient of Φ from that of thecorresponding coefficient at α of the convolution Φ ? K. To justify this, let us first observe that making a

42

change of variables r∞ = x∞ − u∞ so that x∞ = r∞ + u∞ and dr∞ = dx∞, and using the invariance of thepoint pair invariant K under multiplication by the unipotent matrix(

1 −u∞1

)∈ SL2(F∞) ∼= Isom(Hd),

we find that for any nonzero F -integer α we have the identification(s)∫x∞∈I∼=[0,1]d

Φ ? K(x∞ + iy∞)ψ(−αx∞)dx∞

=

∫x∞∈I∼=[0,1]d

∫u∞+it∞∈Hd

K(x∞ + iy∞, u∞ + it∞)Φ (x∞ + iy∞) dµ(u∞ + it∞)ψ(−αx∞)dx∞

=

∫x∞∈I∼=[0,1]d

∫u∞∈I∼=[0,1]d

∫t∞∈F∞,+∼=Rd

>0

K(x∞ + iy∞, u∞ + it∞)Φ(u∞ + it∞)du∞dt∞t2∞

ψ(−αx∞)dx∞

=

∫r∞∈I∼=[0,1]d

∫u∈I∼=[0,1]d

∫t∞∈F∞,+∼=Rd

>0

K(r∞ + u∞ + iy∞, u∞ + it∞)Φ(u∞ + it∞)dt∞t2∞

du∞ψ(−α(r∞ + u∞))dr∞

=

∫r∞∈I∼=[0,1]d

∫u∈I∼=[0,1]d

∫t∞∈F∞,+∼=Rd

>0

K(r∞ + iy∞, it∞)Φ(u∞ + it∞)dt∞t2∞

du∞ψ(−α(r∞ + u∞))dr∞

=

∫t∞∈F∞,>0

∼=Rd>0

(∫r∞∈I∼=[0,1]d

K(r∞ + iy∞, it∞)ψ(−αr∞)dr∞

)(∫u∈I∼=[0,1]d

Φ(u∞ + it∞)ψ(−αu∞)du∞

)dt∞t2∞

.

Hence, writing the inner Fourier coefficients at α in this latter expression more simply as

ρK(α; iy∞; it∞) =

∫r∞∈I∼=[0,1]d

K(r∞ + iy∞, it∞)ψ(−αr∞)dr∞

and

ρΦ(α; it∞) =

∫u∈I∼=[0,1]d

Φ(u∞ + it∞)ψ(−αu∞)du∞,

we arrive at the expression∫x∞∈I∼=[0,1]d

Φ ? K(x∞ + iy∞)ψ(−αx∞)dx∞ =

∫t∞∈F∞,>0

∼=Rd>0

ρK(α; iy∞; it∞)ρΦ(α; it∞)dt∞t2∞

.(53)

Now, observe that this expression (53) implies that the Fourier-Whittaker coefficient of Φ ? K(iy∞) at αon the left hand side can be viewed as the Mellin transform

f∗(s) =

∫t∞∈F∞,+∼=Rd

>0

f(t∞)ts∞dt∞t∞

at s = −1 of the function f(t∞) := ρK(α; iy∞, it∞)ρΦ(α; it∞). Hence by the inversion theorem, we have forsome suitable choice of real parameter σ > 0 the relation

ρK(α; iy∞, it∞)ρΦ(α; it∞) =

∫<(s)=σ

f∗(s)t−s∞ds

2πi(54)

Applying the standard Sobolev norms arguments to bound the coefficient of Φ ? K at α via spectral decom-position, we obtain a bound for the value f?(−1).

4.4.4. Setup for the case of n = 2. When n = 2, the projection operator Pn1 is trivial, and we simply considerthe form Φ = Pn1ϕ · θf,p = ϕ · θf,p for case (A) or Φ = Pn1ϕ · Pm1 ϕ′ = ϕ · ϕ′ for case (B) of the main theorem.Corollary 4.7 implies in either case that the right hand side of the corresponding spectral decomposition (28)of Φ converges in the Sobolev norm topology, with ||Φ||B �B 1 for each integer B ≥ 1. That is, the argumentof Lemma 4.8 justifies expanding the Φ spectrally with respect to the basis B of smooth forms introducedabove. We shall then consider this spectral decomposition in the integral presentation of Proposition 3.4

43

for (A) or Proposition 3.5 for (B) to derive bounds for the corresponding shifted convolution problem(s).Essentially, we shall consider the expansions of the non-constant coefficients given by the unipotent integrals

I =

∫AF /F

Φ

((1Y∞

x

0 1

))ψ(−αx)dx =

∑τ∈B〈Φ, ϕτ 〉

∫AF /F

ϕτ

((1Y∞

x

0 1

))ψ(−αx)dx

in this case, where for each basis element ϕτ ∈ B, writing ρϕτ to denote the nonarchimedean parts of theFourier-Whittaker coefficients, with cτ the corresponding L-function coefficients, we have the relations∫

AF /F

ϕτ

((1Y∞

x

0 1

))ψ(−αx)dx = ρτ (α)Wϕτ

(α

Y∞

)=cτ (α)

|α| 12Wϕτ

(α

Y∞

).

Remark Note that we could also express this decomposition in terms of the normalized Whittaker functionsW ?

Φτdescribed above, at least for certain spectral parameters νi = (νi,j)j .

4.4.5. Setup for the generic case of n ≥ 2. Suppose now that we are in the generic case of all dimensionsn ≥ 2. To be clear, recall that we shall give two proofs of the estimate for the special case of n = 2. Inthe generic case of all dimensions n ≥ 2, we decompose the L2-automorphic form Φ defined in (30) abovespectrally, i.e. after lifting to the metaplectic cover G(AF ) when needed for case (A), so that we may treatcases (A) and (B) of Proposition 4.2 uniformly. Recall that the integral presentations derived in Proposition

4.2 work differently in this setting because of the definition of the lifting Pn1ϕ, and in particular that we fixa smooth partition of unity (16) with corresponding dyadic subdivisions (19) and (20). Recall as well thatwe reduce via the choice of global weight functions W and Wj to the corresponding finite truncated sumsover local weight functions (23) and (24). We shall take this reduction from Proposition 4.2 for granted inthe discussion below, including the implicit choices of local weight functions Wl and Wlj for these estimates(as will do to prove the main theorem). Given a totally positive archimedean idele Y∞ ∈ F×∞ of idele norm|Y∞| = Y contained in our chosen fundamental domain of Theorem 2.5, we consider the Fourier-Whittakercoefficient at a given nonzero F -integer α of the function

Φ

((Y∞

1

)).

Proposition 4.2 shows that these coefficients describe the shifted convolution sums we wish to bound. Now,we saw in (31) and Proposition 4.9 that we can decompose this form Φ spectrally, and in particular that thecoefficients in the spectral expansion of Φ can be bounded in terms of the inner product 〈Pφ,Φ〉 with somesuitably chosen Poincare series Pφ, i.e. which after unfolding can be calculated to extract an upper bound interms of the chosen real parameter Y = |Y∞| > |α|. Moreover, we saw that these spectral coefficients can bebounded uniformly in terms of the spectral parameters of the automorphic forms in the smooth orthonormalbasis B. Hence, we justify bounding the decomposition of Φ in terms of the more general orthonormal basisB′ described above, and in particular for any x∞ ∈ I ∼= [0, 1]d ⊂ F∞ the decomposition

Φ

((Y∞ x∞

1

))=∑i

〈Φ, fi〉 · fi((

Y∞ x∞1

))+∑ξ

〈Φ, θξ〉 · θξ((

Y∞ x∞1

))

+∑$

∫<(s)=1/2

〈Φ, E$(∗, s)〉 · E$((

Y∞ x∞1

), s

)ds

2πi.

Taking unipotent integrals of each side (again, cf. Proposition 4.2) then gives us the relevant decomposition∫I∼=[0,1]d

Φ

((Y∞ x∞

1

))ψ(−αx∞)dx∞

=∑i

〈Φ, fi〉 ·∫I∼=[0,1]d

fi

((Y∞ x∞

1

))ψ(−αx∞)dx∞ +

∑ξ

〈Φ, θξ〉 ·∫I∼=[0,1]d

θξ

((Y∞ x∞

1

))ψ(−αx∞)dx∞

+∑$

∫<(s)=1/2

〈Φ, E$(∗, s)〉 ·∫I∼=[0,1]d

E$

((Y∞ x∞

1

), s

)ψ(−αx∞)dx∞

ds

2πi.

That is, Proposition 4.2 shows that the left hand side of this latter decomposition can be viewed as anintegral presentation for the shifted convolution problem for GLn(AF )-coefficients at shifts of the quadratic

44

form f(a) = f(a1, . . . , ak) in case (A), or as linear shifts coming from some other GLm(AF )-automorphicform in case (B). In particular, it will do to bound the Fourier-Whittaker coefficient at α of Φ in either caseto derive estimates for the corresponding shifted convolution problem.

4.4.6. Derivation of bounds. We now derive bounds for the shifted convolution problem for both the stan-dard case of n = 2 and the generic case of n ≥ 2 in tandem. We shall supply separate arguments in each caseowing to the distinct nature of the integral presentations, although we derive similar final bounds in all cases.

Decompositions setup. Recall that we write B to denote the fixed orthonormal basis of smooth forms(hence eigenvectors for the associated Laplacian operator) for the corresponding Hilbert space, with B′ afixed orthonormal basis for the more general space of L2-automorphic forms which are not necessarily smooth(and hence not necessarily eigenvectors for the associated Laplacian operator). Before going on, let us recallthat the result of Proposition 4.9 for the generic case of dimension n ≥ 2 implies that the spectral coefficientsof Φ decomposed with respect to B′ can be bounded uniformly in terms of these spectral parameters of thebasis of smooth forms B. That is, recall that we consider on the one hand the expansion

Φ =∑i

〈Φ, fi〉 · fi +∑ξ

〈Φ, θξ〉 · θξ +∑$

∫<(s)=1/2

〈Φ, E$(∗, s)〉 · E$(?, s)ds

2πi

of Φ in terms of the orthonormal basis B′. On the other hand, the proof of Proposition 4.9 in effect allowsus to approximate Φ by the smooth form defined by the corresponding linear combination

ΦS :=∑i

〈Φ, φi〉 · φ+∑ξ

〈Φ, ϑξ〉+∑$

∫<(s)=1/2

〈Φ, E$(?, s)〉 · E$(?, s)ds

2πi

of forms the smooth orthonormal basis B. To be clear, we can expand the (smooth) Poincare series Pφintroduced in the proof of Proposition 4.9 above in terms of either orothonormal basis as in (51) to deducethat the corresponding spectral coefficients (including those for B′) are bounded uniformly in terms of thespectral coefficients of the forms in B. We can then compare Parseval-style spectral decompositions of thebounded inner product 〈Φ, Pφ〉 = 〈ΦS , Pφ〉 � 1 as in (52) above to deduce the claim. This is a consequenceof having bounds for the spectral coefficients of Pφ expanded with respect to B′ together with the boundsderived above in the proof Proposition 4.9 for the inner product(s). Thus in the arguments that follow,we shall bound this smooth projection ΦS of Φ in terms of the basis B, as this is seen to suffice to derivebounds for the shifted convolution sums described in Proposition 4.2 above. Thus, we reduce to boundingthe Fourier-Whittaker coefficient

I :=

∫I∼=[0,1]d⊂F∞

Φ

((1Y∞

x∞1

))ψ(−αx∞)dx∞

=∑i

〈Φ, φi〉∫I∼=[0,1]d⊂F∞

φi

((1Y∞

x∞1

))ψ(−αx∞)dx∞

+∑ξ

〈Φ, ϑξ〉∫I∼=[0,1]d⊂F∞

ϑξ

((1Y∞

x∞1

))ψ(−αx∞)dx∞

+∑$

∫<(s)=1/2

〈Φ, E$(?, s)〉∫I∼=[0,1]d⊂F∞

E$((

1Y∞

x∞1

), s

)ψ(−αx∞)dx∞

ds

2πi

45

in the standard case of n = 2, and to bounding the coefficient (as described in Proposition 4.2)

J :=

∫I∼=[0,1]d⊂F∞

ΦS

((Y∞ x∞

1

))ψ(−αx∞)dx∞

=∑i

〈Φ, φi〉∫I∼=[0,1]d⊂F∞

φi

((Y∞ x∞

1

))ψ(−αx∞)dx∞

+∑ξ

〈Φ, ϑξ〉∫I∼=[0,1]d⊂F∞

ϑξ

((Y∞ x∞

1

))ψ(−αx∞)dx∞

+∑$

∫<(s)=1/2

〈Φ, E$(?, s)〉∫I∼=[0,1]d⊂F∞

E$((

Y∞ x∞1

), s

)ψ(−αx∞)dx∞

ds

2πi

in the generic case of n ≥ 2.To bound these unipotent integrals I and J , we separate out the contributions from residual forms BRES

from our fixed smooth orthonormal basis B, and write BNRES to denote the complement given by nonresidualforms (i.e. cuspidal forms and Eisenstein series). We write ϕτ again to denote a generic element of B. Notethat this is only shorthand for the more precise setup described in Lemma 4.8 and Proposition 4.9 above.Given a form ϕτ ∈ B in the smooth basis, viewing ϕτ as a Laplacian eigenvector, let us write κτ = (κτ,j)

dj=1

to denote the weight of ϕτ , and ντ = (ντ,j)dj=1 the spectral parameter of ϕτ .

Bounds for the archimedean Whittaker functions. Recall that the Whittaker functions Wϕτ appear-ing in our spectral expansions with respect to B can be viewed as d-fold products over indices 1 ≤ j ≤ dof the classical Whittaker functions Wκτ,j ,ντ,j (y) described above for Proposition 4.4 and Corollary 4.5. Inparticular (cf. [33, Lemma 6.2]), we deduce from Proposition 4.4 and Corollary 4.5 that we have the followingbounds in y ∈ R>0 with y → 0: For some constant A > 0 and any choice of ε > 0, we have:

• If ϕτ corresponds to a principal or complementary series, then for each j,

Wκτ,j ,ντ,j (y)

Γ(

12 + κτ,j + ντ,j

) �ε y12−

θ02 −ε (1 + |κτ,j |+ |ντ,j |)A ||ϕτ ||.

• If ϕτ corresponds to a holomorphic series, then for each j,

Wκτ,j ,ντ,j (y)√Γ(

12 + κτ,j + ντ,j

)Γ(

12 + κτ,j + ντ,j

) �ε y12−

θ02 −ε (1 + |κτ,j |+ |ντ,j |)A ||ϕτ ||.

Similarly, for y →∞, we have for some choice of constant A > 0 and any choice of ε the bounds

• If ϕτ corresponds to a principal or complementary series, then for each j,

Wκτ,j ,ντ,j (y)

Γ(

12 + κτ,j + ντ,j

) �ε y12 +

θ02 +ε (1 + |κτ,j |+ |ντ,j |)A ||ϕτ ||.

• If ϕτ corresponds to a holomorphic series, then for each j,

Wκτ,j ,ντ,j (y)√Γ(

12 + κτ,j + ντ,j

)Γ(

12 + κτ,j + ντ,j

) �ε y12 +

θ02 +ε (1 + |κτ,j |+ |ντ,j |)A ||ϕτ ||.

In both regimes, the exponent 0 ≤ θ0 ≤ 12 denotes the best uniform approximation towards the generalized

Ramanujan conjecture for GL2(AF )-automorphic forms. Note again that by taking the product over indices1 ≤ j ≤ d, we derive bounds for the Whittaker functions Wϕτ appearing in the expansions of Φ and ΦS .

46

Non-residual contributions. We first consider the contributions from the non-residual spectrum BNRES.That is, we now consider the sums defined by

INRES :=∑

ϕτ∈BNRES

〈Φ, ϕτ 〉∫I∼=[0,1]d⊂F∞

ϕτ

((1Y∞

x∞1

))ψ(−αx∞)dx∞

=∑i

〈Φ, φi〉∫I∼=[0,1]d⊂F∞

φi

((1Y∞

x∞1

))ψ(−αx∞)dx∞

+∑$

∫<(s)=1/2

〈Φ, E$(?, s)〉∫I∼=[0,1]d⊂F∞

E$((

1Y∞

x∞1

), s

)ψ(−αx∞)dx∞

ds

2πi

for the standard case of n = 2 and

JNRES :=∑

ϕτ∈BNRES

〈Φ, ϕτ 〉∫I∼=[0,1]d⊂F∞

ϕτ

((Y∞ x∞

1

))ψ(−αx∞)dx∞

=∑i

〈Φ, φi〉∫I∼=[0,1]d⊂F∞

φi

((Y∞ x∞

1

))ψ(−αx∞)dx∞

+∑$

∫<(s)=1/2

〈Φ, E$(?, s)〉∫I∼=[0,1]d⊂F∞

E$((

Y∞ x∞1

), s

)ψ(−αx∞)dx∞

ds

2πi

for the generic case of n ≥ 2.Separating the nonarchimedean and archimedean components of the Whittaker functions appearing in

these sums, and writing 0 ≤ δ0 ≤ 14 to denote the best exponent approximation for bounds for the L-functions

coefficients of half-integral weight forms, we obtain the following bounds (cf. [33, (6.13)]). For any ε > 0,

(55)

INRES =∑

ϕτ∈BNRES

〈Φ, ϕτ 〉 ·cϕτ (α)

|α| 12Wϕτ

(α

Y∞

)

�ε

∑τ∈BNRES

〈Φ, ϕτ 〉

∣∣∣∣ αY∞∣∣∣∣ 12−

θ02 −ε

|α|δ0− 12

d∏j=1

(1 + |κτ,j |+ |ντ,j |)

A

||ϕτ ||,

and similarly

(56)

JNRES =∑

ϕτ∈BNRES

〈Φ, ϕτ 〉 ·cϕτ (α)

|α| 12Wϕτ (αY∞)

�ε

∑τ∈BNRES

〈Φ, ϕτ 〉(|αY∞|

12 +

θ02 +ε |α|δ0− 1

2

) d∏j=1

(1 + |κτ,j |+ |ντ,j |)

A

||ϕτ ||,

.

Note that for this latter bound (56), we use Corollary 4.5 (as described above).Now recall by the theorem of Kohnen-Zagier [25] and more generally Baruch-Mao [1], the exponent δ0 is

equivalent to the exponent in the best exponent approximation towards the generalized Lindelof hypothesisfor GL2(AF )-automorphic forms in the level aspect3. In the standard case of n = 2, taking A ≥ 1 to be anarbitrary integer as in Lemma 4.8 above, we can deduce via the A-fold iterate ∆(A) of a suitable Laplaceoperator ∆ with the Plancherel formula (cf. [33, (6.4)]) that we also have the bound

∑ϕτ∈BNRES

d∏j=1

(1 + |κτ,j |+ |ντ,j |)

2A

||ϕτ ||2 � ||Φ||2SB � 1(57)

3I.e. so that L(1/2, π) �ε c(π)δ0+ε for any GL2(AF )-automorphic representation π of conductor c(π) and any ε > 0.

47

for some integer B = B(A) ≥ 1 depending only on the integer A ≥ 1. Applying the Cauchy-Schwarzinequality to quantity on the right hand side of (55) with the bound

〈Φ, ϕτ 〉 �C

d∏j=1

|ντ,j |

C

of Lemma 4.8 and the archimedean Whittaker bound (57), it is then easy to derive the desired bound forthe non-residual spectrum

INRES =∑

ϕτ∈BNRES

〈Φ, ϕτ 〉 ·cϕτ (α)

|α| 12·Wϕτ

(α

Y∞

)�ε,Π

∣∣∣∣ αY∞∣∣∣∣ 12−

θ02 −ε

|α|δ0− 12 .(58)

In the generic case of n ≥ 2, we argue via Proposition 4.9 as that we can derive the similar bound

∑ϕτ∈BNRES

d∏j=1

(1 + |κτ,j |+ |ντ,j |)

2A

||ϕτ ||2 � ||ΦS ||2SB � 1(59)

for the archimedean Whittaker functions in this expansion. We can then apply the Cauchy-Schwartz in-equality to the quantity on the right hand side of (56) with the corresponding spectral coefficient bounds ofProposition 4.9 and the archimedean Whittaker bound (59) to derive the corresponding bound

JNRES =∑

ϕτ∈BNRES

〈Φ, ϕτ 〉 ·cϕτ (α)

|α| 12·Wϕτ (αY∞)�Π,ε |αY∞|

12 +

θ02 −ε |α|δ0− 1

2 .(60)

Note that each of the bounds (58) and (60) for the non-residual contribution give the same exponents forour final estimate via the integral presentations of Proposition 3.4 and Proposition 4.2 (A) respectively.

Residual contributions. We now consider the residual contributions

IRES :=∑

ϕτ∈BRES

〈Φ, ϕτ 〉∫I∼=[0,1]d⊂F∞

ϕτ

((1Y∞

x∞1

))ψ(−αx∞)dx∞

=∑ξ

〈Φ, ϑξ〉∫I∼=[0,1]d⊂F∞

ϑξ

((1Y∞

x∞1

))ψ(−αx∞)dx∞

=∑

ϕτ∈BRES

〈Φ, ϕτ 〉 ·cϕτ (α)

|α| 12·Wϕτ

(α

Y∞

)for the standard case of n = 2, and

JRES :=∑

ϕτ∈BRES

〈Φ, ϕτ 〉∫I∼=[0,1]d⊂F∞

ϕτ

((Y∞ x∞

1

))ψ(−αx∞)dx∞

=∑ξ

〈Φ, ϑξ〉∫I∼=[0,1]d⊂F∞

ϑξ

((Y∞ x∞

1

))ψ(−αx∞)dx∞

=∑

ϕτ∈BRES

〈Φ, ϕτ 〉 ·cϕτ (α)

|α| 12·Wϕτ (αY∞)

for the generic case of n ≥ 2. To estimate these contributions of the residual spectrum BRES in either case, letus first recall that we fix a smooth weight function W ∈ L2(F×∞) as in the statements of Propositions 3.4 and4.2 which decays rapidly for y∞ ∈ F×∞ with |y∞| → ∞. It is then easy to deduce that both IRES and JRES

are linear in W . Now, we deduce by comparison with of Fourier-Whittaker expansions of metaplectic formsof the same weight k/2 (as determined by the number of variables k ≥ 1 of the quadratic form f) that IRES is

proportional to the scaling factor |Y∞|−k4 , which JRES is proportional to the scaling factor |Y∞|

k4 Note that

we have to multiply IRES out by the scaling factor |Y∞|k4 according to the integral presentation described in

Proposition 3.4 above. On the other hand, we multiply JRES by the scaling factor |Y∞|−k4 according to the

48

integral presentation of Proposition 4.2 (A) above. Thus in either case, we can interpret the contribution ofthe residual spectrum to the shifted convolution sum for (A) as a constant times some linear functional I(W )in the weight function W . We also deduce via inspection of the corresponding Fourier-Whittaker expansionsthat the coefficients appearing in IRES and JRES are identically zero unless the F -integer α is totally positive.

Putting these observations together for the standard case with n = 2, we can derive the crude estimate

IRES =∑

ϕτ∈BRES

〈Φ, ϕτ 〉 ·cϕτ (α)

|α| 12·Wϕτ

(α

Y∞

)= |Y∞|

k4 I(W )MΠ,α,(61)

where I(W ) = IΦ(W ) is a linear functional in the weight function W depending on Φ = ϕθf,p, and MΠ,α ≥ 0is a constant which vanishes unless the F -integer α is totally positive and (as explained below) the symmetric

square L-function L(s,Sym2 Π) of Π has a pole. Multiplying by the factor |Y∞|k4 in our integral presentation

of Proposition 3.4, we can then describe the residual contribution of the shifted convolution sum as

|Y∞|k4 IRES = I(W )MΠ,α.

For the generic case of n ≥ 2 according to Propositions 4.2 (A) and 4.9, we can also derive the crude estimate

|Y∞|−k4 JRES = I(W )MΠ,α,(62)

where I(W ) = IΦ(W ) is a linear functional in the chosen weight function F depending on Φ = Pn1ϕ · θf,p,and MΠ,α ≥ 0 more generally is some constant which vanishes unless the F -integer α is totally positive andthe symmetric square L-function L(s,Sym2 Π) has a pole.

Some remarks on the constant term MΠ,α. Again, the constant MΠ,α ≥ 0 appearing in the residualcontributions vanishes unless the analytic continuation of the Dirichlet series corresponding to the Mellintransform for the constant coefficient of Φ has a pole. In the special case where the quadratic form is givenby f(x) = q(x) = x2, so that θq is the standard genuine metaplectic theta series, this analytic continuation isnone other than the symmetric square L-function L(s,Sym2 Π) of Π. In this special case, the constant termvanishes unless L(s,Sym2 Π) has a pole and the F -integer α is totally positive, using Shimura’s well-knowntheory of the integral presentation in this setting.

To give a more explicit description of the vanishing criterion for this constant term from the residualspectrum in the generic setting of n ≥ 2 with the metaplectic theta series θq corresponding to the quadraticform q(x) = x2, let us first note that the residual forms ϕτ = ϑξ are given as residues of Eisenstein seriesϕτ = ϑξ = Ress=s0 Eτ (?, s) (cf. [33, §4.7]), in addition to being translates of the metaplectic theta seriesθq. As explained in [33, §4.7] for the classical setting, this constant term can also be computed in terms ofresidues at s = 1 of the symmetric square L-function of Π. To explain this connection, let us first considerthe simplest case of the genuine metaplectic theta series θq. We can first calculate the Mellin transformM(ρPn1ϕθq,0

, s) of the constant coefficient ρPn1ϕθq,0(y) in the expansion

Pn1ϕθq((

y x1

))= ρPn1ϕθq,0

(y) +∑γ∈F×

ρPn1ϕθq(γyf )WPn1ϕθq

(γy∞)

of Pn1ϕθq according to Proposition 5.1 below. To be more precise, taking y = (yj)dj=1 ∈ F∞∞,+ to be a totally

positive archimedean idele, we can compute

M(ρPn1ϕθq,0, s) =

∫F×∞,+

ρPn1ϕθq,0(y)|y|sdy× =

∫ ∞0

· · ·∫ ∞

0

ρPn1ϕθq,0((yj)

dj=1

)ys1dy1

y1· · · ysd

dydyd

as

M(ρPn1ϕθq,0, s) =

∑a 6=0

cΠ(a2)

|a2|s+12−(n−2

2 )

∫F×∞,+

ψ(ia2y)Wϕ(a2y)dy×

= 2 ·L?1(2

(s+ 1

2 −(n−2

2

)),Sym2 Π)

L1(4(s+ 1

2 −(n−2

2

)), ωΠ)

∫F×∞,+

ψ(ia2y)Wϕ(a2y)dy×.

Here, L?1(s,Sym2 Π) equals L1(s,Sym2 Π) up to an Euler product which converges absolutely for <(s) > 12 ,

where L1(s,Sym2 Π) denotes the partial series expansion over principal ideals of OF of the symmetric square49

L-function L(s,Sym2 Π) of Π. As well, L1(s, ωΠ) denotes the corresponding partial Dirichlet series of thecentral character ωΠ. The Mellin transforms appearing in these expressions can be computed explicitly interms of the classical Whittaker functions Wκτ,j ,ντ,j (4π|yj |) (Proposition 5.1). A convolution calculation then

allows us to relate the Mellin transform M(ρPn1ϕθq,0, s) to the inner product 〈Pn1ϕθq, E(?, s)〉 of Pn1ϕθq against

a metaplectic Eisenstein series E(?, s) = Eq(?, s) (Lemma 5.2). This in fact gives an analytic continuation

for M(ρPn1ϕθq,0, s), and hence for the partial Dirichlet series L?1(s,Sym2 Π). To be more precise, if we assume

the archimedean local Whittaker function equals Wϕ(y) =∏dj=1Wκj ,νj (4π|yj |), the we obtain the relations

(first for <(s)� 1)

M(ρPn1ϕθq,0, s)

= 2 ·L?1(2(s+ 1

2 − (n−12 )),Sym2 Π

)L1

(4(s+ 1

2 − (n−12 )), ωΠ

) ·d∏j=1

Γ(s+ 3

4 − (n−22 ) + νj

)Γ(s+ 3

4 − (n−22 )− νj

)(4π)s+

14−(n−2

2 )Γ(s+ 5

4 − (n−22 ) + κj

)= 〈Pn1ϕθq, Eq(?, s)〉,

and in the more general case of the metaplectic theta series θf,p described above the relations

M(ρPn1ϕθf,p,0, s)

=∑

a=(a1,...,ak)∈OkF

f(a) 6=0

p(a)cΠ(f(a))

|f(a)|s+ k2−(n−2

2 )·d∏j=1

Γ(s+ 1

2 + k4 − (n−2

2 ) + νj)

Γ(s+ 1

2 + k4 − (n−2

2 )− νj)

(4π)s+k2−(n−2

2 )Γ(s+ k

4 − (n−22 ) + κj

)= 〈Pn1ϕθf,p, Ef,p(?, s)〉.

Returning to the coefficients 〈Φ, ϕτ 〉 = 〈Φ,Ress=s0 Eτ 〉 = 〈Φ, ϑξ〉 appearing in (61) above, we deduce thata given ϕτ ∈ BRES contributes only if the corresponding L-series has a pole. Hence for f = q, we deducefrom the theory of integral presentations of symmetric square L-functions that the residual contributionsIRES and JRES vanishes unless Π is orthogonal (and hence self-dual). In the general case on the quadraticform f , we argue that we can also deduce that the corresponding residual contribution will not vanish if Πis orthogonal. Hence, we characterize the contribution of the residual spectrum IRES or JRES crudely in thisway.

Bounds in the genuine metaplectic case for (A). Putting together (61) and (55) for the standard caseof n = 2 with the scaling factor of Proposition 3.4, we derive the estimate∑

a=(a1,...,ak)∈OkF

p(a) · cΠ(f(a) + α)

|f(a) + α| 12W

(f(a) + α

Y∞

)

= I(W )MΠ,α +OΠ,ε

|Y∞| k4 +ε

∣∣∣∣ αY∞∣∣∣∣ 12−

θ02

|α|δ0− 12

.

In the generic case n ≥ 2, using the integral presentation of Proposition 4.2 (A), we can put together thebounds for the corresponding contributions (62) and (56) for to derive the more general estimate∑

a=(a1,...,ak)∈OkF

p(a) · cΠ(f(a) + α)

|f(a) + α| 12W

(f(a) + α

Y∞

)

= I(W )MΠ,α +OΠ,ε

(|Y∞|−

k4 |αY∞|

12 +

θ02 −ε |α|δ0− 1

2

)= OΠ,f,p,α,W (1) +OΠ,ε

(|Y∞|

2−k4 +

θ02 +ε|α|δ0+

θ02 −ε

).

This gives the corresponding estimate (for odd k) in Theorem 1.1 (A). Notice that we derive an improvementon the standard setup of dimension n = 2 by considering the modified theta series Θf,p, i.e. to absorb the

extra scaling factor of |Y∞|k4 that we would otherwise have to include (cf. e.g. [33, Theorem 1]).

50

Bounds for the non-genuine cases of (A) and (B). To deal with the remaining cases of Proposition3.4 for even k and Proposition 3.5, we can use the same argument with the non-genuine forms Φ = φθf,p,i.e. with f having an even number of variables k ≥ 2, and also with Φ = φφ′. Again, in the generic casen ≥ 2, we consider the functions Φ defined in (30) above for Proposition 4.2. Recall that these functions haveconvergent spectral coefficients by the argument of Proposition 4.9. We also saw in Proposition 4.2 (usingPropositions 3.4, 3.5, and 2.9) that the Fourier-Whittaker coefficients of these restricted extended functionscarry sufficient information about the shifted convolution problem via the underlying mirabolic form that itwill suffice to bound the spectral decomposition of the unipotent integral describing the Fourier-Whittakercoefficient at α of Φ. Here, we decompose the corresponding form Φ as an L2-automorphic form on GL2(AF )in a completely analogous way, and repeat the same argument essentially verbatim. That is, we now considerthe L2-spectral decomposition of Φ in the Hilbert space L2(GL2(F )\GL2(AF ), ξ), as described for instancein [6, §2.2,(8)]:

L2(GL2(F )\GL2(AF ), ω) =⊕π∈Cξ

cuspidal

Vπ ⊕⊕ξ2=ω

residual

Vχ ⊕∫ ∞

0

⊕χ1χ2=ω

χ1χ−12 =|·|2iy on F

diag∞,+

continuous

Vχ1,χ2dy

= L2cusp(GL2(F )\GL2(AF ), ω)⊕ L2

res(GL2(F )\GL2(AF ), ω)⊕ L2cont(GL2(F )\GL2(AF ), ω).

Here, each Vχ denotes the subspace generated by g 7→ χ(det g), with ξ the idele class characters of F forwhich ξ2 = ω. Repeating the arguments above for Φ in this setting, the exponent δ0 is seen easily to bereplaced by θ0 (as each form in the expansion is an L2-automorphic form on GL2(AF )). As well, it is not hardto deduce that special or residual spectrum contributes only if the Rankin-Selberg L-function L(s,Π × π)has a pole. This again comes from the fact that such forms ϑξ from the residual spectrum occur as residuesof Eisenstein series ϑξ = Ress=s0 Eξ(∗, s), so that the inner product 〈Φ, ϑξ〉 = Ress=s0〈Φ, Eξ(∗, s)〉 vanishesunless 〈Φ, Eξ(∗, s)〉 ≈ L(s,Π × π) has a pole. Let us remark that the shifted convolution sum appearing inPropositions 4.2 (B) and 3.5 in particular can be estimated in the style of [6, Theorem 2]. As we explainbelow, this latter bound can be used to derive estimates for the corresponding subconvexity problem forcentral values of GLn(AF )-automorphic L-functions.

5. Remarks on analytic continuation of shifts by quadratic forms

We now explain how to derive the analytic continuation of the Dirichlet series (1), although this is notstrictly necessary for the rest of the work. Let us now return to the setup of Proposition 3.4 above, but withthe constant coefficient α = 0. To be clear, let us fix a pure tensor ϕ = ⊗vϕv ∈ VΠ whose nonarchimedeanlocal components ϕv are essential Whittaker vectors, but whose archimedean component is not yet specified.Let f(a1, . . . , ak) be an F -rational positive definite quadratic form in k variables, with p(a1, . . . , ak) anassociated harmonic polynomial (possibly trivial), and θf,p the corresponding theta series. We shall consider

the automorphic form Pn1ϕθf,p on GL2(AF ), which for x ∈ AF an adele and y = y∞yf ∈ A×F an idele hasthe Fourier-Whittaker expansion

Pn1ϕθf,p((

y x1

))= ρPn1ϕθf,p,0

(y) +∑γ∈F×

ρPn1ϕθf,p(γyf )WPn1ϕθf,p

(γy∞).

Let us now consider the constant coefficient ρPn1ϕθf,p,0(y), and in particular its specialization to an

archimedean idele y∞ ∈ F×∞. Writing F×∞,+∼= Rd

>0 as above to denote the totally positive plane, wecompute the Mellin transform

M(ρPn1ϕθf,p,0, s) =

∫F×∞,+

ρPn1ϕθf,p,0(y∞)|y∞|sdy×∞,

51

which after expanding out the archimedean idele y∞ = (yj)dj=1 = (y∞,j)

dj=1 ∈ F

×∞,+ with each yj = y∞,j ∈

R>0 is defined more explicitly as the d-fold Mellin transform

M(ρPn1ϕθf,p,0, s) =

∫ ∞0

· · ·∫ ∞

0

ρPn1ϕθf,p,0

((yj)

dj=1

)ys1dy1

y1· · · ysd

dydyd

.

This Mellin transform can be computed explicitly as follows, using all of the same notations and conventionsfor (archimedean) Whittaker functions described above.

Proposition 5.1. Let Π = ⊗vΠv be a cuspidal automorphic representation of GLn(AF ) with unitary centralcharacter for n ≥ 2. Again, we write cΠ to denote the L-function coefficients of Π. As well, we write Pn1ϕto denote the image of ϕ under the projection operator Pn1 , so that Pn1ϕ is a cuspidal L2-automorphic formon the mirabolic subgroup P2(AF ) ⊂ GL2(AF ). Let ϕ = ⊗vϕv ∈ VΠ be a pure tensor whose nonarchimedeanlocal components are all essential Whittaker vectors. Let f(a1, . . . , ak) be an F -rational positive definitequadratic form in k many variables, and p(a1, . . . , ak) a harmonic polynomial for f (possibly trivial). Then,

M(ρPn1ϕθf,p,0, s) =

∑a=(a1,...,ak)∈Ok

Ff(a)6=0

p(a)cΠ(f(a))

|f(a)|s+k2−(n−2

2 )

×∫F×∞,+

ψ(if(a)y∞)Wϕ (f(a)y∞) |y∞|s+k4−(n−2

2 )dy×∞.

In particular, if we assume that the archimedean Whittaker function

Wϕ(y∞) = Wϕ

((y∞

1n−1

))with y∞ = (yj)

dj=1 ∈ F

×∞,+∼= Rd

>0 is given by the product

Wϕ(y∞) =

d∏j=1

Wκj ,νj (4π|yj |)

for some d-tuples of complex numbers κ = (κj)dj=1 ∈ Cd and ν = (νj)

dj=1 ∈ Cd, where Wκj ,νj (yj) denotes

the standard Whittaker function described above, then

M(ρPn1ϕθf,p,0, s) =

∑a=(a1,...,ak)∈Ok

Ff(a) 6=0

p(a)cΠ(f(a))

|f(a)|s+k2−(n−2

2 )

×d∏j=1

Γ(s+ 1

2 + k4 −

(n−2

2

)+ νj

)Γ(s+ 1

2 + k4 −

(n−2

2

)− νj

)(4π)s+

k4−(n−2

2 )Γ(s+ 1 + k

4 −(n−2

2

)+ κj

)for <(s) sufficiently large, i.e. for <(s)− k

4 −(n−2

2

)> −1.

Proof. The first claim follows as an easy consequence of the definitions. That is,

M(ρPn1ϕθf,p,0, s) =

∫F×∞,+

ρPn1ϕθf,p,0(y∞) |y∞|sdy×∞,

which after expanding out the coefficient (as done in the proof of Proposition 3.4),

ρPn1ϕθf,p,0(y∞) = |y∞|

k4−(n−2

2 )∑

a=(a1,...,ak)∈OkF

f(a)6=0

p(a)cΠ(f(a))

|f(a)|n−22

ψ(if(a)y∞)Wϕ(f(a)y∞),

then switching the order of summation gives the stated identity. Keep in mind that this is the same as

ρPn1ϕθf,p,0(y∞) = |f(a)y∞|

k4−(n−2

2 )∑

a=(a1,...,ak)∈OkF

f(a)6=0

p(a)cΠ(f(a))

|f(a)| k4ψ(if(a)y∞)Wϕ(f(a)y∞).

To prove the second claim, we compute the remaining d-fold integral Mellin transform as follows. Letus write (τj)

dj=1 to denote the collection of real places of F , i.e. the collection of embeddings τj : F → R.

52

Recall that we fix ψ to be the standard additive character for which ψ(y∞) =∏dj=1 exp(2πiyj). Hence

ψ(if(a)y∞) =∏dj=1 exp(−2πτj(f(a))yj), whence unraveling definitions gives∫

F×∞,+

ψ(if(a)y∞)Wϕ(f(a)y∞)|y∞|s+k4−(n−2

2 )dy×∞

=

d∏j=1

∫ ∞0

e−2πτj(f(a))yjWκj ,νj (4πτj(f(a))yj)ys+ k

4−(n−22 )

j

dyjyj.

Now, to evaluate each of the single-variable Mellin transforms∫ ∞0

e−2πτj(f(a))yjWκj ,νj (4πτj(f(a))yj)ys+ k

4−(n−22 )

j

dyjyj

appearing in this latter expression, we use the formula (25) above. Hence for

<(s) +k

4−(n− 2

2

)> −1,

making a simple change of variables, we find that∫ ∞0

e−2πτj(f(a))yjWκj ,νj (4πτj(f(a))yj)ys+ k

4−(n−22 )

j

dyjyj

= (4πτj(f(a)))−(s+ k4−(n−2

2 )) Γ(s+ 1

2 + k4 −

(n−2

2

)+ νj

)Γ(s+ 1

2 + k4 −

(n−2

2

)− νj

)Γ(s+ 1 + k

4 −(n−2

2

)+ κj

) .

Thus, we compute∫F×∞,+

ψ(if(a)y∞)Wϕ(f(a)y∞)|y∞|s+k4−(n−2

2 )dy×∞

= |f(a)|−s−k4 +(n−2

2 )d∏j=1

Γ(s+ 1

2 + k4 −

(n−2

2

)+ νj

)Γ(s+ 1

2 + k4 −

(n−2

2

)− νj

)(4π)s+

k4−(n−2

2 )Γ(s+ 1 + k

4 −(n−2

2

)+ κj

) ,

from which the claim follows easily. �

We now use this to derive the following semi-classical unfolding calculation. Let Γ ⊂ P2(F∞) be a discretesubgroup under which Pn1ϕ and θf,p are both invariant. Let us also consider

Γ∞ =

{(1 α

1

): α ∈ OF

}.

We then consider the Eisenstein series defined for x ∈ AF and y = yfy∞ ∈ A×F by

E(y, s) = E

((y x

1

), s

)=

{∑Γ∞\Γ |γy|

sif k ≡ 0 mod 2∑

Γ∞\Γ |j (y, γ)|s if k ≡ 1 mod 2,

where j(y, γ) is the (metaplectic) automorphy factor defined by

j(y, γ) = j

((y x

1

), γ

)=

θf,p

(γ

(y x

1

))θf,p

((y x

1

)) ,

and s ∈ C is a complex variable with <(s) > 1. Let us also write HF = AF ∪ iF×∞,+, which we identify with

the mirabolic subgroup of P2(AF ) of matrices

(y∞ x

1

)with x ∈ AF and y∞ ∈ F×∞,+ in the usual way.

Lemma 5.2. We have for <(s) > 1 that

M(ρPn1ϕθf,p,0, s) =

∫∫FF

Pn1ϕθf,p((

y∞ x1

))E

((y∞ x

1

), s

)dxdy×∞,

where FF denotes some fixed fundamental domain for the action of Γ on HF .53

Proof. We have that

M(ρPn1ϕθf,p,0, s) =

∫F×∞,+

ρPn1ϕθf,p,0(y∞)|y∞|sdxdy×∞

=

∫F×∞,+

∫AF /F

Pn1ϕ((

y∞ x1

))θf,p

((y∞ x

1

))|y∞|sdxdy×∞

=

∫∫Γ∞\HF

Pn1ϕ((

y∞ x1

))θf,p

((y∞ x

1

))|y∞|sdxdy×∞,

where Γ∞ acts on HF by translation. Now, we argue that we can fix a set of representatives for this actionof the form ∪γFF , where γ ranges over a set of representatives for the action of Γ modulo Γ∞, and FF is afundamental domain for the action of Γ on HF . Hence, we can expand the latter integral expression as∑

γ∈Γ∞\Γ

∫∫γFF

Pn1ϕ((

y∞ x1

))θf,p

((y∞ x

1

))|y∞|sdxdy×∞,

which after using the automorphy of Pn1ϕ and θf,p with respect to Γ and switching the order of summationis the same as the stated integral∫∫

FF

Pn1ϕ((

y∞ x1

))θf,p

((y∞ x

1

))E

((y∞ x

1

), s

)dxdy×∞.

�

This latter result allows us to derive the following from the well-known analytic properties of the Eisensteinseries E(y, s) in either case on the parity of the number of variables k of the positive definite quadratic formf (see e.g. [30] or [18]).

Corollary 5.3. Fix n ≥ 2, and let Π = ⊗vΠv be a cuspidal GLn(AF )-automorphic representation. Again,we write cΠ to denote the L-function coefficients of Π, so that the Dirichlet series of the standard of L-function of Π can be written as

∑m⊂OF cΠ(m)Nm−s for <(s) > 1. Let f(a1, . . . , ak) be a positive definite

quadratic form in k ≥ 1 many variables, and p(a1, . . . , ak) a harmonic polynomial for f (possibly trivial).Then the Dirichlet series

D(s,Π, f, p) =∑

a1,...,ak∈OFf(a1,...,ak) 6=0

p(a1, . . . , ak)cΠ(f(a1, . . . , ak))

|f(a1, . . . , ak)|s,

defined a priori only for s ∈ C with <(s) � 1, has an analytic continuation to all s ∈ C, and satisfies afunctional equation relating values at s to 1− s.

6. Estimates for twisted GLn(AF )-automorphic L-functions

We now give the proof of Theorem 1.2. That is, we now explain how to use the integral presentationof Proposition 3.5 (as adapted in Proposition 4.2) and the estimates of Theorem 1.1 (2) in the style ofBlomer-Harcos [6, § 3] and Cogdell [15], as well as an earlier work of Cogdell-Piatetski-Shapiro-Sarnak [11](see [6, Remark 2]), to estimate central values of twisted GLn(AF )-automorphic L-functions.

Recall that for <(s) > 1, we write L(s,Π) =∑

m⊂OF cΠ(m)Nm−s to denote the finite part of the standard

L-function Λ(s,Π) =∏v≤∞ L(s,Πv) = L(s,Π∞)L(s,Π) of Π. Hence, the coefficients cΠ are the L-function

coefficients of Π, and in the special case of dimension n = 2 described in [6, §2] have a direct relation to theHecke eigenvalues λΠ. Assume that Π is irreducible. Fix a Hecke character χ of F of conductor q ⊂ OF .Hence, there exist characters χf : (OF /qOF )× −→ S1 and χ∞ : F×∞ −→ S1 such that for any F -integerα ∈ OF coprime to q, we have χ((α)) = χf (α)χ∞(α). Note that we shall use the same notation χ to denotethe corresponding idele class character of F , interchanging these notions freely when the context is clear.

54

6.1. Esquisse. Let us first sketch an outline of the main argument following the heuristic description ofexponents given in Blomer-Harcos [6, §3.1], assuming for simplicity that F = Q, and that Π is everywhereunramified. We shall also ignore epsilons for now. Fixing a real parameter L > 0, and writing l v L todenote the sum over L ≤ l ≤ 2L, the idea is to derive upper and lower bounds for the amplified secondmoment defined over Hecke characters ξ of conductor q,

S :=∑

ξ mod q

∣∣∣∣∣∑lvL

ξ(l)χf (l)

∣∣∣∣∣2

|L(1/2,Π⊗ ξ)|2 .

To begin, observe that we have the lower bound

S � L2|L(1/2,Π⊗ χ)|2.On the other hand, using an approximate functional equation to describe each of the central values appearingin S, opening up squares gives an upper bound of

S � q∑

l1,l2vL

χf (l1)χf (l2)∑

l1m1−l2m2≡0 mod q

cΠ(m1)cΠ(m2)

(m1m2)12

V

(m1

qn2

)V

(m2

qn2

)for a smooth and rapidly decaying function V of y ∈ R>0 which depends only on the archimedean componentsΠ∞, ξ∞, and χ∞, and which by standard arguments can be replaced by a compactly supported function(see Corollary 6.2 (B)). Now, the diagonal term coming from the contribution l1m1− l2m2 = 0 in this latterquantity is seen easily to be bounded above by �Π qL (see (73)). It therefore remains to estimate thecontribution of the non-negligible part of the remaining off-diagonal sum∑

l1,l2vL

χf (l1)χf (l2)∑

1≤h≤qn−22 L

∑m1,m2≥1

l1m1−m2l2=hq

cΠ(m1)cΠ(m2)

(m1m2)12

V

(m1

qn2

)V

(m2

qn2

).

Using a variation of the argument of Propositions 3.5 and 4.2 (B) above (see Proposition 6.3), we can derivefor each inner sum corresponding to the congruence pair l1m2 − l2m2 = hq in this latter expression anintegral presentation of the form∫

I∼=[0,1]⊂RPn1ϕ

((l1Lq

n2 x

1

))Pn1ϕ

((l2Lq

n2 x

1

))ψ(−qhx)dx,

and more intrinsically a presentation as a sum over Fourier-Whittaker coefficients∫I∼=[0,1]⊂R

Φl1,l2

((Lq

n2 x

1

))ψ(−qhx)dx for Φl1,l2 := R(l−1

1 )Pn1ϕ ·R(l−12 )Pn1ϕ,(63)

where R(lj) for j = 1, 2 is the shift operator (cf. [6, (14) and also (15), (38)-(41)]) defined on an automorphicform φ of p ∈ P2(AF ) ⊂ GL2(AF ) by the rule

(R(lj)φ)(p) = φ

(p

(l−1j x

1

)).

Note that these operators are isometries (see [6, §2.9]). Hence, it remains to bound∑l1,l2vL

χf (l1)χf (l2)∑

1≤h≤qn−22 L

∫I∼=[0,1]⊂R

Φl1,l2

((Lq

n2 x

1

))ψ(−hqx)dx.(64)

Decomposing each automorphic form Φ = Φl1,l2 in the sum (64) spectrally according to the discussion aboveProposition 4.9 for the generic case n ≥ 2 as

Φ =∑ϕτ∈B

〈Φ, ϕτ 〉 · ϕτ

then allows us to derive bounds from Theorem 1.1 (B). Roughly speaking, we know that each archimedeanlocal Whittaker coefficient Wϕτ is a multiple of the standard Whittaker functions described above, and hencebounded in the spectral parameter. Using Plancherel’s formula, together with the fact that the Kirillov mapand the operators R(li) for i = 1, 2 are isometries, we know that

∑ϕτ∈B ||Wϕτ ||2 ≈ ||ΦS || � 1. As well,

we know that each Φ = Φl1,l2 has level of size L2. By Weyl’s law, there should be approximately L2 many55

eigenvalues in the interval of constant length L, and so we argue that the sum over ϕτ ∈ B has length of size

L2. Since Π is irreducible, the Rankin-Selberg L-function L(s,Π⊗ χ× Π) does not have a pole, and henceTheorem 1.1 (B) allow us to derive the estimate∑

l1,l2vL

χf (l1)χf (l2)∑

1≤h≤Lqn−22

∑ϕτ∈B

〈Φ, ϕτ 〉∫I∼=[0,1]⊂R

ϕτ

((Lq

n2 x

1

))ψ(−hqx)dx

�Π L2∑

1≤h≤Lqn−22

∑ϕτ∈B

cϕτ (qh)

(qh)12

Wϕτ

((qh) · q n2 L

)

�Π L2 · L2∑

1≤h≤Lqn−22

(qh)θ0−12

(qh

qn2 L

) 12−

θ02

� L2 · L 52 +θ0q

n2 (θ0+ 1

2 )−1.

Using this latter bound to estimate the off-diagonal contribution for S, we find that

|L(1/2,Π⊗ χ)|2 �Π,χ∞ qL−1 + qn2 ( 1

2 +θ0)L52 +θ0 .(65)

Making a suitable choice of parameter L of size qu for some 0 ≤ u ≤ 1 then gives some correspondingbound for the modulus of the central value L(1/2,Π ⊗ χ). For instance, taking L = q1/4−θ0/2 gives us the(surprising) uniform upper bound

L(1/2,Π⊗ χ)�Π,χ∞ q38 +

θ04 + q

n4 ( 1

2 +θ0)+(5−8θ0)

16 .

We now give a more precise derivation of the bound (65), adapting the main line of argument of [6] tothis setting, using a variation of Proposition 3.5 above (see Proposition 6.3) and the bounds of Theorem 1.1(B) in lieu of [6, Theorem 2].

6.2. Reductions via approximate functional equations. To make the sketch above precise, we firstreview how to use the approximate functional equation to reduce the study of the value L(1/2,Π ⊗ ξ)appearing in the definition of S to finite sums analogous to those described in [6, (75)]. Let us writeΛ(s,Π ⊗ χ) = L(s,Π∞ ⊗ χ∞)L(s,Π ⊗ χ) to denote the standard L-function of the GLn(AF )-automorphicrepresentation Π⊗ χ, where the archimedean component L(s,Π∞ ⊗ χ∞) is defined by the product

L(s,Π∞ ⊗ χ∞) =

n∏i=1

∏v|∞

ΓR (s+ χv(−1)µi(Π∞)) .

Here, the second product on the right hand side runs over the real places of F , we use the standard notationΓR(s) = π−

s2 Γ(s2

), we view the Hecke character in terms of its corresponding idele class character χ = ⊗vχv,

and we use the symbols µi(Π∞) to denote the archimedean Satake parameters of Π∞. Hence, the generalizedRamanujan-Petersson conjecture predicts that <(µi(Π∞)) = 0 for each index i, and we have the uniformupper bound

max1≤i≤n

<(µi(Π∞)) ≤ 1

n2 + 1

for any dimension n ≥ 2 thanks to the theorem(s) of Luo-Rudnick-Sarnak [26], [27]. Note that if χ is awide ray class character, equivalently if χ∞ is the trivial character, then we have the useful identificationL(s,Π∞) = L(s,Π∞ ⊗ χ∞). In any case, Λ(s,Π ⊗ χ) has a well-known analytic continuation to all s ∈ C,and satisfies the function equation

(N(f(Π)DnF q

n))s2 Λ(s,Π⊗ χ) = ε(1/2,Π⊗ χ) (N(f(Π)Dn

F qn))

1−s2 Λ(1− s, Π⊗ χ−1),

where f(Π) ⊂ OF denotes the conductor of Π, DF ⊂ OF the discriminant of F , ε(1/2,Π⊗ χ) ∈ S1 the root

number of Λ(s,Π ⊗ χ), and Π = ⊗vΠv the contragredient representation. Following the discussions in [6]

and [21], let us consider the corresponding analytic conductor C(Π ⊗ χ) = C(Π ⊗ χ ⊗ | det |s− 12 )|s=1/2 of

56

Λ(s,Π⊗ χ) at s = 1/2 defined by

C(Π⊗ χ) = N(f(Π)DF qn) · π−nd

n∏i=1

∏v|∞

∣∣∣∣14 +χv(−1)µi(Π∞)

2

∣∣∣∣2 .We have the following uniform approximate functional equation for the central values L(1/2,Π⊗ χ) in thissetting thanks to the main result of Harcos [21].

Theorem 6.1. There exists a complex function V : R≥0 −→ C and a complex number of modulus oneu ∈ S1 depending only on the archimedean local parameters χv(−1)µi(Π∞) for v ranging over real places ofF and 1 ≤ i ≤ n such that

L(1/2,Π⊗ χ) =∑

m⊂OF

cΠ(m)χ(m)

Nm12

V

(Nm

C(Π⊗ χ)12

)+ u · ε(1/2,Π⊗ χ)

∑m⊂OF

cΠ(m)χ(m)

Nm12

V

(Nm

C(Π⊗ χ)12

).

The function V and its derivatives V (j) for each j ≥ 1 satisfy the following uniform decay estimates:

V (y) =

{1 +Oσ (yσ) as y → 0 for 0 < σ < 1/(n2 + 1)

Oσ (y−σ) as y →∞ for any σ > 0,

and

V (j)(y) = Oσ,j(y−σ

)as y →∞ for σ > j − 1/(n2 + 1).

Here, the implied constants depend only on σ, j, m, and the degree d = [F : Q]. As well, the region0 < σ < 1/(n2 + 1) in the first estimate can be widened to 0 < σ < 1/2 if we known that <(µi(Π∞)) = 0 foreach index 1 ≤ i ≤ n.

Proof. See [21, Theorem 1]. To be more precise about the definition of V (y), let H(s) be an entire functionof s ∈ C which is bounded as H(s) �σ,A (1 + |s|)−A for any choice A > 0, where σ = <(s). Assume that

H(0) = 1, and also that H(s) = H(−s) = H(−s). Note that such a function can be realized at the Mellintransform H(s) =

∫∞0h(x)xs dxx of a smooth function h : R>0 → R having total mass one with respect to

the measure dx/x which satisfies the functional equation h(x) = h(x−1), and whose derivatives decay fasterthan any negative power of x. Consider the function defined by

K(s,Π∞ ⊗ χ∞)

=1

2

(N(DF f(Π)qn)

C(Π⊗ χ)12

)s· L(1/2 + s,Π∞ ⊗ χ∞)

L(1/2− Π∞ ⊗ χ−1∞ )

· L(1/2, Π∞ ⊗ χ−1∞ )

L(1/2,Π∞ ⊗ χ∞)+

1

2C(Π⊗ χ)

s2 .

The cutoff function V (y) is then defined explicitly by the contour integral

V (y) =

∫<(s)=σ

K(s,Π∞ ⊗ χ∞)H(s)(yC(Π⊗ χ)

12

)−s ds2πi

.

Moreover, u ∈ S1 is given by the ratio u = L(1/2, Π∞ ⊗ χ−1∞ )/L(1/2,Π∞ ⊗ χ∞).

�

The result can be used to derive the following standard preliminary estimates.

Corollary 6.2. We have the following estimates for the central value L(1/2,Π⊗ χ).

(A) We have for any choice of ε > 0 the uniform convexity bound

L(1/2,Π⊗ χ)�ε C(Π⊗ χ)14 +ε,

where the implied constant depends only on ε, n, and d.

57

(B) Writing V ?(y) to denote the restriction of V (y) to the compact interval [2−1, 2], we have for anychoice of ε > 0 the estimate

L(1/2,Π⊗ χ)�Π,χ∞,ε Nqε maxY≤C(Π⊗χ)

12+ε

∣∣∣∣∣ ∑m⊂OF

cΠ(m)χ(m)

Nm12

V ?(

Nm

Y

)∣∣∣∣∣ .Proof. See [21, Corollary 2] and [6, (75)] (cf. [7, § 5.1]). �

Hence for some constant c = c(Π, χ∞, ε) > 0, we can begin with the latter bound

L(1/2,Π⊗ χ)�Π,χ∞,ε Nqε maxY≤cNq

n2

+ε

∣∣∣∣∣ ∑m⊂OF

cΠ(m)χ(m)

Nm12

V ?(

Nm

Y

)∣∣∣∣∣ .(66)

To estimate this latter quantity in the style outlined above (following [6, §3.3]), we must first split in them-sum into its corresponding narrow ideal class components. Let us therefore fix a narrow ideal class of F ,together with an integer representative y coprime to the conductor q whose norm we can and do assume isbounded by Ny�ε Nqε. It will do to estimate the corresponding sum over totally positive nonzero ideals∑

0�γ∈yγ mod O×

F,+

cΠ(γy−1)χ(γy−1)

N(γy−1)12

V ?(

Nγ

Y

)(67)

for Y �ε Nqn2 +ε, where O×F,+ ⊂ O

×F denotes the subgroup of totally positive units. Note that we shall

always take sums over nonzero ideals in the subsequent discussion, but that we suppress this condition fromthe notations for simplicity.

Recall that we write (τj)dj=1 to denote the collection of real embeddings of F . Let us fix a fundamental

domain F0 for the action of O×F,+ on the hyperboloid

h ={y ∈ F×∞,+ : Ny = |y| = 1

},

where again F×∞,+ ⊂ F×∞ denotes the totally positive sub-plane. We assume that this action is fixed so thatits image under the map

F×∞,+ −→ Rd, y 7−→ (log τ1(y), · · · , log τd(y))

is a fundamental parallotope for the image of the totally positive units O×F,+ under the same map. Note thatthe cone

F = F diag∞,+F0

is a fundamental domain for the action of the O×F,+ on F×∞,+. To use these choices of fundamental domains

to describe our sum (67) in a convenient way for estimates, let us first fix a smooth and compactly supportedfunction G0 : h −→ C satisfying ∑

u∈O×F,+

G0(uy) = 1, for any y ∈ h.

We extend this function G0 to all y ∈ F×∞,+ by defining

G(y) = G0

(y

Ny1d

).

Note that the support of the hyperboloid function G0 is contained in some box [c1, c2]d ⊂ F×∞,+, and that

the support of G is contained in the cone F diag∞,+[c1, c2]d of this box. Using these conventions, we can rewrite

(67) as the finite sum ∑0�γ∈y

cΠ(γy−1)χ(γy−1)

N(γy−1)12

G(γ)V ?(

Nγ

Y

).

To be clear, since y is a lattice in F∞, this latter sum is seen easily to be finite from the support of thechosen function G. More precisely, it vanishes outside of the box [ 1

2c1Y1d , 2c2Y

1d ]d. Let us now fix a smooth

58

function W : F×∞ −→ C with support contained in the box [ 13c1, 3c2]d, and such that W (y) = 1 on [ 1

2c1, 2c2]d.

Making these choices, we can express the latter sum as an integral over vectors w = (wj)dj=1 ∈ (iR)d, which

in particular allows us to express (67) equivalently as

χ(y−1)∑

0�γ∈y

cΠ(γy−1)χ((γ))

N(γy−1)F (γ)V ?

(Nγ

Y

)W

(γ

Y1d

)

=χ(y−1)χ∞(Y

1d )

(2πi)d

∫(iR)d

V(w)∑

0�γy

cΠ(γy−1)χf (γ)

N(γy−1)12

Ww

(γ

Y1d

)dw,

(68)

where

V(w) =

∫F×∞,+

G(y)V ? (|y|)χ∞(y)

d∏j=1

ywjj d×y

and for y = (yj)dj=1 ∈ F

×∞,+ (and also for y = (τj(y))dj=1 with y ∈ R≥0 as above),

Ww(y) = W (y)

d∏j=1

y−wjj .

Before going on, we record that the functions G(y)V ?(|y|) = G(y)V ?(Ny) and W (y) appearing in this

latter expression are smooth and compactly supported. Moreover, since χ∞(y) =∏dj=1 y

sjj for some fixed

vector s = (sj)dj=1 ∈ (iR)d, it is easy to see that we have the following bounds as functions of w = (wj)

dj=1 ∈

(iR)d, for any choices of constant A > 0 and integer d-tuple µ = (µj)dj=1 ∈ Zd≥0:

V(w)�A,χ∞ N((1 + |wj |)dj=1)−A = |(1 + |wj |)dj=1|−A(69)

and

∂µ1y1 · · · ∂

µdydWw(y)�µ

d∏j=1

(1 + |wj |)µj .(70)

Let us now fix a vector w = (wj)dj=1 ∈ (iR)d in the integral on the right hand side of (68), deferring the

integration until the last step in the argument. Given a character ξ : (OF /qOF )× → S1, we then considerthe sum defined by

Lξ(w) =∑

0�γ∈y

cΠ(γy−1)ξ(γ)

N(γy−1)12

Wv

(γ

Y1d

),(71)

so that Lχf (w) is the sum appearing in the integrand on the right hand side of (68). The next step is to formand estimate an amplified second moment for each of these sums Lχf (w), using a variation of Proposition3.5 to describe the off-diagonal term.

6.3. Amplified second moments. Let us now focus on the sum (71), which observe has support contained

in the box [ 13c1Y

1d , 3c2Y

1d ]d. Note as well that the cone C ⊂ F×∞,+ of this box is independent of Y , and

covered by a finite number of O×F,+-translates of the fundamental domain F . Hence, fixing a parameter

L ≥ log(Nq), and writing l v L to denote the sum over totally positive principal prime ideals l - q ⊂ OFin the interval L ≤ Nl ≤ 2L, we consider the amplified second moment defined by taking the sum over allcharacters ξ of (OF /qOF )×,

S =∑

ξ mod q

∣∣∣∣∣∑lvL

ξ(l)χf (l)

∣∣∣∣∣2

|Lξ(w)|2 .

It is easy to check that for any ε > 0, we have the lower bound

# {l - q ⊂ OF principal, prime, and totally positive with L ≤ Nl ≤ 2L} �ε Nq−εL.

59

This implies the lower bound S �ε Nq−εL2|Lχf (w)|2, and hence the upper bound

∣∣Lχf (w)∣∣2 �ε

Nqε

L2

∑ξ mod q

∣∣∣∣∣∣∣Lξ(w)∑

l∈OF∩FlvL

ξ(l)χf (l)

∣∣∣∣∣∣∣2

,

which by Plancherel’s formula for (OF /qOF )× is the same as

∣∣Lχf (w)∣∣2 �ε

ϕ(q)Nqε

L2

∑x∈(OF /qOF )×

∣∣∣∣∣∣∣∑

l∈OF∩FlvL

χf (l)∑γ∈y∩C

lγ≡x mod q

cΠ(γy−1)

N(γy−1)12

Ww

(γ

Y1d

)∣∣∣∣∣∣∣2

.

Here, we put ϕ(q) = #(OF /qOF )×. Now, extending the summation to all classes x mod q and opening upthe square, we derive the more explicit upper bound∣∣Lχf (w)

∣∣2 �εNq1+ε

L2

∑l1,l2∈OF∩Fl1,l2vL

χf (l1)χf (l2)

×∑

l1γ1−l2γ2∈qγ1,γ2∈y∩C

cΠ(γ1y−1)cΠ(γ2y

−1)

N(γ1γ2y−2)12

Ww

(γ1

Y1d

)Ww

(γ2

Y1d

).

(72)

Let us first consider the diagonal term in this latter expression (72) coming from the contribution ofl1γ1 − lγ2 = 0. This contribution is seen easily to be bounded above for any ε > 0 (uniformly in the choiceof vector w ∈ (iR)d) by the quantity

�εNq1+ε

L2

∑l∈OF∩FlvL

∑γ∈y∩CNγ≈Y

|cΠ(γy−1)|2

N(γy−1)# {(l′, γ′) ∈ (OF ∩ F)× (y ∩ C) : l′γ′ ≡ lγ} .

Now, it is easy to see that∑l∈OF∩FlvL

∑γ∈y∩CNγ≈Y

# {(l′, γ′) ∈ (OF ∩ F)× (y ∩ C) : l′γ′ ≡ lγ} �ε (LY )ε.

On the other hand, we have for any Y ∈ R>0 the well-known bound∑m⊂OFNm≤Y

|cΠ(m)|2 ≤ CΠY

as Y → ∞ for some constant CΠ > 0 depending only on Π (see e.g. [26, (14)]). Using these estimates, wesee that the diagonal contribution is bounded above by

�εNq1+ε

L2# {l ⊂ OF : l v L}

∑m⊂OFNm�Y

|cΠ(m)|2

Nm�ε,Π

Nq1+ε

L.(73)

Let us now consider the heart of the matter, which is the remaining off-diagonal contribution to thequantity in the right hand side of (72). Here again, we follow the reduction steps and setup of [6, §3.3]closely. Fix a box [c3, c4]d ∈ F×∞,+ containing the fundamental domain F0. It is easy to see that the totally

positive principal ideals l1, l2 in the sum are contained in the box l1, l2 ∈ [c3L1d , 2c4L

1d ]d, and the totally

positive F -integers γ1, γ2 in the box γ1, γ2 ∈ [ 13c1Y

1d , 3c2Y

1d ]d, so that

l1γ1 − l2γ2 ∈ K := [−6c2c4(LY )1d , 6c2c4(LY )

1d ]d.

Using this definition of box region K, we can describe the inner sum corresponding to a given pair l1, l2 inthe off-diagonal sum of (72) more explicitly in terms of congruences modulo F -integers as∑

α∈qy∩K

∑γ1,γ2∈y

l1γ1−l2γ2=α

cΠ(γ1y−1)cΠ(γ2y

−1)

N(γ1γ2y−2)12

Ww,1

(l1γ1

(LY )1d

)Ww,2

(l2γ2

(LY )1d

),(74)

60

where for i = 1, 2 we use the functions Ww,i : F×∞ → C defined on y ∈ F×∞ by

Ww,i(y) =

{Ww

(l−1i L

1d y)

if y ∈ F×∞,+0 otherwise.

Note that these functions Ww,i are smooth and supported on [ 13c1c3, 6c2c4]d, and moreover (as functions of

the vector w = (wj)dj=1 ∈ (iR)d) bounded via (70) as

∂µ1y1 · · · ∂

µdydWw,i �µ

d∏j=1

(1 + |wj |)µj

for any d-tuple µ = (µj)dj=1 ∈ Z≥0. We now derive an integral presentation for this sum (74) in the style

of [6, Theorem 2, (115)], using a variation of Propositions 3.5 and 4.2 above with the projection operatorPn1 . Hence, we first revert to the adelic setup above, fixing a pure tensor ϕ = ⊗vϕv ∈ VΠ as in Proposition3.5. Recall that the projected form Pn1ϕ defines a cuspidal L2-automorphic form on the mirabolic subgroupP2(AF ) ⊂ GL2(AF ) having the Fourier-Whittaker described in Proposition 2.1 and Corollary 2.3, and alsothat we use the strong approximation and Iwasawa decomposition to lift to an L2-automorphic form in thespace L2(GL2(F )\GL2(AF ),1) as in Definition 2.6, Proposition 2.7, and Proposition 2.7. Given a nonzeroF -integer t, and using the same notation to denote a fixed finite idele representative t ∈ A×F,f , let us now

consider the operator R(t) defined on Pn1ϕ for a mirabolic matrix p ∈ P2(AF ) by

R(t)Pn1ϕ(p) = Pn1ϕ(p

(t−1

1

)).

Hence, for x ∈ AF an adele, and y = yfy∞ ∈ A×F an idele, we deduce from Proposition 2.1 or more simplyProposition 2.3 that we have the Fourier expansion

R(t)Pn1ϕ((

y x1

))=∣∣yt−1

∣∣−(n−22 ) ∑

γ∈F×

cΠ(γyf t

−1)

|γyf t−1|n−12

Wϕ (γy∞)ψ(γx).

This expansion allows us to derive the following key integral presentations.

Proposition 6.3. Let Πi for i = 1, 2 be cuspidal automorphic representations of GLn(AF ). Fix pure tensorsϕi ∈ VΠi whose nonarchimedean local components are essential Whittaker vectors. Fix a nonzero integralideal y ⊂ OF , and let us use the same notation to denote a fixed finite idele representative y ∈ A×F,f . Fixnonzero F -integers l1, l2, and α. Assume that li and α are coprime for i = 1, 2. Let y = yfy∞ be an idelehaving some specified archimedean component y∞ = Y −1

∞ , and nonarchimedean component yf = y−1. Then,the Fourier-Whittaker coefficient at α of the L2-automorphic form Φ = R(l1)Pn1ϕ1 · R(l2)Pn1ϕ2 on P2(AF )has the expansion

∫AF /F

R(l1)Pn1ϕ1R(l2)Pn1ϕ2

((y x

1

))ψ(−αx)dx

= |yY∞|n−2∑

γ1,γ2∈yl1γ1−l2γ2=α

cΠ1(γ1y

−1)cΠ2(γ2y

−1)

|γ1γ2y−2|n−12

Wϕ1

(γ1l1Y∞

)Wϕ2

(γ2l2Y∞

).

61

Proof. We open up Fourier-Whittaker expansions and evaluate via orthogonality of additive characters onAF /F as usual to find that∫

AF /F

R(l1)Pn1ϕ1 ·R(l2)Pn1ϕ2

(((yY∞)−1 x

1

))ψ(−αx)dx

=

∫AF /F

Pn1ϕ1

(((l1yY∞)−1 x

1

))Pn1ϕ2

(((l2yY∞)−1 x

1

))ψ(−αx)dx

= |l1yY∞|n−22

∑γ1∈F×

cΠ1(γ1l

−11 y−1)

|γ1l−11 y−1|n−1

2

Wϕ1

(γ1

Y∞

)|l2yY∞|

n−22

∑γ2∈F×

cΠ2(γ2l

−12 y−1)

|γ2l−12 y−1|n−1

2

Wϕ2

(γ2

Y∞

)×∫AF /F

ψ(γ1x− γ2x− αx)dx

= |yY∞|n−2∑

γ1,γ2∈F×γ1−γ2=α

cΠ1(γ1l−11 y−1)

|γ1l−11 y−1|n−1

2

cΠ2(γ2l−12 y−1)

|γ2l−12 y−1|n−1

2

Wϕ1

(γ1

Y∞

)Wϕ2

(γ2

Y∞

)

= |yY∞|n−2∑

γ1,γ2∈F×l1γ1−l2γ2=α

cΠ(γ1y−1)cΠ(γ2y

−1)

|γ1γ2y−2|n−12

Wϕ1

(γ1l1Y∞

)Wϕ2

(γ2l2Y∞

).

Since the latter sum is supported only on F -integers in y, we derive the identity. �

Corollary 6.4. Taking Y∞ ∈ F×∞ of idele norm |Y∞| = (Y L)1d with Π1 = Π and Π2 = Π, let us choose pure

tensors ϕi ∈ VΠi in such a way that the corresponding archimedean local Whittaker functions Wϕi : F×∞ → Cfor i = 1, 2 satisfy

Wϕi(y∞) =∣∣l−1i y∞

∣∣n−22 Ww,i (y∞) =

{|l−1i y∞|

n−22 Ww

(l−1i y∞L

1d

)if y∞ ∈ F×∞,+

0 otherwise.

Then, we obtain the integral presentation∫AF /F

R(l1)Pn1ϕ1 ·R(l2)Pn1ϕ2

(((y(Y∞L)

1d )−1 x

1

))ψ(−αx)dx

= |y|n−2∑

γ1,γ2∈yl1γ1−l2γ2=α

cΠ(γ1y−1)cΠ(γ2y

−1)

N(γ1γ2y−2)12

Ww,1

(l1γ1

(Y L)1d

)Ww,2

(l2γ2

(Y L)1d

).

Proof. The stated identity is easy to check after making a direct substitution. �

We also have the following less direct Corollary for the Fourier-Whittaker coefficient of the corresponding

extended form Φ following Proposition 2.9 and Proposition 4.2 (B) above. Note that we fix a smoothpartition of unity (16) and dyadic discussion as in the discussion above, and in particular reduce to workingwith compactly supported weight function as in Proposition 4.2 (B). Taking this standard reduction forgranted, let us drop the additional notations from the discussion that follows for simplicity.

Corollary 6.5. Let Y∞ = (Y∞,j)dj=1 ∈ F∞× be a totally positive idele with idele norm |Y∞| = (Y L)

1d

exceeding that of the chosen nonzero F -integer α ∈ OF . Let us again consider Π1 = Π and Π2 = Π, andchoose pure tensors ϕi ∈ VΠi in such a way that the corresponding archimedean local Whittaker functionsWϕi : F×∞ → C for i = 1, 2 satisfy

Wϕi(y∞) =∣∣l−1i y∞

∣∣n−22 Ww,i

(1

|y∞|

)=

|l−1i y∞|

n−22 Ww

(1

|l−1i y∞|L

1d

)if y∞ ∈ F×∞,+

0 otherwise.

Let us now write Φl1,l2 to denote the L2-automorphic form on the mirabolic subgroup P2(AF ) defined by

Φl1,l2 = R(l−11 )P

n1ϕ1R(l−1

2 )Pn1ϕ2. However, let us also write Φl1,l2 ∈ L2(GL2(F )\GL2(AF ),1)K to denote the

62

extension of Φl1,l2 to an L2-automorphic form on GL2(AF ) via Theorem 2.5, Definition 2.6, and Proposition2.7. Then, we obtain the integral presentation∫

I∼=[0,1]d⊂F∞Φl1,l2

((y(Y∞L)

1d x∞

1

))ψ(−αx∞)dx∞

=

∫I∼=[0,1]d⊂F∞

Φl1,l2

((y(Y∞L)

1d x∞

1

))ψ(−αx∞)dx∞

= |y|n−2∑

γ1,γ2∈yl1γ1−l2γ2=α

cΠ(γ1y−1)cΠ(γ2y

−1)

N(γ1γ2y−2)12

Ww,1

(l1γ1

(Y L)1d

)Ww,2

(l2γ2

(Y L)1d

).

In particular, we may decompose this L2-automorphic form Φl1,l2 on GL2(AF ), as its spectral coefficientsare bounded by Proposition 4.9 (or the subsequent discussion of convolution with smoothing kernals) above.

Proof. We deduce the result from Proposition 6.3 and Corollary 6.4 using Proposition 2.9, using a minorvariation of the argument given for Proposition 4.2 (B) above. �

Now, Corollaries 6.4 and 6.5 allow us to express (74) as a sum over Fourier-Whittaker coefficients of theautomorphic forms Φl1,l2 = R(l1)Pn1ϕ1 · R(l2)Pn1ϕ2 on P2(AF ), and more generally (for the generic case of

n ≥ 2) of the L2-automorphic form Φl1,l2 on GL2(AF ). In particular, for the generic case of n ≥ 2 (distinctfrom the standard case of n = 2 as treated by Blomer-Harcos [6]), we have that

|y|2−n∑

l1,l2vL

∑α∈qy∩K

∫I∼=[0,1]d⊂F∞

Φl1,l2

((y(Y L)

1d x∞

1

))ψ(−αx∞)dx∞.(75)

In particular, we may decompose this form spectrally via the argument of Proposition 4.9 above to derivebounds of the form described in Theorem 1.1 (B) to obtain a suitable bound this sum (75), i.e. so as toderive a suitable bound for the off-diagonal contribution in (72) above. In this way, we derive the followingkey estimate. Recall that we write 0 ≤ θ0 ≤ 1/2 to denote the best known approximation towards thegeneralized Ramanujan-Petersson conjecture for GL2(AF )-automorphic forms, with θ0 = 0 conjectured, andθ0 = 7/64 known thanks to the theorem of Blomer-Brumley [5].

Theorem 6.6. Assume that our initial cuspidal GLn-automorphic form Π = ⊗vΠv is irreducible. The sumLχf (w) defined in (71) is bounded above for any ε > 0 as

|Lχf (w)|2 �Π,ε Nq1+εL−1 + Nqn2 ( 1

2 +θ0)+εL52 +θ0+ε.

Proof. Taking for granted the reductions leading up to the diagonal bound (73) and the sum defined by thebox region K in (74), we reduce to bounding the sum∑

l1,l2vL

∑α∈qy∩K

∑γ1,γ2∈y

l1γ1−l2γ2=α

cΠ(γ1y−1)cΠ(γ1y

−1)

N(γ1γ2y−2)12

Ww,1

(l1γ1

(LY )1d

)Ww,2

(l2γ2

(LY )1d

),

which by (75) can be written equivalently as a sum of Fourier-Whittaker coefficients

|y|2−n∑

l1,l2vL

∑α∈qy∩K

∫I∼=[0,1]d⊂F∞

Φl1,l2

((y(Y L)

1d x∞

1

))ψ(−αx∞)dx∞.(76)

Decomposing each of the smooth and compactly supported L2-automorphic forms Φl1,l2 on GL2(AF ) spec-trally in the style of the proofs of the shifted convolution sums estimates of Theorem 1.1 above, we thusreduce to bounding the corresponding sums in the decomposition

|y|2−n∑

l1,l2vL

∑α∈y∩K

∑ϕτ∈B

〈Φl1,l2 , ϕτ 〉∫I∼=[0,1]d⊂F∞

ϕτ

((y(Y∞L)

1d x∞

1

))ψ(−αx∞)dx∞

= |y|2−n∑

l1,l2vL

∑α∈qy∩K

∑ϕτ∈B

〈Φl1,l2 , ϕτ 〉 ·cτ (αn−1)

N(αy−1)12

Wϕτ

(α(LY )

12

).

63

Again, the spectral coefficients 〈Φl1,l2 , ϕτ 〉 in this decomposition can be bounded via Proposition 4.9 above,and we use the shorthand notation introduced in the proof of Theorem 1.1 above for smooth basis elementsϕτ ∈ B, i.e. for simplicity of exposition we do not write out the contribution from the continuous spectrum

explicitly. Note again that we could also take the convolution of Φl1,l2 with a smoothing kernel, as indicatedin the remark after Proposition 4.9, to reduce to estimating the sum on right hand side of this expression.

Now, recall that we choose the narrow class representative y so that Ny �ε Nqε. Using Weyl’s law asdescribed above (cf. [6, §3.1]), we argue that the basic sum over ϕτ ∈ B in this latter expression can beviewed as having approximately L2 many terms. Putting these observations together, we then use the proofof Theorem 1.1 (B) to argue that the latter sum is bounded above for any choice of ε > 0 by the quantity

�Π,ε NqεL2 · L2∑

α∈qy∩KN(αy−1)θ0−

12

(Nα

Y L

) 12−

θ02 −ε

.

To estimate the α-sum in this latter expression, we observe (cf. [6, pp. 43-46]) that it can be expressed as a

sum over integral ideals m ⊂ OF of absolute norm bounded by Nm ≤ (LY )/N(qη) ≤ LNqn−22 . Hence, the

sum (76) is bounded above by

�Π,ε NqεL4 · (LY )θ02 −

12 +εNq

θ02 (LNq

n−22 )1+

θ02 = NqεL4L

12 +θ0+εNq

n2 ( 1

2 +θ0)−1.

It follows that the corresponding off-diagonal term in (72) is bounded above by

�Π,εNq1+ε

L2·NqεL4L

12 +θ0+εNq

n2 ( 1

2 +θ0)−1 = Nqn2 ( 1

2 +θ0)+εL52 +θ0+ε.

Putting this together with the diagonal bound (73) then implies the stated estimate. �

Using this, we derive the following estimate for the central value L(1/2,Π⊗ χ).

Corollary 6.7. Let Π = ⊗Πv be an irreducible cuspidal GLn(AF )-automorphic representation, and χ aHecke character of F of conductor q ⊂ OF . We have for any choice of parameters L = Nqu with 0 ≤ u ≤ 1and ε > 0 the estimate

|L(1/2,Π⊗ χ)|2 �Π,χ∞,ε Nq1+εL−1 + Nqn2 ( 1

2 +θ0)+εL52 +θ0+ε.

For instance, taking u = 1/4− θ0/2 gives the estimate

L(1/2,Π⊗ χ)�Π,χ∞,ε Nq38 +

θ04 + Nq

n4 ( 1

2 +θ0)+5−8θ0

16 +ε.

Taking u = (1− 6θ0)/(14− 4θ0) when n = 3 gives the estimate

L(1/2,Π⊗ χ)�Π,χ∞,ε Nq13+2θ0

2(14−4θ0)+ε

+ Nq34 ( 1

2 +θ0)+(5−28θ0−12θ20)

4(14−4θ0) ,

and taking u = 0 when n ≥ 4 gives the estimate

L(1/2,Π⊗ χ)�Π,χ∞,ε Nqn4 ( 1

2 +θ0)+ε.

Proof. Following the argument of [6, §3.3], we use the bound of Theorem 6.6 above in the integral (68),together with the estimates (69) and (70) for the vector-valued functions V(w) and Ww(y) respectively, wederive the corresponding bound∣∣∣∣∣∣∣∣

∑0�γ∈y

γ mod O×F,+

cΠ(γy−1)χ(γy−1)

N(γy−1)12

V ?(

Nγ

Y

)∣∣∣∣∣∣∣∣2

�ε,Π,χ∞

Nq1+ε

L+ Nq

n2 ( 1

2 +θ0)+εL52 +θ0+ε

for the modulus squared of the sum (67), which by (66) suffices to to deduce the corresponding bound forthe modulus squared of the central value L(1/2,Π⊗ χ). �

64

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