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Advances in Mathematics

Volume 426, 1 August 2023, 109099
Advances in Mathematics

Beyond the Weyl barrier for GL(2) exponential sums

https://doi.org/10.1016/j.aim.2023.109099Get rights and content

Abstract

In this paper, we use the Bessel δ-method, along with new variants of the van der Corput method in two dimensions, to prove non-trivial bounds for GL(2) exponential sums beyond the Weyl barrier. More explicitly, for sums of GL(2) Fourier coefficients twisted by e(f(n)), with length N and phase f(n)=Nβlogn/2π or anβ, non-trivial bounds are established for β<1.63651..., which is beyond the Weyl barrier at β=3/2.

Introduction

Let gSk(D,ξ) be a holomorphic cusp newform of level D, weight k, nebentypus character ξ, with the Fourier expansiong(z)=n=1λg(n)n(k1)/2e(nz),e(z)=e2πiz, for Imz>0.

In this paper, we consider the following smoothed exponential sumSf(N)=n=1λg(n)e(f(n))V(nN), where the weight function VCc(0,) and the phase function f is of the form:f(x)=Nβϕ(x/N), with β>1+ε for an arbitrarily small ε>0, andϕ(x)={logx2π,axβ, for a fixed real number a0. For the logarithm case, if one lets N=t1/β, then Sf(N)=NitSg(N,t) withSg(N,t)=n=1λg(n)nitV(nN). For the monomial case, β is also considered fixed, and Sa,β(N) is often used to denote the exponential sum:Sa,β(N)=n=1λg(n)e(anβ)V(nN).

Thanks to the Rankin–Selberg theory, we know that |λg(n)|'s obey the Ramanujan conjecture on average: An application of the Cauchy–Schwarz inequality followed by (1.6) yields the trivial bound

.

In [1], for the range 1ε<β<3/2ε, with the aid of a so-called Bessel δ-method, the following non-trivial ‘Weyl bound’ is proven: This extends a result of Jutila [12] for modular forms g of level D=1. The primary purpose of this paper is to break the upper ‘Weyl barrier’ at β=3/2. For this we have the following theorem.

Theorem 1.1

Let N>1. Let a0 be a fixed real number. Let V(x)Cc(0,) be a smooth function with support in [1,2] and derivatives

for every j=0,1,2,.... Let gSk(D,ξ) and λg(n) be its Fourier coefficients.

(1) We have if t139219+εNt79115, and if t608995+εNt27914311.

(2) We have if β[115/79,219/139){3/2}, and if β[4311/2791,995/608){8/5,37/23,66/41,29/18,50/31,21/13,34/21,13/8}.

Note that 115/79=1.45569..., 219/139=1.57554..., 4311/2791=1.54461..., and 995/608=1.63651.... Therefore the Weyl barrier at 3/2=1.5 is extended to 1.63651....

Our idea is to use the two-dimensional stationary phase method to transform the off-diagonal sum in the Bessel δ-method to certain double exponential sums, and then develop two new van der Corput methods of exponent pairs to treat this type of sums with ‘almost separable’ phase. More explicitly, if (κ,λ) is such an exponent pair, then we may prove

Definition 1.2

For an exponent pair (κ,λ) we define its β-barrier byβ(κ,λ)=4λ4κ14λ2.

The bound in (1.12) is better than the trivial bound N if and only if β does not exceed the barrier β(κ,λ), so we seek (κ,λ) with β-barrier as large as possible.

By using the one-dimensional van der Corput method in the trivial manner, one may already extend the Weyl barrier to a β-barrier at 59/38=1.55263.... Next, by our first van der Corput method, the exponent pair (7/188,327/376) yields the β-barrier at 219/139=1.57554... as above. Further, by our second van der Corput method, the exponent pair (359/3758,2791/3758) yields the β-barrier at 995/608=1.63651.... See Remark 6.10, §§6.3, 6.4, 6.6, and 6.7 for detailed discussions.

Our secondary object is to improve the ‘Weyl bound’ in (1.7) for 1<β<3/2. However, the quantity of improvement is not our main concern.

Theorem 1.3

Let notation be as above. Let q be a positive integer. Set Q=2q and defineβ1=219139,βq=1+9Q7+9qQ(q=2,3,...). We have for β[βq+1,βq), with β1+1/(q+1) in the monomial case.

The estimate in (1.14) is a consequence of our first van der Corput method, and may be considered as ‘sub-Weyl’ for 1<β<1.57554.... Note that when q=1, (1.14) amounts to (1.8) and (1.10) in Theorem 1.1. Our second van der Corput method, though stronger in principle, does not always work for β<1.54461.... See §6.7.

Theorem 1.3 may be further improved by the Vinogradov method if β is close to 1.

Theorem 1.4

There is an absolute constant c>0 such that for 1<β4/3, with β1+1/q (q=3,4,...) in the monomial case.

Finally, for the non-generic case when ϕ(x)=ax1+1/q, we can still attain a sub-Weyl bound by the Weyl method.

Theorem 1.5

Let notation be as above. For q=2,3,... set Q=2q. We have if q is odd, and if q is even.

Reduction of the sub-Weyl subconvex problem  Let L(s,g) be the L-function associated to the holomorphic newform g. The functional equation and the Phragmén–Lindelöf principle imply the t-aspect convex bound By the approximate functional equation, where N is dyadic. The Weyl bound in (1.7) reads

. By substituting this into (1.18) and choosing θ=2/3+ε, we obtain the Weyl subconvex bound: which was first proven by Good [6] in the full-level case D=1. Any bound of the type with ρ>0, is a sub-Weyl subconvex bound. Here we make a first step towards this sub-Weyl subconvexity problem.

Theorem 1.6

For any given δ>0, there exists ρ>0 such that with N dyadic.

Proof

Choose θ=608/995+ε. Then Theorem 1.1, Theorem 1.3, Theorem 1.4 ensure the existence of ρ>0 so that whenever tθ<Nt1δ. Thus (1.19) follows immediately on inserting (1.20) into (1.18). Q.E.D.

Theorem 1.6 manifests that to get a sub-Weyl subconvex bound for L(s,g) it suffices to prove sub-Weyl bounds for Sg(N,t) with N in the transition range t1δ<N<t1+ε.

Remarks

In the monomial case, by suitable adaptations, the forgoing results remain valid when a in f(x)=axβ is not necessarily fixed (see [1, (1.15)]), and more generally, when the function fF1γ(N,Nβ) as in Definition 6.1.

In view of the work [4], the form g can also be a Hecke–Maass newform.

Notation

By

or F=O(G) we mean that |F|cG for some constant c>0, and by FG we mean that
and
. We write
or F=Og,ϕ,(G) if the implied constant c depends on g, ϕ,. For notational simplicity, in the case ϕ(x)=axβ, we shall not put ϕ, a or β in the subscripts of
and O.

Let p always stand for prime. The notation nN or pP is used for integers or primes in the dyadic segment [N,2N] or [P,2P], respectively.

We adopt the usual ε-convention of analytic number theory; the value of ε may differ from one occurrence to another.

Acknowledgments

We thank Brian Conrey and the referee for their comments.

Section snippets

Setup

Throughout this paper, we assume 1+ε<β<5/3 and set T=Nβ, so thatN1+ε<T<N5/3.

We start with the following result from [1, §4], which is a consequence of applications of the Voronoï summation formula along with the Bessel δ-identity.

Proposition 2.1

Let U(x),V(x)Cc(0,) be supported in [1,2], with U(x)0 and

for every j=0,1,2.... Define CU=(1+i)/U˜(3/4), with U˜ the Mellin transform of U. For a fixed newform gSk(D,ξ), let λg(n) be its Fourier coefficients, and let ηg denote its Atkin–Lehner

Stationary phase lemmas

The following two lemmas respectively are consequences (or special cases) of Theorems 7.7.1 and 7.7.5 in two dimensions in Hörmander's book [7]. In the following, we use the standard abbreviations 1=/x1 and 2=/x2.

Lemma 3.1

Let KR2 be a compact set and X be an open neighborhood of K. Let k be a non-negative integer. If uCck(K), fCk+1(X), and f is real valued, then for λ>0 we have|Ku(x)e(λf(x))dx|Cλkj1+j2ksup|1j12j2u||f|2kj1j2, where C is bounded when f stays in a bounded set in Ck+1(X).

Lemma 3.2

Basic analytic lemmas

In this section, we prove some simple analytic lemmas which will be used for analyzing the stationary point in §5.1 and also the phase functions in the B-processes of the two van der Corput methods in §§6.2, 6.5. For simplicity, we shall not be concerned here with the domain of functions, as long as they are defined on compact subsets of R or R2.

We start with Faà di Bruno's formula (see [11]) and its two-dimensional generalization in a less precise form.

Lemma 4.1

For smooth functions f(x) and x(y) we have

Treating the sum S(v;n,p1,p2)

Since w and v will play a minor role in what follows, we shall write

, ϕ(x1,x2)=ϕ(x1,x2;w;v), and δ(x1,x2)=δ(x1,x2;w;v); see (2.15)–(2.18) for their definitions. We stress that all the implied constants in the sequel will be independent on the values of w and v.

Recall thatϕ(x1,x2)=ϕ(x1)y1x1ϕ(x2)+y2x2+δ(x1,x2). Firstly, we haveϕ(x1,x2)=(ϕ(x1)y1,ϕ(x2)+y2)+δ(x1,x2), andϕ(x1,x2)=(ϕ(x1)ϕ(x2))+δ(x1,x2). Subsequently, we shall denote δ=K2/T|w| and let δ be sufficiently small.

The van der Corput methods for almost separable double exponential sums

The exponential sum Sψ2(N,T) in Proposition 2.2 has phase function containing a separable main term Tψ(Ny2/T)Tψ(Ny1/T), with ψ(y)=logy/2π or byα, along with a ‘mixing’ error term Nω(Ny1/T,Ny2/T)—exponential sums of this type will be called almost separable. Note that T=Mα and N=Mα1 if we set M=T/N.

In this section, we shall develop two van der Corput methods for almost separable double exponential sums. They are very much like the method for one-dimensional exponential sums, and in the end we

Proof of Theorem 1.1 and 1.3

For either the logarithm case or the generic monomial case for β1+1/q (q=2,3,...), we have developed in §6 the van der Corput methods of exponent pairs for the type of double sums like Sψ2(N,T). More precisely, on applying Theorem 6.15, Theorem 6.25, we obtain non-trivial estimates of the form for certain exponent pairs (κ,λ)[0,1/2]×[1/2,1] depending on the value of α=logT/log(T/N). As (7.1) is non-trivial, it is necessary thatNκλ+1<T1λε. Substituting (7.1) into (2.26), we obtain where

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The third author is supported by the National Natural Science Foundation of China (Grant No. 12071420).

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