A Bessel delta-method and exponential sums for GL(2)

TL;DR

This paper introduces a Bessel delta-method for exponential sums in GL(2), generalizing previous results and providing a concise proof of the Weyl-type subconvex bound for related L-functions.

Contribution

It presents a new Bessel delta-method for GL(2) exponential sums and extends Jutila's results to more general holomorphic newforms, simplifying the proof of subconvex bounds.

Findings

Generalized exponential sum estimates for holomorphic newforms

Provided a short proof of the Weyl-type subconvex bound in the t-aspect

Introduced a novel Bessel delta-method for GL(2) analysis

Abstract

In this paper, we introduce a simple Bessel $\delta$-method to the theory of exponential sums for $\rm GL_2$. Some results of Jutila on exponential sums are generalized in a less technical manner to holomorphic newforms of arbitrary level and nebentypus. In particular, this gives a short proof for the Weyl-type subconvex bound in the $t$-aspect for the associated $L$-functions.

Equations

$$g(z)=\sum _{n=1}^{\infty }{\text{lambdaup}}_g(n)n^{(k-1)/2}e(nz),e(z)=e^{2\pi iz},$$

$$\sum _{n⩽N}{\text{lambdaup}}_g(n)e(\gamma n){}_g{N}^{1/2}\log (2N),$$

$$S^{♯}(N)=S_f^{♯}(N)=\sum _{N⩽n⩽2N}{\text{lambdaup}}_g(n)e(f(n)),$$

$$f(x)=T\varphi (x/N)+\gamma x,$$

$$S(N)=S_f(N)=\sum _{n=1}^{\infty }{\text{lambdaup}}_g(n)e(f(n))V\left(\frac{n}{N}\right),$$

$$\sum _{n⩽N}|{\text{lambdaup}}_g(n){|}^2{}_{g}N.$$

$$\displaystyle\lambdaup_{g}(n)\Lt n^{\scalebox{0.62}{$\vvepsilon$}}.$$

$$\begin{split}&\hskip 12.5pt\frac{1}{p}\sum_{a(\mathrm{mod}\,p)}e\left(\frac{a(n-r)}{p}\right)\cdot\int_{0}^{\infty}e\left(\frac{2\sqrt{rx}}{p}\right)J_{k-1}\left(\frac{4\pi\sqrt{nx}}{p}\right)U\left(\frac{x}{X}\right)\mathrm{d}x\\ &=\delta({r\equiv n(\mathrm{mod}\,p)})\cdot\delta\big{(}|r-n|<X^{\scalebox{0.62}{$\vvepsilon$}}p{\textstyle\sqrt{N/X}}\big{)}\cdot\text{``some\, factor''}+\text{``error"}\\ &=\delta({r=n})\cdot\text{``some\, factor''}+\text{``error"},\end{split}$$

$${\int }_0^{\infty }{J}_{k-1}(4\pi a\sqrt{x})J_{k-1}(4\pi b\sqrt{x})\mathrm{d}x=\frac{\delta (a-b)}{8{\pi }^{2}b},$$

$$\displaystyle N^{\scalebox{0.62}{$\vvepsilon$}}\varDelta\leqslant T.$$

$$\sum_{n=1}^{\infty}\lambdaup_{g}(n)e(f(n))V\left(\frac{n}{N}\right)\Lt T^{1/3}N^{1/2+\scalebox{0.62}{$\vvepsilon$}}+\frac{N^{1+\scalebox{0.62}{$\vvepsilon$}}}{T^{1/6}},$$

$$\displaystyle\sum_{N\leqslant n\hskip 1.0pt\leqslant N+H}\lambdaup_{g}(n)\,e(f(n)){}_{g,\hskip 1.0pt\phi,\hskip 1.0pt\scalebox{0.62}{$\vvepsilon$}}T^{1/3}N^{1/2+\scalebox{0.62}{$\vvepsilon$}}+\frac{N^{1+\scalebox{0.62}{$\vvepsilon$}}}{T^{1/6}}.$$

$$\displaystyle S_{f}^{\scriptscriptstyle\sharp}(N)=\sum_{N\leqslant n\hskip 1.0pt\leqslant 2N}\lambdaup_{g}(n)\,e(f(n)){}_{g,\hskip 1.0pt\phi,\hskip 1.0pt\scalebox{0.62}{$\vvepsilon$}}T^{1/3}N^{1/2+\scalebox{0.62}{$\vvepsilon$}}+\frac{N^{1+\scalebox{0.62}{$\vvepsilon$}}}{T^{1/6}}.$$

$$\displaystyle S_{f}^{\scriptscriptstyle\sharp}(N){}_{g,\hskip 1.0pt\phi,\hskip 1.0pt\scalebox{0.62}{$\vvepsilon$}}T^{1/3}N^{1/2+\scalebox{0.62}{$\vvepsilon$}},$$

$$\displaystyle S_{\scalebox{0.62}{$\varalpha$},\hskip 1.0pt\beta,\hskip 1.0pt\gamma}^{\scriptscriptstyle\sharp}(N)=\sum_{n\hskip 1.0pt\leqslant N}\lambdaup_{g}(n)\,e(\scalebox{0.87}{$\varalpha$}n^{\hskip 1.0pt\beta}+\gamma n).$$

$$\displaystyle S_{\scalebox{0.62}{$\varalpha$},\hskip 1.0pt\beta,\hskip 1.0pt0}^{\scriptscriptstyle\sharp}(N){}_{g,\hskip 1.0pt\scalebox{0.62}{$\varalpha$},\hskip 1.0pt\beta,\hskip 1.0pt\scalebox{0.62}{$\vvepsilon$}}N^{\frac{1}{2}+\frac{\beta}{3}+\scalebox{0.62}{$\vvepsilon$}}.$$

$$\sum _{n=1}^{\infty }{\text{lambdaup}}_g(n)e(-2\sqrt{qn})V\left(\frac{n}{N}\right)$$

$$\displaystyle S_{\scalebox{0.62}{$\varalpha$},\hskip 1.0pt\beta,\hskip 1.0pt0}^{\scriptscriptstyle\sharp}(N){}_{g,\hskip 1.0pt\scalebox{0.62}{$\varalpha$},\hskip 1.0pt\beta,\hskip 1.0pt\scalebox{0.62}{$\vvepsilon$}}N^{\beta+\scalebox{0.62}{$\vvepsilon$}}+N^{\frac{1}{2}-\frac{\beta}{4}+\scalebox{0.62}{$\vvepsilon$}}.$$

$$\displaystyle S_{\scalebox{0.62}{$\varalpha$},\hskip 1.0pt2,\hskip 1.0pt\gamma}^{\scriptscriptstyle\sharp}(N){}_{g,\hskip 1.0pt\scalebox{0.62}{$\vvepsilon$}}N^{\frac{15}{16}+\scalebox{0.62}{$\vvepsilon$}},$$

$$\displaystyle\sum_{n\hskip 1.0pt\leqslant N}\lambdaup_{g}(n)\,e(\scalebox{0.87}{$\varalpha$}n^{\hskip 1.0pt\beta}+\gamma n){}_{g,\hskip 1.0pt\beta,\hskip 1.0pt\scalebox{0.62}{$\vvepsilon$}}|\scalebox{0.87}{$\varalpha$}|^{\frac{1}{3}}N^{\frac{1}{2}+\frac{\beta}{3}+\scalebox{0.62}{$\vvepsilon$}}+{|\scalebox{0.87}{$\varalpha$}|^{-\frac{1}{6}}}{N^{1-\frac{\beta}{6}+\scalebox{0.62}{$\vvepsilon$}}}.$$

$$\displaystyle\sum_{n\hskip 1.0pt\leqslant N}\lambdaup_{g}(n)\,e(\scalebox{0.87}{$\varalpha$}n^{\hskip 1.0pt\beta}+\gamma n){}_{g,\hskip 1.0pt\scalebox{0.62}{$\varalpha$},\hskip 1.0pt\beta,\hskip 1.0pt\scalebox{0.62}{$\vvepsilon$}}N^{\frac{1}{2}+\frac{\beta}{3}+\scalebox{0.62}{$\vvepsilon$}}+{N^{1-\frac{\beta}{6}+\scalebox{0.62}{$\vvepsilon$}}}.$$

$$\begin{array}{r}L(s,g)=\sum _{n=1}^{\infty }\frac{{\text{lambdaup}}_g(n)}{n^s},\mathrm{Re}s>1.\end{array}$$

$$L\left(1/2+it,g\right){}_{g,\hskip 1.0pt\scalebox{0.62}{$\vvepsilon$}}(1+|t|)^{1/2+\scalebox{0.62}{$\vvepsilon$}}$$

$$L(1/2+it,g){}_{g,\hskip 1.0pt\scalebox{0.62}{$\vvepsilon$}}(1+|t|)^{1/3+\scalebox{0.62}{$\vvepsilon$}},$$

$$L\left(1/2+it,g\right)\Lt(1+|t|)^{1/3+\scalebox{0.62}{$\vvepsilon$}}.$$

$$L(1/2,g\otimes\scalebox{0.9}{$\varchi$}){}_{g,\hskip 1.0pt\scalebox{0.62}{$\vvepsilon$}}q^{3/8+\scalebox{0.62}{$\vvepsilon$}}.$$

$$\begin{array}{r}\sum _{n=1}^{\infty }{\text{lambdaup}}_g(n)e\left(\frac{an}{c}\right)F\left(n\right)=\frac{{\eta }_g\xi (-c)}{c\sqrt{M}}\sum _{n=1}^{\infty }\overline{{\text{lambdaup}}_g(n)}e\left(-\frac{\overline{a}n}{c}\right)\check{F}\left(\frac{n}{c^{2}M}\right).\end{array}$$

$$\check{F}(y)=2\pi i^k{\int }_0^{\infty }F(x)J_{k-1}\left(4\pi \sqrt{xy}\right)\mathrm{d}x$$

$$J_{\text{varnu}}(z)=\sum _{n=0}^{\infty }\frac{(-{)}^n{\left(z/2\right)}^{\text{varnu}+2n}}{n!\Gamma (\text{varnu}+n+1)}.$$

$$J_{\text{varnu}}(x)=\frac{1}{\sqrt{2\pi x}}\left(e^{ix}{W}_{\text{varnu},+}(x)+e^{-ix}{W}_{\text{varnu},-}(x)\right),$$

Full text

1. Introduction

It is a classical problem to estimate exponential sums involving the Fourier coefficients of a modular form. Let $g\in S^{\star}_{k}(M,\xi)$ be a holomorphic cusp newform of level $M$, weight $k$, nebentypus character $\xi$, with the Fourier expansion

[TABLE]

for ${\mathrm{Im}}\,z>0$. For example, it is well-known that for any real $\gamma$ and $N\geqslant 1$,

[TABLE]

with the implied constant depending only on $g$ (see [Iwa, Theorem 5.3]). This is a classical estimate due to Wilton. This type of estimates with uniformity in $\gamma$ was generalized by Stephen D. Miller to cusp forms for ${\mathrm{GL}}_{3}({\mathbb{Z}})$ in [Mil].

In this paper, we consider the following exponential sum (and its variants),

[TABLE]

for a phase function $f$ of the form

[TABLE]

where $\phi$ is real-valued and smooth (see Theorem 1.1), $\gamma$ is real, and $N,T\geqslant 1$ are large parameters. We assume here that $\phi$ is not a linear function, as otherwise the sum is already estimated in (1.1). As usual, we shall be mainly investigating the smoothed exponential sum

[TABLE]

for a certain smooth weight function $V\in C_{c}^{\infty}(0,\infty)$ supported in $[1,2]$ as described in Theorem 1.1.

This type of exponential sums (with $\gamma=0$) for modular forms $g$ of level $M=1$ were first studied by Jutila [Jut1], using Farey fractions, the Voronoï summation formula, and stationary phase analysis. See also [Hux, §10] for an account of Jutila’s method.

Thanks to the Rankin–Selberg theory, we know that $|\lambdaup_{g}(n)|$’s obey the Ramanujan conjecture on average:

[TABLE]

Moreover, by the work of Deligne [Del] and Deligne–Serre [DS] (the latter is for $k=1$), the Ramanujan conjecture for holomorphic cusp forms is now well-known:

[TABLE]

An application of the Cauchy–Schwarz inequality followed by (1.5) yields the trivial bounds $S(N)$, $S^{\scriptscriptstyle\sharp}(N){}_{g}N$. Thus one aims to improve over these trivial bounds or, in other words, to show that there is no correlation between $\lambdaup_{g}(n)$ and $e(f(n))$.

The primary purpose of this paper is to find a $\delta$-method which is analytically richer so that the stationary phase analysis at later stages becomes cleaner. It turns out that an added benefit of our pursuit is a generalization of some results in Jutila’s treatise [Jut1] to modular forms of arbitrary level and nebentypus. An application amongst others is the Weyl-type subconvex bound for the associated $L$-functions in the $t$-aspect.

The main novelty of our work is a simple Bessel $\delta$-method to be described as follows.

A simple Bessel $\delta$-method

As usual, let $e(x)=e^{2\pi ix}$ and let $J_{\varnu}(x)$ be the $J$-Bessel function of order $\varnu$. For a condition $\mathrm{C}$, let $\delta(\mathrm{C})$ denote the Kronecker $\delta$ that detects $\mathrm{C}$.

We fix a smooth bump function $U$ in $C_{c}^{\infty}(0,\infty)$. Our Bessel $\delta$-method is based on the observation that for a prime $p$, some large parameters $N,X$, and integers $r,n\in[N,2N]$, one has

[TABLE]

provided that N<X^{1-\scalebox{0.62}{\vvepsilon}} and $p^{2}<NX$. This is made explicit in Lemma 3.3. The merit of this Bessel $\delta$-identity is that it arises naturally from the Voronoï summation formula, for the Bessel integral may be interpreted as the Hankel transform of $e\left({2\sqrt{rx}}/{p}\right)U(x/X)$.

As explained in §3.3, there is a vague but interesting connection between the Bessel integral above and the formula

[TABLE]

where $\delta(a-b)$ is now the Dirac $\delta$-distribution. Thus the use of $\delta$ is justified from a different perspective.

Main results

Theorem 1.1**.**

Let \scalebox{0.87}{\vvepsilon}>0 be an arbitrarily small constant. Let $N,T,\varDelta>1$ be parameters such that

[TABLE]

Let $V(x)\in C_{c}^{\infty}(0,\infty)$ be a smooth function with support in $[1,2]$. Assume that its total variation $\mathrm{Var}(V)\Lt 1$ and that $V^{(j)}(x){}_{j}\varDelta^{j}$ for $j\geqslant 0$. For $\gamma$ real, and $\phi(x)\in C^{\infty}(1/2,5/2)$ satisfying $|\phi^{\prime\prime}(x)|\Gt 1$ and $\phi^{(j)}(x){}_{j}1$ for $j\geqslant 1$, define $f(x)=T\phi(x/N)+\gamma x$. Let $g\in S^{\star}_{k}(M,\xi)$ and $\lambdaup_{g}(n)$ be its Fourier coefficients. Then

[TABLE]

*with the implied constant depending only on $g$, $\phi$ and

$\vvepsilon$

.*

Corollary 1.2**.**

Let $\phi$, $f$ and $g$ be as above. Let N^{1+\scalebox{0.62}{\vvepsilon}}/T\leqslant H\leqslant N. We have

[TABLE]

As a consequence,

[TABLE]

Jutila’s estimate for $S_{f}^{\scriptscriptstyle\sharp}(N)$, say, for modular forms $g$ of level $M=1$ and for phase functions $f(x)=T\phi(x/N)$ (see [Hux, §10]) is as follows,

[TABLE]

provided that $N^{3/4}<T<N^{3/2}$.

Corollary 1.2 may be regarded as a generalization of Theorem 4.6 of Jutila [Jut1] in several aspects. First of all, the modular form $g$ here is of arbitrary level and nebentypus. Secondly, the estimate in (1.10) is non-trivial as long as N^{\scalebox{0.62}{\vvepsilon}}<T<N^{3/2-\scalebox{0.62}{\vvepsilon}}, while it is assumed in [Jut1] that $N^{3/4}<T<N^{3/2}$. Note that our estimate is weaker than Jutila’s when $N^{3/4}<T<N$. Nevertheless, we are usually more concerned with the case when N^{1-\scalebox{0.62}{\vvepsilon}}<T<N^{3/2-\scalebox{0.62}{\vvepsilon}}, for example, in the subconvexity problem; our estimate is the same as Jutila’s in this case. Thirdly, our phase function $f(x)$ contains an additional linear term $\gamma x$.

Note that we shall be content with the averaged Ramanujan conjecture (1.5) in the proof of Theorem 1.1, while we shall need the Ramanujan conjecture (Deligne’s bound) (1.6) only for the deduction of Corollary 1.2.

For ease of exposition, only holomorphic modular forms are considered here, but our approach also works for Maass forms with some efforts.

Examples

A typical and simple choice of $\phi(x)$ is the power function $\pm x^{\hskip 1.0pt\beta}$ so that f(x)=\scalebox{0.87}{\varalpha}x^{\hskip 1.0pt\beta}+\gamma x (T=|\scalebox{0.87}{\varalpha}|N^{\beta}). Let

[TABLE]

For modular forms $g$ of level $M=1$, there are abundant works on this type of exponential sums in the literature (usually, with $\gamma=0$).

As alluded to above, the first non-trivial bound for S_{\scalebox{0.62}{\varalpha},\hskip 1.0pt\beta,\hskip 1.0pt0}^{\scriptscriptstyle\sharp}(N) was obtained by Jutila (see [Jut1, Theorem 4.6]) for the range $3/4<\beta<3/2$, $\beta\neq 1$, as follows,

[TABLE]

When $\beta=1/2$, \scalebox{0.87}{\varalpha}=-2\sqrt{q} for integer $q>0$, and $\gamma=0$, it was first shown by Iwaniec, Luo and Sarnak [ILS, (C.17)] that the smoothed sum

[TABLE]

has a main term of size $N^{3/4}$.

The first non-trivial bound towards S_{\scalebox{0.62}{\varalpha},\hskip 1.0pt\beta,\hskip 1.0pt\gamma}^{\scriptscriptstyle\sharp}(N) for all $0<\beta<1$ is due to X. Ren and Y. Ye [RY1], who refined the aforementioned result of Iwaniec, Luo and Sarnak for $\beta=1/2$, and proved for $\beta\neq 1/2$ that

[TABLE]

This was improved into N^{1/3+\scalebox{0.62}{\vvepsilon}} in [SW] for $0<\beta<1/2$ (the Maass form case is also considered there). Note that Jutila’s estimate (1.12) is stronger than (1.13) for $3/4<\beta<1$.

It should be mentioned that Q. Sun [Sun] obtained the bound N^{1-\frac{\beta}{2}+\scalebox{0.62}{\vvepsilon}} for S_{\scalebox{0.62}{\varalpha},\hskip 1.0pt\beta,\hskip 1.0pt\gamma}^{\scriptscriptstyle\sharp}(N) in the range $0<\beta\leqslant 1/2$. Her bound was improved into N^{\frac{1}{2}+\frac{\beta}{2}+\scalebox{0.62}{\vvepsilon}} by Godber [God] (for $0<\beta<1$). For $\gamma=0$, these bounds are both weaker than (1.13).

There is also a very distinguishable result—Pitt’s uniform estimate for S_{\scalebox{0.62}{\varalpha},\hskip 1.0pt2,\hskip 1.0pt\gamma}^{\scriptscriptstyle\sharp}(N) with quadratic phase in [Pit],

[TABLE]

where the implied constant depends only on $g$ and

$\vvepsilon$

. The exponent $15/16$ was later improved into $7/8$ by K. Liu and X. Ren [LR].

More generally, one can also consider analogous exponential sums of Fourier coefficients of Maass cusp forms for ${\mathrm{GL}}_{m}$, $m\geqslant 3$. Some similar results for ${\mathrm{GL}}_{3}$ and ${\mathrm{GL}}_{m}$ were obtained later by X. Ren and Y. Ye in [RY2, RY3]. Recently, Kumar et al. [KMS] had some improvement over the results in [RY2], by using the $\delta$-symbol method of Duke–Friedlander–Iwaniec [DFI] together with a conductor-lowering trick which was first introduced by Munshi [Mun1].

A direct consequence of Corollary 1.2 is the following estimates for S_{\scalebox{0.62}{\varalpha},\hskip 1.0pt\beta,\hskip 1.0pt\gamma}^{\scriptscriptstyle\sharp}(N) for modular forms $g\in S^{\star}_{k}(M,\xi)$.

Corollary 1.3**.**

Let $g\in S^{\star}_{k}(M,\xi)$ and $\lambdaup_{g}(n)$ be its Fourier coefficients. For real \scalebox{0.87}{\varalpha},\gamma and $\beta$ with \scalebox{0.87}{\varalpha}\neq 0, $\beta\neq 1$, we have

[TABLE]

In particular,

[TABLE]

Note that the estimate (1.16) is non-trivial for $0<\beta<3/2$. Though weaker for $3/4<\beta<1$, it is the same as Jutila’s estimate (1.12) for $1<\beta<3/2$. At any rate, our estimate is an extension of Jutila’s result (for $1<\beta<3/2$, literally) to modular forms of general level.

Also note that (1.16) is better than Ren and Ye’s estimate (1.13) as long as $\beta>6/7$. However, our bound is worse than theirs for $\beta<6/7$. This is due to the nature and the limitation of our Bessel $\delta$-method or any $\delta$- or circle method. For if $\beta$ is relatively small then e(\scalebox{0.87}{\varalpha}n^{\hskip 1.0pt\beta}) is not quite oscillatory, and it would not benefit much to separate the oscillations of e(\scalebox{0.87}{\varalpha}n^{\hskip 1.0pt\beta}) and $\lambdaup_{g}(n)$ by the $\delta$-method. The approach in [RY1] works far better in this situation, where the Voronoï summation (with modulus $1$) is applied directly, followed by stationary phase arguments.

In [Pit], the $\delta$-method of Duke–Friedlander–Iwaniec, along with Diophantine approximation, is used to prove the estimate in (1.14) in the quadratic case $\beta=2$. However, this approach does not work with fractional $\beta$.

Application: Weyl-type subconvex bound in the $t$-aspect

For $g\in S^{\star}_{k}(M,\xi)$ with Fourier coefficients $\lambdaup_{g}(n)$, the associated $L$-function is given by

[TABLE]

This $L$-series has an analytic continuation to the whole complex plane. The Phragmén–Lindelöf principle implies the $t$-aspect convex bound

[TABLE]

for any \scalebox{0.87}{\vvepsilon}>0. Any improvement on the exponent on the right-hand side of the inequality is referred to as a subconvex bound, and in general it requires significant amount of work to achieve it.

When $M=1$, the following Weyl-type subconvex bound was first proven by Good [Goo],

[TABLE]

by appealing to the spectral theory of automorphic functions. Later, the same bound was obtained by Jutila using his method developped in [Jut1]. See [Meu, Jut2] for the extension of these methods to the Maass-form case.

There has been much progress lately, due to new methods, especially variants of the $\delta$-symbol or circle method become available. For example, Munshi [Mun1] solved the $t$-aspect subconvexity problem for $L$-functions on $\rm GL_{3}$ by adopting Kloosterman’s version of the circle method. He also invented a $\rm GL_{2}$ $\delta$-method and used it in a series of papers [Mun2]–[Mun4] for various subconvexity problems. These methods were applied in [AS, AKMS] to obtain the Weyl bound in the ${\mathrm{GL}}_{2}$ setting. In a recent preprint [Mun5], Munshi was even able to break the long standing Weyl-bound barrier by introducing extra variants into the $\rm GL_{2}$ $\delta$-method approach.

Recently, there are Weyl-type subconvexity results for cusp forms of general level by Booker et al. [BMN] and the first-named author [Agg]. Booker et al. [BMN] generalized Huxley’s treatment of Jutila’s method by using a Voronoï formula with arbitrary additive twists to obtain their result. On the other hand, Aggarwal [Agg] used a simple $\delta$-symbol method and followed Munshi’s approach [Mun1]. This treatment allowed him to use the Voronoï formula of Kowalski–Michel–VanderKam to get the Weyl-type bound, along with an explicit dependence on the level of the cusp form.

By applying Theorem 1.1, with $\phi(x)=-\log x$, we shall derive in §6 the Weyl subconvex bound for $g\in S^{\star}_{k}(M,\xi)$.

Theorem 1.4**.**

Let $g\in S^{\star}_{k}(M,\xi)$. Then

[TABLE]

*with the implied constant depending only on $g$ and

$\vvepsilon$

.*

This work is of the same theme as [Agg], but it is technically simpler here, for our Bessel $\delta$-method is more intimate to the Voronoï summation formula than his trivial $\delta$-method. Moreover, our argument by the Bessel $\delta$-method is very short compared to that by the Jutila method generalized in [BMN].

A motivation of our work is from [AHLS], in which, together with Q. Sun, the first three named authors investigated subconvex bounds for L(1/2,g\otimes\scalebox{0.9}{\varchi}), where

$\varchi$

is a primitive Dirichlet character of prime conductor $q$. They were able to use a ‘trivial’ delta method to give a simpler proof for the Burgess bound in the $q$-aspect,

[TABLE]

The Bessel $\delta$-method is an outcome of our search for a similar simple approach to strong subconvex bounds in the $t$-aspect. It seems natural that the argument of this paper can be combined with the approach in [AHLS] to obtain a uniform subconvexity bound for L(1/2+it,g\otimes\scalebox{0.9}{\varchi}) in both the $q$ and $t$ aspects.

Notation

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

2. The Voronoï summation

Let $S^{\star}_{k}(M,\xi)$ denote the set of primitive newforms of level $M$, weight $k$ and nebentypus $\xi$. We have necessarily $\xi(-1)=(-1)^{k}$. The term “primitive” means that the form is Hecke-normalized so that its Fourier coefficients and Hecke eigenvalues coincide.

The following Voronoï summation formula is a special case of [KMV, Theorem A.4]. Note that $g_{M}=\widebar{g}\in S^{\star}_{k}(M,\widebar{\xi})$ in their notation (see [KMV, Proposition A.1]).

Lemma 2.1** (The Voronoï Summation Formula).**

Let $g$ be a primitive holomorphic newform in $S^{\star}_{k}(M,\xi)$. Let $a,\overline{a},c$ be integers such that $c\geqslant 1$, $(a,c)=1$, $a\overline{a}\equiv 1(\mathrm{mod}\,c)$ and $(c,M)=1$. Let $F(x)\in C_{c}^{\infty}(0,\infty)$. Then there exists a complex number $\eta_{g}$ of modulus $1$ (the Atkin–Lehner pseudo-eigenvalue of $g$) such that

[TABLE]

where $\check{F}(y)$ is the Hankel transform of $F(x)$ defined by

[TABLE]

The Voronoï summation formula in [KMV, Theorem A.4] is more general, where it is only required that $((c,M),M/(c,M))=1$. However, in our setting $c=p$ will be a large prime while $M$ is fixed, so our condition $(c,M)=1$ in Lemma 2.1 is justified. For comparison, we remark that, Jutila’s method requires the $a/c$ to be every fraction, so this Voronoï works only if $M$ is square-free; thus in [BMN], they need a more general Voronoï even without the restriction $((c,M),M/(c,M))=1$.

3. A Bessel $\delta$-method

3.1. Basics of Bessel functions

For complex $\varnu$, let $J_{\varnu}(z)$ be the Bessel function of the first kind ([Wat]), defined by the series

[TABLE]

Moreover, we may write (see [WW, §16.12, 16.3, 17.5] or [Wat, §7.2])

[TABLE]

with

[TABLE]

3.2. Asymptotic of a Bessel integral

For a fixed (non-negative valued) bump function $U\in C_{c}^{\infty}(0,\infty)$, say with support in $[1,2]$, $a,b>0$ and $X>1$, consider the Bessel integral

[TABLE]

By [GR, 6.699 1, 2], we have

[TABLE]

for $b>a>0$ and $-{\mathrm{Re}}\,\varnu<{\mathrm{Re}}\,\mu<3/2$. By appealing to the Gaussian formula (see [MOS, §2.1])

[TABLE]

and the duplication formula for the gamma function, we obtain

[TABLE]

after letting $b\rightarrow a$. Note that the limit $b\rightarrow a$ is legitimate because both the integral on the left and the hypergeometric series on the right are absolutely and uniformly convergent for $-{\mathrm{Re}}\,\varnu<{\mathrm{Re}}\,\mu<1/2$ (see [MOS, §2.1]).

We first consider $I_{k}(a,a;X)$ as defined in (3.4). By Mellin inversion

[TABLE]

where $\widetilde{U}(s)$ denotes the Mellin transform of the function $U$, and $(\sigma)$ stands for the contour ${\mathrm{Re}}\,s=\sigma$ as usual. Applying (3.5) to evaluate the inner integral, we infer that

[TABLE]

for ${3}/{4}<\sigma<(k+1)/2$. Assume that $a^{2}X>1$. By shifting the contour of integration to ${\mathrm{Re}}\,s=0$, say, and collecting the residues at $s=3/4$ and $1/4$, we obtain the following asymptotic for $I_{k}(a,a;X)$.

Lemma 3.1**.**

We have

[TABLE]

with the implied constant depending only on $k$ and $U$.

We now consider $I_{k}(a,b;X)$ as in (3.4) for $a\neq b$. For this, we assume that $b^{2}X>1$ so that $J_{k-1}({4\pi b\sqrt{x}})$ is oscillatory. In view of (3.2) and (3.3), the lemma below is a direct consequence of Lemma A.1.

Lemma 3.2**.**

Suppose that $b^{2}X>1$. Then $I_{k}(a,b;X)=O(X^{-A})$ for any $A\geqslant 0$ if |a-b|\sqrt{X}>X^{\scalebox{0.62}{\vvepsilon}}.

3.3. Remarks on the Bessel integral

After suitable changes, Weber’s second exponential integral formula in [Wat, 13.31 (1)] may be written as

[TABLE]

for $a,b,X>0$. Since $J_{k-1}(4\pi a\hskip-1.0pt\sqrt{x})$ and $e(2a\sqrt{x})$ have the same type of oscillation (see (3.2) or [Wat, 7.21 (1)]), the Weber integral in (3.7) may be viewed as a variant of the Bessel integral in (3.4). However, the exponential function $\exp(-2\pi x/X)$ is not as nice as the compactly supported function $U(x/X)$ from the perspective of Fourier analysis—the Fourier transform of $\exp(-2\pi x/X)$ ($x\in(0,\infty)$) decays at $\infty$ only to the first order.

The connection between the Weber integral and the Dirac $\delta$-distribution might be of its own interest. This justifies the use of $\delta$ in another way.

According to [Wat, 7.23 (2)], we have the asymptotic $I_{k-1}(x)\sim\exp(x)/\sqrt{2\pi x}$ as $x\rightarrow\infty$, so if one let $X\rightarrow\infty$ then the right-hand side of (3.7) is asymptotic to

[TABLE]

where $N(a-b,1/\hskip-1.0pt\sqrt{4\pi X})$ is the Gaussian distribution of variance $1/\hskip-1.0pt\sqrt{4\pi X}$ and $\delta(a-b)$ is the Dirac $\delta$-distribution. Thus the limiting form of (3.7) is

[TABLE]

or

[TABLE]

while this is equivalent to the Hankel inversion formula (see [Wat, 14.3 (3), 14.4 (1)])

[TABLE]

for $F(a)\in C^{\infty}(0,\infty)$ subject to the condition

[TABLE]

3.4. A Bessel $\delta$-method

By Lemma 3.1 and 3.2, we have the following asymptotic $\delta$-identity.

Lemma 3.3**.**

Let $p$ be prime and $N,X>1$ be such that $X>p^{2}/N$ and X^{1-\scalebox{0.62}{\vvepsilon}}>N. Let $r,n$ be integers in the dyadic interval $[N,2N]$. For any $A\geqslant 0$, we have

[TABLE]

where $C_{U}=(1+i)/{\widetilde{U}(3/4)}$, the $\delta({r=n})$ is the Kronecker $\delta$ that detects $r=n$, and the implied constants depend only on $k$, $U$ and $A$.

Proof.

Lemma 3.1 yields the $\delta$-term, while Lemma 3.2 implies that $I_{k}(\hskip-1.5pt{\sqrt{r}}/p,\hskip-1.5pt{\sqrt{n}}/p;X)$ is negligibly small unless |r-n|\leqslant X^{\scalebox{0.62}{\vvepsilon}}p\sqrt{N/X}. On the other hand, the exponential sum in (3.12) gives us $r\equiv n(\mathrm{mod}\,p)$. Consequently, (3.12) follows immediately for X^{\scalebox{0.62}{\vvepsilon}}p\sqrt{N/X}<p as assumed. ∎

Remark 3.4**.**

We should point out that the identity

[TABLE]

plays a key role in the work [AHLS]. In fact, the approach therein is based on the observation:

[TABLE]

where the modulus $c$ is chosen to be c=pq\Gt N^{1+\scalebox{0.62}{\vvepsilon}} and $X={p^{2}q^{2}}/{N}$;*

$\varchi$

is a primitive Dirichlet character modulo $q$.*

Here the Bessel-exponential integral $I_{k}(\hskip-1.5pt{\sqrt{r}}/p,\hskip-1.5pt{\sqrt{n}}/p;X)$ serves the role of “lowering” the conductor of the underlying problem.

4. Application of the Bessel $\delta$-method and the Voronoï summation

We start with separating oscillations by writing

[TABLE]

Applying the $\delta$-method identity (3.12) in Lemma 3.3 and dividing the $a$-sum according as $(a,p)=1$ or not, we have

[TABLE]

with

[TABLE]

[TABLE]

where $V_{\scriptscriptstyle\natural}(x)=C_{U}\eta_{g}\xi(-1)M^{-1/2}\cdot x^{1/4}V(x)$ (recall that $\xi(-1)=(-1)^{k}$) and $\displaystyle\sideset{}{{}^{\star}}{\sum}$ means that the $a$-sum is subject to $(a,p)=1$, and

[TABLE]

Assuming $p>M$, we now apply the Voronoï summation in Lemma 2.1 to the $n$-variable. Recall from (3.4) that

[TABLE]

and the integral may be regarded as a Hankel transform as in (2.2). By applying the (complex conjugation of) Voronoï summation in (2.1) with $c=p$ in the reversed direction, we infer that

[TABLE]

where, as usual, $S(n,r;p)$ is the Kloosterman sum

[TABLE]

Similarly,

[TABLE]

after the Voronoï with modulus $c=1$. Estimating trivially, we find that

[TABLE]

Finally, we introduce an average over primes $p$ in $[P,2P]$ for a large parameter $P$; there are $\asymp P/\log P$ many such $p$’s. The results that we have established are summarized as follows.

Proposition 4.1**.**

Let $V(x)\in C_{c}^{\infty}(0,\infty)$ be supported in $[1,2]$, with $\mathrm{Var}(V)\Lt 1$ and $V^{(j)}(x){}_{j}\varDelta^{j}$ for $j\geqslant 0$. Let parameters N,X,P>N^{\hskip 1.0pt\scalebox{0.62}{\vvepsilon}} be such that

[TABLE]

Let $P^{\star}$ be the number of primes in $[P,2P]$. We have

[TABLE]

with

[TABLE]

where $V_{\scriptscriptstyle\natural}(x)=C_{U}\eta_{g}\xi(-1)M^{-1/2}\cdot x^{1/4}V(x)$ is again supported in $[1,2]$, satisfying $\mathrm{Var}({V_{\scriptscriptstyle\natural}})\Lt 1$ and $V_{\scriptscriptstyle\natural}^{(j)}(x){}_{j}\varDelta^{j}$.

5. Application of the Poisson summation and the Cauchy inequality

In view of Proposition 4.1, to study $S(N)$ it suffices to consider the sum ${S}(N,X,P)$ defined in (4.9). For convenience of our analysis, we let

[TABLE]

with the parameter $K$ to be optimized later. Then the first assumption in (4.7) is justified, while the second assumption N<X^{1-\scalebox{0.62}{\vvepsilon}} amounts to

[TABLE]

5.1. First application of the Poisson summation

Recall that $f(r)=T\phi(r/N)+\gamma r$ (as in (1.3)). By applying the Poisson summation to the $r$-sum in (4.9), we have

[TABLE]

where

[TABLE]

Recall that $y$ ($=n$) $\sim MX$. Thus the derivative of the phase function is

[TABLE]

and hence it is dominated by $N(\gamma-r/p)$ when

[TABLE]

Under this condition, Lemma A.1 implies that $\text{J}\hskip 0.5pt(y,r,p)$ is negligibly small, provided that $\phi^{(j)}(x){}_{j}1$ ($j\geqslant 1$) and that $V_{\scriptscriptstyle\natural}^{(j)}(x){}_{j}\varDelta^{j}$ for \varDelta\leqslant T/N^{\scalebox{0.62}{\vvepsilon}}. Accordingly, set

[TABLE]

So if we assume that

[TABLE]

then we can effectively truncate the sum at |r-\gamma p|\text{ \small\asymp }R, at the cost of a negligible error. Note that (5.5) amounts to the condition (1.7) in Theorem 1.1.

Moreover, the second derivative test in Lemma A.2 immediately yields the following estimate for $\text{J}\hskip 0.5pt(y,r,p)$.

Lemma 5.1**.**

Suppose that $|\phi^{\prime\prime}(x)|\Gt 1$. Then, for $1\leqslant y/MX\leqslant 2$, we have

[TABLE]

Proof.

The second derivative of the phase function in (5.3) is equal to

[TABLE]

By our assumptions, $|\phi^{\prime\prime}(x)|\Gt 1$ and K<T^{1-\scalebox{0.62}{\vvepsilon}}, the estimate above follows easily from Lemma A.2. ∎

Consequently, (4.9) is transformed into

[TABLE]

5.2. Application of the Cauchy inequality and the second Poisson summation

Next we apply Cauchy and the Ramanujan bound on average for the Fourier coefficients ${\lambdaup_{g}(n)}$ as in (1.5). Thus,

[TABLE]

Opening the square and switching the order of summations, the square of the right-hand side is

[TABLE]

Remark 5.2**.**

To keep in mind some representative cases, we notice that the diagonal contribution $(p_{1},r_{1})=(p_{2},r_{2})$ towards ${S}(N,X,P)$ is

[TABLE]

where $R=PT/N$ as in (5.4) and we have used the bound in Lemma 5.1 for $\text{J}\hskip 0.5pt(n,r,p)$.

It is therefore important to introduce the extra average over $p$ as in (4.9), because without it the diagonal contribution would be $O(\hskip-1.5pt\sqrt{pNK})$ instead.

We then apply Poisson summation with modulus $p_{1}p_{2}$ (note that $p_{1}$ and $p_{2}$ need not be distinct) to the $n$-sum in (5.8), getting

[TABLE]

with

[TABLE]

Recall that $\sqrt{NX}=PK$ as in (5.1) and that $\text{J}\hskip 0.5pt(MXy,r,p)$ is defined as in (5.3). We have

[TABLE]

We note that the $a$-sum in (5.9) yields the congruence condition

[TABLE]

where $\overline{r}_{1}$ and $\overline{r}_{2}$ denote the multiplicative inverses of $r_{1}$ and $r_{2}$ modulo $p_{1}$ and $p_{2}$ respectively. Thus the right-hand side of (5.9) is simplified to

[TABLE]

5.3. Analysis of the integral $\text{L}\hskip 0.5pt(x)$

This section is dedicated to the analysis of the integral $\text{L}\hskip 0.5pt(x)$ as defined in (5.10) or (5.11).

By applying Lemma 5.1 to the integral in (5.10), we obtain the following (trivial) estimate.

Lemma 5.3**.**

We have

[TABLE]

Further, we wish to improve the estimate above by examining the triple integral in (5.11).

We first investigate in the lemma below the $y$-integral in the second line of (5.11); it is the integral $\text{K}\hskip 0.5pt(\varw K,x)$ defined as in (5.15), with $\varw=\sqrt{\varv_{1}}P/{p_{1}}-\hskip-1.5pt\sqrt{\varv_{2}}P/{p_{2}}$.

Lemma 5.4**.**

Let K>N^{\scalebox{0.62}{\vvepsilon}}>1. For real $\varw,x$, with $|\varw|\leqslant\hskip-1.0pt\sqrt{2}-1/2<1$, define

[TABLE]

(1).* We have $\text{K}\hskip 0.5pt(\varw K,x)=O(N^{-A})$ if $|x|\geqslant K$.*

(2).* For |x|>N^{\scalebox{0.62}{\vvepsilon}}, we have $\text{K}\hskip 0.5pt(\varw K,x)=O(N^{-A})$ unless $2/3<\varw K/x<\hskip-1.0pt3/2$, say, and for $1/2<\varw K/x<2$, if we let $\lambdaup=K^{2}\varw^{2}/x$ and W(\lambdaup)=W(\lambdaup,x)=e(-\lambdaup)\text{K}\hskip 0.5pt\big{(}\hskip-1.5pt\sqrt{\lambdaup x},x\big{)} then*

[TABLE]

(3).* $\text{K}\hskip 0.5pt(\varw K,0)=W_{0}(2\varw K)$ for some Schwartz function $W_{0}$.*

Proof.

The statements in (1) and the first part of (2) follow from Lemma A.1. (3) is also clear, for $W_{0}$ is the inverse Fourier transform of the function given by $2yU(y^{2})$ for $y>0$ and by [math] for $y\leqslant 0$. It is left to prove (5.16) for $1/4<\lambdaup/x<4$. For this, we change the variable $y$ to $\lambdaup y/x=\varw^{2}K^{2}y/x^{2}$ in (5.15) so that

[TABLE]

Then the estimates in (5.16) follow from Lemma A.4. ∎

Lemma 5.5**.**

Let $N,T,K,P>1$ be parameters with N^{\scalebox{0.62}{\vvepsilon}}<K\Lt T and N^{1+\scalebox{0.62}{\vvepsilon}}<PK. Let $p_{i}\sim P$ and $|r_{i}-\gamma p_{i}|\Lt PT/N$ ($i=1,2$). Suppose that $\phi^{(j)}(\varv)\Lt 1$ for $j=2,3$ and that $|\phi^{\prime\prime}(\varv)|\Gt 1$ for all $\varv\in(1/2,5/2)$. Let the integral $\text{L}\hskip 0.5pt(x)$ be as in (5.11).

(1).* We have $\text{L}\hskip 0.5pt(x)=O(N^{-A})$ if $|x|\geqslant K$.*

(2).* Assume that K^{2}/T>N^{\scalebox{0.62}{\vvepsilon}}. For $K^{2}/T\Lt|x|<K$, we have*

[TABLE]

For $|x|\Lt K^{2}/T$, we have

[TABLE]

(3).* Let $p_{1}=p_{2}=p$. Then*

[TABLE]

Proof.

The statement in (1) is obvious in view of Lemma 5.4 (1).

We then turn to the proof of (2) in the first case when $K^{2}/T\Lt|x|<K$. First of all, by Lemma 5.4 (2), we may write the integral in (5.11) as below,

[TABLE]

where $\varw=\hskip-1.0pt\sqrt{\varv_{1}}P/{p_{1}}-\hskip-1.0pt\sqrt{\varv_{2}}P/{p_{2}}$, $W_{\scriptscriptstyle\natural}(y)=\sqrt{|x|}W(xy^{2})F(y)$ for $W$ defined as in Lemma 5.4 (2) and $F$ a smooth function supported in $[1/2,2]$, with $F\equiv 1$ on \big{[}2/3,3/2\big{]}, and

[TABLE]

In view of (5.16), we have

[TABLE]

By Fourier inversion, we write

[TABLE]

where $\widehat{W}_{\scriptscriptstyle\natural}$ is the Fourier transform of $W_{\scriptscriptstyle\natural}$, satisfying

[TABLE]

Thus, we may further write

[TABLE]

with

[TABLE]

We have

[TABLE]

[TABLE]

[TABLE]

Since $\phi^{\prime\prime}(\varv)\Gt 1$, when $K^{2}/|x|\Lt T$, it is clear that

[TABLE]

for $1\leqslant\varv_{1},\varv_{2}\leqslant 2$ and |\varv|\leqslant N^{\scalebox{0.62}{\vvepsilon}}. We obtain the estimate in (5.17) by applying the two-dimensional second derivative test in Lemma A.3 with $\lambdaup=\rho=T$.

In the second case in (2) when $|x|\Lt K^{2}/T$ is small, the estimate in (5.18) is just (5.14) in Lemma 5.3.

Finally, let us consider (3). The bound $\text{L}\hskip 0.5pt(0)\Lt 1/T$ is already contained in (5.18). Now assume that |r_{1}-r_{2}|>PT/KN^{1-\scalebox{0.62}{\vvepsilon}}. In view of Lemma 5.4 (3), we may write

[TABLE]

where V_{0}(\varw,\varv_{2})=(2p/P)\big{(}p\varw/P+\hskip-1.0pt\sqrt{\varv_{2}}\big{)}V_{\scriptscriptstyle\natural}\big{(}\big{(}p\varw/P+\hskip-1.0pt\sqrt{\varv_{2}}\big{)}^{2}\big{)}\overline{V_{\scriptscriptstyle\natural}(\varv_{2})}, satisfying

[TABLE]

and

[TABLE]

Recall that $\phi^{(j)}(\varv)\Lt 1$ ($j=2,3$). For $|r_{1}-\gamma p|\Lt PT/N$ and |\varw|<N^{\scalebox{0.62}{\vvepsilon}}/K, we have

[TABLE]

and, similarly,

[TABLE]

It follows from |r_{1}-r_{2}|>PT/KN^{1-\scalebox{0.62}{\vvepsilon}} that $|\partial f_{0}(\varw,\varv_{2})/\partial\varv_{2}|\Gt N|r_{2}-r_{1}|/P$. By partial integration (the first derivative test), we infer that \text{L}\hskip 0.5pt(0)\Lt P/KN^{1-\scalebox{0.62}{\vvepsilon}}|r_{1}-r_{2}|, as desired. ∎

5.4. Estimates for ${S}(N,X,P)$

Combining (5.8), (5.9) and (5.13), along with Lemma 5.5, we conclude that

[TABLE]

with

[TABLE]

and

[TABLE]

in correspondence to the cases where $n=0$ and $n\neq 0$ in (5.13), respectively. Since the modular form $g$ is considered fixed, we have absorbed its level $M$ into the implied constant. Note that in the case $n=0$ the congruence condition in (5.12) would imply $p_{1}=p_{2}$ ($=p$) and $r_{1}\equiv r_{2}(\mathrm{mod}\,p)$. Moreover, when applying the estimates (5.17) and (5.18) to $\text{L}\hskip 0.5pt(x)$ with $x={MXn}/{p_{1}p_{2}}$, note that $K^{2}/T\Lt|x|<K$ or $|x|\Lt K^{2}/T$ amounts to $N/T\Lt|n|\Lt N/K$ or $|n|\Lt N/T$, respectively, for $X=P^{2}K^{2}/N$ (see (5.1)). We record here the condition in Lemma 5.5 (2):

[TABLE]

For ${S}_{\mathrm{diag}}^{2}(N,X,P)$, we split the sum over $r_{1}$ and $r_{2}$ according as $r_{1}=r_{2}$ or not,

[TABLE]

and hence

[TABLE]

Recall here that $NX=P^{2}K^{2}$ and $R=PT/N$ as in (5.1) and (5.4).

To deal with ${S}_{\mathrm{off}}^{2}(N,X,P)$, we first note that necessarily $p_{1}\neq p_{2}$. Otherwise, if $p_{1}=p_{2}=p$, then the congruence $n\hskip 1.0pt\equiv\,\overline{r}_{1}p-\overline{r}_{2}p\,(\mathrm{mod}\,p^{2})$ would imply $p|n$. This is impossible, in view of our assumption {N^{1+\scalebox{0.62}{\vvepsilon}}}/{K}<P in (5.2) and the length $N/K$ of the $n$-sum. We now interchange the sum over $n$ and the sums over $r_{1}$, $r_{2}$. Note that for fixed $n$, the congruence $n\equiv\overline{r}_{1}p_{2}-\overline{r}_{2}p_{1}(\mathrm{mod}\,p_{1}p_{2})$ splits into $r_{1}\equiv\overline{n}p_{2}(\mathrm{mod}\,p_{1})$ and $r_{2}\equiv-\overline{n}p_{1}(\mathrm{mod}\,p_{2})$, so

[TABLE]

When $T\geqslant N$ so that $R\geqslant P$, we have

[TABLE]

When $T<N$, the $(R/P)^{2}$ in (5.25) needs to be replaced by $1$. In other words, we lose $(P/R)^{2}=(N/T)^{2}$. However, the loss may be reduced to $N/T$ if we rearrange the sum ${S}_{\mathrm{off}}^{2}(N,X,P)$ as follows

[TABLE]

Thus for $T<N$, we have

[TABLE]

Combining (5.25) and (5.26), we have

[TABLE]

We conclude from (5.20), (5.24) and (5.27) that

[TABLE]

At this point a mild assumption like $P<N^{A}$ is needed so that \log P<N^{\scalebox{0.62}{\vvepsilon}}, where $A$ is a large fixed constant.

5.5. Conclusion

In view of (4.8) in Proposition 4.1 and (5.28), we have

[TABLE]

For the estimate in (1.8) to be non-trivial, we assume that N^{\scalebox{0.62}{\vvepsilon}}<T<N^{3/2-\scalebox{0.62}{\vvepsilon}}. Then

[TABLE]

on choosing $K=T^{2/3}$ and $P=N/T^{1/3}$. The required conditions in (5.1), (5.2) and (5.23) are well justified for our choice of $K$ and $P$. This proves Theorem 1.1.

For Corollary 1.2, define

[TABLE]

Let the smooth function $V$ in Theorem 1.1 be supported on $[1,1+H/N]$ with $V(x)\equiv 1$ on $[1+1/\varDelta,1+H/N-1/\varDelta]$. For this, it is necessary that $\varDelta\geqslant 2N/H$. By the Deligne bound (1.6), we would have

[TABLE]

Then Corollary 1.2 follows from Theorem 1.1 upon choosing \varDelta=T/N^{\scalebox{0.62}{\vvepsilon}}.

6. Proof of the Weyl-type subconvex bound

For $g\in S^{\star}_{k}(M,\xi)$ with Fourier coefficients $\lambdaup_{g}(n)$, let $\widebar{g}\in S^{\star}_{k}(M,\widebar{\xi})$ be its dual form with Fourier coefficients $\lambdaup_{\widebar{g}}(n)=\overline{\lambdaup_{g}(n)}$, and let $\epsilon_{g}$ be the root number of $L(s,g)$ satisfying the functional equation

[TABLE]

with

[TABLE]

We remark that $\epsilon_{g}=i^{k}\eta_{g}$ for the $\eta_{g}$ as in Lemma 2.1. From this one may deduce the following Approximate Functional Equation (see [Har, Theorem 2.5] and [BMN, Lemma 2.1]).

Lemma 6.1** (Approximate Functional Equation).**

Let $F$ be a real-valued smooth function on $(0,\infty)$ satisfying $F(x)+F(1/x)=1$ and with derivatives decaying faster than any negative power of $x$ as $x\rightarrow\infty$. Then

[TABLE]

where $C=C(g,t)$ is the analytic conductor defined by ([Har, (2.4)])

[TABLE]

Let $t>1$ be large. By applying a dyadic partition of unity to the approximate functional equation (6.1), we infer that

[TABLE]

for some N<t^{1+\scalebox{0.62}{\vvepsilon}}, where

[TABLE]

and $V(x)$ is some function in $C_{c}^{\infty}(0,\infty)$ supported on $[1,2]$, satisfying $V^{(j)}(x){}_{j}1$.

Recall that the Rankin–Selberg estimate in (1.5) yields the trivial bound $S(N)\Lt N$. Therefore it suffices to prove the bound S(N)\Lt\sqrt{N}t^{1/3+\scalebox{0.62}{\vvepsilon}} in the range t^{2/3+\scalebox{0.62}{\vvepsilon}}<N<t^{1+\scalebox{0.62}{\vvepsilon}}.

Note that $e(f(n))=N^{it}n^{-it}$ if we choose $\phi(x)=-\log x$, $T=t/2\pi$ and $\gamma=0$ in (1.3). Consequently, Theorem 1.1 implies that for t^{2/3+\scalebox{0.62}{\vvepsilon}}<N<t^{1+\scalebox{0.62}{\vvepsilon}} the sum $S(N)$ has the following bound:

[TABLE]

as desired.

Appendix A Stationary phase

Firstly, we have Lemma 8.1 in [BKY] with some improvements.

Lemma A.1**.**

Let $\textit{w}(x)$ be a smooth function supported on $[a,b]$ and $f(x)$ be a real smooth function on $[a,b]$. Suppose that there are parameters $Q,U,Y,Z,R>0$ such that

[TABLE]

for $i\geqslant 2$ and $j\geqslant 0$, and

[TABLE]

Then for any $A\geqslant 0$ we have

[TABLE]

Proof.

In the proof of Lemma 8.1 in [BKY], one can actually impose an additional condition $\gamma_{2}+\gamma_{3}+...=\varnu-n$ to the inner sum in (8.5) so that the $Y^{{(\varnu-\mu)}/2}$ may be replaced by $Y^{\varnu-n}$ in (8.6) and the sum over $\mu$ should be only up to $2n-\varnu$. In this way, their condition $Y\geqslant 1$ becomes unnecessary and their estimate in (8.3) may be improved as above.

∎

For the reader’s convenience, we record here the one- and two-dimensional second derivative tests (see [Hux, Lemma 5.1.3], [Mun1, Lemma 4]).

Lemma A.2**.**

Let $f(x)$ be a real smooth function on $[a,b]$. Let $\textit{w}(x)$ be a real smooth function supported on $[a,b]$ and let $V$ be its total variation111Since $\textit{w}(x)$ is supported on $[a,b]$, we do not need to add its maximum modulus to $V$ as in [Hux, Lemma 5.1.3].. If $f^{\prime\prime}(x)\geqslant\lambdaup>0$ on $[a,b]$, then

[TABLE]

Lemma A.3**.**

Let $f(x,y)$ be a real smooth function on $[a,b]\times[c,d]$ with

[TABLE]

on the rectangle $[a,b]\times[c,d]$. Let $\textit{w}(x,y)$ be a real smooth function supported on $[a,b]\times[c,d]$ and let

[TABLE]

Then

[TABLE]

with an absolute implied constant.

Finally, the following stationary phase estimate is from [Sog, Theorem 1.1.1].

Lemma A.4**.**

Let $Z>0$ and ${\lambdaup}\geqslant 1$. Let $\textit{w}(x;\lambdaup)$ be a smooth function supported on $[a,b]$ for all $\lambdaup$, and $f(x)$ be a real smooth function on an open neighborhood of $[a,b]$. Suppose that $\lambdaup^{j}\partial_{x}^{i}\partial_{\lambdaup}^{j}\textit{w}(x;\lambdaup){}_{\,i,\,j}Z$ and that $f(x_{0})=f^{\prime}(x_{0})=0$ at a point $x_{0}\in(a,b)$, with $f^{\prime\prime}(x_{0})\neq 0$ and $f^{\prime}(x)\neq 0$ for all $x\in[a,b]\smallsetminus\{x_{0}\}$. Then

[TABLE]

Acknowledgements**.**

We would like to thank Philippe Michel, Qingfeng Sun, and the referee for their valuable comments.