Beyond the Weyl barrier for GL(2) exponential sums
Introduction
Let be a holomorphic cusp newform of level D, weight k, nebentypus character ξ, with the Fourier expansion for .
In this paper, we consider the following smoothed exponential sum where the weight function and the phase function f is of the form: with for an arbitrarily small , and for a fixed real number . For the logarithm case, if one lets , then with For the monomial case, β is also considered fixed, and is often used to denote the exponential sum:
Thanks to the Rankin–Selberg theory, we know that '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 , 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 . The primary purpose of this paper is to break the upper ‘Weyl barrier’ at . For this we have the following theorem.
Theorem 1.1 Let . Let be a fixed real number. Let be a smooth function with support in and derivatives for every . Let and be its Fourier coefficients. (1) We have if , and if . (2) We have if , and if .
Note that , , , and . Therefore the Weyl barrier at is extended to .
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 is such an exponent pair, then we may prove
Definition 1.2 For an exponent pair we define its β-barrier by
The bound in (1.12) is better than the trivial bound N if and only if β does not exceed the barrier , so we seek 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 . Next, by our first van der Corput method, the exponent pair yields the β-barrier at as above. Further, by our second van der Corput method, the exponent pair yields the β-barrier at . 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 . 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 and define We have for , with 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 . Note that when , (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 . 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 such that for , with () in the monomial case.
Finally, for the non-generic case when , we can still attain a sub-Weyl bound by the Weyl method.
Theorem 1.5 Let notation be as above. For set . We have if q is odd, and if q is even.
Reduction of the sub-Weyl subconvex problem Let 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 , we obtain the Weyl subconvex bound: which was first proven by Good [6] in the full-level case . Any bound of the type with , is a sub-Weyl subconvex bound. Here we make a first step towards this sub-Weyl subconvexity problem.
Theorem 1.6 For any given , there exists such that with N dyadic.
Proof Choose . Then Theorem 1.1, Theorem 1.3, Theorem 1.4 ensure the existence of so that whenever . 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 it suffices to prove sub-Weyl bounds for with N in the transition range .
Remarks In the monomial case, by suitable adaptations, the forgoing results remain valid when a in is not necessarily fixed (see [1, (1.15)]), and more generally, when the function as in Definition 6.1. In view of the work [4], the form g can also be a Hecke–Maass newform.
Notation By or we mean that for some constant , and by we mean that and . We write or if the implied constant c depends on g, . For notational simplicity, in the case , we shall not put ϕ, a or β in the subscripts of and O. Let p always stand for prime. The notation or is used for integers or primes in the dyadic segment or , 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 and set , so that
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 be supported in , with and for every . Define , with the Mellin transform of U. For a fixed newform , let be its Fourier coefficients, and let 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 and .
Lemma 3.1 Let be a compact set and X be an open neighborhood of K. Let k be a non-negative integer. If , , and f is real valued, then for we have where C is bounded when f stays in a bounded set in .
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 or .
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 and we have
Treating the sum
Since w and v will play a minor role in what follows, we shall write , , and ; 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 that Firstly, we have and Subsequently, we shall denote and let δ be sufficiently small.
The van der Corput methods for almost separable double exponential sums
The exponential sum in Proposition 2.2 has phase function containing a separable main term , with or , along with a ‘mixing’ error term —exponential sums of this type will be called almost separable. Note that and if we set .
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 (), we have developed in §6 the van der Corput methods of exponent pairs for the type of double sums like . More precisely, on applying Theorem 6.15, Theorem 6.25, we obtain non-trivial estimates of the form for certain exponent pairs depending on the value of . As (7.1) is non-trivial, it is necessary that Substituting (7.1) into (2.26), we obtain where
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