Subconvex bounds on via degeneration to frequency zero
Abstract.
For a fixed cusp form on and a varying Dirichlet character of prime conductor , we prove that the subconvex bound
holds for any . This improves upon the earlier bounds and obtained by Munshi using his variant of the method. The method developed here is more direct. We first express as the degenerate zerofrequency contribution of a carefully chosen summation formula à la Poisson. After an elementary “amplification” step exploiting the multiplicativity of , we then apply a sequence of standard manipulations (reciprocity, Voronoi, Cauchy–Schwarz and the Weil bound) to bound the contributions of the nonzero frequencies and of the dual side of that formula.
2010 Mathematics Subject Classification:
11F66, 11M41Contents
1. Introduction
We consider the problem of bounding , where

is a fixed cusp form on , not necessarily selfdual, and

traverses a sequence of Dirichlet characters of (say) prime conductor tending off to .
Munshi [21] recently established the first subconvex bound in this setting by showing that if satisfies the Ramanujan–Selberg conjecture, then for any fixed , the estimate
(1.1) 
holds for some positive quantity that may depend upon and , but not upon . In the preprint [17], he improves the exponent range to and removes the Ramanujan–Selberg assumption.
A striking feature of his work is the introduction of a novel “ symbol method,” whereby one detects an equality of integers by averaging several instances of the Petersson trace formula. We summarize this approach in Appendix B, referring to [21] and [17] for details, to [19] and [18] for other recent applications of the symbol method, and to [10, §5.5] for general discussion of the spectral decomposition of the symbol.
It is natural to ask about the true strength of the symbol method. How does it compare to the classical symbol method of Duke–Friedlander–Iwaniec [6] and HeathBrown [8]? For which problems does one fail and the other succeed? For which problems are the two methods “identical” or “equivalent”? Can the symbol method be simplified or removed in certain applications?
In pondering such questions, we were able to better understand the arithmetical structure and mechanisms underlying Munshi’s argument and construct a more direct proof of the following quantitative strengthening of Munshi’s bound.
Theorem 1.
The subconvex bound (1.1) holds for any .
The proof is surprisingly short compared to earlier proofs of related estimates. Indeed, we regard the primary novelty of this work as not in the numerical improvement of the exponent but rather in the drastic simplification obtained for the proof of any subconvex bound (1.1).
Our point of departure is a formula (see §3.2), derived via Poisson summation, that expresses in terms of additive characters and twisted Kloosterman sums. We insert this into an approximate functional equation for . After an elementary “amplification” step exploiting the multiplicativity of , we then conclude via standard manipulations. We discuss in Appendix B how we arrived at this approach through a careful study of Munshi’s arguments.
We hope that the technique described here may be applied to many other problems. For instance, it seems natural to ask whether it allows a simplification or generalization of the arguments of [19] for bounding symmetric square functions.
2. Preliminaries
2.1. Asymptotic notation
We work throughout this article with a cusp form on and a sequence of primitive Dirichlet characters to prime moduli , indexed by , with . To simplify notation, we drop the subscripts and write simply and . Our convention is that any object (number, set, function, …) considered below may depend implicitly upon unless we designate it as fixed; it must then be independent of . Thus is understood as fixed, while is not. All assertions are to be understood as holding after possibly passing to some subsequence of the original sequence , and in particular, for sufficiently large.
We define standard asymptotic notation accordingly: or or means that for some fixed , while means for every fixed (for large enough, by convention). We write for . We write to denote that for each fixed . Less standardly, we write or as shorthand for , or equivalently, . Our goal is then to show that
(2.1) 
We say that is inert if it satisfies the support condition
and the value and derivative bounds
2.2. General notation
We write , and denote by a sum over integers . Let . We write and to denote sums over and , respectively. We denote the inverse of by or . We denote by the additive character given by , by the Kloosterman sum, by the normalized Kloosterman sum, by the twisted Kloosterman sum, and by the normalized Gauss sum (of magnitude one).
We define the Fourier coefficients of as in [7], so that for complex numbers with large enough real part, and .
For a condition , we define to be if holds and otherwise. For instance, is if and if .
We denote by the Fourier transform of a Schwartz function on .
For a pair of integers , we denote by and their the greatest common divisor and least common multiple, respectively.
2.3. Voronoi summation formula
By [15] (cf. [2, §4] for the formulation used here), we have for , , and that
(2.2) 
for integral transforms of the shape
where is meromorphic on and holomorphic in the domain , where it satisfies for fixed . (The indices and in (2.2) are implicitly restricted to be positive integers.) Set for some sufficiently small fixed . By shifting the contour to and to , we see that if is inert, then
(2.3) 
for all fixed .
In the special case , we have , and so
2.4. Rankin–Selberg bounds
3. Division of the proof
3.1. Approximate functional equation
Recall our main goal (2.1). By [11, §5.2], we may write
for some with and some smooth functions satisfying for all fixed . By a smooth dyadic partition of unity and the Rankin–Selberg estimate (2.4), it will suffice to show for each and each inert that the normalized sum
satisfies the estimate
(3.1) 
By further application of (2.4), we may and shall assume further that
(3.2) 
The proof of (3.1) will involve positive parameters satisfying
(3.3) 
Thus every integer in is coprime to .
3.2. A formula for
Fix a smooth function on supported in the interval with . Then . Observe that is defined for all integers for which . Set
By Poisson summation, we have
(3.4) 
For , we have . Setting , we deduce by rearranging (3.4) that
(3.5) 
The properties of the sequence to be used in what follows are that it is supported on and satisfies the estimates and .
3.3. “Amplification”
We choose sequences of complex numbers and supported on (say) primes in the intervals and , respectively, so that
(3.6) 
Then
(3.7) 
The properties of and just enunciated, rather than an explicit choice, are all that will be used; one could take, for instance , where denotes the set of primes, and similarly for .
3.4. A formula for
3.5. Main estimates
We prove these in the next two sections.
Proposition 1.
Assume that
(3.8) 
Then
(3.9) 
Remark.
Proposition 2.
Assume that
(3.10) 
for some fixed . Then
(3.11) 
3.6. Optimization
Our goal reduces to establishing that . (By comparison, we note the trivial bounds and .) We achieve this by applying the above estimates with
Then (3.8) is clear, while (3.10) follows from (3.2). The required bound for follows readily from (3.11). We now deduce the required bound for . Note that the first term on the RHS of (3.9) is acceptable thanks to our choice of . Note also from (3.2) that ; from our choice of , it follows that . The bound for then readily simplifies to . (By solving a linear programming problem, we see moreover that these choices give the optimal bound for derivable from the above propositions.)
4. Estimates for
We now prove Proposition 1.
4.1. Reciprocity
Our assumption (3.8) implies that for all with , the function is inert. By the Chinese remainder theorem, we have for . We may thus rewrite
where
4.2. Voronoi
We introduce the notation
so that and . Applying Voronoi summation (§2.3), we obtain
for some smooth functions satisfying for fixed .
4.3. Cleaning up
The Weil bound, the Rankin–Selberg bound (2.4) and the condition give
(4.1) 
If , then (because is prime) , hence by (4.1),
Since the square of the latter is the first term on the RHS of (3.9), the proof of Proposition 1 reduces to that of an adequate bound for the sum
If , then and , hence
(4.2) 
with
Remark.
With slightly more casebycase analysis in the arguments to follow, one can verify that the reduction performed here to the case is unnecessary, hence that the bound (3.9) remains valid in the stated generality even after deleting the first term on its RHS.
4.4. Cauchy–Schwarz
4.5. Application of exponential sum bounds
Opening the square, expanding the definition of and wastefully discarding some summation conditions, we obtain
(4.3) 
where is defined for in the support of by
(4.4) 
with
and
We have for fixed . By a smooth dyadic partition of unity, we may write
(4.5) 
where each function is inert. Substituting (4.5) into (4.4) and applying the incomplete exponential sum estimates recorded in Appendix A, we obtain with
that
Since , the above sum is dominated by the contribution from ; estimating that contribution a bit crudely with respect to , we obtain
(4.6) 
4.6. Diagonal and offdiagonal
To state the estimates to be obtained shortly, we introduce the notation
We estimate separately the contribution of each term on the RHS of (4.6) to via (4.3), splitting off the contribution to the first from terms with . We obtain in this way that
where
(In deriving the estimate involving , we used the slightly wasteful bound .) Noting that iff , we verify using the divisor bound that
These estimates combine to give an adequate estimate for .
5. Estimates for
We now prove Proposition 2.
5.1. Cauchy–Schwarz
Using again the Rankin–Selberg bound (2.4), we obtain
5.2. Elementary exponential sum bounds
Let be fixed but sufficiently small. Since is prime and satisfies the lower bound in (3.3), we know that the integers and are coprime whenever . By the rapid decay of , we may truncate the sum to with negligible error . We then open the square and apply Cauchy–Schwarz, leading us to consider for with
(5.1) 
the sums
(5.2) 
We apply Poisson summation. By the lower bound on in (3.2) and the assumption , we have for some fixed . Thus only the zero frequency after Poisson contributes nonnegligibly, and so with
Opening the Kloosterman sums and executing the sum gives
Our assumptions imply that the quantities are all coprime to , so after a change of variables we arrive at
5.3. Diagonal vs. offdiagonal
We have shown thus far that
where the sum is restricted by the condition (5.1). By our assumption (3.10), the quantities and are congruent modulo precisely when they are equal. By the divisor bound, the number of tuples for which is . Since was arbitrary, we obtain
(5.3) 
By another application of our assumption (3.10), the first term in the latter bound dominates, giving the required bound for .
The proof of our main result (Theorem 1) is now complete.
Appendix A Correlations of Kloosterman sums
The estimates recorded here are unsurprising, but we were unable to find references containing all cases that we require (compare with e.g. [5, 4, 17]).
Lemma 1.
Let be a natural number. Let be congruence classes for which . For each prime , let be a subset of cardinality . Let denote the set of elements for which

the class of modulo belongs to for each , and

.
Define by
Then the exponential sum satisfies
where denotes the number of prime divisors of , without multiplicity.
Proof.
We may assume that for some prime . For , there is nothing to show. For , we appeal either to the Weil bound, to bounds for Ramanujan sums, or to the trivial bound according as , or and , or . We treat the remaining cases by induction on . If , then the conclusion follows by our inductive hypothesis applied to . We may thus assume that . A short calculation gives the identities of rational functions
(A.1) 
Write or , and set