Hybrid subconvexity bounds for
Fix an integer . Let be prime and let be an even integer. For a holomorphic cusp form of weight and full level and a primitive holomorphic cusp form of weight and level , we prove hybrid subconvexity bounds for in the and aspects when for any . These bounds are achieved through a first moment method (with amplification when ).
Subconvexity estimates for Rankin-Selberg -functions have been established in a variety of settings recently with strong motivation coming from equidistribution problems of an arithmetic nature. In general, for an -function associated to an irreducible cuspidal automorphic representation with analytic conductor , one hopes to obtain subconvexity estimates of the form for some when . Though the actual value of does not often matter in applications, establishing such a subconvexity bound for some is non-trivial and requires careful consideration of the arithmetic/algebraic information associated with . The convexity bound , on the other hand, follows purely from standard tools in complex analysis.
The resolution of one equidistribution problem related to central values of Rankin-Selberg -functions, the quantum unique ergodicity conjecture of Rudnick and Sarnak , has thus far required several techniques from analytic number theory and ergodic theory. In many cases, however, the conjecture would follow directly from subconvexity estimates for and . Here we think of as a varying modular form and as a fixed form.
Subconvexity estimates for such -values have proven to be very difficult to establish through current methods and several authors have first given attention to analogous subconvexity problems for Rankin-Selberg -functions in order to possibly better understand the structure behind the symmetric square. For a partial list of related works, see , , , , , , ,  and the references therein. For many Rankin-Selberg -functions, it appears as though the arithmetic/analytic structure of the conductor dictates the method of proof that should be adopted to achieve subconvexity. For example, an amplified moment method is usually required when only one of the forms in the convolution is varying. However, when at least two of the forms are varying, as in the present work, then a moment computation without amplification suffices in certain hybrid ranges. Curiously, the moment method may be avoided all together in cases where the level of the varying form has special structure, for example if the level of the varying form factorizes in a suitable manner .
In the work of Rizwanur Khan  on , a conditional amplifier of long length relative to the conductor was employed in a first moment method in order to establish subconvexity estimates for fixed and varying of prime level . Following ideas seen in  and  (among others), the work of Khan  suggests that the number of points of summation for the unamplified first moment of is insufficient for application to the subconvexity problem and that one would benefit from increasing the complexity of the -function by allowing to vary independently with .
As we demonstrate in this paper, varying the weight of along with the level of increases the conductor to be of size and allows us to establish hybrid subconvexity bounds for in the and aspects when for any . Given the above lower bound for in terms of , this suggests that much more work remains in establishing subconvexity for the case of fixed and varying as required in the holomorphic analogue to the quantum unique ergodicity conjecture. A related situation and hybrid subconvexity bound may be found in (, Corollary 1.5) where the authors consider with all three forms and varying in weights and respectively.
2. Statement of results
Fix an integer and newform of weight and level . Let be a Hecke eigenform of even weight . Let and let be a set of primes in the range not dividing the level . Our choice for will be such that . We will be working with an amplified first moment containing
where is as in (4.15) and the amplifier is given by
When , the Hecke relation yields .
Opening the absolute square and using Hecke multiplicativity gives
In §5, we shall prove the following result.
Suppose are integers, with even, is a prime, and is a Hecke cusp form of weight for . Let be a positive integer. Then under the assumption
we have for any
(2) Setting , we note that the above bound is the Lindelöf on average bound when . Therefore, this is the only range in which amplification is applied.
Inserting the above bound into (2.2) and trivially averaging over and we get
Using the non-negativity of the central -values and the definition of our amplifier according to (2.1) gives
one verifies that the assumption (2.3) on is satisfied and we therefore obtain the following corollary.
For as above and a newform of weight and level , we have
which is extracted from the second line of (2.5). This bound is already able to beat the convexity bound when , and therefore amplification is unnecessary (although the bound from amplification, i.e. the first line of (2.5), also provides a subconvexity bound on the overlapping range ). Thus, the amplification method extends the range of admissible exponents from below by .
3. Sketch of hybrid subconvexity in a simplified case
Let be a holomorphic cusp form of even weight and full level and let be a primitive holomorphic cusp form of even weight and prime level . In order to demonstrate the ideas behind the proofs of our main results, we provide a brief sketch of how one might establish hybrid subconvexity bounds when is large and fixed. For notational convenience, we denote the Dirichlet coefficients of by and the coefficients of by such that a standard approximate functional equation argument will essentially equate our central -value with
for any with and root number .
Our method will be a normalized first moment average over newforms . Therefore, we wish to achieve a better result than the first moment convexity bound
where is as in (4.15). As noted in the previous section, one gains from amplification in certain ranges of relative to , but we omit this component here.
Assume that for our particular choice of and , the space of newforms spans the space of all forms . Write
and consider first . Applying the Petersson trace formula in the average over along with standard Bessel function bounds (4.21) and the Weil bound for Kloosterman sums, one obtains
The Deligne bound for the coefficients of holomorphic forms then gives
Now for , an application of Petersson’s trace formula shows that is essentially equal to
Here we pulled out the divisor in the original -sum and used basic properties of the Kloosterman sums, i.e.
Focusing on the transition range of the Bessel function () for the remainder of the sketch and opening the Kloosterman sum, we see that we must analyze a smoothed version of
An application of Voronoï summation in leads to sums of the form
i.e. we obtain a new -sum of length “conductor divided by the original length of summation” with a summand “dual” to the previous summand. Summing over , one sees that we must consider
Trivially bounding the sum over using Deligne’s bound, the above is bounded by
This bound is better than the first moment convexity bound (3.1) when .
Equating the two bounds we get . Such a choice of satisfies our assumption when . Now assume that
Equating the two bounds we get Such a choice of satisfies our assumption when .
Therefore, one establishes hybrid subconvexity bounds for all with the range of relative to tending to as .
4.1. Holomorphic cusp forms
For a positive integer and an even positive integer , the space of cusp forms of weight for the Hecke congruence group is a finite-dimensional Hilbert space with respect to the Petersson inner product
where denotes the upper half-plane. Every has a Fourier series expansion
For , define the Hecke operator by
Let be the orthogonal set of Hecke-normalized (i.e., ) newforms in . Every is an eigenfunction of all Hecke operators ; let be its eigenvalue of . We have for all . The Hecke eigenvalues are multiplicative, i.e., for any
In particular, (4.1) becomes completely multiplicative when
For any , we have Deligne’s bound
and when is squarefree, it is known that ([11, (2.24)])
4.2. Automorphic -functions
In this section some preliminary results on automorphic -functions are given. We shall particularly focus on the Rankin-Selberg -function for and with squarefree, an even positive integer and a positive integer. A brief calculation of the -factor and the -factor of will be given in §4.2.4.
For the Hecke -function is defined by
This has an Euler product with local factors
where is the principal character modulus . The gamma factor is
The complete product is entire and satisfies the functional equation
with root number where is the eigenvalue of the Atkin-Lehner involution . If is squarefree, then
For , the local factors factor further as
where , are complex numbers with and .
For the symmetric square -function is defined by
where denotes the Riemann zeta function and is defined by
This has Euler product with
The gamma factor is
The complete product is entire and it satisfies the functional equation
We have the convexity bound
where the implied constant depends only on . Moreover, it is known that 
Let and with squarefree, an even positive integer and a positive integer. We define the Rankin-Selberg -function
This has Euler product with local factor
if , and
if . The gamma factor is
The complete product is entire and it satisfies the functional equation
with root number given by
Computation of the -factor and the -factor of
Let be the standard additive character on , namely , and be a normalized unramified additive character of for each prime .
At the real place, the local component , respectively , is the discrete series representation of with weight , respectively .
The Weil group The Weil group of is realized as satisfying and for . Under the local Langlands correspondence the discrete series with weight , , corresponds to the two dimensional representation of given by
The -factor and the -factor . See . With some matrix calculations we have
This implies the formula (4.6) for the -factor , and the -factor at is
4.3. Approximate functional equation
where is a smooth function on defined by
with a positive integer. For large we shift the contour of integration in to , and for small we left shift the contour to with , passing through the pole with residue