Hybrid subconvexity bounds for L\left(\tfrac{1}{2},\textnormal{Sym}^{2}f\otimes g\right)

# Hybrid subconvexity bounds for L(12,Sym2f⊗g)

## Abstract.

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 ).

## 1. Introduction

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 [20], 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 [14], [17], [6], [8], [9], [1], [2], [16] 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 [19].

In the work of Rizwanur Khan [12] 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 [8] and [9] (among others), the work of Khan [12] 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 ([2], 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

 ∑g∈H⋆2κ(P)ω−1g|Ag|2L(12,Sym2f⊗g)

where is as in (4.15) and the amplifier is given by

 Ag:=∑ℓαℓλg(ℓ)

with

 (2.1) αℓ:=⎧⎨⎩λg0(ℓ),ifℓ∈P,−1,ifℓ=p2 with p∈P,0,otherwise.

When , the Hecke relation yields .

Opening the absolute square and using Hecke multiplicativity gives

 (2.2) ∑∑ℓ1,ℓ2αℓ1¯¯¯¯¯¯¯αℓ2∑ℓ3|(ℓ1,ℓ2)∑g∈H⋆2κ(P)ω−1gλg(ℓ1ℓ2ℓ−23)L(12,Sym2f⊗g).

In §5, we shall prove the following result.

###### Theorem 2.1.

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

 (2.3) L⩽k−25P320,

we have for any

 ∑g∈H⋆2κ(P)ω−1g λg(ℓ)L(12,Sym2f⊗g)⋘ε,κ⎛⎝1√ℓ+L227k137P47⎞⎠(kP)ε.

###### Remark 2.2.

(1) The assumptions and (2.3) are a result of technical difficulties in the proof. See Remark 4.1 and 5.1.

(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

 ∑g∈H⋆2κ(P)ω−1g|Ag|2L(12,Sym2f⊗g)⋘ε,κ⎛⎝L+L367k137P47⎞⎠(kP)ε.

Using the non-negativity of the central -values and the definition of our amplifier according to (2.1) gives

 L(12,Sym2f⊗g0)⋘ε,κ(PL+L227k137P37)(kP)ε.

Finally, setting

 (2.4) L={k−1329P429, if P1364

one verifies that the assumption (2.3) on is satisfied and we therefore obtain the following corollary.

###### Corollary 2.3.

For as above and a newform of weight and level , we have

 (2.5) L(12,Sym2f⊗g)⋘ε,κ⎧⎪ ⎪⎨⎪ ⎪⎩k1329P2529(kP)ε, if P1364

###### Remark 2.4.

Note that (2.5) beats the convexity bound when for some . Putting in Theorem 2.1, we arrive at the following bound by non-negativity

 L(12,Sym2f⊗g)⋘ε,κ(P+k137P37)(kP)ε,

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

 D1g(Y)+D2g(Y):=∑n⋘Y√QA(n)λ(n)√n+ε(Sym2f⊗g)∑n⋘√Q/YA(n)λ(n)√n

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

 (3.1) ∑g∈H⋆2κ(P)ω−1gL(12,Sym2f⊗g)⋘Q14+εP

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

 S1(Y)+S2(Y):=∑g∈H⋆2κ(P)ω−1gD1g(Y)+∑g∈H⋆2κ(P)ω−1gD2g(Y)

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

 S1(Y)=1+2π(−1)κ∑c≡0(modP)c>0 1c∑n⋘Y√QA(n)√nS(n,1;c)J2κ−1(4π√nc)⋘κ1+∑c≡0(modP)c>0 1√c∑n⋘Y√Q|A(n)|√n(√nc)2κ−1.

The Deligne bound for the coefficients of holomorphic forms then gives

 (3.2) S1(Y)⋘ε,κmax⎧⎨⎩1,√P(Y√QP2)κ⎫⎬⎭Qε.

Note that the condition or equivalently is necessary for the bound (3.2) above to be better than the first moment convexity bound (3.1).

###### Remark 3.1.

We shall see that such basic analysis of , when is large, is sufficient for establishing at least some hybrid subconvexity bound due to the congruence condition in the sum over . In §5, we improve on the above bound (3.2) in order to establish our main results in §2.

Now for , an application of Petersson’s trace formula shows that is essentially equal to

 ∑c≡0(modP)c>0√Pc∑n⋘√Q/YA(n)√nS(nP,1;c)J2κ−1(4π√nPc)=∑(c,P)=1c>01c√P∑n⋘√Q/YA(n)√nS(nP,1;cP)J2κ−1(4π√nc√P)=−∑(c,P)=1c>01c√P∑n⋘√Q/YA(n)√nS(n,¯¯¯¯P;c)J2κ−1(4π√nc√P).

Here we pulled out the divisor in the original -sum and used basic properties of the Kloosterman sums, i.e.

 S(nP,1;cP)={0, if P|c,S(0,¯¯c;P)S(n,¯¯¯¯P;c)=−S(n,¯¯¯¯P;c), if (c,P)=1.

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

 ∑(c,P)=1c≍Q14/√YP1c√P ∑∑⋆a(modc)e(a¯¯¯¯Pc)∑n⋘√Q/YA(n)√ne(n¯¯¯ac).

An application of Voronoï summation in leads to sums of the form

 ∑(c,P)=1c≍Q14/√YP1c√P ∑∑⋆a(modc)e(a¯¯¯¯Pc)∑n⋘c3k2√Q/YA(n)√nS(a,n;c)√c

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

 ∑(c,P)=1c≍Q14/√YP1√cP∑n⋘k2Q14√YP3A(n)√ne(−nPc).

Trivially bounding the sum over using Deligne’s bound, the above is bounded by

 (3.3) ∑(c,P)=1c≍Q14/√YP1√cP⎛⎝kQ18Y14P34⎞⎠1+ε⋘Q14+εP(k√YP).

This bound is better than the first moment convexity bound (3.1) when .

We now combine the bounds in (3.2) and (3.3). Assume first that

 √P(Y√QP2)κ⩽1

Equating the two bounds we get . Such a choice of satisfies our assumption when . Now assume that

 1⩽√P(Y√QP2)κ

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. Preliminaries

### 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

 ⟨f1,f2⟩:=∫Γ0(N)∖Hf1(z)¯¯¯¯¯¯¯¯¯¯¯¯f2(z)yk−2dxdy,f1,f2∈Sk(N),

where denotes the upper half-plane. Every has a Fourier series expansion

 f(z)=∞∑n=1ψf(n)nk−12e(nz).

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

 (4.1) λf(m)λf(n)=∑d|(m,n)(d,N)=1λf(mn/d2).

In particular, (4.1) becomes completely multiplicative when

 (4.2) λf(m)λf(n)=λf(mn).

For any , we have Deligne’s bound

 |λf(n)|≤τ(n),

and when is squarefree, it is known that ([11, (2.24)])

 λf(n)2=1n, if n|N.

### 4.2. Automorphic L-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

 L(s,f)=∞∑n=1λf(n)n−s.

This has an Euler product with local factors

 Lp(s,f)=(1−λf(p)p−s+\chiup0(p)−2s)−1,

where is the principal character modulus . The gamma factor is

 γ(s,f)=π−sΓ(s2+k−14)Γ(s2+k+14).

The complete product is entire and satisfies the functional equation

 Λ(s,f)=εfΛ(1−s,f),

with root number where is the eigenvalue of the Atkin-Lehner involution . If is squarefree, then

For , the local factors factor further as

 Lp(s,f)=(1−αf(p)p−s)−1(1−βf(p)p−s)−1,

where , are complex numbers with and .

For the symmetric square -function is defined by

 L(s,Sym2f)=ζ(2s)Z(s,f),

where denotes the Riemann zeta function and is defined by

 Z(s,f)=∞∑n=1λf(n2)n−s.

This has Euler product with

 Lp(s,Sym2f)=(1−α2f(p)p−s)−1(1−p−s)−1(1−β2f(p)p−s)−1.

The gamma factor is

 γ(s,Sym2f)=π−3s2Γ(s+12)Γ(s+k−12)Γ(s+k2)

The complete product is entire and it satisfies the functional equation

 Λ(s,Sym2f)=Λ(1−s,Sym2f).

We have the convexity bound

 (4.3) L(σ+it,Sym2f)⋘ε((|t|+1)(|t|+k)2)1−σ2+ε,0⩽σ⩽1,

where the implied constant depends only on . Moreover, it is known that [7]

 (4.4) k−ε⋘εL(1,Sym2f)⋘εkε.

According to [3], is also an -function of some automorphic representation of , and the normalized Fourier coefficients are given by

 AF(1,n)=AF(n,1)=∑mℓ2=nλf(m2),
 AF(m,n)=AF(−m,n)=AF(m,−n)=AF(−m,−n),
 AF(m,0)=AF(0,n)=0,

and the Hecke relations ([18, (7.7)])

 (4.5) AF(m,n)=∑d|(m,n)μ(d)AF(m/d,1)AF(1,n/d).

We have

 L(s,Sym2f)=L(s,F)=∞∑n=1AF(1,n)n−s.

Let and with squarefree, an even positive integer and a positive integer. We define the Rankin-Selberg -function

 L(s,Sym2f⊗g)=∞∑m=1∞∑n=1AF(m,n)λg(n)(m2n)−s.

This has Euler product with local factor

 Lp(s,Sym2f⊗g)=(1−α2f(p)αg(p)p−s)−1(1−αg(p)p−s)−1(1−β2f(p)αg(p)p−s)−1(1−α2f(p)βg(p)p−s)−1(1−βg(p)p−s)−1(1−β2f(p)βg(p)p−s)−1

if , and

 Lp(s,Sym2f⊗g)=(1−α2f(p)λg(p)p−s)−1(1−λg(p)p−s)−1(1−β2f(p)λg(p)p−s)−1

if . The gamma factor is

 (4.6)

The complete product is entire and it satisfies the functional equation

 Λ(s,Sym2f⊗g)=ε(Sym2f⊗g)Λ(1−s,Sym2f⊗g),

with root number given by

 (4.7) ε(Sym2f⊗g)={(−1)κ+1μ(N)λg(N)√N, if k>κ,(−1)κμ(N)λg(N)√N, % if k⩽κ.

It follows from the Hecke relations (4.5) and (4.2) that

 (4.8) L(s,Sym2f⊗g)=∞∑d=1μ(d)d−3s∞∑m=1AF(m,1)m−2s∞∑n=1AF(1,n)λg(dn)n−s=L(2s,Sym2f)∞∑d=1μ(d)d−3s∞∑n=1AF(1,n)λg(dn)n−s=L(2s,Sym2f)LN(3s,g)−1∑(d,N)=1μ(d)d−3s∞∑n=1AF(1,n)λg(dn)n−s,

where denotes the finite Euler product

 LN(s,g)=∏p|N(1−λg(p)p−s)−1.

#### Computation of the γ-factor and the ε-factor of Sym2f⊗g

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

 ρk(reiθ)=(ei(k−1)θe−i(k−1)θ)ρk(j)=((−1)k−11)

The -factor and the -factor . See [13]. With some matrix calculations we have

 Sym2(ρk)⊗ρ2κ=ρ2κ⊕ρ|2k−2κ−1|+1⊕ρ2k+2κ−2.

This implies the formula (4.6) for the -factor , and the -factor at is

 (4.9)

For any prime , the local component is an unramified principle series representation of with trivial central character, so

 (4.10)

Multiplying (4.9) and (4.10) yields the formula (4.7) for the -factor .

### 4.3. Approximate functional equation

In view of (4.8), we have the following approximate functional equation (see [10, Theorem 5.3, Proposition 5.4])

 (4.11)

where is a smooth function on defined by

 12πi∫(3)(cos(πu4A))−24Aγ(12+u,Sym2f⊗g)γ(12,Sym2f⊗g)L(1+2u,Sym2f)LN(32+3u,g)y−uduu,

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