Hybrid Level Aspect Subconvexity for GL(2)×GL(1) Rankin-Selberg L -Functions
aa r X i v : . [ m a t h . N T ] M a r HYBRID LEVEL ASPECT SUBCONVEXITY FOR GL (2) × GL (1) RANKIN-SELBERG L -FUNCTIONS KESHAV AGGARWAL, YEONGSEONG JO, AND KEVIN NOWLAND
Abstract.
Let M be a squarefree positive integer and P a prime number coprime to M such that P ∼ M η with 0 < η < /
5. We simplify the proof of subconvexity bounds for L ( , f ⊗ χ ) when f is a primitiveholomorphic cusp form of level P and χ is a primitive Dirichlet character modulo M . These bounds areattained through an unamplified second moment method using a modified version of the delta method dueto R. Munshi. The technique is similar to that used by Duke-Friedlander-Iwaniec save for the modificationof the delta method. In studying the subconvexity problem for character twists of holomorphic modular forms of full level, Duke-Friedlander-Iwaniec [6] introduced a simple yet powerful decomposition of the delta symbol which detectswhen an integer is zero. The starting point for their method was an amplified second moment average overprimitive Dirichlet characters of a given level. The subconvexity problem for twisted L -functions L ( s, f ⊗ χ )in the conductor-aspect was also solved in their paper for the first time for f a holomorphic cusp form of levelone with subconvexity exponent 1 / − /
22 (following the computation of section 4.3 in [15] for example).Recent works achieving hybrid subconvexity bounds for Rankin-Selberg convolution L -functions of largelevel include Kowalski-Michel-Vanderkam [13], Michel [14] and Harcos-Michel [8]. Michel-Venkatesh [16]solve the subconvexity problem for the L -functions of GL (1) and GL (2) automorphic representations over afixed number field, uniformly in all aspects. Holowinsky-Munshi [10] prove a hybrid level aspect bound forthe L -function coming from the convolution of two holomorphic modular forms of nontrivial levels, one beingsquarefree and the other being prime. Z. Ye [19] relaxed the level conditions in [10] to both levels square-free.Moreover, he used a Large Sieve inequality to establish a subconvexity bound for the full range of levelswhen both forms are holomorphic. The works of Holowinsky-Munshi and Z. Ye relied on an application ofHeath-Brown’s refinement [9] of the classical delta method due to Duke-Friedlander-Iwaniec [6]. Browning-Munshi in [2] introduce a modification of the delta method with factorization moduli to obtain a structuraladvantage.In this paper we follow the work of Duke-Friedlander-Iwaniec [6] who studied the Rankin-Selberg con-volution of a primitive Dirichlet character with a holomorphic modular form for the full modular group.Their paper, as ours, uses a second moment average over primitive Dirichlet characters of a given level.In this paper, we allow for holomorphic forms with a range of permissible but nontrivial levels relative tothe conductor of the Dirichlet character. It is here that we make use of the modified delta method with aconductor lowering trick, based on the work of Munshi in [17].Let f be a primitive holomorphic cusp form of level P and χ a primitive Dirichlet character modulo M .The Rankin-Selberg convolution L ( s, f ⊗ χ ) is given by L ( s, f ⊗ χ ) = X n ≥ λ f ( n ) χ ( n ) n s , at least for Re( s ) sufficiently large. Subconvexity for these Rankin-Selberg L -functions has already beenestablished in [3, 6, 7, 15]. The main point of this paper, however, is to demonstrate how using a modifiedapplication of the delta method simplifies arithmetic structure and lengthens the admissible hybrid range ofthe level parameters M and P . Specifically, we present a simple method which ends with a trivial applicationof the Weil bound for Kloosterman sums and establishes subconvexity for the hybrid range P = M η for0 < η < /
5. Using the classical delta method without the conductor lowering trick and following the same
Mathematics Subject Classification.
Key words and phrases.
Special values of L -functions, Rankin-Selberg convolution, subconvexity, δ -method. process, one would obtain a hybrid range of 0 < η < /
7. In the meanwhile, Blomer and Harcos in theorem2 of [1] establish the following hybrid estimate L (cid:18) , f ⊗ χ (cid:19) ≪ k,ε P + ε M + ε + P + ε M + ε . For P = M η , we obtain that L (cid:18) , f ⊗ χ (cid:19) ≪ k,ε Q + ε (cid:16) Q − η ) − − η η ) (cid:17) , where Q = Q ( f ⊗ χ ) = P M is the size of the conductor of L -function L ( s, f ⊗ χ ). The hybrid range0 < η < < η < /
5. However we emphasize that ourtechnique does not require amplification or Large Sieve inequality and recently this method is adopted in[11] to extend a hybrid subconvexity range bound for L (1 / , g ⊗ h ) where g is a primitive holomorphic cuspform of level M and h is a primitive either holomorphic or Mass cusp form of level P with ( M, P ) = 1, M asquarefree integer, and P a prime.Of course, one has the ability to push the analysis further, in either method, by analyzing the resultingsum of Kloosterman sums through Large Sieve inequality similar to the work of Z. Ye. Again, however,there is an advantage in the modified delta method in that we obtain sums of standard Kloosterman sumsfor the group Γ ( P ) associated to the cusp at ∞ . Without the modified delta method, one would insteadget Kloosterman sums associated to the cusps 0 and ∞ and then more work is required (using the work ofDeshoullier-Iwaniec [5] for example).We provide a sketch of these arguments below and note that our methods may also be applied to analogousRankin-Selberg convolutions. Let
P > k > S k ( P ) be the linear space of holomorphiccusp forms of weight k , level P , and trivial nebentypus. We let Γ ( P ) be the Hecke congruence subgroupsdefined by Γ ( P ) = (cid:26)(cid:18) a bc d (cid:19) ∈ SL (2 , Z ) (cid:12)(cid:12)(cid:12)(cid:12) c ≡ P ) (cid:27) If f ∈ S k ( P ), then f : H → C is holomorphic and satisfies f ( gz ) = ( cz + d ) k f ( z )for every g = (cid:18) a bc d (cid:19) ∈ Γ ( P ) acting as a linear fractional transformation on z in the upper half-plane H .Additionally, f vanishes at every cusp. Such an f has Fourier expansion f ( z ) = ∞ X n =1 ψ f ( n ) n k − e ( nz ) , where e ( x ) := exp(2 πix ), and the Fourier coefficients ψ f ( n ) satisfy ψ f ( n ) ≪ f τ ( n ) n ε , with τ ( n ) the divisor function for ε > S k ( P ) is a finite dimensional Hilbert space with respect to the Petersson inner product. S k ( P ) has anorthogonal basis B k ( P ) which consists of eigenfunctions of all Hecke operators T n such that ( n, P ) = 1,where T n acts on f by the formula T n f ( z ) = 1 √ n X ad = n ( a,P )=1 (cid:16) ad (cid:17) k/ X b (mod d ) f (cid:18) az + bd (cid:19) =: λ f ( n ) f. Such an f is called a Hecke eigen cusp form . The Hecke operators are multiplicative and satisfy ψ f ( m ) λ ( n ) = X d | ( m,n ) ψ f (cid:16) mnd (cid:17) for n coprime with P . In particular, ψ f (1) λ f ( n ) = ψ f ( n ) for ( n, P ) = 1. We normalize such that ψ f (1) = 1and we have λ f ( n ) = ψ f ( n ) for ( n, P ) = 1. There exists a subset B ∗ k ( P ) of B k ( P ) of newforms or primitive YBRID LEVEL ASPECT SUBCONVEXITY FOR GL (2) × GL (1) RANKIN-SELBERG L -FUNCTIONS 3 holomorphic cusp forms which are eigenfunctions of all Hecke operators T n for n ≥ λ f ( n ) = ψ f ( n ).Let f ∈ B ∗ k ( P ) be a newform and χ a primitive Dirichlet character modulo M with ( P, M ) = 1. Let f ⊗ χ be a twisted modular form on H given by the Fourier expansion( f ⊗ χ )( z ) = ∞ X n =1 χ ( n ) λ f ( n ) n k − e ( nz ) . Then f ⊗ χ is a newform of level P M . L − functions Let f ∈ B ∗ k ( P ) be a newform and χ a primitive Dirichlet character modulo M where M and P arecoprime, M is squarefree, and P is a prime. Then the Rankin-Selberg convolution L -function associated to f ⊗ χ is L ( s, f ⊗ χ ) = ∞ X n =1 λ f ( n ) χ ( n ) n s . The associated completed L -function isΛ( s, f ⊗ χ ) = Q s L ∞ ( s, f ⊗ χ ) L ( s, f ⊗ χ ) , where Q = Q ( f ⊗ χ ) = P M and the local factor at infinity L ∞ is a product of gamma functions. Theapproximate functional equation shows that the special value L (1 / , f ⊗ χ ) is given by L (1 / , f ⊗ χ ) = ∞ X n =1 λ f ( n ) χ ( n ) √ n V (cid:18) n √Q (cid:19) + ǫ ( f ⊗ χ ) ∞ X m =1 λ f ( m ) χ ( m ) √ m V (cid:18) m √Q (cid:19) (see chapter 5 of [12]) where V is a smooth function with rapid decay at infinity, and for any positive integer A , the derivatives of V ( y ) satisfy y j V ( j ) ( y ) ≪ k Q ε (1 + y ) − A log(2 + y − )for any ε >
0. We also have the asymptotic V ( y ) = 1 + O (cid:18)(cid:18) y √Q (cid:19) α (cid:19) for α >
0. Let h be a smooth function which is compactly supported on [1 / , /
2] with bounded derivativesand suppose that X runs over 2 ν with ν = − , , , , , . . . . Applying a smooth partition of unity and theasymptotic for V , we are left with | L (1 / , f ⊗ χ ) | ≪ k,ε Q ε ( max Q − δ ≤ X ≤Q
12 + ε | L f ⊗ χ ( X ) | + Q − δ ) , where(1) L f ⊗ χ ( X ) := X n λ f ( n ) χ ( n ) √ n h (cid:16) nX (cid:17) . We prove the hybrid range and subconvexity bounds for L (1 / , f ⊗ χ ). We average over primitive charactersmodulo M through a second moment method to achieve subconvexity bounds. Theorem 2.1 (Second Moment) . Let P and M be coprime positive integers with P prime and M squarefree.Let k be a fixed positive even integer. Set Q = P M . Let h be a smooth function with support in [1 / , / and bounded derivatives. Let ε, δ > and choose any X such that Q − δ ≤ X ≤ Q + ε . If f ∈ B ∗ k ( P ) and χ is a primitive Dirichlet character modulo M , then (2) 1 ϕ ⋆ ( M ) X ⋆χ (mod M ) | L f ⊗ χ ( X ) | ≪ k,ε Q ε Q M · P + δ M − δ ! . KESHAV AGGARWAL, YEONGSEONG JO, AND KEVIN NOWLAND P ⋆ means summation over primitive characters or over integers coprime with the specified modulus,and ϕ ⋆ is the number of primitive multiplicative characters modulo M . We apply the theorem below to F ( x, y ) = h ( x ) h ( y ), f = f , and X = Y for any Q − δ ≤ X ≤ Q + ε to obtain a second moment bound.The first and second terms on the right hand side in (2) come from a zero shift and theorem 2.2 on shiftedconvolution sums, respectively. Notice that if P = 1, then this matches the bound in [6] without amplification. Theorem 2.2 (Shifted Convolution Sums) . Let f , f be newforms in B ∗ k ( P ) . Let M be a positive squarefreeinteger coprime with P . Let r = 0 be an integer coprime with P . Let X, Y ≥ and F a smooth functionsupported on [1 / , / × [1 / , / satisfying x i y j ∂ i ∂x i ∂ j ∂y j F (cid:16) xX , yY (cid:17) ≪ ZZ ix Z jy , for Z > and Z x , Z y ≥ . Then X n X m = n + rM λ f ( n ) λ f ( m ) √ nm F (cid:16) nX , mY (cid:17) ≪ k,ε ( P M r ) ε Z ( Z x Z y ) max { Z x , Z y } P max { X, Y } ( XY ) . (3)The second moment bound which is better by a power of Q than the convexity bound means that wemust have a bound which is better than1 ϕ ⋆ ( M ) X ⋆χ (mod M ) | L f ⊗ χ ( X ) | ≪ Q + ε M .
Therefore the bounds in (2) induce the hybrid subconvexity range P ∼ M η for 0 < η < / Corollary 2.3 (Subconvexity) . Let f and χ be as above with η = log P log M . Then L (1 / , f ⊗ χ ) ≪ k,ε Q + ε Q − η η ) . This produces a subconvex bound for < η < / .Proof. From the reduction in section 1,(4) | L (1 / , f ⊗ χ ) | ≪ k,ε Q ε ( Q − δ + max Q − δ ≤ X ≤Q
12 + ε | L f ⊗ χ ( X ) | ) . To bound this, weaken the bound in the above theorem by just taking the second term in (2) as the firstprovides saving for any P and M having size. This gives | L f ⊗ χ ( X ) | ≪ Q + ε P δ M − δ . Equate the first term and second terms on the right hand side of (4) while replacing P and M with powersof Q by using P = M η . We have P = Q η η and M = Q η . Therefore the optimal choice of δ satisfies − δ δ · η η − − δ ·
12 + η .
Then the saving δ is then explicitly calculated to be δ = − η η ) . (cid:3) We give a quick sketch of the proof which results in the 0 < η < / < η < / L -function to theanalysis of the following sum X n ∼√ P M λ f ( n ) χ ( n ) √ n . YBRID LEVEL ASPECT SUBCONVEXITY FOR GL (2) × GL (1) RANKIN-SELBERG L -FUNCTIONS 5 A second moment average over primitive characters leads us to needing to understand the sum X n ∼√ P M λ f ( n ) √ n X m ∼√ P Mm ≡ n (mod M ) λ f ( m ) √ m . Using the delta symbol, we rewrite the above as X = | r |≤√ P X n ∼√ P M λ f ( n ) √ n X m ∼√ P M λ f ( m ) √ m δ ( m − n + rM, , where the r = 0 term can be bounded trivially. Applying the usual delta method and using Voronoisummation on both the n - and m -sums, one can obtain Kloosterman sums of the form(5) X = | r |≤√ P Q X q ≤ Q q X n ∼ P λ f ( n ) √ n X m ∼ P λ f ( m ) √ m S ( rM, ( n − m ) ¯ P ; q ) . with Q = M / P / the square root of the size of the equation. Applying the Weil bound for each Klooster-man sum leads to the of second moment average being bounded by √ P (cid:16) P M (cid:17) / , such that 0 < P < M / is a range for subconvexity, as a bound of √ P would produce the convexity bound.However, using the conductor lowering trick by instead using δ (( n − m + rM ) /P,
0) with the condition n − m + rM ≡ P ) followed by Voronoi summation as in the previous case, one instead obtains thesum of Kloosterman sums(6) X = | r |≤√ P Q X q ≤ Q qP X n ∼ P λ f ( n ) √ n X m ∼ P λ f ( m ) √ m S ( rM, n − m ; qP ) . The length of the n - and m -sums is the same as before, whereas we have additional saving P in the denom-inator on the q -sum, which comes from detecting the congruence condition n − m + rM ≡ P ). TheWeil bound for Kloosterman sums allows us to have a better bound by P / . However, Q has also changedand is now M / /P / as the equation in the delta symbol has changed and the division by Q – which endsup a division by Q / because of the Weil bound – means we only save P / . Dividing the previous boundby this gives √ P (cid:16) P M (cid:17) / . The subconvexity range has been improved to 0 < P < M / .Let σ be the element in Γ ( P ). If a is a cusp of Γ ( P ), its stabilizer is defined by Γ a = { σ ∈ Γ ( P ) | σ · a = a } . In particular Γ ∞ = (cid:26) ± (cid:18) b (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) b ∈ Z (cid:27) . Let a and b be two cusps of Γ ( P ). Let σ a ∈ SL (2 , R ) be ascaling matrix such that σ a · ∞ = a and σ − a Γ a σ a = Γ ∞ . We define similarly for σ b . The Kloosterman sumattached to two cusps a and b is defined in [5] as S ab ( α, β ; c ) = X a ∗ c d ∈ Γ ∞ \ σ − a Γ ( P ) σ b / Γ ∞ e (cid:18) αa + βdc (cid:19) . Returning back to our analysis, we obtain the standard Kloosterman sum associated simply to the cusp at ∞ for the group Γ ( P ) in (6). Previously, P was attached to ( n − m ) and not the modulus q , which gave aKloosterman sum associated with the cusps at 0 and ∞ for Γ ( P ) in (5).Finally, we remark that we can improve the hybrid range further by using a Large Sieve inequality asin [5] to estimate the sums of Kloosterman sums instead of bounding them individually. Even using theKuznetsov formula with a Large Sieve inequality would improve the subconvexity estimate if not the rangeof subconvexity. One could also introduce an amplification to improve the subconvexity bound. Sinceour purpose is simply to demonstrate the utility of the modified delta method in improving the range ofsubconvexity and simplifying the Kloosterman sum structure, we do not continue the argument in theseways. KESHAV AGGARWAL, YEONGSEONG JO, AND KEVIN NOWLAND
Before beginning the proof of the theorem, we collect several lemmas which will be used in the proof. Thecrux of the work is an application of the delta method, which we state below. The delta method was used in[6]. This is a decomposition of the δ -symbol via a character sum. We use the version given by Heath-Brown[9]. Lemma 4.1 (The delta method, [9]) . For any
Q > there exist c Q > and a smooth function g ( x, y ) defined on (0 , ∞ ) × R , such that δ ( n,
0) = c Q Q ∞ X q =1 X ⋆a mod q e (cid:18) anq (cid:19) g (cid:18) qQ , nQ (cid:19) . The constant c Q satisfies c Q = 1 + O A ( Q − A ) for any A > . Moreover, g ( x, y ) ≪ x − for all y , and g ( x, y ) is non-zero only for x ≤ max { , | y |} . If x ≤ and | y | ≤ x , then x i ∂ i ∂x i g ( x, y ) ≪ x − , ∂∂y g ( x, y ) = 0 . If | y | > x , then x i y j ∂ i ∂x j ∂ j ∂y j g ( x, y ) ≪ x − . Let k be a positive integer. We recall elementary properties of J -Bessel functions as can be seen in [18].The J -Bessel function is defined by J k ( x ) = e ix W k ( x ) + e − ix W k ( x )where W k ( x ) = e i ( π k − π ) Γ( k + ) r πx Z ∞ e − y (cid:18) y (cid:18) iy x (cid:19)(cid:19) k − dy. Applying the asymptotic expansion for Whittaker functions W for x ≫
1, we have x j W ( j ) k ( x ) ≪ k,j x ) with j ≥
0. Using the Taylor series expansion for x ≪
1, we obtain the bound x j J ( j ) k ( x ) ≪ k,j x k with j ≥
0. The J -Bessel function is used in the integral transform found in Voronoi summation. Lemma 4.2 (Voronoi summation, [13]) . Let ( a, q ) = 1 and let h be a smooth function compactly supportedin (0 , ∞ ) . Let f be a holomorphic newform of level P and weight k . Set P := P/ ( P, q ) . Then there existsa complex number η of modulus 1 (depending on a, q, f ) and a newform f ∗ of the same level P and sameweight k such that X n λ f ( n ) e (cid:18) n aq (cid:19) h ( n ) = 2 πηq √ P X n λ f ∗ ( n ) e (cid:18) − n aP q (cid:19) Z ∞ h ( y ) J k − (cid:18) π √ nyq √ P (cid:19) dy. In general, given e (¯ a/q ) the overline on the numerator a indicates the multiplicative inverse modulo thedenominator q . In order to truncate and bound the sums which result after Voronoi summation, we use thefollowing lemma. Lemma 4.3.
Let k, M, P be positive integers with k ≥ and let r be a nonzero integer. Take Q > and X, Y ≥ . For any a, b > , define J ( a, b ; c ) := Z ∞ Z ∞ √ xy F (cid:16) xX , yY (cid:17) g (cid:18) qcQ , x − y + rMP Q (cid:19) J k − (4 πa √ x ) J k − (4 πb √ y ) dxdy, where g (cid:16) qcQ , x − y + rMP Q (cid:17) is the function in lemma 4.1 and F is a smooth function supported in [1 / , / × [1 / , / with partial derivatives satisfying x i y j ∂ i ∂x i ∂ j ∂y j F ( x, y ) ≪ ZZ ix Z jy YBRID LEVEL ASPECT SUBCONVEXITY FOR GL (2) × GL (1) RANKIN-SELBERG L -FUNCTIONS 7 for some Z > , Z x , Z y ≫ . We have J ( a, b ; c ) ≪ Z √ XY Qqc a √ X ) / b √ Y ) / (cid:20) a √ X (cid:18) Z x + XqcQP (cid:19)(cid:21) i (cid:20) b √ Y (cid:18) Z y + YqcQP (cid:19)(cid:21) j . (7) Also, (8) J ( a, b ; c ) ≪ Zab (1 + a √ X ) / (1 + b √ Y ) / Qqc min n Z x b √ Y , Z y a √ X o ( XY | r | M QqP ) ε . Proof.
Starting with the change of variables x xX and y yY and then integrating by parts and usingthe J -Bessel function bound gives (7). For example, integrating by parts once in the x integral leads to J ( a, b ; c ) ≪ Z √ XY Qqc a √ X ) / b √ Y ) / (cid:20) a √ X ( Z x + XI ) (cid:21) , where I := Z / / Z / / | xX − yY + rM | >qQP | xX − yY + rM | dxdy. The condition on the integral comes from where the g function is nonzero. However, we have | xX − yY + rm | − ≪ ( qcQP ) − , which gives (7) for i = 1 and j = 0. Repeated integration by parts gives the sameresult for higher values of i and j .For (8), we treat the integral I differently. Let u = xX − yY rM to get I ≪ X Z / / Z ( X + Y + | r | M ) qQP/ u dudy ≪ ( XY | r | M qQP ) ε X .
Doing the same thing with i = 0 and j = 1 and taking the minimum of the two bounds finishes the proof. (cid:3) Let ε, δ > X such that Q / − δ ≤ X ≤ Q / ε . Define S f ( X ) := 1 ϕ ⋆ ( M ) X ⋆χ (mod M ) | L f ⊗ χ ( X ) | , where L f ⊗ χ ( X ) is given in (1). The proof of the reduction to shifted sums is similar to [6]. We open thesquare and write the primitive characters in terms of Gauss sums to obtain S f ( X ) = 1 M ϕ ⋆ ( M ) X ⋆χ (mod M ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X b (mod M ) ¯ χ ( b ) X n λ f ( n ) √ n e (cid:18) nbM (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Adding the nonprimitive characters to this sum produces the summation over all characters modulo M . Bythe orthogonality of multiplicative characters, S f ( X ) ≤ ϕ ( M ) M ϕ ⋆ ( M ) X ⋆b (mod M ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n λ f ( n ) √ n e (cid:18) nbM (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We extend the summation to all residue classes M by adding the residues which are not coprime with M ,and open the square. Using the orthogonality of additive characters, S f ( X ) ≪ M ε X n λ f ( n ) √ n h (cid:16) nX (cid:17) X m ≡ n (mod M ) λ f ( m ) √ m h (cid:16) mX (cid:17) . Write m = n + rM and note that r ≪ XM ≤ Q
12 + ε M . The diagonal term m = n satisfies X n λ f ( n ) n h (cid:16) nX (cid:17) ≪ Q ε . KESHAV AGGARWAL, YEONGSEONG JO, AND KEVIN NOWLAND
We are left to consider the off-diagonal terms R f ( X ) := X = | r |≪ Q / εM X n λ f ( n ) √ n h (cid:16) nX (cid:17) X m = n + rM λ f ( m ) √ m h (cid:16) mX (cid:17) . Let
X, Y ≥
1. Motivated by the reduction of the second moment problem to bounding the shiftedconvolution sums, we now take f , f to newforms in B ∗ k ( P ) and consider S f ,f ( X, Y ) := X n X m = n + rM λ f ( n ) λ f ( m ) √ nm F (cid:16) nX , mY (cid:17) , where r is a nonzero integer coprime to P (valid in our specific application, where | r | ≪ P / ε ). In thissection we establish theorem 2.2. We start with detecting the equation m = n + rM in S f ,f ( X, Y ). To do this, we note that n − m + rM = 0is equivalent to ( n − m + rM ) /P = 0 and n − m + rM ≡ P . Therefore, S f ,f ( X, Y ) =
X X n − m + rM ≡ P ) λ f ( n ) λ f ( m ) √ nm F (cid:16) nX , mY (cid:17) δ (cid:18) n − m + rMP , (cid:19) . Using lemma 4.1 to detect ( n − m + rM ) /P = 0 and a sum of additive characters to detect the congruencegives S f ,f ( X, Y ) = X n λ f ( n ) √ n X m λ f ( m ) √ m F (cid:16) nX , mY (cid:17) P X b (mod P ) e (cid:18) b ( n − m + rM ) P (cid:19) × c Q Q ∞ X q =1 X ⋆a (mod q ) e (cid:18) a ( n − m + rM ) P q (cid:19) g (cid:18) qQ , n − m + rMP Q (cid:19) = c Q P Q ∞ X q =1 X ⋆a (mod q ) X b (mod P ) e (cid:18) rM ( a + bq ) qP (cid:19) X n λ f ( n ) √ n e (cid:18) n ( a + bq ) qP (cid:19) × X m λ f ( m ) √ m e (cid:18) − m ( a + bq ) qP (cid:19) F (cid:16) nX , mY (cid:17) g (cid:18) qQ , n − m + rMP Q (cid:19) . From lemma 4.1, g (cid:16) qQ , n − m + rMP Q (cid:17) = 0 for qQ ≤ max n , | n − m + rM | P Q o . We want to choose Q such that themax is always 1. To do this, note that2 | n − m + rM | P Q ≤ X + Y + | r | MP Q . For there to be any solutions to m − n + rM = 0, we need | r | ≤ ( X + Y ) /M . Therefore it is sufficient tochoose Q such that 8 max { X, Y } P Q ≤ . Set Q := 8 max { X,Y } P so that the outer q -sum only extends to Q . We write γ = a + bq . Since the a - and b -sums yield a complete set of residue γ modulo qP with ( γ, q ) = 1, S f ,f ( X, Y ) reduces to c Q P Q Q X q =1 X γ (mod qP )( γ,q )=1 e (cid:18) rM γqP (cid:19) X n λ f ( n ) √ n e (cid:18) nγqP (cid:19) X m λ f ( m ) √ m e (cid:18) − mγqP (cid:19) F (cid:16) nX , mY (cid:17) g (cid:18) qQ , n − m + rMP Q (cid:19) . YBRID LEVEL ASPECT SUBCONVEXITY FOR GL (2) × GL (1) RANKIN-SELBERG L -FUNCTIONS 9 We apply Voronoi summation to S f ,f ( X, Y ) in the m -sum, then in the n -sum. Since we are dealingwith forms of prime level P , we break the q -sum above as S f ,f ( X, Y ) = S + T , where S is the sum over( q, P ) = 1 and T is the sum over P | q .6.2.1 ( q, P ) = 1 caseAssume that ( q, P ) = 1. We split the γ -sum in S as S = S + S , where S is the sum over γ for which( P, γ ) = 1 and S is the sum over γ for which P | γ . When ( P, γ ) = 1, we get the sum S := c Q P Q Q X q =1( q,P )=1 X ⋆γ (mod qP ) e (cid:18) rM γqP (cid:19) X n λ f ( n ) √ n e (cid:18) nγqP (cid:19) × X m λ f ( m ) √ m e (cid:18) − mγqP (cid:19) F (cid:16) nX , mY (cid:17) g (cid:18) qQ , n − m + rMP Q (cid:19) . For each q , applying lemma 4.2 first in the n -sum and then in the m -sum results in1 P q X n λ f ∗ ( n ) e (cid:18) − nγP q (cid:19) X m λ f ∗ ( m ) e (cid:18) mγP q (cid:19) J (cid:18) √ nP q , √ mP q ; 1 (cid:19) , where J is again the function given in lemma 4.3. This produces a Kloosterman sum modulo P q :1 Q P Q X q =1( q,P )=1 q X n λ f ∗ ( n ) X m λ f ∗ ( m ) J (cid:18) √ nP q , √ mP q ; 1 (cid:19) S ( rM, m − n ; P q ) . Using the first bound of the J -function in lemma 4.3 allows us to truncate the n - and m -sums to the ranges n ≤ T := P q X (cid:18) Z x + XqQP (cid:19) , m ≤ T := P q Y (cid:18) Z y + YqQP (cid:19) . Applying the Weil bound and the second bound in lemma 4.3, we see that S ≪ Q ε Q P Q X q =1( q,P )=1 q X n ≤ T X m ≤ T Z √ nP q √ mP q (cid:16) √ nXP q (cid:17) (cid:16) √ mYP q (cid:17) × Qq min ( Z x √ mYP q , Z y √ nXP q ) ( rM, m − n, P q ) ( P q ) . We bound the minimum by taking the geometric mean and simplify to S ≪ Q ε Z ( Z x Z y ) QP Q X q =1( q,P )=1 q X n ≤ T X m ≤ T nm ) ( rM, m − n, P q ) ≪ Q ε Z ( Z x Z y ) QP Q X q =1( q,P )=1 q ( T T ) ( rM, m − n, P q ) ≪ Q ε Z ( Z x Z y ) XY ) P Q Q X q =1( q,P )=1 q (cid:18) Z x + XqQP (cid:19) (cid:18) Z y + YqQP (cid:19) ( rM, m − n, P q ) . Notice that ( rM, m − n, P q ) ≤ ( rM, P q ) ≤ ( rM, q ), since r and P are coprime, and P does not divide M .Rewriting q as qd with d = ( rM, q ) gives S ≪ Q ε Z ( Z x Z y ) XY ) P Q X d | rM d X q ≤ Q/d ( qd ) (cid:18) Z x + XqdQP (cid:19) (cid:18) Z y + YqdQP (cid:19) ≪ Q ε Z ( Z x Z y ) XY ) P Q X d | rM d (cid:18) Z x + XQ P (cid:19) (cid:18) Z y + YQ P (cid:19) ≪ Q ε r ε Z ( Z x Z y ) XY ) P Q (cid:18) Z x + XQ P (cid:19) (cid:18) Z y + YQ P (cid:19) ≪ Q ε r ε Z ( Z x Z y ) XY ) P Q (cid:18) max { Z x , Z y } + max { X, Y } Q P (cid:19) . With Q = 8 max { X,Y } P , we obtain S ≪ Q ε r ε Z ( Z x Z y ) max { Z x , Z y } P max { X, Y } ( XY ) . This matches the bound in the statement of theorem 2.2.Next we deal with the sum where P | γ . S := c Q P Q Q X q =1( q,P )=1 X γ (mod qP )( γ,q )=1 P | γ e (cid:18) rM γqP (cid:19) X n λ f ( n ) √ n e (cid:18) nγqP (cid:19) × X m λ f ( m ) √ m e (cid:18) − mγqP (cid:19) F (cid:16) nX , mY (cid:17) g (cid:18) qQ , n − m + rMP Q (cid:19) = c Q P Q Q X q =1( q,P )=1 X ⋆γ (mod q ) e (cid:18) rM γq (cid:19) X n λ f ( n ) √ n e (cid:18) nγq (cid:19) × X m λ f ( m ) √ m e (cid:18) − mγq (cid:19) F (cid:16) nX , mY (cid:17) g (cid:18) qQ , n − m + rMP Q (cid:19) . Applying Voronoi summation in the m - and n -sums gives1 P Q Q X q =1 q X n λ f ∗ ( n ) X m λ f ∗ ( m ) J (cid:18) √ nq √ P , √ mq √ P ; 1 (cid:19) S ( rM, ( m − n ) ¯ P ; q ) . The n - and m -sums can be truncated to n ≤ T := P q X (cid:18) Z x + XqQP (cid:19) , m ≤ T := P q Y (cid:18) Z y + YqQP (cid:19) . Repeating the estimation process used for S with these parameters, one is led to S ≪ Q ε r ε Z ( Z x Z y ) max { Z x , Z y } P max { X, Y } ( XY ) . This is better than the bound of theorem 2.2.
YBRID LEVEL ASPECT SUBCONVEXITY FOR GL (2) × GL (1) RANKIN-SELBERG L -FUNCTIONS 11 P | q caseFinally, observing that γ is coprime with P q , the remaining sum over q , when P | q is T := c Q P Q Q X q =1 P | q X ⋆γ (mod P q ) e (cid:18) rM γP q (cid:19) X n λ f ( n ) √ n e (cid:18) nγP q (cid:19) × X m λ f ( m ) √ m e (cid:18) − mγP q (cid:19) F (cid:16) nX , mY (cid:17) g (cid:18) qQ , n − m + rMP Q (cid:19) . After we rewrite q as qP , Voronoi summation in the m -sum then the n -sum gives1 Q P X q ≤ Q/P q X n X m λ f ∗ ( n ) λ f ∗ ( m ) J (cid:18) √ nP q , √ mP q ; P (cid:19) S ( rM, m − n ; P q ) . Lemma 4.3 implies that we can truncate the n - and m -sums to the ranges n ≤ T := P q X (cid:18) Z x + XqQP (cid:19) , m ≤ T := P q Y (cid:18) Z y + YqQP (cid:19) . Using the Weil bound for Kloosterman sums as in the previous case, we see that
T ≪ Q ε r ε Z ( Z x Z y ) max { Z x , Z y } P max { X, Y } ( XY ) . This bound on T is better than the bound on S . This completes the proof of theorem 2.2. Acknowledgments.
This work was completed as an extension of a topics course given by Roman Holowin-sky at The Ohio State University in Spring 2015. We are very grateful to Roman Holowinsky for a carefulreading of the draft. The authors would also like to thank Fei Hou for identifying an error in an earlierversion of this manuscript.
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