Subconvexity Bound for Hecke character L-Functions of Imaginary quadratic Number fields

# Subconvexity Bound for Hecke character L-Functions of Imaginary quadratic Number fields

###### Abstract.

Let be an imaginary number field, be a split odd prime and be a Hecke character of conductor . Let be the associated -function. We prove the Burgess bound in -aspect and a hybrid bound in conductor aspect,

 L(1/2+it,ψ)≪D,ε(1+|t|)3/8+εp1/8

for . In Appendix A, we present the ideas for an elementary proof of Voronoi summation formula for holomorphic cusp forms with CM and squarefree level. This is done by exploiting the lattice structure of ideals in number fields. Voronoi summation for such cusp forms is given by Kowalski, Michel and Vanderkam . We hope that our method of proof can extend their result to any CM cusp form in and arbitrary additive twist. We encounter quadratic and quartic Gauss sums in the process. We shall present the calculations for the general case in the next version of the paper.

Keshav Aggarwal

## 1. Introduction

Let be an imaginary quadratic number field with squarefree. The units in the ring of integers are the roots of unity . Let be an odd split prime, say . By Dedekind’s theorem, the polynomial splits into linear factors. Let be generated by over . A possible isomorphism is given by (when ) or (when ).

Let be a Hecke character of conductor and weight . Then is given by,

 ψ(a+b√−D)=χ(a+bd)(a+b√−D√a2+b2D)r, when −D≡2,3mod4ψ(a+b√−D2)=χ(a+bd)¯¯¯¯χ(2)(a+b√−D√a2+b2D)r, when −D≡1mod4.

where is a primitive Dirichlet character and for .

The -function associated with the Hecke character is given by

 L(s,ψ)=∑a is integal  coprime with pψ(a)(Na)s for Re(s)>1.

Hecke gave a functional equation for this -function and extended it meromorphically to all of the complex plane. Phragmen-Lindelöf principle implies that - the convexity bound. The aim of this paper is to prove the following result.

###### Theorem 1.1.

Let be an imaginary number field with squarefree. Let be an odd split prime, . Let be a Hecke character with weight and conductor prime , and be the associated -function. Then we have

 L(1/2+it,ψ)≪D,ε(1+|t|)3/8+εp1/8+ε

for .

The first such result was obtained by Kaufman  and Sohne , who obtained

 L(1/2+it,ψ)≪(1+|t|)d/6+ε.

Here is a Hecke character of an arbitrary number field and is the degree of . Sohne actually got a stronger result which implies subconvexity in level aspect when the conductor of has a small factor. Diaconu and Garrett  proved a -aspect subconvexity bound for -functions attached to cuspforms for . Their argument uses asymptotics with error term with a power saving for second integral moments over spectral families of twists by Hecke characters . Michel and Venkatesh  used spectral theory and gave a subconvexity bound uniformly in all aspects for -functions attached to cusp forms for and their twists . Wu  followed  and used an amplification technique to get a subconvexity bound uniformly in both conductor and -aspect,

 L(1/2+it,ψ)≪εQ1/4−(1−2θ)/16+ε.

Here is any bound towards the Ramanujan-Petersson conjecture. Very recently, Booker, Milinovich and Ng  proved a -aspect subconvexity bound for -functions of cusp forms ,

 L(1/2+it,f)≪(1+|t|)1/3log|t|.

They thus obtain a bound of strength comparable to Good’s bound for the full modular group. A key innovation in their proof is a general form of Voronoi summation that applies to all fractions, even when the level is not squarefree.

We also require a Voronoi summation formula for any fraction and any level. We worked in parallel to find a way of proof of Voronoi summation formula for holomorphic cusp forms with complex multiplication in . This is discussed in Appendix A. This also happens to be the key innovation in our proof. Though our bound is not as strong as that of , we hope to be able to achieve that by more careful analysis as done in . Moreover, our result gives a hybrid bound in level aspect. We must also note that our method of proof utilizes the algebraic structure of number fields, and we are hopeful to extend this Voronoi type formula to arbitrary number fields. In particular, by following our ideas, one can get a Voronoi formula for the Hecke-Maass cusp forms attached to Hecke charaters of real quadratic fields with arbitrary additive twists. In our knowledge, a Voronoi formula in such generality is yet unknown.

The proof of Theorem 1.1 closely follows the method of proof for -aspect subconvexity for -functions of full level . We have to additionally care about the level and nebentypus.

We shall use Kloosterman’s version of the circle method to detect when an integer equals . Let be any real number. We have,

 (1.1) δ(n=0)=2Re∫10∑∑∑∗1≤q≤Q

for . Here and the on the inner sum means that . is the multiplicative inverse of . The application of the circle method will not be sufficient, and we will apply a conductor lowering trick in both the and the level aspect.

By approximate functional equation,

 L(1/2+it,ψ)≪D,εsupN≪X1+εS(N)N1/2+O((tp)−2017)

where

 (1.2) S(N):=∑a⊂OKψ(a)(Na)−itV(NaN).

It therefore suffices to estimate . Let . We follow Munshi’s approach and write (1.2) as

 (1.3) S(N)=∑n≥1∑m≥1λ(n)m−itV(nN)U(mN)δ(n−mp=0)δ(n≡mmod p).

We use Kloosterman’s delta method with Munshi’s modification  to detect equality. Then where

 (1.4) S±(N)=1K∫10∫R1p∑αmod p∑∑∑∗1≤q

Here, is the unique multiplicative inverse of inside the interval and need not be coprime with . and are smooth functions supported on and respectively. is equal to on . Due to the symmetry between and , the estimates on both are exactly the same modulo some change of signs. Therefore we only consider . Separating the and -sums,

 (1.5)

It will turn out that the optimal choice of is (and thus lowering the conductor by ).

We will take

 (1.6) t3/4p1/4≪N

In this range, we will establish the following bound.

###### Proposition 1.2.

For and , we have

 (1.7) S+(N)N1/2≪t1/2+εp1/4(K1/2p1/4N1/2+1K1/4p1/4).

Same bound holds for , and consequently for . The optimal choice of is therefore . With this choice of , . For , the trivial bound is sufficient. This follows by applying Cauchy’s inequality to the -sum followed by Lemma 2.2 (Ramanujan bound on average). Theorem 1.1 then follows from Lemma 2.2 and Proposition 1.2.

Our writing style is expository and we shall justify our approach in various remarks.

### 1.1. Proof Sketch of Theorem 1.1

We briefly explain the steps of the proof and provide heuristics in this subsection. The calculations are parallel to our previous work . The circle method is used to separate the sums on and , and we arrive at (1.5). Trivial estimate gives . For simplicity let . We are required to save and a little more in a sum of the form

 ∫2KK1p∑bmod p∑q≍Q⋆⋆∑Q

The sum over has ‘conductor’ . Roughly speaking, the conductor takes into account the arithmetic modulus , with the size of oscillation of the analytic weight. If we assume , then the size of the oscillation is , so the extra oscillation of does not hurt us here. Poisson summation changes the length of summation to , and contributes a factor of along with a congruence condition mod and an oscillatory integral. The oscillatory integral saves us . In all, we will save in this step. So far the saving is independent of . The next step is to apply Voronoi summation to the -sum. We need to save in a sum of the form

 ∫2KK∑q≍Q∑(m,q)=1|m|≪Qpt/N((t+v)aqp(x−ma))−i(t+v)∑n≍Nλ(n)e(nmqp)nive(−nxaqp)dv,

where is the unique multiplicative inverse of in the range . Since the -sum involves Fourier coefficients, the ‘conductor’ for the -sum would be . The new length of sum would be . Voronoi summation would contribute a factor of , a dual additive twist and an oscillatory weight function. The oscillation in the weight function would save us . In all, we will save . If is large, we are actually making the bound worse. We are therefore left to save in . Using stationary phase analysis, we will be able to save in the integral over . At this point, seems to be hurting more than helping. The final step is to get rid of the oscillations by using the Cauchy inequality and then changing the structure using Poisson summation. After Cauchy, the sum roughly looks like

 [∑n≪Kp∣∣∣∑q≍Q|m|≍Qpt/N(m,q)=1e(−nmqp)∫K−Kn−iτg(q,m,τ)dτ∣∣∣2]1/2

where is an oscillatory weight function of size . The next steps would be to open the absolute value squared and, apply Poisson to the -sum and analyze the -integral. The -integral gives us a saving of . After Cauchy and Poisson summation, we will save in the diagonal term and in the off-diagonal term. The saving over the convexity bound in the diagonal terms is . The saving over the convexity bound from the off-diagonal terms is . We will therefore get maximum saving when , that is . This gives us a saving of over the - aspect convexity bound of . Matching this with the trivial bound for gives us the Burgess-type bound in the -aspect and a subconvex bound in the level aspect for .

## 2. Gl(2) Voronoi formula and Stationary Phase Method

Let be a CM holomorphic cusp form of weight (an integer) and level . Let be the Fourier coefficients of . The nebentypus is induced from a character of level dividing . For coprime integers and , let , and be coprime with . Let be a smooth function compactly supported on , and let be its Mellin transform. An application of the functional equation of , followed by unwinding the integral and shifting the contour gives the Voronoi summation formula .

###### Lemma 2.1.

When ,

 (2.1) ∑n≥1λ(n)e(nmq)F(n)=1qχL1(−¯¯¯¯¯m)χL2(−q)η(L2)√L2∑n≥1λL2(n)e(−n¯¯¯¯¯¯¯¯¯¯¯mL2q)×∫∞0F(x)[2πikJk−1(4π√nxq√L2)]dx.

By arguments given in Appendix A, the Voronoi formula in the case is not much different. Let . Then,

 (2.2) ∑n≥1λ(n)e(nmq)F(n)=1qχL1(−¯¯¯¯¯m)χL2(−q1)η(L2)√L2∑n≥1λL2(n)e(−n¯¯¯¯¯¯¯¯¯¯¯¯¯2mL2q1)×∫∞0F(x)[2πikJk−1(4π√nxq√L2)]dx.

For our calculations, we take a step back and use the following representation of as an inverse Mellin transform,

 (2.3) Jk−1(x)=1212πi∫(σ)(x2)−sΓ(s/2+(k−1)/2)Γ(1−s/2+(k−1)/2) for 0<σ<1.

We would also need the following bound, which is the Ramanujan conjecture on average. It follows from standard properties of Rankin-Selberg -functions and is well known.

###### Lemma 2.2.

We have,

 ∑n≤x|λ(n)|2≪f,εx1+ε.

We would need that stationary phase analysis done in . The contents of this section are given in Lemmas 2.3 - 2.5 of . For completeness, we mention those here.

###### Lemma 2.3.

Suppose and are smooth real valued functions satisfying

 (2.4) f(i)(x)≪Θf/Ωif,g(j)(x)≪1/Ωjg

for and . Suppose . Define

 (2.5) I=∫bag(x)e(f(x))dx.
1. Suppose and do not vanish in . Let . Then we have

 (2.6) I≪ΘfΩ2fΛ3⎛⎝1+ΩfΩg+Ω2fΩ2gΛΘf/Ωf⎞⎠.
2. Suppose changes sign from negative to positive at the unique point . Let . Further suppose that (2.4) holds for and

 (2.7) f(2)(x)≫Θf/Ω2f

holds. Then

 (2.8) I=g(x0)e(f(x0)+1/8)√f′′(x0)+O⎛⎜⎝Ω4fΘ2fκ3+ΩfΘ3/2f+Ω3fΘ3/2fΩ2g⎞⎟⎠.

We will also need a second derivative bound for integrals in two variables. Let

 (2.9) I(2)=∫ba∫dcg(x,y)e(f(x,y))dydx.

with and smooth real valued functions. Let supp. Let be such that inside the support of the integral,

 (2.10) f(2,0)(x,y)≫r21,f(0,2)(x,y)≫r22,f(2,0)(x,y)f(0,2)(x,y)−[f(1,1)(x,y)]2≫r21r22,

where . Then we have (see ),

 I(2)≪1r1r2.

Define the total variance of by

 var(g):=∫ba∫dc∣∣∣∂2∂x∂yg(x,y)∣∣∣dydx.

Integration by parts along with the above bound gives us the following.

###### Lemma 2.4.

Suppose are as above and satisfy condition (2.10). Then we have

 I(2)≪var(g)r1r2.

### 2.1. An integral of interest

Following Munshi , let be a smooth real valued function with supp and . Define

 (2.11) W†(r,s)∫∞0W(x)e(−rx)xs−1dx

where and . This integral is of the form (2.5) with

 g(x)=W(x)xσ−1 and f(x)=−rx+12πβlogx.

Then,

 f′(x)=−r+12πβx and f(j)(x)=(−1)j(j−1)!12πβxj

for . The unique stationary phase occurs at . Note that we can write

 (2.12) f′(x)=β2π(1x−1x0)=r(x0x−1).

Applying Lemma 2.3 appropriately to , we get the following.

###### Lemma 2.5.

Let be a smooth real valued function with supp and . Let and . We have

 (2.13) W†(r,s)=√2πe(1/8)√−βW(β2πr)(β2πr)σ(β2πer)iβ+Oa,b,σ(min{|β|−3/2,|r|−3/2}).

We also have

 (2.14) W†(r,s)=Oa,b,j,σ⎛⎝min⎧⎨⎩(1+|β||r|)j,(1+|r||β|)j⎫⎬⎭⎞⎠.

## 3. Application of Dual summation formulas

### 3.1. Poisson summation to the m-sum

The -sum is given by

 ∑m≥1m−i(t+v)e(−m¯a−mαqqp+mxaqp)U(mN).

Breaking the -sum into congruence classes modulo by changing variables , we get

 ∑βmod qpe(−β¯¯¯a−βαqqp)∑m∈Z(β+mqp)−i(t+v)e((β+mqp)xaqp)U(β+mqpN).

Poisson summation to the -sum gives

 ∑βmod qpe(−β¯¯¯a−βαqqp)∑m∈Z∫R(β+yqp)−i(t+v)e((β+yqp)xaqp)U(β+yqpN)e(−my)dy.

Letting and executing the complete character sum , we arrive at

 (3.1) N1−i(t+v)∑m∈Zm≡¯a+αq(mod qp)∫Ru−i(t+v)e(N(x−ma)aqpu)U(u)du

The above integral equals

 (3.2) U†(N(ma−x)aqp,1−i(t+v)).

Since , we have and . Therefore is determined , and the -sum in the expression of vanishes. Everything together,

 (3.3) S+(N)=1K∫10∫RV(vK)1≤q≤Q

We first observe that can occur only when and bounds on from lemma 2.5 give arbitrary saving as soon as has a size in , i.e. as soon as for any . With the assumption , lemma 2.5 gives arbitrary saving for . The sum becomes

 (3.4) S+(N)=N1−itK∫10∫RN−iv∑1≤q

We next split the sum into dyadic segments

where

 (3.5) S(N,C)=∫10∫RN−i(t+v)V(vK)∑C

### 3.2. Voronoi summation to the n-sum

Let , for . When , the level of the cusp form is . When , the level is . In the former case, and . In the latter case, and . In the latter case, and we can use the Voronoi summation formula in . However in the former case, depending on the congruence class of and . We therefore take up the case where  is insufficient. The calculations for are similar, that’s why we present the details of the case .

For ease of notation, let and . Then and . if and otherwise. Due to lemma 2.1, the calculations for the cases and are similar, so it suffices to work with one of the cases. When ,

 ∑n≥1λ(n)e(nmqp)nive(−nxaqp)V(nN)=2πikN1+iv4πiqp1−δχL1(−¯¯¯¯¯mpδ)χL2(−qp1−δ)η(4qDqpδ)√4qDqpδ∑n≥1λL2(n)e⎛⎝−n¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯m4qDqqp1−δ⎞⎠×∫∞0yive(−yNxaqp)V(y)∫(σ)(2π√Nnyqp1−δ/2√4qDq)−sγk(s)dsdy.

where

 (3.6) γk(s)=(2π)−sΓ(s/2+(k−1)/2)Γ(1−s/2+(k−1)/2).

Since , has no poles in the region , and we can shift the -integral to without picking up poles. This allows us to interchange the and integrals.

 (3.7) ∑n≥1λ(n)e(nmqp)nive(−nxaqp)V(nN)=2πikN1+iv4πiqp1−δχL1(−¯¯¯¯¯mpδ)χL2(−qp1−δ)η(L2)√4qDqpδ∑n≥1λL2(n)e⎛⎝−n¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯m4qDqqp1−δ⎞⎠×∫(σ)(2π√Nnqp1−δ/2√4qDq)−sγk(s)V†(Nxaqp,1−s/2+iv)ds.

The bound on gives

 (3.8) V†(xNaqp,1−s/2+iv)≪jmin⎧⎨⎩1,(1+|Nx/aqp||v−τ/2|)j⎫⎬⎭.

We can therefore shift the -integral to for large and get arbitrary saving for large . With arguments similar to Remark 3.2 of , we will get arbitrary saving for . So the optimal choice of . Thus the introduction of divisibility by in (1.3) is a conductor lowering trick. For smaller values of , we shift the integral to . Note that .

Assuming , we get arbitrary saving for due to the bounds on . Thus we can restrict the integral to by defining a smooth partition of unity on this set. Let for be smooth bump functions satisfying for all . For , let the support of be in and for (resp. ), let the support of be in (resp ). Finally, we require that