hybrid bounds for quadratic weyl sums and arithmetic ... · d;q be the set of positive de nite,...

HYBRID BOUNDS FOR QUADRATIC WEYL SUMS AND ARITHMETICAPPLICATIONS

SHENG-CHI LIU AND RIAD MASRI

Abstract. Let D < 0 be an odd fundamental discriminant and q be a prime number whichsplits in Q(

√D). Given a suitable smooth function f supported on [X, 2X] for X ≥ 1, we

establish a uniform bound in X,D and q for∑c≡0 (mod q)

Wh(D; c)f(c),

where

Wh(D; c) :=∑

b (mod 2c)

b2≡D (mod 4c)

e

(hb

2c

), h ∈ Z, e(z) := e2πiz

is the Weyl sum for roots of the quadratic congruence x2 ≡ D (mod 4c). We use thisresult to study several problems of arithmetic interest, including “level-aspect” versions ofequidistribution of quadratic roots, and the asymptotic distribution of traces of CM valuesof weakly holomorphic modular functions of level q. By work of Zagier and Bruinier-Funke,the generating functions for these traces are weight 3/2 weakly holomorphic modular formsof level 4q satisfying Kohnen’s plus space condition.

1. Introduction and statement of results

Let D < 0 be a discriminant and define the quadratic Weyl sum

Wh(D; c) :=∑

b (mod 2c)b2≡D (mod 4c)

e

(hb

2c

), h ∈ Z, e(z) := e2πiz.

In many arithmetic problems one needs a bound for these sums as the modulus c varies insome range, for example

Wh(f,D, q) :=∑

c≡0 (mod q)

Wh(D; c)f(c) (1.1)

where q ≥ 1 is an integer and f is a smooth function supported on [X, 2X] for X ≥ 1. Themain result of this paper is the following hybrid bound for (1.1).

Theorem 1.1. Let D < 0 be an odd fundamental discriminant and q be a prime numberwhich splits in Q(

√D). Let f : R → C be a C∞ function supported on [X, 2X] for X ≥ 1

which satisfies

f (j) � X−j, j = 0, 1, . . .

For X � |D|1/2, we have

Wh(f,D, q)�ε |h|(q|D|X|h|)ε min{A(X,D), B(X,D, q), C(X,D, q)},1

2 SHENG-CHI LIU AND RIAD MASRI

where

A(X,D) := X3/4|D|1/16, B(X,D, q) := X1/2

(1 +

X1/4

q1/4|D|1/8

)(1 +|D|1/4

q1/2

),

C(X,D, q) := X1/2

(1 +

X1/4

q1/2|D|1/8

)(1 +|D|1/4

q1/8

).

A bound for (1.1) with a power saving in X was first established by Hooley [H] and laterimproved by Bykovski [B]. In these papers D is fixed and q = 1, however, for many arithmeticproblems it is crucial to have a wide range of uniformity in at least two of the variables X,Dand q. Examples of this occur in the groundbreaking work of Duke, Friedlander and Iwaniec[DFI] on the equidistribution of quadratic roots to prime moduli, and the work of Toth [T]which gives an analog of this result for positive discriminants. Other examples where suchuniformity is required occur in Duke, Friedlander and Iwaniec’s recent paper [DFI2], wherethey established a strong uniform bound for (1.1) (their method works for both positive andnegative discriminants). A central role was played by their important work on bilinear formswith Kloosterman fractions [DFI3].

In this paper we study (1.1) from the perspective of period formulas and mean-values ofL–functions. Our approach is influenced in many ways by [DFI], and our recent joint workwith Matt Young [LMY] on hybrid subconvexity and equidistribution of Heegner points inthe level aspect. To prove Theorem 1.1, we first express Wh(f,D, q) as the trace of a certainweight zero, smooth Poincare series for Γ0(q) over the Heegner points of discriminant D.After spectrally decomposing the Poincare series and calculating the spectral coefficients, weare led to estimating, up to a very small error term, an expression of the form∑

|tg |�(qX|D|1/2|h|)ερg(−h)φ(tg)WD,g + continuous spectrum contribution

where g runs over an orthonormal basis of Maass cusp forms for Γ0(q) with spectral parametertg. Here ρg(h) is the h-th Fourier coefficient of g, φ(t) is the integral transform

φ(t) :=

∫ ∞0

φ(u)Kit(u)u−3/2du

with

φ(u) := f

(π|h|

√|D|

u

),

and WD,g is the trace of g over the Heegner points of discriminant D. A formula of Wald-spurger [W] and Zhang [Zh] relates |WD,g|2 to L(g, 1/2)L(g × χD, 1/2) where χD is the

Kronecker symbol associated to Q(√D). Various applications of Holder’s inequality are

possible here (see section 7 for further discussion), and we are led naturally to estimatingmean-values of different families of L–functions. For example, with the choice of exponents4, 2, 4 in Holder’s inequality, we are led to estimating the mean-values

M1 :=∑

|tg |≤(qX|D|1/2|h|)ε|ρg(h)|4|φ(tg)|4, M2 :=

∑|tg |≤(qX|D|1/2|h|)ε

L(g × χD, 12)

L(sym2g, 1),

3

M3 :=∑

|tg |≤(qX|D|1/2|h|)εL(g, 1

2)2,

where the sums are over Maass newforms g for Γ0(q). We estimate M1 using the Kuznetsovtrace formula, and we estimate M3 using the spectral large sieve inequality (see [DI]). Weestimate M2 in two different ways, first using a hybrid subconvexity bound of Blomer-Harcos[BH], and second using the following estimate proved in [LMY, Theorem 1.5] for q a primewith (q,D) = 1 and M ≥ 1,∑

|tg |≤M

L(g × χD, 12)

L(sym2g, 1)�ε (qM2 + |D|1/2)(|D|Mq)ε. (1.2)

As mentioned earlier, Duke, Friedlander and Iwaniec [DFI2] established a strong uni-form bound for (1.1) using methods quite different from those in this paper and gave manyinteresting arithmetic applications. One application they gave was to the equidistributionmodulo 1 of roots of the quadratic congruence x2 ≡ D (mod c) as X, |D|, q →∞ (see [DFI2,Theorem 1.3]). Here we give some examples of results in this direction one can obtain byemploying the bounds in Theorem 1.1. In particular, these bounds yield a wide range ofuniformity in q, which has significance for an analog of Linnik’s problem on the least primein an arithmetic progression.

Let I ⊂ [0, 1] be a fixed subinterval of length `(I) > 0, and define

NI(X,D, q) := |{b (mod 2c) : 1 ≤ c ≤ X, c ≡ 0 (mod q), b2 ≡ D (mod 4c),b

2c∈ I}|.

If X = |D|1/2, so that the minimum in Theorem 1.1 is A(X,D), we obtain the following

Theorem 1.2. For D and q as in Theorem 1.1, we have

NI(|D|1/2, D, q) ∼12

π2

L(χD, 1)

q + 1|D|1/2`(I)

as q, |D| → ∞ subject to the condition q ≤ |D|1/16−ε.

In particular, we obtain: given q and I, for any |D|1/2 � q8+ε we have NI(|D|1/2, D, q) > 0.This is a natural analog of Linnik’s problem on the least prime in an arithmetic progression:given q and a residue class a (mod q) with (a, q) = 1, for any X � qL one has π(X, a, q) > 0where

π(X, a, q) := |{p ≤ X : p a prime, p ≡ a (mod q)}|and L is the famous Linnik constant (see [IK, chapter 18]).

On the other hand, if X = |D|, so that the minimum in Theorem 1.1 is B(X,D, q), weobtain the following

Theorem 1.3. For D and q as in Theorem 1.1, we have

NI(|D|, D, q) ∼12

π2

L(χD, 1)

q + 1|D|`(I)

as q, |D| → ∞ subject to the condition |D|1/12 ≤ q ≤ |D|1/2−ε.

We can also establish “sparse equidistribution” analogs of these results in which the subin-terval I is allowed to shrink as a function of X,D and q. For example, if X = |D|1/2 and qis fixed, we obtain the following


Theorem 1.4. Let D and q be as in Theorem 1.1. If ID ⊂ [0, 1] is a subinterval of length`(ID) = |D|−η with η > 1/48, we have

NID(|D|1/2, D, q) ∼ 12

π2

L(χD, 1)

q + 1|D|1/2−η

as |D| → ∞.

In particular, we obtain: given ID with `(ID)� |D|− 148

+ε, we have NID(|D|1/2, D, q) > 0.In a somewhat different direction, we will use Theorem 1.1 to study the distribution of

traces of singular values of weakly holomorphic modular functions of level q. To describethis result, let QD,q be the set of positive definite, integral binary quadratic forms

Q(X, Y ) = aX2 + bXY + cY 2

of discriminant b2 − 4ac = D < 0 with a ≡ 0 (mod q). The group Γ∗0(q) generated by Γ0(q)and the Fricke involution acts on QD,q with finite quotient. Let M !

0(Γ∗0(q)) be the space ofweakly holomorphic modular forms of weight zero for Γ∗0(q). Such a form f is holomorphicon the complex upper half-plane H, meromorphic in the cusps of Γ∗0(q), and has a Fourierexpansion in the cusp at ∞ of the form

f(z) =

Nf∑n=1

af (−n)e(−nz) +∞∑n=0

af (n)e(nz)

for some integer Nf ≥ 1. Define the trace

Tr∗D,q(f) :=∑

Q∈QD,q/Γ∗0(q)

f(τQ)

#Γ∗0(q)Q,

where

τQ =−b+

√D

2a

is the root of Q(X, 1) in H and Γ∗0(q)Q is the stabilizer of Q in Γ∗0(q).The classical modular j-function

j(z) := e(−z) + 744 + 196884 · e(z) + · · ·

is a weakly holomorphic modular form of weight zero for Γ0(1) whose values j(τQ) arealgebraic integers called “singular moduli”. Let J := j − 744. Zagier [Z2] proved that thegenerating function for traces of singular moduli,

e(−z)− 2−∑D<0

TrD,1(J)e(|D|z),

is a weight 3/2 weakly holomorphic modular form for Γ0(4) satisfying Kohnen’s plus spacecondition. Bruinier and Funke [BF] generalized this result to forms f ∈ M !

0(Γ∗0(q)). Inparticular, if q is a prime (or q = 1) and f has constant term af (0) = 0, they proved thatthe generating function∑

D<0

Tr∗D,q(f)e(|D|z) +∑n≥1

(σ1(n) + qσ1(n/q))af (−n)−∑m≥1

∑n≥1

maf (−mn)e(−m2z)

5

is a weight 3/2 weakly holomorphic modular form for Γ0(4q) satisfying Kohnen’s plus spacecondition. Here σ1(0) = −1/24, σ1(n) =

∑t|n t for n ∈ Z≥0 and σ1(x) = 0 for x /∈ Z≥0. In

particular, if q = 1 and f = J , they recovered Zagier’s result.We are interested in the asymptotic distribution of the traces Tr∗D,q(f). For traces of

singular moduli, Bruinier, Jenkins and Ono [BJO] established the Rademacher type exactformula

TrD,1(J) = −24H(D) + 2∑c∈Z+

W1(D; c) sinh

(π√|D|c

),

where

H(D) :=∑

Q∈QD,1/Γ0(1)

1

#Γ0(1)Q

is the Hurwitz class number. Based on this exact formula, they conjectured that

TrD,1(J) = −24H(D) + 2∑

1≤c<√|D|/3

W1(D; c) sinh

(π√|D|c

)+ o(H(D))

as |D| → ∞ through fundamental discriminants. This conjecture was proved by Duke [D]using the equidistribution of CM points.

For traces of weakly holomorphic forms f ∈M !0(Γ∗0(q)), Choi, Jeon, Kang and Kim [CJKK]

established the exact formula

Tr∗D,q(f) = −24H∗q (D)∑h>0

af (−h)cq(h) + 2∑h>0

af (−h)∑

c≡0 (mod q)

Wh(D; c) sinh

(πh√|D|c

),

where

H∗q (D) :=∑

Q∈QD,q/Γ∗0(q)

1

#Γ∗0(q)Q

is the “level q” Hurwitz class number and

cq(h) := − qα+1

q + 1σ1(h/qα) + σ1(h), qα‖h.

We will use Theorem 1.1 to establish the following asymptotic formula for Tr∗D,q(f) witha power saving in D.

Theorem 1.5. For D and q as in Theorem 1.1 and f ∈M !0(Γ∗0(q)), we have

Tr∗D,q(f) = −24H∗q (D)∑h>0

af (−h)cq(h) + 2∑h>0

af (−h)∑

1≤c<√|D|

c≡0 (mod q)

Wh(D; c) sinh

(πh√|D|c

)

+Of,ε(|D|7/16+ε)

as |D| → ∞.


Acknowledgments. We thank Matt Young for several helpful discussions and the refereefor a very careful reading of the manuscript leading to an improved exposition. The secondauthor was partially supported by the NSF grant DMS-1162535 during the preparation ofthis work.

2. Heegner points

Let D < 0 be an odd fundamental discriminant and q be a prime number which splits inQ(√D). Let QD,q be the set of positive definite, integral binary quadratic forms

Q(X, Y ) = [aQ, bQ, cQ](X, Y ) = aQX2 + bQXY + cQY

2

of discriminant b2Q − 4aQcQ = D with aQ ≡ 0 (mod q). The group Γ0(q) acts on QD,q by

Q = [aQ, bQ, cQ] 7→ Qσ = [aσQ, bσQ, c

σQ],

where for σ =

(α βγ δ

)∈ Γ0(q) we have

aσQ = aQα2 + bQαβ + cQβ

2,

bσQ = 2aQαγ + bQ(αδ + βγ) + 2cQβδ,

cσQ = aQγ2 + bQγδ + cQδ

2.

Given a solution r (mod 2q) of r2 ≡ D (mod 4q) (there are 2 such solutions since q is aprime), we define the subset of forms

QD,q,r := {Q = [aQ, bQ, cQ] ∈ QD,q : bQ ≡ r (mod 2q)}.

Then Γ0(q) also acts on QD,q,r, and we have the decomposition (see [GKZ, p. 507])

QD,q/Γ0(q) =⋃

r (mod 2q)r2≡D (mod 4q)

QD,q,r/Γ0(q). (2.1)

To each form Q ∈ QD,q we associate the root

τQ =−bQ +

√D

2aQ∈ H.

This is compatible with the group action in the sense that στQ = τQσ for σ ∈ Γ0(q). Fix aset of representatives for the Γ0(q)-equivalence classes of forms in QD,q and define the set ofHeegner points of discriminant D,

ΛD,q := {τQ : Q ∈ QD,q/Γ0(q)}.

Given r and QD,q,r as above, each set

ΛD,q,r := {τQ : Q ∈ QD,q,r/Γ0(q)}

is a Gal(H/K)-orbit of Heegner points of discriminant D, where H is the Hilbert class field

of K = Q(√D) (see [GZ, pp. 235-236]).

7

3. Preliminaries on Maass forms

In this section we review some facts we will need concerning Maass forms (see e.g. [I] and[B, section 2]). Let f1, f2 : H→ C be Γ0(q)-invariant functions and Y0(q) be a fundamentaldomain for Γ0(q). Define the Petersson inner product

〈f1, f2〉q :=

∫Y0(q)

f1(z)f2(z)dxdy

y2, (3.1)

provided the integral is absolutely convergent.The hyperbolic Laplacian is defined by

∆ := −y2

(∂2

∂x2+

∂2

∂y2

).

Given a Maass form g with Laplace eigenvalue λg, let

tg :=√λg − 1/4 ∈ R ∪ (−1/2, 1/2)i

be the spectral parameter. Let Bq be an orthonormal basis of Hecke-Maass newforms ofweight 0 for Γ0(q), and B1 be a basis of Hecke-Maass cusp forms of weight 0 for SL2(Z)which is orthonormal with respect to the inner product (3.1). One has the following upperbounds (see [I, section 11.1])

#{g ∈ Bq : |tg| ≤ T} � qT 2 and #{g ∈ B1 : tg ≤ T} � T 2.

Given g ∈ Bq ∪ B1, let λg(n) be the n-th Hecke eigenvalue. Then λg(−n) = ±λg(n)depending on whether g is even or odd, and λg(n) satisfies

λg(n)� n12−δ (3.2)

for some δ > 0. A newform g ∈ Bq is an eigenfunction for the Fricke involution z 7→ −1/qz,and one has

λg(q) = ε(g)q−1/2 (3.3)

where ε(g) = ±1 is the Fricke eigenvalue.Given g ∈ B1, define (see [ILS, Proposition 2.6])

gq(z) :=

(1−

qλ2g(q)

(q + 1)2

)−1/2(g(qz)− q1/2λg(q)

q + 1g(z)

), (3.4)

and let B∗1 := {gq : g ∈ B1}. Then an orthonormal basis for the subspace of cusp forms inL2(Y0(q)) is given by

B := Bq ∪ B1 ∪ B∗1.A form g ∈ Bq ∪ B1 has the Fourier expansion

g(z) =√y∑n6=0

ρg(n)Kitg(2πny)e(nx),

where ρg(n) = ρg(1)λg(n) and (see [B, eq. (2.9)])

|ρg(1)| =(

2 cosh(πtg)

L(sym2g, 1)

)1/2

×

(

q2

q+1

)−1/2

, g ∈ Bq(q + 1)−1/2, g ∈ B1.

(3.5)


Similarly, gq ∈ B∗1 has the Fourier expansion

gq(z) =√y∑n6=0

ρgq(n)Kitgq (2πny)e(nx),

where ρgq(n) = ρg(1)λgq(n) and

λgq(n) :=

(1−

qλ2g(q)

(q + 1)2

)−1/2(q1/2λg(n/q)−

q1/2λg(q)

q + 1λg(n)

)(3.6)

for g ∈ B1, with the convention λg(x) = 0 for x ∈ Q\Z.Using (3.2)–(3.6) and the Hoffstein-Lockhart [HL] bound

L(sym2g, 1)�ε (|tg|q)−ε, (3.7)

we obtain

ρg(n)�ε |n|1/2|tg|εq−12

+εeπ2|tg |, g ∈ B. (3.8)

For the Eisenstein series Ea(z, s) associated to a cusp a of Γ0(q), one has the Fourierexpansion (see [I, section 3.4])

Ea(z,12

+ it) = δa=∞y1/2+it + φa(

12

+ it)y1/2−it +√y∑n 6=0

τa(n, t)Kit(2πny)e(nx),

where φa(s) is a certain meromorphic function and τa(n, t) = ρa(1, t)ηa(n, t), where (see [B,eq. (2.13)])

|ρa(1, t)| =(

4 cosh(πt)

q|ζ(q)(1 + 2it)|

)1/2

, (3.9)

η∞(n, t) :=η(n, t)

q1/2+it− q1/2η(n/q, t), η0(n, t) := η(n, t)− q−itη(n/q, t), (3.10)

η(n, t) :=∑ad=|n|

(a/d)it,

with the convention η(x, t) = 0 for x ∈ Q\Z.Using (3.9) and (3.10), we obtain

τa(n, t)�ε |tn|εeπ2|t|. (3.11)

Finally, recall that for a function f in L2(Y0(q)), one has the spectral expansion

f(z) =〈f, 1〉q

vol(Y0(q))+∑g∈B

〈f, g〉qg(z) +∑a

∫ ∞−∞〈f, Ea(·, 1

2+ it)〉qEa(z,

12

+ it)dt

4π, (3.12)

which converges in the norm topology (see [I, section 7.3]). If, for example, f is smooth andcompactly supported, then (3.12) converges pointwise absolutely and uniformly on compactsets.

9

4. Traces of Poincare series

Let φ : R≥0 → C be a C∞ function with compact support and define the Poincare series

Ph,φ(z) :=∑

σ∈Γ∞\Γ0(q)

φ(2π|h|Im(σz))e(−hRe(σz)), z ∈ H,

which is absolutely convergent. We will need the following standard identity (see [By, Lemma5] and [DFI, section 2]).

Proposition 4.1. Let D < −4 be an odd fundamental discriminant. Then∑c≡0 (mod q)

Wh(D; c)φ

(π|h|

√|D|

c

)=

∑Q∈QD,q/Γ0(q)

Ph,φ(τQ).

Proof. For c ≡ 0 (mod q) we have∑Q∈QD,q/Γ0(q)

∑σ∈Γ∞\Γ0(q)

Im(στQ)=

√|D|2c

e(−hRe(στQ)) =∑

Q∈QD,q/Γ∞Im(τQ)=

√|D|2c

e(−hRe(τQ))

=∑

bQ (mod 2c)

b2Q≡D (mod 4c)

e

(hbQ2c

)

= Wh(D; c),

where we used that the stabilizer of Q in Γ0(q) is {±I} for D < −4. It follows that∑Q∈QD,q/Γ0(q)

Ph,φ(τQ) =∑

Q∈QD,q/Γ0(q)

∑σ∈Γ∞\Γ0(q)

φ(2π|h|Im(στQ))e(−hRe(στQ))

=∑

c≡0 (mod q)

φ

(π|h|

√|D|

c

) ∑Q∈QD,q/Γ0(q)

∑σ∈Γ∞\Γ0(q)

Im(στQ)=

√|D|2c

e(−hRe(στQ))

=∑

c≡0 (mod q)

Wh(D; c)φ

(π|h|

√|D|

c

).

�

5. Spectral expansion of Wh(f,D, q)

Let f : R→ C be a C∞ function supported on [X, 2X] for X ≥ 1 which satisfies

f (j) � X−j, j = 0, 1, . . .

and define

φ(u) := f

(π|h|

√|D|

u

). (5.1)

Then φ is a C∞ function supported on [Y −1, 2Y −1] for

Y −1 =π|h|

√|D|

2X,


and using Faa di Bruno’s formula [F], for example, one obtains the bound

φ(j) � Y j, j = 0, 1, . . .

By Proposition 4.1 with φ defined as in (5.1), we have

Wh(f,D, q) :=∑

c≡0 (mod q)

Wh(D; c)f(c) =∑

τ∈ΛD,q

Ph,φ(τ). (5.2)

Then spectrally expanding Ph,φ(z) as in (3.12) and substituting in (5.2) yields

Wh(f,D, q) = h(D)〈Ph,φ, 1〉qvol(Y0(q))

+∑g∈B

〈Ph,φ, g〉qWD,g

+∑a

∫ ∞−∞〈Ph,φ, Ea(·, 1

2+ it)〉qWD,a(t)

dt

4π,

where h(D) is the class number of K and the hyperbolic Weyl sums are defined by

WD,g :=∑

τ∈ΛD,q

g(τ) and WD,a(t) :=∑

τ∈ΛD,q

Ea(τ,1

2+ it).

Unfolding the Poincare series gives

〈Ph,φ, 1〉q = 0.

Similarly, unfolding gives (see for example [IK, Chapter 16])

〈Ph,φ, g〉q = (2π|h|)12ρg(−h)φ(tg)

and

〈Ph,φ, Ea(·,1

2+ it)〉q = (2π|h|)

12 τa(−h, t)φ(t),

where φ is the integral transform

φ(t) :=

∫ ∞0

φ(u)Kit(u)u−3/2du. (5.3)

Finally, combining the preceding calculations yields

Wh(f,D, q) = (2π|h|)12

∑g∈B

ρg(−h)φ(tg)WD,g (5.4)

+ (2π|h|)12

∑a

∫ ∞−∞

τa(−h, t)φ(t)WD,a(t)dt

4π.

6. Period integral formulas

Given g ∈ B, define the hyperbolic Weyl sum

WD,g,r :=∑

τ∈ΛD,q,r

g(τ).

Since ΛD,q,r is a Gal(H/K)-orbit of Heegner points of discriminant D, it follows from aformula of Waldspurger [W] and Zhang [Zh] that

WD,g,r = θg,D|D|1/4

q1/2

L(g × χD, 12)1/2L(g, 1

2)1/2

L(sym2g, 1)1/2if g ∈ Bq ∪ B1, (6.1)

11

where θg,D is some complex number satisfying |θg,D| ≤ 10. Similarly,

WD,gq ,r = θg,D|D|1/4

q1/2

L(g × χD, 12)1/2L(g, 1

2)1/2

L(sym2g, 1)1/2if gq ∈ B∗1,

where on the right hand side of the identity, g ∈ B1 is the Maass form in the definition ofgq (see (3.4)). The deduction of these formulas from Waldsburger/Zhang can be found in[LMY, Lemma 5.1], for example.

A similar formula for the Eisenstein series was established by Duke, Friedlander andIwaniec [DFI4, equation (10.30)],

WD,a(t) = θD,q,t|D|1/4

q1/2

|ζ(12

+ it)||L(χD,12

+ it)||ζ(1 + 2it)|

, (6.2)

where θD,q,t is some complex number satisfying |θD,q,t| ≤ 10.

7. contribution of the discrete spectrum

In this section we estimate the sum over the Maass forms B in (5.4).We will use the following estimates repeatedly in our analysis (see e.g. [DI] and [DFI]).

Lemma 7.1. Let φ : R≥0 → C be a C∞ function supported on [Y −1, 2Y −1] which satisfies

φ(j) � Y j, j = 0, 1, . . .

The following estimates hold for the integral transform φ(t) defined in (5.3).

(1) For t ∈ R,

φ(t)� Y 1/2 log(Y + 6).

(2) For t ∈ iR with 0 < |t| < 1/2,

φ(t)� Y 1/2 log(Y + 6)(Y it + Y −it).

(3) For t ∈ R with |t| ≥ 1 and Y � 1,

φ(t)� Y 1/2(1 + |t|)−Ae−π2|t|, A = 0, 1, . . .

First we estimate the contribution of the oldforms.

Lemma 7.2. For X � |D|1/2 we have∑g∈B1∪B∗1

ρg(−h)φ(tg)WD,g �ε |h|1/2(q|D|X|h|)εq−1X1/2|D|1/6.

Proof. Since WD,g grows polynomially in tg, by (3.8) and Lemma 7.1 we may impose thetruncation tg ≤ (q|D|Y |h|)ε with an error term which is O((q|D|Y |h|)−1000) to obtain∑

g∈B1∪B∗1

ρg(−h)φ(tg)WD,g �ε |h|1/2(q|D|Y |h|)εY 1/2 log(Y + 6)q−12

+ε∑

tg≤(q|D|Y |h|)ε|WD,g|

for Y � 1. By (6.1) and the Conrey-Iwaniec [CI] bound L(g×χD, 1/2)�ε |D|1/3(q|D|Y |h|)εfor tg ≤ (q|D|Y |h|)ε, we obtain WD,g �ε q−1/2|D|5/12(q|D|Y |h|)ε. Since there are �(q|D|Y |h|)2ε oldforms with tg ≤ (q|D|Y |h|)ε, upon substituting the bound Y � X|D|−1/2 wecomplete the proof. �

Next we estimate the contribution of the newforms.


Lemma 7.3. For X � |D|1/2 we have∑g∈Bq

ρg(−h)φ(tg)WD,g �ε |h|1/2(q|D|X|h|)ε min{A(X,D), B(X,D, q), C(X,D, q)}, (7.1)

where

A(X,D) := X3/4|D|1/16, B(X,D, q) := X1/2

(1 +

X1/4

q1/4|D|1/8

)(1 +|D|1/4

q1/2

),

C(X,D, q) := X1/2

(1 +

X1/4

q1/2|D|1/8

)(1 +|D|1/4

q1/8

).

Proof. Since WD,g grows polynomially in tg, by (3.8) and Lemma 7.1 we may impose thetruncation |tg| ≤ (q|D|Y |h|)ε with an error term which is O((q|D|Y |h|)−1000) to obtain∑

g∈Bq

ρg(−h)φ(tg)WD,g �ε

∑|tg |≤(q|D|Y |h|)ε

|ρg(h)||φ(tg)||WD,g|

for Y � 1. By (2.1) we have

WD,g � |WD,g,r|.

Then from (6.1) we obtain∑g∈Bq

ρg(−h)φ(tg)WD,g �ε|D|1/4

q1/2

∑|tg |≤(q|D|Y |h|)ε

|ρg(h)||φ(tg)|L(g × χD, 1

2)1/2L(g, 1

2)1/2

L(sym2g, 1)1/2.

Various applications of Holder’s inequality are possible here. For example, applyingHolder’s inequality with exponents 4, 2, 4 yields∑

g∈Bq


q1/2M

1/41 M

1/22 M

1/43 ,

where

M1 :=∑

|tg |≤(q|D|Y |h|)ε|ρg(h)|4|φ(tg)|4, M2 :=

∑|tg |≤(q|D|Y |h|)ε

L(g × χD, 12)

L(sym2g, 1),

M3 :=∑

|tg |≤(q|D|Y |h|)εL(g, 1

2)2.

We estimate M3 using a variant of the approximate functional equation [LY, Lemma 2.4]and the following spectral large sieve inequality which was first derived by Deshouillers andIwaniec [DI] in a slightly weaker form.

Theorem 7.4 ([IK], Theorem 7.24). Let T ≥ 1 and N ≥ 1. For any sequence of complexnumbers {an}Nn=1, we have

∑g∈B|tg |≤T

1

L(sym2g, 1)

∣∣∣∣∣N∑n=1

anλg(n)

∣∣∣∣∣2

�(qT 2 +N log(N)

) N∑n=1

|an|2. (7.2)

13

By [LY, Lemma 2.4], there exists a function W (x), depending on Q := q(q|D|Y |h|)ε andε only, such that W (x) is supported on x ≤ Q1/2+ε and satisfies

xjW (j) � 1,

where the implied constant depends on j and ε only (not on Q), and for which∣∣L(g, 12)∣∣2 � Qε

∫ log(Q)

− log(Q)

∣∣∣∣∣∑n≥1

λg(n)

n12

+ivW (n)

∣∣∣∣∣2

dv +O(Q−100), (7.3)

where the implied constant depends on ε, W , and the degree of L(g, s) only.We insert (7.3) into the average over g with |tg| � (q|D|Y |h|)ε, then apply the spectral

large sieve inequality (7.2) and the bound (see [I2])

L(sym2g, 1)�ε (|tg|q)ε

to obtain

M3 �ε q(q|D|Y |h|)ε.We estimate M2 in two different ways. If we apply the Blomer-Harcos [BH] bound L(g ×

χD, 1/2) �ε q1/2|D|3/8(q|D|Y |h|)ε for |tg| ≤ (q|D|Y |h|)ε, the bound (3.7), and multiply by

the number of newforms which is � q(q|D|Y |h|)2ε, we obtain

M2 �ε (q|D|Y |h|)εq3/2|D|3/8.On the other hand, using the estimate (1.2) we obtain

M2 �ε (q|D|Y |h|)ε(q + |D|1/2).

Taken together, these estimates yield

M2 �ε (q|D|Y |h|)ε min(q3/2|D|3/8, q + |D|1/2).

By Proposition 8.2, we have

M1 �ε |h|2+ε(q−1+εY 2 log4(Y + 6) + q−2+εY 2(Y + Y −1 + log4(Y + 6)) log2(Y + 6)

).

Alternatively, applying Holder’s inequality with exponents 2, 4, 4 yields∑g∈Bq


q1/2N

1/21 N

1/42 N

1/43 ,

where

N1 :=∑

|tg |≤(q|D|Y |h|)ε|ρg(h)|2|φ(tg)|2, N2 :=

∑|tg |≤(q|D|Y |h|)ε

L(g × χD, 12)2

L(sym2g, 1),

N3 :=∑

|tg |≤(q|D|Y |h|)ε

L(g, 12)2

L(sym2g, 1).

We estimate N3 and N2 using the spectral large sieve inequality as above to obtain

N3 �ε q(q|D|Y |h|)ε

and

N2 �ε (q + q1/2|D|)(q|D|Y |h|)ε.


Note that to estimate N2, we take Q := q|D|2(q|D|Y |h|)ε.By Proposition 8.1, we have

N1 �ε Y log2(Y + 6) + |h|1+εq−1+εY log2(Y + 6)(Y 1/2 + Y −1/2).

Finally, by combining the estimates for M1,M2 and M3 (resp. N1, N2 and N3) and sub-stituting the bounds Y � X|D|−1/2 and q < X, we obtain (7.1) after a straightforwardcalculation.

�

8. Application of the Kuznetsov formula

We begin by showing how to quickly deduce an estimate for N1 from [DFI, eq. (23)].

Proposition 8.1. For Y � 1 we have

N1 �ε Y log2(Y + 6) + |h|1+εq−1+εY log2(Y + 6)(Y 1/2 + Y −1/2). (8.1)

Proof. Let

H(t) :=1

cosh(πt)h(t)(Y 2it + Y −2it + L2)

where h(t) = 3/(1 + t2)(4 + t2) and L = log(Y + Y −1 + 6), and define the sum

Q :=∑g∈B

H(tg) |ρg(h)|2 +∑a

∫ ∞−∞

H(t) |τa(h, t)|2dt

4π.

Duke, Friedlander and Iwaniec [DFI, eq. (23)] used the Kuznetsov trace formula to show

Q� L2 + L2(Y + Y −1)1/2h1/2q−1(h, q)1/2τ(hq). (8.2)

By Lemma 7.1 and positivity, we have

N1 :=∑g∈Bq

|tg |≤(q|D|Y |h|)ε

|ρg(h)|2|φ(tg)|2 � Y Q

for Y � 1. The estimate (8.1) now follows immediately from (8.2). �

In the following proposition we adapt the argument in [DFI] to estimate the fourth momentM1.

Proposition 8.2. For Y � 1 we have

M1 �ε |h|2+ε(q−1+εY 2 log4(Y + 6) + q−2+εY 2(Y + Y −1 + log4(Y + 6)) log2(Y + 6)

). (8.3)

Proof. First assume that (h, q) = 1. Using the Hecke relations ([I, eq. (8.39)])

λg(m)λg(n) =∑`|(m,n)

λg(mn/`2),

a short calculation yields

|ρg(h)|4 = |ρg(1)|2∑d|h

∑k|h

ρg(h2/d2)ρg(h2/k2).

Since (see (3.5))

|ρg(1)|2 � q−1+εeπ|tg |,

15

by Lemma 7.1 and positivity we have

M1 :=∑g∈Bq

|tg |≤(q|D|Y |h|)ε

|ρg(h)|4|φ(tg)|4 �ε q−1+εY 2

∑d|h

∑k|h

∑g∈B

ρg(h2/d2)ρg(h2/k2)H∗(tg)

for Y � 1, where

H∗(t) :=1

cosh(πt)(1 + t4)−1(Y 4it + Y −4it + log4(Y + 6)).

The function H∗(t) clearly satisfies the conditions in [IK, eq. (15.19)]. Therefore by theKuznetsov trace formula [IK, Theorem 16.3] we have∑

g∈B

ρg(h2/d2)ρg(h2/k2)H∗(tg) +

∑a

∫ ∞−∞

τa(h2/d2, t)τa(h2/k2, t)H∗(t)

dt

4π

= δ(h2/d2, h2/k2)H∗0 +∑

c≡0 (mod q)

c−1S(h2/d2, h2/k2; c)H∗(

4πh2

dkc

),

where S(m,n; c) is the Kloosterman sum,

H∗0 :=1

π2

∫ ∞−∞

sinh(πt)H∗(t)tdt

and

H∗(x) :=2i

π

∫ ∞−∞

J2it(x)H∗(t)tdt.

Estimating trivially yields

H∗0 � log4(Y + 6).

We next estimate the sum of Kloosterman sums. Make the change of variables s = 1/2+itand shift the line of integration to σ + it with 1/2 < σ < 1 to obtain

H∗(x) =2i

π

∫(σ)

J2s−1(x)H∗ ((s− 1/2)i) (s− 1/2) ds.

By [GR, eq. 8.411.6] and Stirling’s formula,

J2s−1(x)� eπ|s|x2σ−1.

We also have

H∗((s− 1/2)i)� |s|−4e−π|s|(Y 2−4σ + Y 4σ−2 + log4(Y + 6)

).

These estimates then yield

H∗(x)� x2σ−1(Y 2−4σ + Y 4σ−2 + log4(Y + 6)

). (8.4)

By [IK, eq. (16.50)] we have∑c≡0 (mod q)

c−1−ω|S(m,n; c)| � (ω − 1/2)−2τ((m,n))(m,n, q)1/2τ(q)q−12−ω (8.5)


for 1/2 < ω ≤ 1. Then from (8.4) and (8.5) we obtain∑c≡0 (mod q)

c−1S(h2/d2, h2/k2; c)H∗(

4πh2

dkc

)

�(h2

dk

)2σ−1 (Y 4σ−2 + Y 2−4σ + log4(Y + 6)

)× (σ − 3/4)−2τ((h2/d2, h2/k2))(h2/d2, h2/k2, q)1/2τ(q)q

12−2σ

� |h|2+εq−1+ε(Y + Y −1 + log4(Y + 6)

)log2(Y + 6),

where for the last inequality we substituted

σ =3

4+

1

2 log(Y + 6).

By (3.11),

τa(h2/`2, t)�ε |ht|εe

π2|t|,

hence we obtain∑a

∫ ∞−∞

τa(h2/d2, t)τa(h2/k2, t)H∗(t)

dt

4π�ε |h|ε log4(Y + 6).

Combining the preceding estimates yields (8.3) for (h, q) = 1.If (h, q) > 1, write h = qαn for some integer α ≥ 1 and integer n coprime to q. Then by

the Hecke relations ([I, (8.39)]) λg(qαn) = λg(q

α)λg(n) = λg(q)αλg(n) and λg(q) = ±q−1/2

(see (3.3)), we have

|ρg(h)|4 = q−2α|ρg(n)|4,and we reduce to the case already considered (with a sharper bound in q). �

9. Contribution of the continous spectrum

In this section we estimate the sum over the cusps a in (5.4).

Lemma 9.1. For X � |D|1/2 we have∑a

∫ ∞−∞


4π�ε (q|D|X|h|)εq−1/2X1/2|D|1/6.

Proof. Since WD,a(t) grows polynomially in t, by (3.11) and Lemma 7.1 we may impose thetruncation |t| ≤ (q|D|Y |h|)ε with an error term which is O((q|D|Y |h|)−1000) to obtain∫ ∞

−∞τa(−h, t)φ(t)WD,a(t)

dt

4π�ε (q|D|Y |h|)εY 1/2 log(Y + 6)

∫|t|≤(q|D|Y |h|)ε

|WD,a(t)|dt

for Y � 1. By (6.2), the convexity bound for ζ(1/2 + it), the Conrey-Iwaniec [CI] boundL(χD, 1/2 + it) �ε |D|1/6(q|D|Y |h|)ε for |t| ≤ (q|D|Y |h|)ε, and a standard lower bound for|ζ(1 + 2it)|, we obtain WD,a(t)�ε q

−1/2|D|5/12(q|D|Y |h|)ε. Thus∫ ∞−∞


4π�ε Y

1/2 log(Y + 6)q−1/2|D|5/12(q|D|Y |h|)ε.

Upon substituting the bound Y � X|D|−1/2, we complete the proof. �

17

10. Proof of Theorem 1.1

Theorem 1.1 follows by combining the spectral decomposition (5.4) with the estimates inLemmas 7.2, 7.3 and 9.1.

11. Proof of Theorems 1.2 and 1.3

We proceed as in [DFI2, section 12] to prove the following proposition, which impliesTheorems 1.2 and 1.3.

Proposition 11.1. Let D, q and f be as in Theorem 1.1. Let φ : R≥0 → C be a C∞ functionsupported on a fixed subinterval I ⊂ [0, 1] of length `(I) > 0 and

Nφ(f,D, q) :=∑

c≡0 (mod q)

f(c)∑

b (mod 2c)b2≡D (mod 4c)

φ

(b

2c

).


Nφ(f,D, q) = f(0)φ(0)12

π2

L(χD, 1)

q + 1+Oε ((q|D|X)ε min{A(X,D), B(X,D, q), C(X,D, q)}) ,

where

f(ξ) :=

∫ ∞−∞

f(u)e(ξu)du

(resp. φ) is the Fourier transform of f (resp. φ).

Proof. Integrating by parts A-times yields

φ(h) =

∫ ∞−∞

φ(u)e(hu)du� |h|−A.

Then by Poisson summation and Theorem 1.1, we have

Nφ(f,D, q) =∑h∈Z

φ(h)Wh(f,D, q)

= φ(0)∑

c≡0 (mod q)

f(c)ρ(c) +Oε ((q|D|X)ε min{A(X,D), B(X,D, q), C(X,D, q)}) ,

where

ρ(c) := #{b (mod 2c) : b2 ≡ D (mod 4c)}.Define the L–series

L(s) :=∑

c≡0 (mod q)

ρ(c)c−s.

By Lemma 11.2 we have

L(s) =2

1 + qsζ(s)L(χD, s)

ζ(2s),

which has a simple pole at s = 1 with residue

R :=12

π2

L(χD, 1)

q + 1.


By Mellin inversion, ∑c≡0 (mod q)

f(c)ρ(c) =1

2πi

∫(σ)

f(s)L(s)ds for σ > 1,

where

f(s) :=

∫ ∞0

f(u)us−1du

is the Mellin transform of f . Shifting the contour to (1/2), we pick up the residue R toobtain ∑

c≡0 (mod q)

f(c)ρ(c) = f(0)12

π2

L(χD, 1)

q + 1+

1

2πi

∫(1/2)

f(s)ζ(s)L(χD, s)

ζ(2s)

2

1 + qsds.

Integrating by parts A-times yields

f(s)� X1/2|s|−A for σ = 1/2.

Then by the Conrey-Iwaniec [CI] bound L(χD, 1/2 + it)�ε (1 + |t|)B|D|1/6+ε we obtain∑c≡0 (mod q)

f(c)ρ(c) = f(0)12

π2

L(χD, 1)

q + 1+Oε(q

−1/2X1/2|D|1/6+ε).

�

The following lemma is a minor variant of the classical formula∞∑c=1

ρ(c)c−s =ζ(s)L(χD, s)

ζ(2s)

due to Dirichlet (see e.g. [Z, Proposition 3, (i)]).

Lemma 11.2. For D and q as in Theorem 1.1, we have

L(s) =2

1 + qsζ(s)L(χD, s)

ζ(2s).

Proof. Observe that

L(s) =∑

c≡0 (mod q)

ρ(c)c−s = Lq(s)∏p-qD

Lp(s)∏p|D

Lp(s),

where

Lq(s) :=ρ(q)

qs+ρ(q2)

q2s+ · · ·

and

Lp(s) := 1 +ρ(p)

ps+ρ(p2)

p2s+ · · · .

A calculation yields

Lq(s) = q−s(1 + χD(q))(1− q−s)−1

19

and

Lp(s) =

{(1− p−s)−1(1 + χD(p)p−s), if p - qD(1− p−s)−1(1− p−2s), if p|D.

The result now follows. �


We proceed as in [DFI2, section 12] to prove the following proposition, which impliesTheorem 1.4.

Proposition 12.1. Let D, q and f be as in Theorem 1.1. Let ID ⊂ [0, 1] be a subintervalof length `(ID) = |D|−η for some η > 0 and φ : R≥0 → C be a C∞ function supported on IDwhich satisfies

φ(j) � |D|ηj, j = 0, 1, . . .


Nφ(f,D, q) = f(0)φ(0)12

π2

L(χD, 1)

q + 1

+Oε

(|D|2η(q|D|X)ε min{A(X,D), B(X,D, q), C(X,D, q)}

).

Proof. Integrating by parts A-times yields

φ(h)�(|D|η

|h|

)A.

Then proceeding as in the proof of Proposition 11.1, we have

Nφ(f,D, q) =∑h∈Z

φ(h)Wh(f,D, q)

= φ(0)∑

c≡0 (mod q)

f(c)ρ(c) +∑

|h|≤|D|η+εφ(h)Wh(f,D, q) +O(|D|−1000)

= f(0)φ(0)12

π2

L(χD, 1)

q + 1

+Oε

(|D|2η(q|D|X)ε min{A(X,D), B(X,D, q), C(X,D, q)}

).

�


We proceed as in [DFI2, section 14] to prove Theorem 1.5. Let {ω`}∞`=0 be a smooth

partition of unity such that each constituent ω` is supported on [Y`, 2Y`] with Y` := 2`/2√|D|

and satisfies

ω(j)` (y)� y−j, j = 0, 1, . . .

Then the function

f`(y) := sinh

(πh√|D|y

)ω`(y)


satisfies

f(j)` (y)�

√|D|y

y−j, j = 0, 1, . . .

It follows from Theorem 1.1 with f = f` and X = Y` that∑c≥√|D|

c≡0 (mod q)

Wh(D; c) sinh

(πh√|D|c

)�

∞∑`=0

|Wh(f`, D, q)| �ε

∞∑`=0

√|D|Y`

Y3/4` |D|

1/16(q|D|Y`)ε

�ε |D|7/16+ε.

�

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Department of Mathematics, Washington State University, Pullman, WA 99164-3113E-mail address: [email protected]

Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, TX77843-3368

E-mail address: [email protected]

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