- Email: [email protected]
The asymptotic distribution of traces of Maass–Poincaré series
Advances in Mathematics 226 (2011) 3724–3759 www.elsevier.com/locate/aim
The asymptotic distribution of traces of Maass–Poincaré series Amanda Folsom a , Riad Masri b,∗ a Department of Mathematics, Yale University, New Haven, CT 06520, United States b Department of Mathematics, Texas A&M University, College Station, TX 77843, United States
Received 22 February 2010; accepted 1 November 2010 Available online 13 November 2010 Communicated by George E. Andrews
Abstract We establish an asymptotic formula with a power savings in the error term for traces of CM values of a family of Maass–Poincaré series which contains the modular j -function as a special case. By work of Borcherds (1998) [2], Zagier (2002) [31], and Bringmann and Ono (2007) [4], these traces are Fourier coefficients of half-integral weight weakly holomorphic modular forms and Maass forms. © 2010 Elsevier Inc. All rights reserved. Keywords: Asymptotic distribution; CM points; Maass–Poincaré series; Traces; Fourier coefficients
1. Introduction and statements of results Let j (z) be the classical modular j -function for SL2 (Z), j (z) = q −1 + 744 + 196 884q + 21 493 760q 2 + · · · , where q = e2πiz . The values of j (z) at CM points in the complex upper half-plane H are known as singular moduli. These are algebraic integers which play a fundamental role in number theory. For example, they generate class fields of imaginary quadratic fields, and parameterize isomorphism classes of elliptic curves with complex multiplication. * Corresponding author.
E-mail addresses: [email protected] (A. Folsom), [email protected] (R. Masri). 0001-8708/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2010.11.001
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3725
In the influential paper [31], Zagier proved that traces of singular moduli are Fourier coefficients of a half-integral weight modular form and used this to give another proof of a famous theorem of Borcherds [2] on infinite product expansions of integer weight modular forms with Heegner divisor. Zagier’s results have inspired a large number of works in recent years. An overview of some of these works can be found in the survey articles of Ono [26,27]. In this paper we will study the asymptotic distribution of “twisted traces” of CM values of a family of Maass–Poincaré series which contains the j -function as a special case. Let m ∈ Z with m = 0, and let s ∈ C. Define the Maass–Poincaré series 1
Fm (z, s) := 2π|m|s− 2
1 Im(γ z) 2 Is− 1 2π|m| Im(γ z) e m Re(γ z) , 2
γ ∈Γ∞ \SL2 (Z)
Re(s) > 1,
where Γ∞ < SL2 (Z) is the subgroup of elements stabilizing the cusp at ∞, Iν is the Bessel function of order ν, and e(z) := e2πiz . This Poincaré series was first studied by Niebur in [25], and it has since appeared in many different contexts (see e.g. [4–7,12,13,24,18]). Niebur [25] proved that Fm (z, s) has an analytic continuation to Re(s) = 1, and showed that for each m ∈ Z− , Fm (z, 1) = j|m| (z) + 24σ |m| ,
(1.1)
where j|m| (z) is the unique modular function satisfying j|m| (z) = q −|m| + O(q) and σ (|m|) :=
||m|
is the divisor function. In particular, j1 (z) = j (z) − 744.
The twisted traces of CM values of Fm (z, s) are defined as follows. Let −D < 0 be an odd fundamental discriminant. Let QD be the set of positive definite, primitive, integral binary quadratic forms Q(X, Y ) = [a, b, c] = aX 2 + bXY + cY 2 of discriminant b2 − 4ac = −D. Then SL2 (Z) acts on QD in the usual way. Let ΛD (1) be the set of Heegner points of discriminant −D on the modular curve X√0 (1). There are exactly h(−D) such points, where h(−D) is the ideal class number of K = Q( −D ). Furthermore, there is a bijection QD /SL2 (Z) → ΛD (1) given by [Q] → zQ where zQ =
√ −b + −D 2a
is the unique root in H of the dehomogenized form Q(X, 1) = aX 2 + bX + c. Let d > 0 be an odd fundamental discriminant (possibly 1) coprime to D. Define the genus character χd : QdD /SL2 (Z) → {±1}
3726
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
by χd (Q) = χd (a, b, c) :=
d n
where n is any integer coprime to d which is represented by Q = [a, b, c] ∈ QdD . Here (d/·) is the usual Kronecker symbol. This definition is independent of the choice of n (see Section 4). The “twisted trace” of Fm (z, s) is defined by
Trd Fm (·, s); D :=
χd (Q)Fm (zQ , s).
zQ ∈ΛdD (1)
Zagier [31] proved that the traces Trd (j1 ; D) are Fourier coefficients of a weight 3/2 weakly holomorphic modular form for Γ0 (4). Generalizing many results in [31], Bringmann and Ono [4] proved that the traces Trd (Fm (·, s); D) are Fourier coefficients of certain half-integral weight Maass forms (see also [12, Proposition 6]). For example, if s ∈ Z+ is an odd integer, one has bs (−d, D) =
(−1)
s+1 2
d
D
s−1 2
s 2
Trd F−1 (·, s); D ,
where the numbers bs (−d, D) are Fourier coefficients of a weight s + 1/2 Maass form Ps (−d; z) ∈ M +
s+ 12
Γ0 (4)
in Kohnen’s plus-space with Fourier expansion at ∞ of the form Ps (−d; z) = q −d +
bs (−d, n)q n .
n0 −n≡0,1 mod 4
In light of these results, it is natural to study the asymptotic distribution of the traces Trd (Fm (·, s); D) as D → ∞. For traces of singular moduli, this problem is closely related to the classical observation that the number eπ
√
163
= 262 537 412 640 768 743.9999999999992 . . .
is nearly an integer (see [3,26]). In [8], Bruinier, Jenkins, and Ono conjectured that a certain “perturbed” average of the traces Tr1 (j1 ; D) − Gred (D) − Gold (D) → −24 h(−D) as D → ∞. See [26] for an explanation of why the number −24 appears in this limit. This conjecture was studied by Duke [10] from a somewhat different perspective. Using the equidistribution of Heegner points on X0 (1) and a regularization of the pole at ∞ of j (z), he established the following result which implies the conjecture of Bruinier, Jenkins, and Ono,
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
1 1 Tr1 (j|m| ; D) − h(−D) 2
√ 0
3727
√ 4π|m| D → −24 SD |m|, c exp c
as D → ∞, where SD (|m|, c) is the exponential sum 2|m|x . e c
SD |m|, c :=
x 2 ≡−D (mod c)
We now state our main results. For n ∈ Z+ , define the twisted exponential sum SD,d (n, c) :=
χd
x 2 ≡−dD (mod c)
c x 2 + dD 2nx , x, e , 4 c c
and for m ∈ Z− and s ∈ Z+ , define the function D,d gm,s (c) := |m|s−1
s−1 √ − dD j (s − 1 + j )! . 2π|m|c (s − 1 − j )!j ! j =0
Let F be the standard fundamental domain for SL2 (Z). For Y > 0, define the truncated domain FY := z ∈ F : Im(z) Y , and for m ∈ Z− , define the “regularized” integral
Fm (z, s) dμ := lim Fm (z, s) dμ, reg
Y →∞
FY
where dμ = (3/π) dx dy/y 2 is the normalized hyperbolic measure on the open modular curve Y0 (1) = SL2 (Z)\H. We will establish the following asymptotic formula with a power savings in the error term for the traces Trd (Fm (·, s); D) as dD → ∞. An outline of the proof is given in Section 2. Theorem 1.1. Let m ∈ Z− , s ∈ Z+ , and let −D < 0 and d > 0 be odd, coprime fundamental discriminants. Then there exists an effective constant a > 0 such that for all δ < 1/16 and 0 < b < δ/a, the following asymptotic formulas hold: (1) If d = 1 then 1 1 Tr1 Fm (·, s); D − h(−D) 2
= reg
√ 0
SD,1
c≡0 (mod 4)
Fm (z, s) dμ + O D −(δ−ab) + O D −b
√ D,1 4π|m| D |m|, c gm,s (c) exp c
3728
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
as D → ∞, where
Fm (z, s) dμ = reg
0, −24σ (|m|)|m|s−1 ,
if s = 1, if s 2.
(2) If d > 1 then 1 1 Trd Fm (·, s); D − h(−dD) 2
0
√ 2 dD 1+(dD)−b
√ D,d 4π|m| dD SD,d |m|, c gm,s (c) exp c
c≡0 (mod 4)
= O (dD)−(δ−ab) + O (dD)−b as dD → ∞.
If we let s = 1 in Theorem 1.1 and use the relation (1.1), we obtain the following asymptotic formula with a power savings in the error term for the traces Trd (j|m| ; D) as dD → ∞. Corollary 1.2. Let m ∈ Z− , and let −D < 0 and d > 0 be odd, coprime fundamental discriminants. Then there exists an effective constant a > 0 such that for all δ < 1/16 and 0 < b < δ/a, the following asymptotic formulas hold: (1) If d = 1 then 1 1 Tr1 (j|m| ; D) − h(−D) 2
√
0
√ 4π|m| D SD,1 |m|, c exp c
= −24σ |m| + O D −(δ−ab) + O D −b as D → ∞. (2) If d > 1 then 1 1 Trd (j|m| ; D) − h(−dD) 2
0
√ 2 dD 1+(dD)−b
√ 4π|m| dD SD,d |m|, c exp c
c≡0 (mod 4)
= O (dD)−(δ−ab) + O (dD)−b as dD → ∞.
We conclude by noting that in [14], we used techniques related to those in the proof of Theorem 1.1 to study the asymptotic distribution of the partition function p(n), which counts the
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3729
number of partitions of a positive integer n. Let m ∈ Z with m = 0, and let s ∈ C. For N ∈ Z+ , define the Γ0 (N)-invariant Maass–Poincaré series 1
FmN (z, s) := 2π|m|s− 2
1 Im(γ z) 2 Is− 1 2π|m| Im(γ z) e m Re(γ z) , 2
γ ∈Γ∞ \Γ0 (N )
Re(s) > 1
where Γ∞ < Γ0 (N ) is the subgroup of elements stabilizing the cusp at ∞ of X0 (N ). In [5], Bringmann and Ono established the following arithmetic reformulation of Rademacher’s [28] exact formula for p(n), p(n) =
1 Dn
zQ ∈ΛDn (6)
6 χ12 (Q)F−1 (zQ , 2),
where Dn := 24n − 1 is square-free, χ12 := (12/·) is the Legendre symbol, and ΛDn (6) is the set of Heegner points of discriminant −Dn on X0 (6) (see Section 3). In [14], we used this formula and the equidistribution of Galois orbits of Heegner points on the modular curve X0 (6) to obtain a new asymptotic formula for p(n) with an effective error term which sharpens the classical bounds of Hardy and Ramanujan, Rademacher, and Lehmer on the error term in Rademacher’s exact formula for p(n). The estimates in Lemmas 9.1 and 10.1 of this paper are crucial to the proofs in [14], so this paper can in some respects be viewed as a companion to [14]. 2. Outline of the proof of Theorem 1.1 In this section we outline the proof of Theorem 1.1. In Section 5 we establish an asymptotic formula for twisted traces of smooth, Γ0 (N )-invariant functions which are allowed to grow moderately in the cusps of X0 (N ) (see Theorem 5.1). For example, if N = 1 suppose that F : H → C is a smooth, SL2 (Z)-invariant function such that for all n ∈ Z0 , n F (z) − c · y α = O e−c1 ·y
as y → ∞
(2.1)
for some c ∈ C, c1 > 0 and α 1/2. Here n is the n-th iterate of the hyperbolic Laplacian = −y 2 ∂x2 + ∂y2 . We show there exists an integer n0 > 0 such that for all δ < 1/16, 1 Trd (F ; D) = δd,1 h(−dD)
F (z) dμ + Oδ n0 FT0 2 + 1 (dD)−δ
(2.2)
F
as dD → ∞. Here δd,1 is the Kronecker diagonal symbol, T0 > 1 is a fixed cutoff parameter, and FT0 := F − ηT0 where ηT0 : H → C is a smooth, SL2 (Z)-invariant function with growth coinciding precisely with that of F for y T0 . The Fourier expansion of the Maass–Poincaré series Fm (z, s) is of the form Fm (z, s) = O ec2 ·y + O y 1−s + O e−c3 ·y as y = Im(z) → ∞
3730
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
where c2 , c3 are positive constants. In Section 6 we use a variant of an argument of Duke [10] to construct a smooth, SL2 (Z)-invariant Poincaré series Pm,ε for each ε > 0 such that the regularized function Fm,s,ε (z) := Fm (z, s) − Pm,ε (z, s) satisfies the growth condition (2.1) with α := 1 − s. In Section 7 we substitute Fm,s,ε into (2.2) to obtain the “preliminary” asymptotic formula 1 Trd Fm,s (·, s); D − Trd Pm,ε (·, s); D h(−dD)
= δd,1 Fm,s,ε (z) dμ + Oδ n0 Fm,s,ε,T0 2 + 1 (dD)−δ F
as dD → ∞. We then show that Trd (Pm,ε (·, s); D) equals the main term in Theorem 1.1 plus an explicit error term depending on ε and dD. See Eq. (7.3), and Sections 9 and 11. In Section 8 we show that for all ε > 0,
Fm,s,ε (z) dμ = Fm (z, s) dμ, F
reg
and in Section 12 we use a Borcherds-type integration to evaluate this regularized integral. In Section 10.1 we establish the upper bound n 0 Fm,s,ε,T ε −A 0 2 as ε → 0 for some fixed positive integer A depending on n0 . Finally, by choosing ε to be a sufficiently small negative power of dD, we obtain the final asymptotic formula with a power savings in the error term. 3. Heegner points on X0 (N) In this section we review some facts concerning Heegner points on the modular curve X0 (N ). Let N be a positive integer, and let −D < 0 be an odd√ fundamental discriminant coprime to N such that every prime divisor p of N is split in K = Q( −D ). Let QD,N be the set of positive definite, primitive, integral binary quadratic forms Q(X, Y ) = [a, b, c] = aX 2 + bXY + cY 2 of discriminant b2 − 4ac = −D with N |a. The set QD,N is stable under the action of Γ0 (N ). Let ΛD (N ) be the set of Heegner points of discriminant −D on X0 (N ). There is a bijection QD,N /Γ0 (N ) → ΛD (N ) given by [Q] → zQ where √ −b + −D zQ = 2N a is the unique root in H of the dehomogenized form Q(X, 1) = N aX 2 + bX + c.
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3731
Fix a solution r mod 2N of r 2 ≡ −D mod 4N . Note that there are exactly 2t (N ) such solutions r, where t (N ) is the number of distinct prime divisors of N . Define the subset of forms QD,N,r = Q = [a, b, c] ∈ QD,N : b ≡ r mod 2N . The set QD,N,r is also stable under the action of Γ0 (N ). There is a decomposition (see [15, p. 507]) QD,N /Γ0 (N ) =
QD,N,r /Γ0 (N ).
(3.1)
r mod 2N r 2 ≡−D mod 4N
The natural map QD,N,r /Γ0 (N ) → QD,1 /SL2 (Z) is a bijection which makes the set QD,N,r /Γ0 (N ) into a group of order h(−D) via the√Gauss law of composition on QD,1 /SL2 (Z). Here, h(−D) is the ideal class number of K = Q( −D ). Moreover, by class field theory QD,N,r /Γ0 (N ) ∼ = CLK ∼ = Gal(H /K) where CLK is the ideal class group of K and H is the Hilbert class field of K. For details concerning these facts, see [9]. The set ΛD (N ) is divided into 2t (N ) simple, transitive Gal(H /K)-orbits of size h(−D) (see [16, pp. 235–236]). Define the Galois orbit σ Or := zQ : σ ∈ Gal(H /K) , r where [Qr ] is any class in QD,N,r /Γ0 (N ). Then one has a bijection QD,N,r /Γ0 (N ) → Or .
(3.2)
4. Generalized genus characters In this section we briefly review the definition of generalized genus characters. For more details concerning these facts, see [15, pp. 508–510]. Let N be a positive integer. Let −D < 0 and d > 0 be odd, coprime fundamental discriminants such that both D and d are squares modulo 4N . For Q = [N a, b, c] ∈ QdD,N set d , χd (Q) := n where n is an integer coprime to d represented by the form [N1 a, b, N2 c] for some decomposition N = N1 N2 , Ni > 0. Such an n exists, and the value of ( dn ) is independent of the choice of N1 , N2 and n. The function χd is Γ0 (N )-invariant and thus defines a function χd : QdD,N /Γ0 (N ) → {±1}.
3732
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
Furthermore, χd restricts to a real ideal class group character on QdD,N,r /Γ0 (N ) for each r mod 2N such that r 2 ≡ −dD mod 4N . 5. Asymptotics for twisted traces Let F : H → C be a C ∞ , Γ0 (N )-invariant function. We say that F has cuspidal growth of power α for some α ∈ R if for every cusp a of Γ0 (N ) there exists a constant ca ∈ C (possibly equal to 0) such that for each integer a 0, a F (σa z) − ca y α = O e−cy as y = Im(z) → ∞ for some c > 0. Here σa ∈ SL2 (R) is a scaling matrix such that σa (∞) = a. Let Y0 (N ) = Γ0 (N )\H be the open modular curve, and let dμ be the normalized hyperbolic measure on Y0 (N). Theorem 5.1. Let N be a fixed positive integer. Let −D < 0 and d > 0 be odd, coprime fundamental discriminants such that D and d are √ squares modulo 4N , dD is coprime to N , and every prime divisor p of N is split in K = Q( −dD ). Let F : H → C be a C ∞ , Γ0 (N )-invariant function with cuspidal growth of power α for some α 1/2. Then there exists an integer a0 > 0 such that for all δ < 1/16,
1 2t (N ) h(−dD)
= δd,1
χd (Q)F (zQ )
Q∈QdD,N /Γ0 (N )
F (z) dμ + ON,δ a0 FT0 2 + 1 (dD)−δ
Y0 (N )
as dD → ∞. Here χd : QdD,N /Γ0 (N ) → {±1} is a generalized genus character as defined in Section 4, and FT0 is a regularized version of F where T0 1 is a fixed cutoff parameter (see Eq. (5.9)). Remark 5.2. Theorem 5.1 should be viewed as consisting of two cases: d = 1 and d > 1. What separates these cases is that when the genus character χd is trivial, the integral in the formula survives, while if χd is nontrivial, the integral is killed by orthogonality. For the proof it is more convenient to combine these cases, so we have used the diagonal symbol δd,1 . Proof of Theorem 5.1. We begin by constructing a C ∞ , Γ0 (N )-invariant cutoff function with growth coinciding with that of F in the cusps a of X0 (N ), and which vanishes on the Heegner points ΛdD (N ) for each sufficiently large cutoff parameter (see also [21]). Lemma 5.3. Let T > 1. There exists a C ∞ , Γ0 (N )-invariant function ηT : H → C such that 0, ηT (σa z) =
ca y α χ(y/T ), ca y α ,
1 < y < T, T y 2T , y > 2T ,
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3733
for each cusp a of X0 (N ), where χ : R+ → [0, 1] is a C ∞ function such that χ(t) =
0, t < 1, 1, t > 2.
Proof. Let χ be as in the statement of the lemma, and define ψT ∈ C0∞ (R+ ) by t ψT (t) := t χ . T α
Then define ηT by (see [19, (3.10)]) ηT (z) :=
ca · Ea (z|ψT ),
(5.1)
a
where
Ea (z|ψT ) :=
ψT Im σa−1 γ z .
γ ∈Γa \Γ0 (N )
Now, by [19, (3.17)] with m = 0, one has the Fourier expansion Ea (σb z|ψT ) = δab ψT (y) +
e(nx)
c>0
n∈Z
Sab (0, n; c)
ψT R
c−2 y e(−nt) dt, t 2 + y2
(5.2)
where δab is the Kronecker diagonal symbol and Sab (0, n; c) is the exponential sum Sab (0, n; c) =
a ∗
cd
∈B\σa−1 Γ0 (N )σb /B
na , e c
where B denotes the group of integral translations B=
1 b : b∈Z 1
(see [19, (2.23)]). This, combined with the inequality ∗ min c > 0: c
∗ ∗
∈ σa−1 Γ0 (N )σb
1
(5.3)
for all cusps a, b of X0 (N ) (see [19, (2.28)–(2.31)]), shows that ηT has the properties stated in the lemma. 2 Lemma 5.4. For T
points ΛdD (N ).
√ dD, the function ηT (z) defined by (5.1) vanishes on the Heegner
3734
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
√ Proof. By Minkowski’s theorem, every ideal class of K = Q( −dD ) contains a primitive integral ideal A of norm NK/Q (A)
√ dD.
(5.4)
Fix a solution r mod 2N of r 2 ≡ −dD mod 4N , and write √ b + −dD A = a, , a = NK/Q (A), b ∈ Z, 2 with b ≡ r mod 2N,
b2 ≡ −dD mod 4N a.
Then √ −b + −dD zA = 2N a is the Heegner point corresponding to Q(X, Y ) = N aX 2 + bXY + cY 2 . For γ ∈ Γ0 (N ) and a cusp a of X0 (N ), write a b σa−1 γ = ∈ SL2 (R). c d Then Im σa−1 γ zA =
√ dD Im(zA ) 1 .
2
2 |c zA + d | |c zA + d | 2N
(5.5)
If c = 0, then d = 1 (see [19, (2.15)–(2.17)]) and we have Im σa−1 γ zA
√ dD . 2N
(5.6)
On the other hand, if c = 0 we write √ dDc bc
+d +i , c zA + d = − 2N a 2N a
so that by (5.4), 2 √
dDc 2 c zA + d 2 = − bc + d + 2N a 2N a (c )2 dD 4N 2 a 2 (c )2
. 4N 2
(5.7)
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3735
It follows from (5.5) and (5.7) that Im σa−1 γ zA
√ √ √ 4N 2 dD dD 1 2 2N dD,
2 |c zA + d | 2N (c ) 2N
(5.8)
where for the last inequality we used (5.3). √ Since ψT (y) = 0 for y < T , we see from the inequalities (5.6) and (5.8) that for T dD, ψT Im σa−1 γ zA = 0 for all γ ∈ Γa \Γ0 (N ). It follows from (5.1) that ηT (zA ) = 0.
2
Now, define the “regularized” function FT (z) := F (z) − ηT (z). Then by Lemma 5.4, to prove Theorem 5.1 it suffices to prove the following proposition. Proposition 5.5. For T
√
dD, we have
1 2t (N ) h(−dD)
χd (Q)FT (zQ )
Q∈QdD,N /Γ0 (N )
F (z) dμ + ON,δ a0 FT0 2 + 1 (dD)−δ
= δd,1 Y0 (N )
as dD → ∞. Proof. Let T T0 1. Here T0 is a fixed cutoff parameter which is independent of dD. We introduce T0 in order to decompose FT into a sum of two functions so that we can isolate the contribution of ηT to the spectral decomposition. Consider the decomposition FT (z) = FT0 (z) + η˜ T (z) where η˜ T (z) := ηT0 (z) − ηT (z). We will first show that for all δ1 < 1/16,
1 2t (N ) h(−dD)
= δd,1 Y0 (N )
as dD → ∞.
χd (Q)FT0 (zQ )
Q∈QdD,N /Γ0 (N )
F (z) dμ − δd,1 Y0 (N )
ηT0 (z) dμ + O a0 FT0 2 (dD)−δ1
(5.9)
3736
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
By definition of ηT0 and our assumption that F has cuspidal growth of power α, we have a FT0 (σa z) = O e−cy as y → ∞ for each integer a 0. The spectral decomposition of L2 (Y0 (N )) with respect to the hyperbolic Laplacian yields the expansion ∞
FT0 (z) = FT0 , 12 +
FT0 , un 2 un (z)
n=1
1 1 1 FT0 , Ea ·, + it Ea z, + it dt + 4π 2 2 2 a R
which converges pointwise absolutely and uniformly on compact subsets of Y0 (N ) since a FT0 is C ∞ with exponential decay in each cusp a of Y0 (N ). Here u0 (z) = 1 is the constant eigenfunction for corresponding to the eigenvalue λ0 = 0, {un (z)}∞ n=1 is an orthonormal basis of Maass cusp forms satisfying un = λn un for n ∈ Z+ where the eigenvalues λn = 1/4 + tn2 are ordered so that 0 < λ1 λ2 · · · , and Ea (z, s) is the real-analytic Eisenstein series
Ea (z, s) =
s Im σa−1 γ z ,
z ∈ H, Re(s) > 1.
γ ∈Γa \Γ0 (N )
Summing the spectral expansion yields
χd (Q)FT0 (zQ ) =
Q∈QdD,N /Γ0 (N )
+
∞
χd (Q)
Q∈QdD,N /Γ0 (N )
FT0 , un 2 Wn +
n=1
1 4π a
R
FT0 (z) dμ
Y0 (N )
1 FT0 , Ea ·, + it Wa (t) dt 2 2
where Q∈QdD,N /Γ0 (N )
t (N ) h(−dD), χd (Q) = 2 0,
d = 1, d > 1,
and the twisted hyperbolic Weyl sums are defined by
Wn :=
χd (Q)un (zQ )
Q∈QdD,N /Γ0 (N )
and Wa (t) :=
Q∈QdD,N /Γ0 (N )
1 χd (Q)Ea zQ , + it . 2
Here FT0 ∈ L1 (Y0 (N )) because y α ∈ L1 ([C, ∞), dy/y 2 ) for all α < 1 and C > 0.
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3737
Using (3.1) we have
Wn =
χd (Q)un (zQ ).
(5.10)
r mod 2N Q∈QdD,N,r /Γ0 (N ) r 2 ≡−dD mod 4N
Furthermore, using the bijection (3.2) we have
χd (Q)un (zQ ) =
Q∈QdD,N,r /Γ0 (N )
σ . χd (σ )un zQ r
σ ∈Gal(H /K)
As a consequence of deep work of Waldspurger [29] and Zhang [32], one has an identity of the form
σ ∈Gal(H /K)
√ σ 2 d 1 −D 1 χd (σ )un zQr = Cun dDL un ⊗ , L un ⊗ , · 2 · 2
(5.11)
where A Cun 1 + |tn |2 1 for some fixed constant A1 > 0, and if denotes a fundamental discriminant (to be distinguished from our notation for the hyperbolic Laplacian), then L(un ⊗ ( · ), s) is the quadratic twist of the L-function of un by the Dirichlet character ( · ). Here we used the factorization of the Rankin– Selberg L-function d −D , s L un ⊗ ,s L(un ⊗ Θχd , s) = L un ⊗ · · where Θχd is the theta series associated to the genus character χd . This√follows as a consequence of Kronecker’s factorization of the class group L-function of K = Q( −dD ) associated to the genus character χd , L(χd , s) = L
d −D ,s L ,s . · ·
Blomer and Harcos [1] established the following deep subconvexity bound, valid for all δ2 < 1/8 and some fixed constant A2 > 0, A 1 1 , 1 + |tn | 2 | | 2 −δ2 . L un ⊗ · 2
(5.12)
After combining (5.11) and (5.12) with Siegel’s theorem, 1
h(−dD) (dD) 2 − ,
(5.13)
3738
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
one finds using (5.10) that for all δ3 < 1/16, 1 2t (N ) h(−dD)
A Wn 1 + |tn | 3 (dD)−δ3 .
(5.14)
To estimate Wa (t) we proceed as above to obtain the identity
Wa (t) =
r mod 2N Q∈QdD,N,r /Γ0 (N ) r 2 ≡−dD mod 4N
1 χd (Q)Ea zQ , + it . 2
(5.15)
Again, using the bijection (3.2) we have Q∈QdD,N,r /Γ0 (N )
1 χd (Q)Ea zQ , + it = 2
σ ∈Gal(H /K)
1 σ χd (σ )Ea zQr , + it . 2
(5.16)
Following the argument in [17, Section 6], one can reduce the estimate of (5.16) to an analogous estimate for σ ∈Gal(H /K)
σ 1 χd (σ )E z , + it 2
where E(z, s) is the Eisenstein series for SL2 (Z) and ΛdD (1) = {zσ : σ ∈ Gal(H /K)} is the set of Heegner points of discriminant −dD on the modular curve X0 (1). By a classical formula of Dirichlet one has an identity of the form (see [16, p. 248])
σ ∈Gal(H /K)
O× = K 2
2 σ 1 χd (σ )E z , + it 2
√ 2 2 d 1 dD L −D , 1 + it , L , + it 2 · 2 · 2
(5.17)
where we have used the fact that L(Θχd , s) = L(χd , s) and employed Kronecker’s factorization a second time. By Blomer and Harcos [1] one has the following subconvexity bound, valid for any δ4 < 1/8 and some fixed constant A4 > 0, 2 L , 1 + it 1 + |t| A4 | | 12 −δ4 . · 2
(5.18)
After combining (5.17) and (5.18) with Siegel’s theorem (5.13), one finds using (5.15) that for all δ5 < 1/16, 1 2t (N ) h(−dD)
A Wa (t) 1 + |t| 5 (dD)−δ5 .
(5.19)
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3739
Because a FT0 is C ∞ with exponential decay in each cusp a of X0 (N ), a repeated application of Stokes’ theorem (see e.g. [19, Lemma 1.18]) yields the following identities, valid for each integer a 0, −a a FT0 , un 2 FT0 , un 2 = 1/4 + tn2 and −a a 1 1 = 1/4 + t 2 . FT0 , Ea ·, + it FT0 , Ea ·, + it 2 2 2 2 Moreover, by the Parseval formula (see [20, (15.17)]), 2 ∞ 1 a a FT , Ea ·, 1 + it dt = a FT 2 . FT , un 2 + 0 0 0 2 2 4π 2 2 a n=1
R
Then by the Cauchy–Schwartz inequality, one finds that for some sufficiently large integer a0 > 0, ∞ ∞ A3 a FT , un 2 1 + |tn | A3 = 0 FT , un (1 + |tn |) 0 0 2 (1/4 + t 2 )a0 n n=1 n=1 ∞ ∞ 2 (1 + |tn |)2A3 a 0 FT0 , un 2 · (1/4 + tn2 )2a0 n=1 n=1 a0 FT0 2 ,
and 1 FT , Ea ·, 1 + it 1 + |t| A5 dt 0 4π 2 2 a R
1 (1 + |t|)A5 1 a0 ·, + it = F , E T a 0 (1/4 + t 2 )a0 dt 4π 2 2 a R
2
1 1 (1 + |t|)2A5 a0 FT , Ea ·, + it dt · dt 0 4π 2 (1/4 + t 2 )2a0 2 a R
R
a0 FT0 2 .
By combining these estimates with (5.14) and (5.19), we find that for all δ6 < 1/16, ∞
1 FT , un 2 Wn a0 FT (dD)−δ6 0 0 2 t (N ) 2 h(−dD) n=1
3740
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
and 1 1 t (N ) 2 h(−dD) a 4π
FT , Ea ·, 1 + it dt W (t) a 0 2 2
R
a0 FT0 2 (dD)−δ6 .
(5.20)
We conclude from these estimates that for all δ7 < 1/16,
1 2t (N ) h(−dD)
= δd,1
χd (Q)FT0 (zQ )
Q∈QdD,N /Γ0 (N )
ηT0 (z) dμ + O a0 FT0 2 (dD)−δ7
F (z) dμ − δd,1
Y0 (N )
Y0 (N )
as dD → ∞. To finish the proof of Proposition 5.5, it suffices to show that for all δ8 < 1/16, 1 2t (N ) h(−dD)
χd (Q)η˜ T (zQ ) = δd,1
Q∈QdD,N /Γ0 (N )
ηT0 (z) dμ
Y0 (N )
α−1 + O (dD)−δ8 + O (dD) 2 √ as T dD and dD → ∞. By combining the definition of ηT in (5.1) with [19, (7.12)], [19, Theorem 11.3] and [19, (7.13)], one has η˜ T (z) = η˜ T , 12 +
1 1 1 1 ψˆ T0 + it − ψˆ T + it Ea z, + it dt ca 2π a 2 2 2
(5.21)
R
where (see [19, (3.13)]) ˆ ψ(s) :=
∞
ψ(y)y −(s+1) dy.
0
Summing over (5.21) yields
χd (Q)η˜ T (zQ )
Q∈QdD,N /Γ0 (N )
=
χd (Q)η˜ T , 12
Q∈QdD,N /Γ0 (N )
+
1 1 1 ψˆ T0 + it − ψˆ T + it Wa (t) dt. ca 2π a 2 2 R
(5.22)
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3741
√ Let dD T dD. Then using Lemma 5.6 (proved below) and an argument similar to that in the proof of (5.20), we obtain the following estimate, valid for any δ9 < 1/16,
1 1 1 1 dt log(dD)(dD)−δ9 . ˆ ˆ ψ + it − ψ + it W |c | (t) a T T a 0 2 2 2t (N ) h(−dD) 2π a R
Moreover, for T
√ dD a straightforward estimate yields α−1 ηT , 12 = O (dD) 2 .
Finally, by combining these estimates with (5.22), we conclude that for δ10 < 1/16, 1 2t (N ) h(−dD)
√ dD T D and any
χd (Q)η˜ T (zQ ) = δd,1
Q∈QdD,N /Γ0 (N )
ηT0 (z) dμ
Y0 (N )
α−1 + O (dD)−δ10 + O (dD) 2 as dD → ∞.
2
Lemma 5.6. For each B > 0 we have
ψˆ T 1 + it − ψˆ T 1 + it 1 + |t| B dt log(T ). 0 2 2
R
Proof. Because χ(y/T0 ) − χ(y/T ) is supported in (T0 , 2T ), we have the identity
fT (t) := ψˆ T0
2T 3 1 1 − it − ψˆ T − it = χ(y/T0 ) − χ(y/T ) y it+α− 2 dy. 2 2 T0
First assume that |t| 1. Because the k-th derivative χ (k) (y) is supported in (1, 2), integrating by parts k-times yields
k
(−1)
k−1 j =0
1 it + + (α − 1) + j fT (t) 2
it+ 1 +(α−1) 1 = T0 2 − T it+ 2 +(α−1)
2 1
Now, we have the estimate
3
χ (k) (y)y it+α− 2 +k dy.
3742
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
k−1 k−1 1 1 k + (1 − α) + j + 1 |t|k , it + + (α − 1) + j (−1) 2 2 j =0
j =0
and the estimate it+ 12 +(α−1) 1 α− 1 T − T it+ 2 +(α−1) 2T 2 . 0
0
By combining the preceding facts we obtain 1
(k) 1 |2k+α− 2 | fT (t) 2T α− 2 max1y2 |χ (y)| |t|−k . 0 k−1 1 1 |k + α − | ( + (1 − α) + j + 1) 2 j =0 2
Because B > 0 is fixed and k 1 is arbitrary, it follows that
1 fT (t) 1 + |t| B dt T α− 2 .
0
|t|1
Next assume that |t| < 1. Since α 1/2 we have the estimate fT (t) 2 sup χ(y) y∈R+
2T
y −1 dy log(T ).
T0
Then because (1 + |t|)B 1 for |t| < 1, it follows that
fT (t) 1 + |t| B dt log(T ).
2
|t|<1
6. Poincaré series The following proposition gives the Fourier expansion of Fm (z, s) in the cusp at ∞ of Γ = SL2 (Z) (see [25]). Proposition 6.1. The Poincaré series Fm (z, s) has a Fourier expansion in the cusp at ∞ of Γ of the form 1 1 Fm (z, s) = 2π|m|s− 2 y 2 Is− 1 2π|m|y e(mx) + cm,s y 1−s 2 1 1 + 4π|m|s− 2 b(m, n; s)y 2 Ks− 1 2π|n|y e(nx), n=0
where cm,s :=
4π 1+s σ2s−1 (|m|) , (2s − 1)Γ (s)ζ (2s)
2
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3743
and b(m, n; s) =
∞ S(m, n; c)
c
c=1
·
√
|mn| ), c √ J2s−1 ( 4π c|mn| ),
I2s−1 ( 4π
mn < 0, mn > 0,
where S(m, n; c) is the Kloosterman sum, and Iν , Jν and Kν are the Bessel functions of order ν. We now construct a family of Poincaré series which regularize Fm (z, s) in the cusp. Let λ : R → [0, 1] be a C ∞ function such that λ(t) =
0, t 0, 1, t 1.
Let m ∈ Z− , s ∈ Z+ , and for ε > 0 define the function s−1 t −1 ψm,s,ε (t) := λ κm,j (s)t −j ε
j =0
where κm,j (s) :=
(−1)j |m|s−j −1 (s − 1 + j )! . (4π)j (s − 1 − j )!j !
Then ψm,s,ε : R → [0, 1] is C ∞ and satisfies
t 1, −j , t > 1 + ε. κ (s)t m,j j =0
0, ψm,s,ε (t) = s−1 Finally, define the Poincaré series
Pm,ε (z, s) :=
ψm,s,ε (Im γ z)e(mγ z).
γ ∈Γ∞ \Γ
Proposition 6.2. For y > 1 + ε we have Fm (z, s) − Pm,ε (z, s) = cm,s y 1−s + e−2π|m|y e(mx)(−1)s
s−1 (−1)j y −j κm,j (s) j =0
1
+ 2π|m|s− 2
n=0
e−2π|n|y e(nx)b(m, n; s)
1 s−1 (s − 1 + j )!|n|−j − 2 y −j
j =0
j !(s − 1 − j )!(4π)j
.
3744
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
Proof. The Poincaré series Pm,ε (z, s) has a Fourier expansion of the form (see [19, p. 60]) Pm,ε (z, s) = ψm,s,ε (y)e(mz) +
e(nx)
∞
S(m, n; c)
c=1
n∈Z
ψm,s,ε R
c−2 y −mc−2 e − nt dt. t + iy t 2 + y2
(6.1)
If y 1 then c−2 y 1 t 2 + y2 for all c ∈ Z+ and t ∈ R. It follows from the definition of ψm,s,ε that if y 1, Pm,ε (z, s) = ψm,s,ε (y)e(mz), and if y > 1 + ε, Pm,ε (z, s) =
s−1
κm,j (s)y −j e(mz).
(6.2)
j =0
Now, the Bessel functions In+ 1 and Kn+ 1 of half-integral order have expansions given as 2 2 follows (see [30, Section 3.71]): for non-negative integers n we have
y
1/2
n n (−1)j (n + j )! (n + j )! 1 y n+1 −y In+ 1 (y) = √ + (−1) e e 2 j !(n − j )!(2y)j j !(n − j )!(2y)j 2π j =0 j =0
and y
1/2
1 n (n + j )! π 2 −y Kn+ 1 (y) = e . 2 2 j !(n − j )!(2y)j j =0
Substitute these expansions into the Fourier expansion of Fm (z, s) given in Proposition 6.1 and use the fact that m ∈ Z− and s ∈ Z+ to obtain Fm (z, s) =
s−1
κm,j (s)y −j e(mz) + cm,s y 1−s + e−2π|m|y e(mx)(−1)s
j =0
s−1 (−1)j y −j κm,j (s) j =0
1
+ 2π|m|s− 2
n=0
e−2π|n|y e(nx)b(m, n; s)
1 s−1 (s − 1 + j )!|n|−j − 2 y −j
j =0
j !(s − 1 − j )!(4π)j
Finally, by combining (6.3) and (6.2), we find that for y > 1 + ε,
.
(6.3)
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3745
Fm (z, s) − Pm,ε (z, s) = cm,s y 1−s + e−2π|m|y e(mx)(−1)s
s−1
(−1)j y −j κm,j (s)
j =0 1
+ 2π|m|s− 2
e−2π|n|y e(nx)b(m, n; s)
n=0
1 s−1 (s − 1 + j )!|n|−j − 2 y −j
j =0
j !(s − 1 − j )!(4π)j
2
.
7. Proof of Theorem 1.1 We will deduce Theorem 1.1 from the following theorem, whose proof will require further technical results from Sections 8–10. Theorem 7.1. Let m ∈ Z− , s ∈ Z+ , and let −D < 0 and d > 0 be odd, coprime fundamental discriminants. Then there exists an effective constant c > 0 such that for all δ < 1/16 and 0 < b < δ/c, 1 h(−dD)
= δd,1
χd (Q)Fm (zQ , s) −
zQ ∈ΛdD (1)
s−1 j =0
κm,j (s)
−j
χd (Q) Im(zQ )
e(mzQ )
Im(zQ )>1+(dD)−b
Fm (z, s) dμ + O (dD)−(δ−bc) + O (dD)−b
reg
as dD → ∞. Proof. Let ε < 1/4, and define the function Fm,s,ε (z) := Fm (z, s) − Pm,ε (z, s) where Pm,ε (z, s) is the Poincaré series defined in Section 6. Then Fm,s,ε (z) is C ∞ and Γ -invariant, and by Proposition 6.2 we see that Fm,s,ε (z) has cuspidal growth of power α = 1−s. Here we emphasize that this growth is uniform in ε for y > 5/4, and thus the same choice of cutoff parameter T0 = 2 and corresponding cutoff function η2 given by ⎧ 1 < y 2, ⎨ 0, 1−s χ(y/2), 2 < y < 4, y c η2 (z) = m,s ⎩ y 4, cm,s y 1−s , works for each Fm,s,ε (z). We now substitute Fm,s,ε (z) into Theorem 5.1 with the choices N = 1 and T0 = 2, and find that for all δ < 1/16, 1 h(−dD)
χd (Q) Fm (zQ , s) − Pm,ε (zQ , s)
zQ ∈ΛdD (1)
= δd,1 Y0 (1)
Fm (z, s) − Pm,ε (z, s) dμ + O a0 Fm,s,ε,2 2 + 1 (dD)−δ
(7.1)
3746
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
as dD → ∞, where Fm,s,ε,2 (z, s) := Fm (z, s) − Pm,ε (z, s) − η2 (z). Because ψm,s,ε (y) = 0 for y 1, it follows from the definitions of Pm,ε (z, s) and ψm,s,ε that for dD sufficiently large,
χd (Q)Pm,ε (zQ , s) =
zQ ∈ΛdD (1)
χd (Q)ψm,s,ε (Im zQ )e(mzQ )
Im(zQ )>1
=
χd (Q)ψm,s,ε (Im zQ )e(mzQ )
1
+
s−1
κm,j (s)
j =0
χd (Q) Im(zQ )−j e(mzQ ).
Im(zQ )>1+ε
For notational convenience define 1 Rε (dD) := h(−dD) −
s−1
κm,j (s)
j =0
χd (Q)Fm (zQ , s)
zQ ∈ΛdD (1) −j
χd (Q) Im(zQ )
e(mzQ ) .
Im(zQ )>1+ε
Then we can write (7.1) in the equivalent form
Rε (dD) = δd,1
Fm (z, s) − Pm,ε (z, s) dμ + O a0 Fm,s,ε,2 2 + 1 (dD)−δ
Y0 (1)
+
1 h(−dD)
χd (Q)ψm,s,ε (Im zQ )e(mzQ ).
1
By Proposition 8.2 and Lemma 10.1, we have
Rε (dD) = δd,1
Fm (z, s) dμ + O ε −a1 (dD)−δ + O (dD)−δ
reg
+
1 h(−dD)
χd (Q)ψm,s,ε (Im zQ )e(mzQ )
1
for some sufficiently large integer a1 > 0 (depending on a0 ). Now, we have the estimate 2πm(1+ε) χ (Q)ψ (Im z )e(mz ) #Λε (dD) d m,s,ε Q Q sup ψm,s,ε (t) e 1
t∈R+
(7.2)
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3747
where Λε (dD) := zQ ∈ ΛdD (1): 1 < Im(zQ ) 1 + ε . By combining (7.2) with Lemma 9.1 and Siegel’s theorem (5.13), we find that 1 h(−dD)
χd (Q)ψm,s,ε (Im zQ )e(mzQ ) = O(ε) + O ε −a2 (dD)−δ
(7.3)
1
for some sufficiently large integer a2 > 0. It follows from the preceding analysis that for all δ < 1/16,
Rε (dD) = δd,1
Fm (z, s) dμ + O ε −a1 (dD)−δ + O (dD)−δ + O(ε) + O ε −a2 (dD)−δ .
reg
Let c = max{a1 , a2 } and choose b > 0 such that b < δ/c. Because ε is independent of dD in the preceding estimates, we can set ε = (dD)−b . Then
Rε (dD) = δd,1
Fm (z, s) dμ + O (dD)−(δ−bc) + O (dD)−b
reg
as dD → ∞.
2
Proof of Theorem 1.1. Theorem 1.1 now follows by combining Theorem 7.1 with Propositions 11.1 and 12.1. 2 8. Proof of Proposition 8.2 In this section we prove Proposition 8.2 (see also [11]). For ε > 0 define the cutoff function ψm,s,ε (t), t Y, Y ψm,s,ε (t) := 0, t > Y, and the associated cutoff Poincaré series Pm,ε,Y (z, s) :=
Y ψm,s,ε (Im γ z)e(mγ z).
γ ∈Γ∞ \Γ
We will need the following lemma. Lemma 8.1. For ε > 0, we have Pm,ε (z, s) = Pm,ε,Y (z, s) for all z ∈ FY .
3748
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
Proof. We need to show that for Im(z) Y ,
Y ψm,s,ε (Im γ z)e(mγ z) =
γ ∈Γ∞ \Γ
(8.1)
ψm,s,ε (Im γ z)e(mγ z).
γ ∈Γ∞ \Γ
Y By definition of ψm,s,ε we have
Y ψm,s,ε (Im γ z)e(mγ z) =
γ ∈Γ∞ \Γ
ψm,s,ε (Im γ z)e(mγ z).
γ ∈Γ∞ \Γ Im(γ z)Y
Moreover, since Im(z) Y , which forces the identity matrix I into the first summand inside the brackets in (8.2), we have
ψm,s,ε (Im γ z)e(mγ z) =
γ ∈Γ∞ \Γ
γ ∈Γ∞ \Γ Im(γ z)Y
+
ψm,s,ε (Im γ z)e(mγ z).
(8.2)
γ ∈Γ∞ \Γ Im(γ z)>Y γ =I
Let SY := γ ∈ Γ∞ \Γ : Im(γ z) > Y, γ = I . Then ψm,s,ε (Im γ z)e(mγ z) sup ψm,s,ε (t) max e2πm Im(γ z) #SY . γ ∈SY + t∈R
γ ∈SY
Suppose that Y 10. Then by [19, Lemma 2.10], #SY < 10/Y 1, which implies that #SY = 0. Thus
ψm,s,ε (Im γ z)e(mγ z) = 0,
γ ∈Γ∞ \Γ Im(γ z)>Y γ =I
and (8.1) holds for Y 10. However, since FY ⊂ F10 for Y < 10, the lemma follows. Proposition 8.2. For ε > 0 we have
Fm (z, s) − Pm,ε (z, s) dμ = lim
Y →∞
Y0 (1)
FY
Fm (z, s) dμ.
2
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3749
Proof. By Lemma 8.1 and an unfolding argument, we find that for all Y > 0,
Pm,ε (z, s) dμ =
FY
Pm,ε,Y (z, s) dμ F
3 = π
Y
dy ψm,s,ε (y)e2πmy 2 y
1 ·
0
e2πimx dx = 0. 0
Then because Fm (z, s) − Pm,ε (z, s) ∈ L1 (Y0 (1)), it follows that
Fm (z, s) dμ = lim
lim
Y →∞
Y →∞
FY
Fm (z, s) dμ − FY
= lim
Y →∞
=
Pm,ε (z, s) dμ
FY
Fm (z, s) − Pm,ε (z, s) dμ
FY
Fm (z, s) − Pm,ε (z, s) dμ.
2
Y0 (1)
Note that the regularized integral in Proposition 8.2 will be evaluated in Section 12. 9. Proof of Lemma 9.1 In this section we prove Lemma 9.1 (see also [10, pp. 248–249]). Lemma 9.1. For ε < 1/4 we have 7 #Λε (dD) 4εh(−dD) + O ε −a2 (dD) 16 + for some sufficiently large integer a2 > 0 and all > 0. Proof. For each ε > 0, one can construct a C ∞ function φε : R → [0, 1] which is supported in the interval (1 − ε, 1 + 2ε), which equals 1 on the interval [1, 1 + ε], and which satisfies the bound a d max a φε (t) ε −a t∈(1−ε,1+2ε) dt for all a ∈ Z+ . Define the incomplete Eisenstein series gε (z) :=
φε Im(γ z) .
γ ∈Γ∞ \Γ
Then using that φε (y) = 1 for 1 y 1 + ε, we obtain the decomposition
3750
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
gε (zQ ) = #Λε (dD) +
zQ ∈ΛdD (1)
zQ ∈ΛdD (1)\Λε (dD)
+
φε Im(zQ )
φε Im(γ zQ ) .
zQ ∈ΛdD (1) γ ∈Γ∞ \Γ γ =I
Since φε 0 it follows that
#Λε (dD)
gε (zQ ).
zQ ∈ΛdD (1)
The real-analytic Eisenstein series
E(z, s) =
Im(γ z)s ,
Re(s) > 1
γ ∈Γ∞ \Γ
has a meromorphic continuation to C with a simple pole at s = 1 with residue 3/π . Therefore we obtain from [19, (7.12)] the expansion gε (z) =
3 1 φˆ ε (1) + π 2π
φˆ ε
R
1 1 + it E z, + it dt 2 2
where φˆ ε (s) :=
∞
φε (y)y −(s+1) dy.
0
Summing the expansion yields zQ ∈ΛdD (1)
gε (zQ ) =
3 1 φˆ ε (1)h(−dD) + π 2π
φˆ ε
R
1 + it 2
zQ ∈ΛdD (1)
1 E zQ , + it dt. 2
Since φε is supported in (1 − ε, 1 + 2ε) and φε 1, we find that for ε < 1/4, φˆ ε (1) =
∞
φε (y)y −2 dy
0
1 1−ε
1+ε 1+2ε
+ + y −2 dy 1
1+ε
ε ε +ε+ (1 − ε)2 (1 + ε)2 < 2ε + ε + ε = 4ε.
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3751
Therefore 3 φˆ ε (1)h(−dD) 4εh(−dD). π By a variant of the argument used to prove (5.19), one finds that zQ ∈ΛdD (1)
A 7 1 E zQ , + it 1/4 + t 2 6 (dD) 16 + 2
for some fixed constant A6 > 0 and all > 0. Thus
φˆ ε 1 + it 2
R
zQ ∈ΛdD (1)
7 1 E zQ , + it dt (dD) 16 + 2
R
φˆ ε 1 + it 1/4 + t 2 A6 dt. 2
Let a ∈ Z+ . Then integrating by parts a-times and using the fact that the support of φε contained in (1 − ε, 1 + 2ε) ⊂ [3/4, 3/2] for ε < 1/4, we obtain
(a)
φˆ ε
∞ 3 1 + it = φˆ ε (y)y −( 2 +it) dy 2 0
−1
a 3 3 − + it + j = (−1) φε(a) (y)y −( 2 +it)+a dy 2 ∞
a
j =1
0
for all t ∈ R. Thus for ε < 1/4, −1 1+2ε
a 3 (a) − 3 +a φˆ ε 1 + it + it − j φ (y)y 2 dy ε 2 2 j =1
1−ε
1+2ε −1
a 3 (a) 3 φε (y) max y − 2 +a dy 2 + it − j y∈(1−ε,1+2ε) j =1
1−ε
−1
3/2 a 3 3 −a y − 2 +a dy 2 + it − j ε j =1
a ε −a
3/4 a
3 2
j =1
−1 + it − j .
is
3752
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
We now see that for some sufficiently large integer a1 > 0 (depending on A6 ), we have
φˆ ε 1 + it 2
R
,a1 ε
−a1
zQ ∈ΛdD (1)
(dD)
7 16 +
1 E zQ , + it dt 2
−1
a1 3 + it − j 1/4 + t 2 A6 dt 2
R j =1 7
,a1 ε −a1 (dD) 16 + .
2
10. Proof of Lemma 10.1 Lemma 10.1. For ε < 1/4 we have a 0 Fm,s,ε,2 = Oa ε −a2 0 2 for some sufficiently large integer a2 > 0 depending on a0 . Proof. Recall that Fm,s,ε,2 (z) := Fm (z, s) − Pm,ε (z, s) − η2 (z), where the cutoff function η2 is given by ⎧ ⎨ 0, η2 (z) = cm,s y 1−s χ(y/2), ⎩ cm,s y 1−s ,
1 < y 2, 2 < y < 4, y 4.
For notational convenience we write the Fourier expansion (6.1) as Pm,ε (z, s) = ψm,s,ε (y)e(mz) + fm,s,ε (x, y), where fm,s,ε (x, y) :=
n∈Z
e(nx)
∞
S(m, n; c)
c=1
ψm,s,ε R
c−2 y −mc−2 e − nt dt. t + iy t 2 + y2
Clearly we have a 0 Fm,s,ε,2 2 2
∞ 1
√
3/2 0
a 0 Fm,s,ε,2 (z)2 dx dy . y2
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3753
If we let T = 2 in the Fourier expansion (5.2), we see that ψ2
c−2 y t 2 + y2
=0
√ √ for y 3/2, so in fact η2 (z) = 0 in the larger range 3/2 y 2. Then using the definitions of ψm,s,ε , η2 , and Proposition 6.2, we find that ⎧ F (z, s) − fm,s,ε (x, y), ⎪ ⎪ ⎪ m ⎪ ⎪ ⎨ Fm (z, s) − ψm,s,ε (y)e(mz), Fm,s,ε,2 (z) = cm,s y 1−s + O(e−cy ), ⎪ ⎪ ⎪ c y 1−s (1 − χ(y/2)) + O(e−cy ), ⎪ ⎪ ⎩ m,s −cy O(e ),
√ 3/2 y < 1, 1 y 1 + ε, 1 + ε < y 2, 2 < y < 4, y 4.
Recalling the explicit form of the O(e−cy ) terms in Proposition 6.2, it follows that
∞ 1 √
a 0 Fm,s,ε,2 (z)2 dx dy y2
3/2 0
1 1 = √
a 0 Fm (z, s) − fm,s,ε (x, y) 2 dx dy y2
3/2 0
1+ε 1 + 1
a 0 Fm (z, s) − ψm,s,ε (y)e(mz) 2 dx dy + O(1). y2
0
By linearity of a0 and the triangle inequality, we have
1+ε 1 I1 := 1
0
1+ε 1 1
a 0 Fm (z, s) − ψm,s,ε (y)e(mz) 2 dx dy y2 a 0 Fm (z, s)2 dx dy y2
0
1+ε 1 +2 1
0
1+ε
a 0 ψm,s,ε (y)2 dy . y2
+ 1
Recall that
a 0 Fm (z, s) · a0 ψm,s,ε (y) dx dy y2
3754
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
ψm,s,ε (t) := λ
s−1 t −1 κm,j (s)t −j , ε j =0
where λ : R → [0, 1] is a C ∞ function such that λ(t) =
0, t 0, 1, t 1.
Then it is clear that 2a d 0 max 2a ψm,s,ε (y) = O ε −2a0 . y∈[1,1+ε] dy 0 Therefore (using that ε < 1/4), I1 = O(1) + O ε −2a0 + O ε −4a0 . Similarly, we have
1 1 I2 := √
3/2 0
1 1 √
a 0 Fm (z, s) − fm,s,ε (x, y) 2 dx dy y2 a 0 Fm (z, s)2 dx dy y2
3/2 0
1 1 +2 √
3/2 0
1 1 + √
Observe that for y
a 0 Fm (z, s) · a0 fm,s,ε (x, y) dx dy y2
a 0 fm,s,ε (x, y)2 dx dy . y2
3/2 0
√ 3/2 and c 2, we have
y 2 √ 1. c2 (t 2 + y 2 ) c2 3 √ Since ψm,s,ε (u) = 0 for u 1, it follows that for y 3/2 and c 2, ψm,s,ε Therefore, for y
c−2 y 2 t + y2
= 0.
√ 3/2 the function fm,s,ε (x, y) simplifies to
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
fm,s,ε (x, y) =
e(nx)S(m, n; 1)
n∈Z
ψm,s,ε R
y −m e − nt dt, t + iy t 2 + y2
i.e., only the term with c = 1 remains. Define the function gm,y,ε (t) := ψm,s,ε
y −m e . t + iy t 2 + y2
Then by definition of ψm,s,ε we have dk gm,y,ε (t) = 0 |t|→∞ dt k lim
for all integers k 0. Integrating by parts (2a0 + 2)-times yields
1 (2πin)2a0 +2
gm,y,ε (t)e(−nt) dt = R
(2a0 +2) gm,y,ε (t)e(−nt) dt.
R
In fact, since
t2
y 1 + y2
⇔
|t|
"
y(1 − y),
we have (2a0 +2) (t) = 0 gm,y,ε
for |t|
√ √ y(1 − y). Thus in the range 3/2 y 1 we obtain
fm,s,ε (x, y) =
n∈Z
e(nx) S(m, n; 1) (2πin)2a0 +2
1
(2a0 +2) gm,y,ε (t)e(−nt) dt.
−1
We now analyze a0 fm,s,ε (x, y) and show that I2 = O(1) + O ε −(2a0 +2) + O ε −2(2a0 +2) . For clarity we first give the argument for a0 = 1. We find that
3755
3756
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
e(nx) −y ∂x2 + ∂y2 fm,s,ε (x, y) = −y 2 S(m, n; 1) (2πin)2
1
2
n∈Z
(4) gm,y,ε (t)e(−nt) dt
−1
e(nx) ∂ 2 (4) −y S(m, n; 1) g (t)e(−nt) dt. 4 (2πin) ∂y 2 m,y,ε 1
2
n∈Z
−1
Since (see [19, (2.37)]) S(m, n; 1) 1, it follows that 2 ∂ (4) (4) 2 fm,s,ε (x, y) y max gm,y,ε (t) + y max 2 gm,y,ε (t). t∈[−1,1] t∈[−1,1] ∂y 2
It is clear from the definition of gm,y,ε that (4) max gm,y,ε (t) = Oy ε −4
t∈[−1,1]
and 2 ∂ (4) max 2 gm,y,ε (t) = Oy ε −4 . t∈[−1,1] ∂y Hence fm,s,ε (x, y) = Oy ε −4 . Finally, it follows that I2 = O(1) + O ε −4 + O ε −8 . The preceding argument generalizes in a straightforward way to higher derivatives, and thus we conclude that I2 = O(1) + O ε −(2a0 +2) + O ε −2(2a0 +2) .
2
11. Exponential sums In this section we express the main term in Theorem 7.1 as a twisted exponential sum. Let n, c ∈ Z+ , and , δ ∈ Z. Following [22], we define the twisted exponential sum S ,δ (n, c) :=
x 2 ≡−δ (mod c)
χd
x 2 + δ 2nx c , x, e . 4 c c
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3757
Proposition 11.1. Let m ∈ Z− , let s, c ∈ Z+ , and let −D < 0 and d > 0 be odd, coprime fundamental discriminants. Then s−1
κm,j (s)
j =0
χd (Q) Im(zQ )−j e(mzQ )
Im(zQ )>1+(dD)−b
√ D,d 4π|m| dD , SD,d |m|, c gm,s (c) exp c
1 = 2
0
√ 2 dD 1+(dD)−b
c≡0 (mod 4)
where D,d gm,s (c) := |m|s−1
s−1 √ − dD j (s − 1 + j )! . 2π|m|c (s − 1 − j )!j ! j =0
Proof. Each Q ∈ QdD is of the form cX 2 + xXY + aY 2 ∈ Z[X, Y ] for integers c, x and a. A classical parameterization due to Gauss implies that those zQ with Im(zQ ) > 1 + (dD)−b √ are given by integer pairs (c, x) with 0 < c < dD/2(1 + (dD)−b ) and x 2 ≡ −dD (mod 4c). Letting 4c → c, we find that s−1
κm,j (s)
j =0
χd (Q) Im(zQ )−j e(mzQ )
Im(zQ )>1+(dD)−b
1 = κm,j (s) 2 j =0
√ 4π|m| dD exp c
s−1
0
√ 2 dD 1+(dD)−b
c≡0 (mod 4)
×
χd
x 2 ≡−dD (mod c)
=
1 2
√ 0
c x 2 + dD , x, 4 c
√ 2 dD −j e 2|m|x/c c
√ 4π|m| dD D,d . gm,s (c)SD,d |m|, c exp c
2
c≡0 (mod 4)
12. Evaluation of the regularized integral In this section we evaluate the regularized integral appearing in Theorem 7.1. Proposition 12.1. Let m ∈ Z− and s ∈ Z+ . Then
lim
Y →∞
FY
Fm (z, s) dμ =
0, −24σ (|m|)|m|s−1 ,
s = 1, s 2.
3758
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
Proof. We proceed using the method of Lerche, Schellekens, and Warner [23] (see also [2,11]). First observe that one has the identity ∂(E2 (z) −
3 π Im(z) )
∂z
=
3i , 2π Im(z)2
where E2 (z) := 1 − 24
∞
σ1 (n)q n ,
q := e(z)
n=1
and z = x + iy. Because dz dz = 2i dx dy and Fm (z, s) is Γ -invariant, it follows from Stokes’ theorem that 1 2 +iY
lim
Y →∞
Fm (z, s) dμ = lim
FY
Y →∞ − 12 +iY
3 dx. Fm (x + iY, s) E2 (x + iY ) − πY
(12.1)
Using the expansion given in Proposition 6.1 for the forms Fm (x + iY, s), we find after multiplying that 3 Fm (x + iY, s) E2 (x + iY ) − πY
4π 1+s σ2s−1 (|m|)Y 1−s 1 1 3 = −24σ |m| 2π|m|s− 2 e−2π|m|Y Y 2 Is− 1 2π|m|Y + 1− 2 (2s − 1)Γ (s)ζ (2s) πY 1 + ak (m, Y, s)e(kx) + O . Y k∈Z k=0
Next, using the expansion for the Bessel function given in the proof of Proposition 6.1, we find that the integral (12.1) equals ∞ s−1 −j 3 1 4π 1+s σ2s−1 (|m|)Y 1−s 1− +O . + O Y lim −24σ |m| |m| 1+ Y →∞ (2s − 1)Γ (s)ζ (2s) πY Y j =1
The result follows upon taking the limit as Y → ∞, where if s = 1 we use the fact that ζ (2) = π 2 /6. 2 Acknowledgments We would like to thank Anton Deitmar, Gergely Harcos, Dennis Hejhal, Jim Kelliher, Ken Ono, Steve Wainger, Tong Hai Yang, and Matt Young for very helpful discussions regarding this work. We would also like to thank the referee for a very careful reading of the manuscript leading to many improvements in exposition.
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3759
References [1] V. Blomer, G. Harcos, Hybrid bounds for twisted L-functions, J. Reine Angew. Math. 621 (2008) 53–79. [2] R. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998) 491–562. [3] R. Borcherds, What is Moonshine?, in: Proceedings of the International Congress of Mathematicians, vol. I, Berlin, 1998, in: Doc. Math., extra vol. I, 1998, pp. 607–615. [4] K. Bringmann, K. Ono, Arithmetic properties of coefficients of half-integral weight Maass–Poincaré series, Math. Ann. 337 (2007) 591–612. [5] K. Bringmann, K. Ono, An arithmetic formula for the partition function, Proc. Amer. Math. Soc. 135 (2007) 3507– 3514. [6] K. Bringmann, K. Ono, Lifting cusp forms to Maass forms with an application to partitions, Proc. Natl. Acad. Sci. USA 104 (10) (2007) 3725–3731. [7] J.H. Bruinier, Borcherds Products on O(2, l) and Chern Classes of Heegner Divisors, Lecture Notes in Math., vol. 1780, Springer-Verlag, Berlin, 2002, viii+152 pp. [8] J. Bruinier, P. Jenkins, K. Ono, Hilbert class polynomials and traces of singular moduli, Math. Ann. 334 (2006) 373–393. [9] H. Darmon, Heegner points, Heegner cycles, and congruences, in: Elliptic Curves and Related Topics, in: CRM Proc. Lecture Notes, vol. 4, Amer. Math. Soc., Providence, RI, 1994, pp. 45–59. [10] W. Duke, Modular functions and the uniform distribution of CM points, Math. Ann. 334 (2006) 241–252. [11] W. Duke, An introduction to the Linnik problems, in: Equidistribution in Number Theory, an Introduction, in: NATO Sci. Ser. II Math. Phys. Chem., vol. 237, Springer, Dordrecht, 2007, pp. 197–216. [12] W. Duke, P. Jenkins, Integral traces of singular values of weak Maass forms, Algebra Number Theory 2 (2008) 573–593. [13] J.D. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math. 293–294 (1977) 143– 203. [14] A. Folsom, R. Masri, Equidistribution of Heegner points and the partition function, Math. Ann. 348 (2010) 289–317. [15] B. Gross, W. Kohnen, D. Zagier, Heegner points and derivatives of L-series. II, Math. Ann. 278 (1987) 497–562. [16] B. Gross, D. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986) 225–320. [17] G. Harcos, P. Michel, The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points. II, Invent. Math. 163 (2006) 581–655. [18] D. Hejhal, The Selberg Trace Formula for PSL2 (R), Lecture Notes in Math., vol. 1001, Springer, 1983. [19] H. Iwaniec, Spectral Methods of Automorphic Forms, Grad. Stud. Math., vol. 53, Amer. Math. Soc./Revista Matematica Iberoamericana, Providence, RI/Madrid, 2002. [20] H. Iwaniec, E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., vol. 53, Amer. Math. Soc., Providence, RI, 2004, xii+615 pp. [21] B.D. Kim, R. Masri, T.H. Yang, Nonvanishing of Hecke L-functions and the Bloch–Kato conjecture, Math. Ann., doi:10.1007/s00208-010-0521-7, in press. [22] W. Kohnen, Fourier coefficients of modular forms of half-integral weight on Γ0 (4), Math. Ann. 271 (2) (1985) 237–268. [23] W. Lerche, A.N. Schellekens, N.P. Warner, Lattices and strings, Phys. Rep. 177 (1989) 1–140. [24] A. Miller, A. Pixton, Arithmetic traces of non-holomorphic modular invariants, Int. J. Number Theory 6 (1) (2010) 69–87. [25] D. Niebur, A class of nonanalytic automorphic functions, Nagoya Math. J. 52 (1973) 133–145. [26] K. Ono, Singular moduli generating functions for modular curves and surfaces, in: Analytic Number Theory, in: Clay Math. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 2007, pp. 185–206. [27] K. Ono, Unearthing the visions of a master: harmonic Maass forms and number theory, Harvard-MIT Current Developments in Mathematics, International Press, 2008, preprint. [28] H. Rademacher, On the expansion of the partition function in a series, Ann. of Math. 44 (1943) 416–422. [29] J.-L. Waldspurger, Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie, Compos. Math. 54 (1985) 173–242. [30] G.N. Watson, A Treatise on the Theory of Bessel Functions, second ed., Cambridge University Press, 1966. [31] D. Zagier, Traces of singular moduli, in: Motives, Polylogarithms and Hodge Theory, Part I, Irvine, CA, 1998, in: Int. Press Lect. Ser., vol. 3, I, Int. Press, Somerville, MA, 2002, pp. 211–244. [32] S. Zhang, Gross–Zagier formula for GL2 , Asian J. Math. 5 (2001) 183–290.
The asymptotic distribution of traces of Maass–Poincaré series Amanda Folsom a , Riad Masri b,∗ a Department of Mathematics, Yale University, New Haven, CT 06520, United States b Department of Mathematics, Texas A&M University, College Station, TX 77843, United States
Received 22 February 2010; accepted 1 November 2010 Available online 13 November 2010 Communicated by George E. Andrews
Abstract We establish an asymptotic formula with a power savings in the error term for traces of CM values of a family of Maass–Poincaré series which contains the modular j -function as a special case. By work of Borcherds (1998) [2], Zagier (2002) [31], and Bringmann and Ono (2007) [4], these traces are Fourier coefficients of half-integral weight weakly holomorphic modular forms and Maass forms. © 2010 Elsevier Inc. All rights reserved. Keywords: Asymptotic distribution; CM points; Maass–Poincaré series; Traces; Fourier coefficients
1. Introduction and statements of results Let j (z) be the classical modular j -function for SL2 (Z), j (z) = q −1 + 744 + 196 884q + 21 493 760q 2 + · · · , where q = e2πiz . The values of j (z) at CM points in the complex upper half-plane H are known as singular moduli. These are algebraic integers which play a fundamental role in number theory. For example, they generate class fields of imaginary quadratic fields, and parameterize isomorphism classes of elliptic curves with complex multiplication. * Corresponding author.
E-mail addresses: [email protected] (A. Folsom), [email protected] (R. Masri). 0001-8708/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2010.11.001
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3725
In the influential paper [31], Zagier proved that traces of singular moduli are Fourier coefficients of a half-integral weight modular form and used this to give another proof of a famous theorem of Borcherds [2] on infinite product expansions of integer weight modular forms with Heegner divisor. Zagier’s results have inspired a large number of works in recent years. An overview of some of these works can be found in the survey articles of Ono [26,27]. In this paper we will study the asymptotic distribution of “twisted traces” of CM values of a family of Maass–Poincaré series which contains the j -function as a special case. Let m ∈ Z with m = 0, and let s ∈ C. Define the Maass–Poincaré series 1
Fm (z, s) := 2π|m|s− 2
1 Im(γ z) 2 Is− 1 2π|m| Im(γ z) e m Re(γ z) , 2
γ ∈Γ∞ \SL2 (Z)
Re(s) > 1,
where Γ∞ < SL2 (Z) is the subgroup of elements stabilizing the cusp at ∞, Iν is the Bessel function of order ν, and e(z) := e2πiz . This Poincaré series was first studied by Niebur in [25], and it has since appeared in many different contexts (see e.g. [4–7,12,13,24,18]). Niebur [25] proved that Fm (z, s) has an analytic continuation to Re(s) = 1, and showed that for each m ∈ Z− , Fm (z, 1) = j|m| (z) + 24σ |m| ,
(1.1)
where j|m| (z) is the unique modular function satisfying j|m| (z) = q −|m| + O(q) and σ (|m|) :=
||m|
is the divisor function. In particular, j1 (z) = j (z) − 744.
The twisted traces of CM values of Fm (z, s) are defined as follows. Let −D < 0 be an odd fundamental discriminant. Let QD be the set of positive definite, primitive, integral binary quadratic forms Q(X, Y ) = [a, b, c] = aX 2 + bXY + cY 2 of discriminant b2 − 4ac = −D. Then SL2 (Z) acts on QD in the usual way. Let ΛD (1) be the set of Heegner points of discriminant −D on the modular curve X√0 (1). There are exactly h(−D) such points, where h(−D) is the ideal class number of K = Q( −D ). Furthermore, there is a bijection QD /SL2 (Z) → ΛD (1) given by [Q] → zQ where zQ =
√ −b + −D 2a
is the unique root in H of the dehomogenized form Q(X, 1) = aX 2 + bX + c. Let d > 0 be an odd fundamental discriminant (possibly 1) coprime to D. Define the genus character χd : QdD /SL2 (Z) → {±1}
3726
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
by χd (Q) = χd (a, b, c) :=
d n
where n is any integer coprime to d which is represented by Q = [a, b, c] ∈ QdD . Here (d/·) is the usual Kronecker symbol. This definition is independent of the choice of n (see Section 4). The “twisted trace” of Fm (z, s) is defined by
Trd Fm (·, s); D :=
χd (Q)Fm (zQ , s).
zQ ∈ΛdD (1)
Zagier [31] proved that the traces Trd (j1 ; D) are Fourier coefficients of a weight 3/2 weakly holomorphic modular form for Γ0 (4). Generalizing many results in [31], Bringmann and Ono [4] proved that the traces Trd (Fm (·, s); D) are Fourier coefficients of certain half-integral weight Maass forms (see also [12, Proposition 6]). For example, if s ∈ Z+ is an odd integer, one has bs (−d, D) =
(−1)
s+1 2
d
D
s−1 2
s 2
Trd F−1 (·, s); D ,
where the numbers bs (−d, D) are Fourier coefficients of a weight s + 1/2 Maass form Ps (−d; z) ∈ M +
s+ 12
Γ0 (4)
in Kohnen’s plus-space with Fourier expansion at ∞ of the form Ps (−d; z) = q −d +
bs (−d, n)q n .
n0 −n≡0,1 mod 4
In light of these results, it is natural to study the asymptotic distribution of the traces Trd (Fm (·, s); D) as D → ∞. For traces of singular moduli, this problem is closely related to the classical observation that the number eπ
√
163
= 262 537 412 640 768 743.9999999999992 . . .
is nearly an integer (see [3,26]). In [8], Bruinier, Jenkins, and Ono conjectured that a certain “perturbed” average of the traces Tr1 (j1 ; D) − Gred (D) − Gold (D) → −24 h(−D) as D → ∞. See [26] for an explanation of why the number −24 appears in this limit. This conjecture was studied by Duke [10] from a somewhat different perspective. Using the equidistribution of Heegner points on X0 (1) and a regularization of the pole at ∞ of j (z), he established the following result which implies the conjecture of Bruinier, Jenkins, and Ono,
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
1 1 Tr1 (j|m| ; D) − h(−D) 2
√ 0
3727
√ 4π|m| D → −24 SD |m|, c exp c
as D → ∞, where SD (|m|, c) is the exponential sum 2|m|x . e c
SD |m|, c :=
x 2 ≡−D (mod c)
We now state our main results. For n ∈ Z+ , define the twisted exponential sum SD,d (n, c) :=
χd
x 2 ≡−dD (mod c)
c x 2 + dD 2nx , x, e , 4 c c
and for m ∈ Z− and s ∈ Z+ , define the function D,d gm,s (c) := |m|s−1
s−1 √ − dD j (s − 1 + j )! . 2π|m|c (s − 1 − j )!j ! j =0
Let F be the standard fundamental domain for SL2 (Z). For Y > 0, define the truncated domain FY := z ∈ F : Im(z) Y , and for m ∈ Z− , define the “regularized” integral
Fm (z, s) dμ := lim Fm (z, s) dμ, reg
Y →∞
FY
where dμ = (3/π) dx dy/y 2 is the normalized hyperbolic measure on the open modular curve Y0 (1) = SL2 (Z)\H. We will establish the following asymptotic formula with a power savings in the error term for the traces Trd (Fm (·, s); D) as dD → ∞. An outline of the proof is given in Section 2. Theorem 1.1. Let m ∈ Z− , s ∈ Z+ , and let −D < 0 and d > 0 be odd, coprime fundamental discriminants. Then there exists an effective constant a > 0 such that for all δ < 1/16 and 0 < b < δ/a, the following asymptotic formulas hold: (1) If d = 1 then 1 1 Tr1 Fm (·, s); D − h(−D) 2
= reg
√ 0
SD,1
c≡0 (mod 4)
Fm (z, s) dμ + O D −(δ−ab) + O D −b
√ D,1 4π|m| D |m|, c gm,s (c) exp c
3728
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
as D → ∞, where
Fm (z, s) dμ = reg
0, −24σ (|m|)|m|s−1 ,
if s = 1, if s 2.
(2) If d > 1 then 1 1 Trd Fm (·, s); D − h(−dD) 2
0
√ 2 dD 1+(dD)−b
√ D,d 4π|m| dD SD,d |m|, c gm,s (c) exp c
c≡0 (mod 4)
= O (dD)−(δ−ab) + O (dD)−b as dD → ∞.
If we let s = 1 in Theorem 1.1 and use the relation (1.1), we obtain the following asymptotic formula with a power savings in the error term for the traces Trd (j|m| ; D) as dD → ∞. Corollary 1.2. Let m ∈ Z− , and let −D < 0 and d > 0 be odd, coprime fundamental discriminants. Then there exists an effective constant a > 0 such that for all δ < 1/16 and 0 < b < δ/a, the following asymptotic formulas hold: (1) If d = 1 then 1 1 Tr1 (j|m| ; D) − h(−D) 2
√
0
√ 4π|m| D SD,1 |m|, c exp c
= −24σ |m| + O D −(δ−ab) + O D −b as D → ∞. (2) If d > 1 then 1 1 Trd (j|m| ; D) − h(−dD) 2
0
√ 2 dD 1+(dD)−b
√ 4π|m| dD SD,d |m|, c exp c
c≡0 (mod 4)
= O (dD)−(δ−ab) + O (dD)−b as dD → ∞.
We conclude by noting that in [14], we used techniques related to those in the proof of Theorem 1.1 to study the asymptotic distribution of the partition function p(n), which counts the
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3729
number of partitions of a positive integer n. Let m ∈ Z with m = 0, and let s ∈ C. For N ∈ Z+ , define the Γ0 (N)-invariant Maass–Poincaré series 1
FmN (z, s) := 2π|m|s− 2
1 Im(γ z) 2 Is− 1 2π|m| Im(γ z) e m Re(γ z) , 2
γ ∈Γ∞ \Γ0 (N )
Re(s) > 1
where Γ∞ < Γ0 (N ) is the subgroup of elements stabilizing the cusp at ∞ of X0 (N ). In [5], Bringmann and Ono established the following arithmetic reformulation of Rademacher’s [28] exact formula for p(n), p(n) =
1 Dn
zQ ∈ΛDn (6)
6 χ12 (Q)F−1 (zQ , 2),
where Dn := 24n − 1 is square-free, χ12 := (12/·) is the Legendre symbol, and ΛDn (6) is the set of Heegner points of discriminant −Dn on X0 (6) (see Section 3). In [14], we used this formula and the equidistribution of Galois orbits of Heegner points on the modular curve X0 (6) to obtain a new asymptotic formula for p(n) with an effective error term which sharpens the classical bounds of Hardy and Ramanujan, Rademacher, and Lehmer on the error term in Rademacher’s exact formula for p(n). The estimates in Lemmas 9.1 and 10.1 of this paper are crucial to the proofs in [14], so this paper can in some respects be viewed as a companion to [14]. 2. Outline of the proof of Theorem 1.1 In this section we outline the proof of Theorem 1.1. In Section 5 we establish an asymptotic formula for twisted traces of smooth, Γ0 (N )-invariant functions which are allowed to grow moderately in the cusps of X0 (N ) (see Theorem 5.1). For example, if N = 1 suppose that F : H → C is a smooth, SL2 (Z)-invariant function such that for all n ∈ Z0 , n F (z) − c · y α = O e−c1 ·y
as y → ∞
(2.1)
for some c ∈ C, c1 > 0 and α 1/2. Here n is the n-th iterate of the hyperbolic Laplacian = −y 2 ∂x2 + ∂y2 . We show there exists an integer n0 > 0 such that for all δ < 1/16, 1 Trd (F ; D) = δd,1 h(−dD)
F (z) dμ + Oδ n0 FT0 2 + 1 (dD)−δ
(2.2)
F
as dD → ∞. Here δd,1 is the Kronecker diagonal symbol, T0 > 1 is a fixed cutoff parameter, and FT0 := F − ηT0 where ηT0 : H → C is a smooth, SL2 (Z)-invariant function with growth coinciding precisely with that of F for y T0 . The Fourier expansion of the Maass–Poincaré series Fm (z, s) is of the form Fm (z, s) = O ec2 ·y + O y 1−s + O e−c3 ·y as y = Im(z) → ∞
3730
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
where c2 , c3 are positive constants. In Section 6 we use a variant of an argument of Duke [10] to construct a smooth, SL2 (Z)-invariant Poincaré series Pm,ε for each ε > 0 such that the regularized function Fm,s,ε (z) := Fm (z, s) − Pm,ε (z, s) satisfies the growth condition (2.1) with α := 1 − s. In Section 7 we substitute Fm,s,ε into (2.2) to obtain the “preliminary” asymptotic formula 1 Trd Fm,s (·, s); D − Trd Pm,ε (·, s); D h(−dD)
= δd,1 Fm,s,ε (z) dμ + Oδ n0 Fm,s,ε,T0 2 + 1 (dD)−δ F
as dD → ∞. We then show that Trd (Pm,ε (·, s); D) equals the main term in Theorem 1.1 plus an explicit error term depending on ε and dD. See Eq. (7.3), and Sections 9 and 11. In Section 8 we show that for all ε > 0,
Fm,s,ε (z) dμ = Fm (z, s) dμ, F
reg
and in Section 12 we use a Borcherds-type integration to evaluate this regularized integral. In Section 10.1 we establish the upper bound n 0 Fm,s,ε,T ε −A 0 2 as ε → 0 for some fixed positive integer A depending on n0 . Finally, by choosing ε to be a sufficiently small negative power of dD, we obtain the final asymptotic formula with a power savings in the error term. 3. Heegner points on X0 (N) In this section we review some facts concerning Heegner points on the modular curve X0 (N ). Let N be a positive integer, and let −D < 0 be an odd√ fundamental discriminant coprime to N such that every prime divisor p of N is split in K = Q( −D ). Let QD,N be the set of positive definite, primitive, integral binary quadratic forms Q(X, Y ) = [a, b, c] = aX 2 + bXY + cY 2 of discriminant b2 − 4ac = −D with N |a. The set QD,N is stable under the action of Γ0 (N ). Let ΛD (N ) be the set of Heegner points of discriminant −D on X0 (N ). There is a bijection QD,N /Γ0 (N ) → ΛD (N ) given by [Q] → zQ where √ −b + −D zQ = 2N a is the unique root in H of the dehomogenized form Q(X, 1) = N aX 2 + bX + c.
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3731
Fix a solution r mod 2N of r 2 ≡ −D mod 4N . Note that there are exactly 2t (N ) such solutions r, where t (N ) is the number of distinct prime divisors of N . Define the subset of forms QD,N,r = Q = [a, b, c] ∈ QD,N : b ≡ r mod 2N . The set QD,N,r is also stable under the action of Γ0 (N ). There is a decomposition (see [15, p. 507]) QD,N /Γ0 (N ) =
QD,N,r /Γ0 (N ).
(3.1)
r mod 2N r 2 ≡−D mod 4N
The natural map QD,N,r /Γ0 (N ) → QD,1 /SL2 (Z) is a bijection which makes the set QD,N,r /Γ0 (N ) into a group of order h(−D) via the√Gauss law of composition on QD,1 /SL2 (Z). Here, h(−D) is the ideal class number of K = Q( −D ). Moreover, by class field theory QD,N,r /Γ0 (N ) ∼ = CLK ∼ = Gal(H /K) where CLK is the ideal class group of K and H is the Hilbert class field of K. For details concerning these facts, see [9]. The set ΛD (N ) is divided into 2t (N ) simple, transitive Gal(H /K)-orbits of size h(−D) (see [16, pp. 235–236]). Define the Galois orbit σ Or := zQ : σ ∈ Gal(H /K) , r where [Qr ] is any class in QD,N,r /Γ0 (N ). Then one has a bijection QD,N,r /Γ0 (N ) → Or .
(3.2)
4. Generalized genus characters In this section we briefly review the definition of generalized genus characters. For more details concerning these facts, see [15, pp. 508–510]. Let N be a positive integer. Let −D < 0 and d > 0 be odd, coprime fundamental discriminants such that both D and d are squares modulo 4N . For Q = [N a, b, c] ∈ QdD,N set d , χd (Q) := n where n is an integer coprime to d represented by the form [N1 a, b, N2 c] for some decomposition N = N1 N2 , Ni > 0. Such an n exists, and the value of ( dn ) is independent of the choice of N1 , N2 and n. The function χd is Γ0 (N )-invariant and thus defines a function χd : QdD,N /Γ0 (N ) → {±1}.
3732
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
Furthermore, χd restricts to a real ideal class group character on QdD,N,r /Γ0 (N ) for each r mod 2N such that r 2 ≡ −dD mod 4N . 5. Asymptotics for twisted traces Let F : H → C be a C ∞ , Γ0 (N )-invariant function. We say that F has cuspidal growth of power α for some α ∈ R if for every cusp a of Γ0 (N ) there exists a constant ca ∈ C (possibly equal to 0) such that for each integer a 0, a F (σa z) − ca y α = O e−cy as y = Im(z) → ∞ for some c > 0. Here σa ∈ SL2 (R) is a scaling matrix such that σa (∞) = a. Let Y0 (N ) = Γ0 (N )\H be the open modular curve, and let dμ be the normalized hyperbolic measure on Y0 (N). Theorem 5.1. Let N be a fixed positive integer. Let −D < 0 and d > 0 be odd, coprime fundamental discriminants such that D and d are √ squares modulo 4N , dD is coprime to N , and every prime divisor p of N is split in K = Q( −dD ). Let F : H → C be a C ∞ , Γ0 (N )-invariant function with cuspidal growth of power α for some α 1/2. Then there exists an integer a0 > 0 such that for all δ < 1/16,
1 2t (N ) h(−dD)
= δd,1
χd (Q)F (zQ )
Q∈QdD,N /Γ0 (N )
F (z) dμ + ON,δ a0 FT0 2 + 1 (dD)−δ
Y0 (N )
as dD → ∞. Here χd : QdD,N /Γ0 (N ) → {±1} is a generalized genus character as defined in Section 4, and FT0 is a regularized version of F where T0 1 is a fixed cutoff parameter (see Eq. (5.9)). Remark 5.2. Theorem 5.1 should be viewed as consisting of two cases: d = 1 and d > 1. What separates these cases is that when the genus character χd is trivial, the integral in the formula survives, while if χd is nontrivial, the integral is killed by orthogonality. For the proof it is more convenient to combine these cases, so we have used the diagonal symbol δd,1 . Proof of Theorem 5.1. We begin by constructing a C ∞ , Γ0 (N )-invariant cutoff function with growth coinciding with that of F in the cusps a of X0 (N ), and which vanishes on the Heegner points ΛdD (N ) for each sufficiently large cutoff parameter (see also [21]). Lemma 5.3. Let T > 1. There exists a C ∞ , Γ0 (N )-invariant function ηT : H → C such that 0, ηT (σa z) =
ca y α χ(y/T ), ca y α ,
1 < y < T, T y 2T , y > 2T ,
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3733
for each cusp a of X0 (N ), where χ : R+ → [0, 1] is a C ∞ function such that χ(t) =
0, t < 1, 1, t > 2.
Proof. Let χ be as in the statement of the lemma, and define ψT ∈ C0∞ (R+ ) by t ψT (t) := t χ . T α
Then define ηT by (see [19, (3.10)]) ηT (z) :=
ca · Ea (z|ψT ),
(5.1)
a
where
Ea (z|ψT ) :=
ψT Im σa−1 γ z .
γ ∈Γa \Γ0 (N )
Now, by [19, (3.17)] with m = 0, one has the Fourier expansion Ea (σb z|ψT ) = δab ψT (y) +
e(nx)
c>0
n∈Z
Sab (0, n; c)
ψT R
c−2 y e(−nt) dt, t 2 + y2
(5.2)
where δab is the Kronecker diagonal symbol and Sab (0, n; c) is the exponential sum Sab (0, n; c) =
a ∗
cd
∈B\σa−1 Γ0 (N )σb /B
na , e c
where B denotes the group of integral translations B=
1 b : b∈Z 1
(see [19, (2.23)]). This, combined with the inequality ∗ min c > 0: c
∗ ∗
∈ σa−1 Γ0 (N )σb
1
(5.3)
for all cusps a, b of X0 (N ) (see [19, (2.28)–(2.31)]), shows that ηT has the properties stated in the lemma. 2 Lemma 5.4. For T
points ΛdD (N ).
√ dD, the function ηT (z) defined by (5.1) vanishes on the Heegner
3734
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
√ Proof. By Minkowski’s theorem, every ideal class of K = Q( −dD ) contains a primitive integral ideal A of norm NK/Q (A)
√ dD.
(5.4)
Fix a solution r mod 2N of r 2 ≡ −dD mod 4N , and write √ b + −dD A = a, , a = NK/Q (A), b ∈ Z, 2 with b ≡ r mod 2N,
b2 ≡ −dD mod 4N a.
Then √ −b + −dD zA = 2N a is the Heegner point corresponding to Q(X, Y ) = N aX 2 + bXY + cY 2 . For γ ∈ Γ0 (N ) and a cusp a of X0 (N ), write a b σa−1 γ = ∈ SL2 (R). c d Then Im σa−1 γ zA =
√ dD Im(zA ) 1 .
2
2 |c zA + d | |c zA + d | 2N
(5.5)
If c = 0, then d = 1 (see [19, (2.15)–(2.17)]) and we have Im σa−1 γ zA
√ dD . 2N
(5.6)
On the other hand, if c = 0 we write √ dDc bc
+d +i , c zA + d = − 2N a 2N a
so that by (5.4), 2 √
dDc 2 c zA + d 2 = − bc + d + 2N a 2N a (c )2 dD 4N 2 a 2 (c )2
. 4N 2
(5.7)
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3735
It follows from (5.5) and (5.7) that Im σa−1 γ zA
√ √ √ 4N 2 dD dD 1 2 2N dD,
2 |c zA + d | 2N (c ) 2N
(5.8)
where for the last inequality we used (5.3). √ Since ψT (y) = 0 for y < T , we see from the inequalities (5.6) and (5.8) that for T dD, ψT Im σa−1 γ zA = 0 for all γ ∈ Γa \Γ0 (N ). It follows from (5.1) that ηT (zA ) = 0.
2
Now, define the “regularized” function FT (z) := F (z) − ηT (z). Then by Lemma 5.4, to prove Theorem 5.1 it suffices to prove the following proposition. Proposition 5.5. For T
√
dD, we have
1 2t (N ) h(−dD)
χd (Q)FT (zQ )
Q∈QdD,N /Γ0 (N )
F (z) dμ + ON,δ a0 FT0 2 + 1 (dD)−δ
= δd,1 Y0 (N )
as dD → ∞. Proof. Let T T0 1. Here T0 is a fixed cutoff parameter which is independent of dD. We introduce T0 in order to decompose FT into a sum of two functions so that we can isolate the contribution of ηT to the spectral decomposition. Consider the decomposition FT (z) = FT0 (z) + η˜ T (z) where η˜ T (z) := ηT0 (z) − ηT (z). We will first show that for all δ1 < 1/16,
1 2t (N ) h(−dD)
= δd,1 Y0 (N )
as dD → ∞.
χd (Q)FT0 (zQ )
Q∈QdD,N /Γ0 (N )
F (z) dμ − δd,1 Y0 (N )
ηT0 (z) dμ + O a0 FT0 2 (dD)−δ1
(5.9)
3736
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
By definition of ηT0 and our assumption that F has cuspidal growth of power α, we have a FT0 (σa z) = O e−cy as y → ∞ for each integer a 0. The spectral decomposition of L2 (Y0 (N )) with respect to the hyperbolic Laplacian yields the expansion ∞
FT0 (z) = FT0 , 12 +
FT0 , un 2 un (z)
n=1
1 1 1 FT0 , Ea ·, + it Ea z, + it dt + 4π 2 2 2 a R
which converges pointwise absolutely and uniformly on compact subsets of Y0 (N ) since a FT0 is C ∞ with exponential decay in each cusp a of Y0 (N ). Here u0 (z) = 1 is the constant eigenfunction for corresponding to the eigenvalue λ0 = 0, {un (z)}∞ n=1 is an orthonormal basis of Maass cusp forms satisfying un = λn un for n ∈ Z+ where the eigenvalues λn = 1/4 + tn2 are ordered so that 0 < λ1 λ2 · · · , and Ea (z, s) is the real-analytic Eisenstein series
Ea (z, s) =
s Im σa−1 γ z ,
z ∈ H, Re(s) > 1.
γ ∈Γa \Γ0 (N )
Summing the spectral expansion yields
χd (Q)FT0 (zQ ) =
Q∈QdD,N /Γ0 (N )
+
∞
χd (Q)
Q∈QdD,N /Γ0 (N )
FT0 , un 2 Wn +
n=1
1 4π a
R
FT0 (z) dμ
Y0 (N )
1 FT0 , Ea ·, + it Wa (t) dt 2 2
where Q∈QdD,N /Γ0 (N )
t (N ) h(−dD), χd (Q) = 2 0,
d = 1, d > 1,
and the twisted hyperbolic Weyl sums are defined by
Wn :=
χd (Q)un (zQ )
Q∈QdD,N /Γ0 (N )
and Wa (t) :=
Q∈QdD,N /Γ0 (N )
1 χd (Q)Ea zQ , + it . 2
Here FT0 ∈ L1 (Y0 (N )) because y α ∈ L1 ([C, ∞), dy/y 2 ) for all α < 1 and C > 0.
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3737
Using (3.1) we have
Wn =
χd (Q)un (zQ ).
(5.10)
r mod 2N Q∈QdD,N,r /Γ0 (N ) r 2 ≡−dD mod 4N
Furthermore, using the bijection (3.2) we have
χd (Q)un (zQ ) =
Q∈QdD,N,r /Γ0 (N )
σ . χd (σ )un zQ r
σ ∈Gal(H /K)
As a consequence of deep work of Waldspurger [29] and Zhang [32], one has an identity of the form
σ ∈Gal(H /K)
√ σ 2 d 1 −D 1 χd (σ )un zQr = Cun dDL un ⊗ , L un ⊗ , · 2 · 2
(5.11)
where A Cun 1 + |tn |2 1 for some fixed constant A1 > 0, and if denotes a fundamental discriminant (to be distinguished from our notation for the hyperbolic Laplacian), then L(un ⊗ ( · ), s) is the quadratic twist of the L-function of un by the Dirichlet character ( · ). Here we used the factorization of the Rankin– Selberg L-function d −D , s L un ⊗ ,s L(un ⊗ Θχd , s) = L un ⊗ · · where Θχd is the theta series associated to the genus character χd . This√follows as a consequence of Kronecker’s factorization of the class group L-function of K = Q( −dD ) associated to the genus character χd , L(χd , s) = L
d −D ,s L ,s . · ·
Blomer and Harcos [1] established the following deep subconvexity bound, valid for all δ2 < 1/8 and some fixed constant A2 > 0, A 1 1 , 1 + |tn | 2 | | 2 −δ2 . L un ⊗ · 2
(5.12)
After combining (5.11) and (5.12) with Siegel’s theorem, 1
h(−dD) (dD) 2 − ,
(5.13)
3738
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
one finds using (5.10) that for all δ3 < 1/16, 1 2t (N ) h(−dD)
A Wn 1 + |tn | 3 (dD)−δ3 .
(5.14)
To estimate Wa (t) we proceed as above to obtain the identity
Wa (t) =
r mod 2N Q∈QdD,N,r /Γ0 (N ) r 2 ≡−dD mod 4N
1 χd (Q)Ea zQ , + it . 2
(5.15)
Again, using the bijection (3.2) we have Q∈QdD,N,r /Γ0 (N )
1 χd (Q)Ea zQ , + it = 2
σ ∈Gal(H /K)
1 σ χd (σ )Ea zQr , + it . 2
(5.16)
Following the argument in [17, Section 6], one can reduce the estimate of (5.16) to an analogous estimate for σ ∈Gal(H /K)
σ 1 χd (σ )E z , + it 2
where E(z, s) is the Eisenstein series for SL2 (Z) and ΛdD (1) = {zσ : σ ∈ Gal(H /K)} is the set of Heegner points of discriminant −dD on the modular curve X0 (1). By a classical formula of Dirichlet one has an identity of the form (see [16, p. 248])
σ ∈Gal(H /K)
O× = K 2
2 σ 1 χd (σ )E z , + it 2
√ 2 2 d 1 dD L −D , 1 + it , L , + it 2 · 2 · 2
(5.17)
where we have used the fact that L(Θχd , s) = L(χd , s) and employed Kronecker’s factorization a second time. By Blomer and Harcos [1] one has the following subconvexity bound, valid for any δ4 < 1/8 and some fixed constant A4 > 0, 2 L , 1 + it 1 + |t| A4 | | 12 −δ4 . · 2
(5.18)
After combining (5.17) and (5.18) with Siegel’s theorem (5.13), one finds using (5.15) that for all δ5 < 1/16, 1 2t (N ) h(−dD)
A Wa (t) 1 + |t| 5 (dD)−δ5 .
(5.19)
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3739
Because a FT0 is C ∞ with exponential decay in each cusp a of X0 (N ), a repeated application of Stokes’ theorem (see e.g. [19, Lemma 1.18]) yields the following identities, valid for each integer a 0, −a a FT0 , un 2 FT0 , un 2 = 1/4 + tn2 and −a a 1 1 = 1/4 + t 2 . FT0 , Ea ·, + it FT0 , Ea ·, + it 2 2 2 2 Moreover, by the Parseval formula (see [20, (15.17)]), 2 ∞ 1 a a FT , Ea ·, 1 + it dt = a FT 2 . FT , un 2 + 0 0 0 2 2 4π 2 2 a n=1
R
Then by the Cauchy–Schwartz inequality, one finds that for some sufficiently large integer a0 > 0, ∞ ∞ A3 a FT , un 2 1 + |tn | A3 = 0 FT , un (1 + |tn |) 0 0 2 (1/4 + t 2 )a0 n n=1 n=1 ∞ ∞ 2 (1 + |tn |)2A3 a 0 FT0 , un 2 · (1/4 + tn2 )2a0 n=1 n=1 a0 FT0 2 ,
and 1 FT , Ea ·, 1 + it 1 + |t| A5 dt 0 4π 2 2 a R
1 (1 + |t|)A5 1 a0 ·, + it = F , E T a 0 (1/4 + t 2 )a0 dt 4π 2 2 a R
2
1 1 (1 + |t|)2A5 a0 FT , Ea ·, + it dt · dt 0 4π 2 (1/4 + t 2 )2a0 2 a R
R
a0 FT0 2 .
By combining these estimates with (5.14) and (5.19), we find that for all δ6 < 1/16, ∞
1 FT , un 2 Wn a0 FT (dD)−δ6 0 0 2 t (N ) 2 h(−dD) n=1
3740
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
and 1 1 t (N ) 2 h(−dD) a 4π
FT , Ea ·, 1 + it dt W (t) a 0 2 2
R
a0 FT0 2 (dD)−δ6 .
(5.20)
We conclude from these estimates that for all δ7 < 1/16,
1 2t (N ) h(−dD)
= δd,1
χd (Q)FT0 (zQ )
Q∈QdD,N /Γ0 (N )
ηT0 (z) dμ + O a0 FT0 2 (dD)−δ7
F (z) dμ − δd,1
Y0 (N )
Y0 (N )
as dD → ∞. To finish the proof of Proposition 5.5, it suffices to show that for all δ8 < 1/16, 1 2t (N ) h(−dD)
χd (Q)η˜ T (zQ ) = δd,1
Q∈QdD,N /Γ0 (N )
ηT0 (z) dμ
Y0 (N )
α−1 + O (dD)−δ8 + O (dD) 2 √ as T dD and dD → ∞. By combining the definition of ηT in (5.1) with [19, (7.12)], [19, Theorem 11.3] and [19, (7.13)], one has η˜ T (z) = η˜ T , 12 +
1 1 1 1 ψˆ T0 + it − ψˆ T + it Ea z, + it dt ca 2π a 2 2 2
(5.21)
R
where (see [19, (3.13)]) ˆ ψ(s) :=
∞
ψ(y)y −(s+1) dy.
0
Summing over (5.21) yields
χd (Q)η˜ T (zQ )
Q∈QdD,N /Γ0 (N )
=
χd (Q)η˜ T , 12
Q∈QdD,N /Γ0 (N )
+
1 1 1 ψˆ T0 + it − ψˆ T + it Wa (t) dt. ca 2π a 2 2 R
(5.22)
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3741
√ Let dD T dD. Then using Lemma 5.6 (proved below) and an argument similar to that in the proof of (5.20), we obtain the following estimate, valid for any δ9 < 1/16,
1 1 1 1 dt log(dD)(dD)−δ9 . ˆ ˆ ψ + it − ψ + it W |c | (t) a T T a 0 2 2 2t (N ) h(−dD) 2π a R
Moreover, for T
√ dD a straightforward estimate yields α−1 ηT , 12 = O (dD) 2 .
Finally, by combining these estimates with (5.22), we conclude that for δ10 < 1/16, 1 2t (N ) h(−dD)
√ dD T D and any
χd (Q)η˜ T (zQ ) = δd,1
Q∈QdD,N /Γ0 (N )
ηT0 (z) dμ
Y0 (N )
α−1 + O (dD)−δ10 + O (dD) 2 as dD → ∞.
2
Lemma 5.6. For each B > 0 we have
ψˆ T 1 + it − ψˆ T 1 + it 1 + |t| B dt log(T ). 0 2 2
R
Proof. Because χ(y/T0 ) − χ(y/T ) is supported in (T0 , 2T ), we have the identity
fT (t) := ψˆ T0
2T 3 1 1 − it − ψˆ T − it = χ(y/T0 ) − χ(y/T ) y it+α− 2 dy. 2 2 T0
First assume that |t| 1. Because the k-th derivative χ (k) (y) is supported in (1, 2), integrating by parts k-times yields
k
(−1)
k−1 j =0
1 it + + (α − 1) + j fT (t) 2
it+ 1 +(α−1) 1 = T0 2 − T it+ 2 +(α−1)
2 1
Now, we have the estimate
3
χ (k) (y)y it+α− 2 +k dy.
3742
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
k−1 k−1 1 1 k + (1 − α) + j + 1 |t|k , it + + (α − 1) + j (−1) 2 2 j =0
j =0
and the estimate it+ 12 +(α−1) 1 α− 1 T − T it+ 2 +(α−1) 2T 2 . 0
0
By combining the preceding facts we obtain 1
(k) 1 |2k+α− 2 | fT (t) 2T α− 2 max1y2 |χ (y)| |t|−k . 0 k−1 1 1 |k + α − | ( + (1 − α) + j + 1) 2 j =0 2
Because B > 0 is fixed and k 1 is arbitrary, it follows that
1 fT (t) 1 + |t| B dt T α− 2 .
0
|t|1
Next assume that |t| < 1. Since α 1/2 we have the estimate fT (t) 2 sup χ(y) y∈R+
2T
y −1 dy log(T ).
T0
Then because (1 + |t|)B 1 for |t| < 1, it follows that
fT (t) 1 + |t| B dt log(T ).
2
|t|<1
6. Poincaré series The following proposition gives the Fourier expansion of Fm (z, s) in the cusp at ∞ of Γ = SL2 (Z) (see [25]). Proposition 6.1. The Poincaré series Fm (z, s) has a Fourier expansion in the cusp at ∞ of Γ of the form 1 1 Fm (z, s) = 2π|m|s− 2 y 2 Is− 1 2π|m|y e(mx) + cm,s y 1−s 2 1 1 + 4π|m|s− 2 b(m, n; s)y 2 Ks− 1 2π|n|y e(nx), n=0
where cm,s :=
4π 1+s σ2s−1 (|m|) , (2s − 1)Γ (s)ζ (2s)
2
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3743
and b(m, n; s) =
∞ S(m, n; c)
c
c=1
·
√
|mn| ), c √ J2s−1 ( 4π c|mn| ),
I2s−1 ( 4π
mn < 0, mn > 0,
where S(m, n; c) is the Kloosterman sum, and Iν , Jν and Kν are the Bessel functions of order ν. We now construct a family of Poincaré series which regularize Fm (z, s) in the cusp. Let λ : R → [0, 1] be a C ∞ function such that λ(t) =
0, t 0, 1, t 1.
Let m ∈ Z− , s ∈ Z+ , and for ε > 0 define the function s−1 t −1 ψm,s,ε (t) := λ κm,j (s)t −j ε
j =0
where κm,j (s) :=
(−1)j |m|s−j −1 (s − 1 + j )! . (4π)j (s − 1 − j )!j !
Then ψm,s,ε : R → [0, 1] is C ∞ and satisfies
t 1, −j , t > 1 + ε. κ (s)t m,j j =0
0, ψm,s,ε (t) = s−1 Finally, define the Poincaré series
Pm,ε (z, s) :=
ψm,s,ε (Im γ z)e(mγ z).
γ ∈Γ∞ \Γ
Proposition 6.2. For y > 1 + ε we have Fm (z, s) − Pm,ε (z, s) = cm,s y 1−s + e−2π|m|y e(mx)(−1)s
s−1 (−1)j y −j κm,j (s) j =0
1
+ 2π|m|s− 2
n=0
e−2π|n|y e(nx)b(m, n; s)
1 s−1 (s − 1 + j )!|n|−j − 2 y −j
j =0
j !(s − 1 − j )!(4π)j
.
3744
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
Proof. The Poincaré series Pm,ε (z, s) has a Fourier expansion of the form (see [19, p. 60]) Pm,ε (z, s) = ψm,s,ε (y)e(mz) +
e(nx)
∞
S(m, n; c)
c=1
n∈Z
ψm,s,ε R
c−2 y −mc−2 e − nt dt. t + iy t 2 + y2
(6.1)
If y 1 then c−2 y 1 t 2 + y2 for all c ∈ Z+ and t ∈ R. It follows from the definition of ψm,s,ε that if y 1, Pm,ε (z, s) = ψm,s,ε (y)e(mz), and if y > 1 + ε, Pm,ε (z, s) =
s−1
κm,j (s)y −j e(mz).
(6.2)
j =0
Now, the Bessel functions In+ 1 and Kn+ 1 of half-integral order have expansions given as 2 2 follows (see [30, Section 3.71]): for non-negative integers n we have
y
1/2
n n (−1)j (n + j )! (n + j )! 1 y n+1 −y In+ 1 (y) = √ + (−1) e e 2 j !(n − j )!(2y)j j !(n − j )!(2y)j 2π j =0 j =0
and y
1/2
1 n (n + j )! π 2 −y Kn+ 1 (y) = e . 2 2 j !(n − j )!(2y)j j =0
Substitute these expansions into the Fourier expansion of Fm (z, s) given in Proposition 6.1 and use the fact that m ∈ Z− and s ∈ Z+ to obtain Fm (z, s) =
s−1
κm,j (s)y −j e(mz) + cm,s y 1−s + e−2π|m|y e(mx)(−1)s
j =0
s−1 (−1)j y −j κm,j (s) j =0
1
+ 2π|m|s− 2
n=0
e−2π|n|y e(nx)b(m, n; s)
1 s−1 (s − 1 + j )!|n|−j − 2 y −j
j =0
j !(s − 1 − j )!(4π)j
Finally, by combining (6.3) and (6.2), we find that for y > 1 + ε,
.
(6.3)
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3745
Fm (z, s) − Pm,ε (z, s) = cm,s y 1−s + e−2π|m|y e(mx)(−1)s
s−1
(−1)j y −j κm,j (s)
j =0 1
+ 2π|m|s− 2
e−2π|n|y e(nx)b(m, n; s)
n=0
1 s−1 (s − 1 + j )!|n|−j − 2 y −j
j =0
j !(s − 1 − j )!(4π)j
2
.
7. Proof of Theorem 1.1 We will deduce Theorem 1.1 from the following theorem, whose proof will require further technical results from Sections 8–10. Theorem 7.1. Let m ∈ Z− , s ∈ Z+ , and let −D < 0 and d > 0 be odd, coprime fundamental discriminants. Then there exists an effective constant c > 0 such that for all δ < 1/16 and 0 < b < δ/c, 1 h(−dD)
= δd,1
χd (Q)Fm (zQ , s) −
zQ ∈ΛdD (1)
s−1 j =0
κm,j (s)
−j
χd (Q) Im(zQ )
e(mzQ )
Im(zQ )>1+(dD)−b
Fm (z, s) dμ + O (dD)−(δ−bc) + O (dD)−b
reg
as dD → ∞. Proof. Let ε < 1/4, and define the function Fm,s,ε (z) := Fm (z, s) − Pm,ε (z, s) where Pm,ε (z, s) is the Poincaré series defined in Section 6. Then Fm,s,ε (z) is C ∞ and Γ -invariant, and by Proposition 6.2 we see that Fm,s,ε (z) has cuspidal growth of power α = 1−s. Here we emphasize that this growth is uniform in ε for y > 5/4, and thus the same choice of cutoff parameter T0 = 2 and corresponding cutoff function η2 given by ⎧ 1 < y 2, ⎨ 0, 1−s χ(y/2), 2 < y < 4, y c η2 (z) = m,s ⎩ y 4, cm,s y 1−s , works for each Fm,s,ε (z). We now substitute Fm,s,ε (z) into Theorem 5.1 with the choices N = 1 and T0 = 2, and find that for all δ < 1/16, 1 h(−dD)
χd (Q) Fm (zQ , s) − Pm,ε (zQ , s)
zQ ∈ΛdD (1)
= δd,1 Y0 (1)
Fm (z, s) − Pm,ε (z, s) dμ + O a0 Fm,s,ε,2 2 + 1 (dD)−δ
(7.1)
3746
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
as dD → ∞, where Fm,s,ε,2 (z, s) := Fm (z, s) − Pm,ε (z, s) − η2 (z). Because ψm,s,ε (y) = 0 for y 1, it follows from the definitions of Pm,ε (z, s) and ψm,s,ε that for dD sufficiently large,
χd (Q)Pm,ε (zQ , s) =
zQ ∈ΛdD (1)
χd (Q)ψm,s,ε (Im zQ )e(mzQ )
Im(zQ )>1
=
χd (Q)ψm,s,ε (Im zQ )e(mzQ )
1
+
s−1
κm,j (s)
j =0
χd (Q) Im(zQ )−j e(mzQ ).
Im(zQ )>1+ε
For notational convenience define 1 Rε (dD) := h(−dD) −
s−1
κm,j (s)
j =0
χd (Q)Fm (zQ , s)
zQ ∈ΛdD (1) −j
χd (Q) Im(zQ )
e(mzQ ) .
Im(zQ )>1+ε
Then we can write (7.1) in the equivalent form
Rε (dD) = δd,1
Fm (z, s) − Pm,ε (z, s) dμ + O a0 Fm,s,ε,2 2 + 1 (dD)−δ
Y0 (1)
+
1 h(−dD)
χd (Q)ψm,s,ε (Im zQ )e(mzQ ).
1
By Proposition 8.2 and Lemma 10.1, we have
Rε (dD) = δd,1
Fm (z, s) dμ + O ε −a1 (dD)−δ + O (dD)−δ
reg
+
1 h(−dD)
χd (Q)ψm,s,ε (Im zQ )e(mzQ )
1
for some sufficiently large integer a1 > 0 (depending on a0 ). Now, we have the estimate 2πm(1+ε) χ (Q)ψ (Im z )e(mz ) #Λε (dD) d m,s,ε Q Q sup ψm,s,ε (t) e 1
t∈R+
(7.2)
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3747
where Λε (dD) := zQ ∈ ΛdD (1): 1 < Im(zQ ) 1 + ε . By combining (7.2) with Lemma 9.1 and Siegel’s theorem (5.13), we find that 1 h(−dD)
χd (Q)ψm,s,ε (Im zQ )e(mzQ ) = O(ε) + O ε −a2 (dD)−δ
(7.3)
1
for some sufficiently large integer a2 > 0. It follows from the preceding analysis that for all δ < 1/16,
Rε (dD) = δd,1
Fm (z, s) dμ + O ε −a1 (dD)−δ + O (dD)−δ + O(ε) + O ε −a2 (dD)−δ .
reg
Let c = max{a1 , a2 } and choose b > 0 such that b < δ/c. Because ε is independent of dD in the preceding estimates, we can set ε = (dD)−b . Then
Rε (dD) = δd,1
Fm (z, s) dμ + O (dD)−(δ−bc) + O (dD)−b
reg
as dD → ∞.
2
Proof of Theorem 1.1. Theorem 1.1 now follows by combining Theorem 7.1 with Propositions 11.1 and 12.1. 2 8. Proof of Proposition 8.2 In this section we prove Proposition 8.2 (see also [11]). For ε > 0 define the cutoff function ψm,s,ε (t), t Y, Y ψm,s,ε (t) := 0, t > Y, and the associated cutoff Poincaré series Pm,ε,Y (z, s) :=
Y ψm,s,ε (Im γ z)e(mγ z).
γ ∈Γ∞ \Γ
We will need the following lemma. Lemma 8.1. For ε > 0, we have Pm,ε (z, s) = Pm,ε,Y (z, s) for all z ∈ FY .
3748
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
Proof. We need to show that for Im(z) Y ,
Y ψm,s,ε (Im γ z)e(mγ z) =
γ ∈Γ∞ \Γ
(8.1)
ψm,s,ε (Im γ z)e(mγ z).
γ ∈Γ∞ \Γ
Y By definition of ψm,s,ε we have
Y ψm,s,ε (Im γ z)e(mγ z) =
γ ∈Γ∞ \Γ
ψm,s,ε (Im γ z)e(mγ z).
γ ∈Γ∞ \Γ Im(γ z)Y
Moreover, since Im(z) Y , which forces the identity matrix I into the first summand inside the brackets in (8.2), we have
ψm,s,ε (Im γ z)e(mγ z) =
γ ∈Γ∞ \Γ
γ ∈Γ∞ \Γ Im(γ z)Y
+
ψm,s,ε (Im γ z)e(mγ z).
(8.2)
γ ∈Γ∞ \Γ Im(γ z)>Y γ =I
Let SY := γ ∈ Γ∞ \Γ : Im(γ z) > Y, γ = I . Then ψm,s,ε (Im γ z)e(mγ z) sup ψm,s,ε (t) max e2πm Im(γ z) #SY . γ ∈SY + t∈R
γ ∈SY
Suppose that Y 10. Then by [19, Lemma 2.10], #SY < 10/Y 1, which implies that #SY = 0. Thus
ψm,s,ε (Im γ z)e(mγ z) = 0,
γ ∈Γ∞ \Γ Im(γ z)>Y γ =I
and (8.1) holds for Y 10. However, since FY ⊂ F10 for Y < 10, the lemma follows. Proposition 8.2. For ε > 0 we have
Fm (z, s) − Pm,ε (z, s) dμ = lim
Y →∞
Y0 (1)
FY
Fm (z, s) dμ.
2
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3749
Proof. By Lemma 8.1 and an unfolding argument, we find that for all Y > 0,
Pm,ε (z, s) dμ =
FY
Pm,ε,Y (z, s) dμ F
3 = π
Y
dy ψm,s,ε (y)e2πmy 2 y
1 ·
0
e2πimx dx = 0. 0
Then because Fm (z, s) − Pm,ε (z, s) ∈ L1 (Y0 (1)), it follows that
Fm (z, s) dμ = lim
lim
Y →∞
Y →∞
FY
Fm (z, s) dμ − FY
= lim
Y →∞
=
Pm,ε (z, s) dμ
FY
Fm (z, s) − Pm,ε (z, s) dμ
FY
Fm (z, s) − Pm,ε (z, s) dμ.
2
Y0 (1)
Note that the regularized integral in Proposition 8.2 will be evaluated in Section 12. 9. Proof of Lemma 9.1 In this section we prove Lemma 9.1 (see also [10, pp. 248–249]). Lemma 9.1. For ε < 1/4 we have 7 #Λε (dD) 4εh(−dD) + O ε −a2 (dD) 16 + for some sufficiently large integer a2 > 0 and all > 0. Proof. For each ε > 0, one can construct a C ∞ function φε : R → [0, 1] which is supported in the interval (1 − ε, 1 + 2ε), which equals 1 on the interval [1, 1 + ε], and which satisfies the bound a d max a φε (t) ε −a t∈(1−ε,1+2ε) dt for all a ∈ Z+ . Define the incomplete Eisenstein series gε (z) :=
φε Im(γ z) .
γ ∈Γ∞ \Γ
Then using that φε (y) = 1 for 1 y 1 + ε, we obtain the decomposition
3750
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
gε (zQ ) = #Λε (dD) +
zQ ∈ΛdD (1)
zQ ∈ΛdD (1)\Λε (dD)
+
φε Im(zQ )
φε Im(γ zQ ) .
zQ ∈ΛdD (1) γ ∈Γ∞ \Γ γ =I
Since φε 0 it follows that
#Λε (dD)
gε (zQ ).
zQ ∈ΛdD (1)
The real-analytic Eisenstein series
E(z, s) =
Im(γ z)s ,
Re(s) > 1
γ ∈Γ∞ \Γ
has a meromorphic continuation to C with a simple pole at s = 1 with residue 3/π . Therefore we obtain from [19, (7.12)] the expansion gε (z) =
3 1 φˆ ε (1) + π 2π
φˆ ε
R
1 1 + it E z, + it dt 2 2
where φˆ ε (s) :=
∞
φε (y)y −(s+1) dy.
0
Summing the expansion yields zQ ∈ΛdD (1)
gε (zQ ) =
3 1 φˆ ε (1)h(−dD) + π 2π
φˆ ε
R
1 + it 2
zQ ∈ΛdD (1)
1 E zQ , + it dt. 2
Since φε is supported in (1 − ε, 1 + 2ε) and φε 1, we find that for ε < 1/4, φˆ ε (1) =
∞
φε (y)y −2 dy
0
1 1−ε
1+ε 1+2ε
+ + y −2 dy 1
1+ε
ε ε +ε+ (1 − ε)2 (1 + ε)2 < 2ε + ε + ε = 4ε.
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3751
Therefore 3 φˆ ε (1)h(−dD) 4εh(−dD). π By a variant of the argument used to prove (5.19), one finds that zQ ∈ΛdD (1)
A 7 1 E zQ , + it 1/4 + t 2 6 (dD) 16 + 2
for some fixed constant A6 > 0 and all > 0. Thus
φˆ ε 1 + it 2
R
zQ ∈ΛdD (1)
7 1 E zQ , + it dt (dD) 16 + 2
R
φˆ ε 1 + it 1/4 + t 2 A6 dt. 2
Let a ∈ Z+ . Then integrating by parts a-times and using the fact that the support of φε contained in (1 − ε, 1 + 2ε) ⊂ [3/4, 3/2] for ε < 1/4, we obtain
(a)
φˆ ε
∞ 3 1 + it = φˆ ε (y)y −( 2 +it) dy 2 0
−1
a 3 3 − + it + j = (−1) φε(a) (y)y −( 2 +it)+a dy 2 ∞
a
j =1
0
for all t ∈ R. Thus for ε < 1/4, −1 1+2ε
a 3 (a) − 3 +a φˆ ε 1 + it + it − j φ (y)y 2 dy ε 2 2 j =1
1−ε
1+2ε −1
a 3 (a) 3 φε (y) max y − 2 +a dy 2 + it − j y∈(1−ε,1+2ε) j =1
1−ε
−1
3/2 a 3 3 −a y − 2 +a dy 2 + it − j ε j =1
a ε −a
3/4 a
3 2
j =1
−1 + it − j .
is
3752
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
We now see that for some sufficiently large integer a1 > 0 (depending on A6 ), we have
φˆ ε 1 + it 2
R
,a1 ε
−a1
zQ ∈ΛdD (1)
(dD)
7 16 +
1 E zQ , + it dt 2
−1
a1 3 + it − j 1/4 + t 2 A6 dt 2
R j =1 7
,a1 ε −a1 (dD) 16 + .
2
10. Proof of Lemma 10.1 Lemma 10.1. For ε < 1/4 we have a 0 Fm,s,ε,2 = Oa ε −a2 0 2 for some sufficiently large integer a2 > 0 depending on a0 . Proof. Recall that Fm,s,ε,2 (z) := Fm (z, s) − Pm,ε (z, s) − η2 (z), where the cutoff function η2 is given by ⎧ ⎨ 0, η2 (z) = cm,s y 1−s χ(y/2), ⎩ cm,s y 1−s ,
1 < y 2, 2 < y < 4, y 4.
For notational convenience we write the Fourier expansion (6.1) as Pm,ε (z, s) = ψm,s,ε (y)e(mz) + fm,s,ε (x, y), where fm,s,ε (x, y) :=
n∈Z
e(nx)
∞
S(m, n; c)
c=1
ψm,s,ε R
c−2 y −mc−2 e − nt dt. t + iy t 2 + y2
Clearly we have a 0 Fm,s,ε,2 2 2
∞ 1
√
3/2 0
a 0 Fm,s,ε,2 (z)2 dx dy . y2
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3753
If we let T = 2 in the Fourier expansion (5.2), we see that ψ2
c−2 y t 2 + y2
=0
√ √ for y 3/2, so in fact η2 (z) = 0 in the larger range 3/2 y 2. Then using the definitions of ψm,s,ε , η2 , and Proposition 6.2, we find that ⎧ F (z, s) − fm,s,ε (x, y), ⎪ ⎪ ⎪ m ⎪ ⎪ ⎨ Fm (z, s) − ψm,s,ε (y)e(mz), Fm,s,ε,2 (z) = cm,s y 1−s + O(e−cy ), ⎪ ⎪ ⎪ c y 1−s (1 − χ(y/2)) + O(e−cy ), ⎪ ⎪ ⎩ m,s −cy O(e ),
√ 3/2 y < 1, 1 y 1 + ε, 1 + ε < y 2, 2 < y < 4, y 4.
Recalling the explicit form of the O(e−cy ) terms in Proposition 6.2, it follows that
∞ 1 √
a 0 Fm,s,ε,2 (z)2 dx dy y2
3/2 0
1 1 = √
a 0 Fm (z, s) − fm,s,ε (x, y) 2 dx dy y2
3/2 0
1+ε 1 + 1
a 0 Fm (z, s) − ψm,s,ε (y)e(mz) 2 dx dy + O(1). y2
0
By linearity of a0 and the triangle inequality, we have
1+ε 1 I1 := 1
0
1+ε 1 1
a 0 Fm (z, s) − ψm,s,ε (y)e(mz) 2 dx dy y2 a 0 Fm (z, s)2 dx dy y2
0
1+ε 1 +2 1
0
1+ε
a 0 ψm,s,ε (y)2 dy . y2
+ 1
Recall that
a 0 Fm (z, s) · a0 ψm,s,ε (y) dx dy y2
3754
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
ψm,s,ε (t) := λ
s−1 t −1 κm,j (s)t −j , ε j =0
where λ : R → [0, 1] is a C ∞ function such that λ(t) =
0, t 0, 1, t 1.
Then it is clear that 2a d 0 max 2a ψm,s,ε (y) = O ε −2a0 . y∈[1,1+ε] dy 0 Therefore (using that ε < 1/4), I1 = O(1) + O ε −2a0 + O ε −4a0 . Similarly, we have
1 1 I2 := √
3/2 0
1 1 √
a 0 Fm (z, s) − fm,s,ε (x, y) 2 dx dy y2 a 0 Fm (z, s)2 dx dy y2
3/2 0
1 1 +2 √
3/2 0
1 1 + √
Observe that for y
a 0 Fm (z, s) · a0 fm,s,ε (x, y) dx dy y2
a 0 fm,s,ε (x, y)2 dx dy . y2
3/2 0
√ 3/2 and c 2, we have
y 2 √ 1. c2 (t 2 + y 2 ) c2 3 √ Since ψm,s,ε (u) = 0 for u 1, it follows that for y 3/2 and c 2, ψm,s,ε Therefore, for y
c−2 y 2 t + y2
= 0.
√ 3/2 the function fm,s,ε (x, y) simplifies to
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
fm,s,ε (x, y) =
e(nx)S(m, n; 1)
n∈Z
ψm,s,ε R
y −m e − nt dt, t + iy t 2 + y2
i.e., only the term with c = 1 remains. Define the function gm,y,ε (t) := ψm,s,ε
y −m e . t + iy t 2 + y2
Then by definition of ψm,s,ε we have dk gm,y,ε (t) = 0 |t|→∞ dt k lim
for all integers k 0. Integrating by parts (2a0 + 2)-times yields
1 (2πin)2a0 +2
gm,y,ε (t)e(−nt) dt = R
(2a0 +2) gm,y,ε (t)e(−nt) dt.
R
In fact, since
t2
y 1 + y2
⇔
|t|
"
y(1 − y),
we have (2a0 +2) (t) = 0 gm,y,ε
for |t|
√ √ y(1 − y). Thus in the range 3/2 y 1 we obtain
fm,s,ε (x, y) =
n∈Z
e(nx) S(m, n; 1) (2πin)2a0 +2
1
(2a0 +2) gm,y,ε (t)e(−nt) dt.
−1
We now analyze a0 fm,s,ε (x, y) and show that I2 = O(1) + O ε −(2a0 +2) + O ε −2(2a0 +2) . For clarity we first give the argument for a0 = 1. We find that
3755
3756
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
e(nx) −y ∂x2 + ∂y2 fm,s,ε (x, y) = −y 2 S(m, n; 1) (2πin)2
1
2
n∈Z
(4) gm,y,ε (t)e(−nt) dt
−1
e(nx) ∂ 2 (4) −y S(m, n; 1) g (t)e(−nt) dt. 4 (2πin) ∂y 2 m,y,ε 1
2
n∈Z
−1
Since (see [19, (2.37)]) S(m, n; 1) 1, it follows that 2 ∂ (4) (4) 2 fm,s,ε (x, y) y max gm,y,ε (t) + y max 2 gm,y,ε (t). t∈[−1,1] t∈[−1,1] ∂y 2
It is clear from the definition of gm,y,ε that (4) max gm,y,ε (t) = Oy ε −4
t∈[−1,1]
and 2 ∂ (4) max 2 gm,y,ε (t) = Oy ε −4 . t∈[−1,1] ∂y Hence fm,s,ε (x, y) = Oy ε −4 . Finally, it follows that I2 = O(1) + O ε −4 + O ε −8 . The preceding argument generalizes in a straightforward way to higher derivatives, and thus we conclude that I2 = O(1) + O ε −(2a0 +2) + O ε −2(2a0 +2) .
2
11. Exponential sums In this section we express the main term in Theorem 7.1 as a twisted exponential sum. Let n, c ∈ Z+ , and , δ ∈ Z. Following [22], we define the twisted exponential sum S ,δ (n, c) :=
x 2 ≡−δ (mod c)
χd
x 2 + δ 2nx c , x, e . 4 c c
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3757
Proposition 11.1. Let m ∈ Z− , let s, c ∈ Z+ , and let −D < 0 and d > 0 be odd, coprime fundamental discriminants. Then s−1
κm,j (s)
j =0
χd (Q) Im(zQ )−j e(mzQ )
Im(zQ )>1+(dD)−b
√ D,d 4π|m| dD , SD,d |m|, c gm,s (c) exp c
1 = 2
0
√ 2 dD 1+(dD)−b
c≡0 (mod 4)
where D,d gm,s (c) := |m|s−1
s−1 √ − dD j (s − 1 + j )! . 2π|m|c (s − 1 − j )!j ! j =0
Proof. Each Q ∈ QdD is of the form cX 2 + xXY + aY 2 ∈ Z[X, Y ] for integers c, x and a. A classical parameterization due to Gauss implies that those zQ with Im(zQ ) > 1 + (dD)−b √ are given by integer pairs (c, x) with 0 < c < dD/2(1 + (dD)−b ) and x 2 ≡ −dD (mod 4c). Letting 4c → c, we find that s−1
κm,j (s)
j =0
χd (Q) Im(zQ )−j e(mzQ )
Im(zQ )>1+(dD)−b
1 = κm,j (s) 2 j =0
√ 4π|m| dD exp c
s−1
0
√ 2 dD 1+(dD)−b
c≡0 (mod 4)
×
χd
x 2 ≡−dD (mod c)
=
1 2
√ 0
c x 2 + dD , x, 4 c
√ 2 dD −j e 2|m|x/c c
√ 4π|m| dD D,d . gm,s (c)SD,d |m|, c exp c
2
c≡0 (mod 4)
12. Evaluation of the regularized integral In this section we evaluate the regularized integral appearing in Theorem 7.1. Proposition 12.1. Let m ∈ Z− and s ∈ Z+ . Then
lim
Y →∞
FY
Fm (z, s) dμ =
0, −24σ (|m|)|m|s−1 ,
s = 1, s 2.
3758
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
Proof. We proceed using the method of Lerche, Schellekens, and Warner [23] (see also [2,11]). First observe that one has the identity ∂(E2 (z) −
3 π Im(z) )
∂z
=
3i , 2π Im(z)2
where E2 (z) := 1 − 24
∞
σ1 (n)q n ,
q := e(z)
n=1
and z = x + iy. Because dz dz = 2i dx dy and Fm (z, s) is Γ -invariant, it follows from Stokes’ theorem that 1 2 +iY
lim
Y →∞
Fm (z, s) dμ = lim
FY
Y →∞ − 12 +iY
3 dx. Fm (x + iY, s) E2 (x + iY ) − πY
(12.1)
Using the expansion given in Proposition 6.1 for the forms Fm (x + iY, s), we find after multiplying that 3 Fm (x + iY, s) E2 (x + iY ) − πY
4π 1+s σ2s−1 (|m|)Y 1−s 1 1 3 = −24σ |m| 2π|m|s− 2 e−2π|m|Y Y 2 Is− 1 2π|m|Y + 1− 2 (2s − 1)Γ (s)ζ (2s) πY 1 + ak (m, Y, s)e(kx) + O . Y k∈Z k=0
Next, using the expansion for the Bessel function given in the proof of Proposition 6.1, we find that the integral (12.1) equals ∞ s−1 −j 3 1 4π 1+s σ2s−1 (|m|)Y 1−s 1− +O . + O Y lim −24σ |m| |m| 1+ Y →∞ (2s − 1)Γ (s)ζ (2s) πY Y j =1
The result follows upon taking the limit as Y → ∞, where if s = 1 we use the fact that ζ (2) = π 2 /6. 2 Acknowledgments We would like to thank Anton Deitmar, Gergely Harcos, Dennis Hejhal, Jim Kelliher, Ken Ono, Steve Wainger, Tong Hai Yang, and Matt Young for very helpful discussions regarding this work. We would also like to thank the referee for a very careful reading of the manuscript leading to many improvements in exposition.
A. Folsom, R. Masri / Advances in Mathematics 226 (2011) 3724–3759
3759
References [1] V. Blomer, G. Harcos, Hybrid bounds for twisted L-functions, J. Reine Angew. Math. 621 (2008) 53–79. [2] R. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998) 491–562. [3] R. Borcherds, What is Moonshine?, in: Proceedings of the International Congress of Mathematicians, vol. I, Berlin, 1998, in: Doc. Math., extra vol. I, 1998, pp. 607–615. [4] K. Bringmann, K. Ono, Arithmetic properties of coefficients of half-integral weight Maass–Poincaré series, Math. Ann. 337 (2007) 591–612. [5] K. Bringmann, K. Ono, An arithmetic formula for the partition function, Proc. Amer. Math. Soc. 135 (2007) 3507– 3514. [6] K. Bringmann, K. Ono, Lifting cusp forms to Maass forms with an application to partitions, Proc. Natl. Acad. Sci. USA 104 (10) (2007) 3725–3731. [7] J.H. Bruinier, Borcherds Products on O(2, l) and Chern Classes of Heegner Divisors, Lecture Notes in Math., vol. 1780, Springer-Verlag, Berlin, 2002, viii+152 pp. [8] J. Bruinier, P. Jenkins, K. Ono, Hilbert class polynomials and traces of singular moduli, Math. Ann. 334 (2006) 373–393. [9] H. Darmon, Heegner points, Heegner cycles, and congruences, in: Elliptic Curves and Related Topics, in: CRM Proc. Lecture Notes, vol. 4, Amer. Math. Soc., Providence, RI, 1994, pp. 45–59. [10] W. Duke, Modular functions and the uniform distribution of CM points, Math. Ann. 334 (2006) 241–252. [11] W. Duke, An introduction to the Linnik problems, in: Equidistribution in Number Theory, an Introduction, in: NATO Sci. Ser. II Math. Phys. Chem., vol. 237, Springer, Dordrecht, 2007, pp. 197–216. [12] W. Duke, P. Jenkins, Integral traces of singular values of weak Maass forms, Algebra Number Theory 2 (2008) 573–593. [13] J.D. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math. 293–294 (1977) 143– 203. [14] A. Folsom, R. Masri, Equidistribution of Heegner points and the partition function, Math. Ann. 348 (2010) 289–317. [15] B. Gross, W. Kohnen, D. Zagier, Heegner points and derivatives of L-series. II, Math. Ann. 278 (1987) 497–562. [16] B. Gross, D. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986) 225–320. [17] G. Harcos, P. Michel, The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points. II, Invent. Math. 163 (2006) 581–655. [18] D. Hejhal, The Selberg Trace Formula for PSL2 (R), Lecture Notes in Math., vol. 1001, Springer, 1983. [19] H. Iwaniec, Spectral Methods of Automorphic Forms, Grad. Stud. Math., vol. 53, Amer. Math. Soc./Revista Matematica Iberoamericana, Providence, RI/Madrid, 2002. [20] H. Iwaniec, E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., vol. 53, Amer. Math. Soc., Providence, RI, 2004, xii+615 pp. [21] B.D. Kim, R. Masri, T.H. Yang, Nonvanishing of Hecke L-functions and the Bloch–Kato conjecture, Math. Ann., doi:10.1007/s00208-010-0521-7, in press. [22] W. Kohnen, Fourier coefficients of modular forms of half-integral weight on Γ0 (4), Math. Ann. 271 (2) (1985) 237–268. [23] W. Lerche, A.N. Schellekens, N.P. Warner, Lattices and strings, Phys. Rep. 177 (1989) 1–140. [24] A. Miller, A. Pixton, Arithmetic traces of non-holomorphic modular invariants, Int. J. Number Theory 6 (1) (2010) 69–87. [25] D. Niebur, A class of nonanalytic automorphic functions, Nagoya Math. J. 52 (1973) 133–145. [26] K. Ono, Singular moduli generating functions for modular curves and surfaces, in: Analytic Number Theory, in: Clay Math. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 2007, pp. 185–206. [27] K. Ono, Unearthing the visions of a master: harmonic Maass forms and number theory, Harvard-MIT Current Developments in Mathematics, International Press, 2008, preprint. [28] H. Rademacher, On the expansion of the partition function in a series, Ann. of Math. 44 (1943) 416–422. [29] J.-L. Waldspurger, Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie, Compos. Math. 54 (1985) 173–242. [30] G.N. Watson, A Treatise on the Theory of Bessel Functions, second ed., Cambridge University Press, 1966. [31] D. Zagier, Traces of singular moduli, in: Motives, Polylogarithms and Hodge Theory, Part I, Irvine, CA, 1998, in: Int. Press Lect. Ser., vol. 3, I, Int. Press, Somerville, MA, 2002, pp. 211–244. [32] S. Zhang, Gross–Zagier formula for GL2 , Asian J. Math. 5 (2001) 183–290.