Singular invariants and coefficients of harmonic weak Maass forms of weight 5/2

Nickolas Andersen 1
  • 1 Department of Mathematics, University of Illinois, Urbana, IL 61801, USA
Nickolas Andersen

Abstract

We study the coefficients of a natural basis for the space of harmonic weak Maass forms of weight 5/2 on the full modular group. The non-holomorphic part of the first element of this infinite basis encodes the values of the partition function p(n). We show that the coefficients of these harmonic Maass forms are given by traces of singular invariants. These are values of non-holomorphic modular functions at CM points or their real quadratic analogues: cycle integrals of such functions along geodesics on the modular curve. The real quadratic case relates to recent work of Duke, Imamoḡlu, and Tóth on cycle integrals of the j-function, while the imaginary quadratic case recovers the algebraic formula of Bruinier and Ono for the partition function.

1 Introduction

A harmonic weak Maass form of weight k (for brevity, we will drop the adjective “weak” throughout this paper) is a real analytic function on the upper-half plane which transforms like a modular form of weight k, is annihilated by the weight k hyperbolic Laplacian

Δk:=-y2(2x2+2y2)+iky(x+iy),τ=x+iy

and has at most linear exponential growth at the cusps. Such a form h has a natural decomposition h=h++h- into the holomorphic part h+ (also called a mock modular form) and the non-holomorphic part h-. Let ξk denote the differential operator

ξk:=2iykτ¯¯.

The function g:=ξkh=ξkh-, the so-called shadow of h+, is a weakly holomorphic modular form (a modular form whose poles, if any, are supported at the cusps) of weight 2-k. See [9, 24] for background on harmonic Maass forms.

A natural problem is to determine what arithmetic data, if any, is encoded in the coefficients of mock modular forms. Although such coefficients are not well understood in general, significant progress has been made recently in certain cases. For instance, Ramanujan’s mock theta functions (whose coefficients encode combinatorial information) are mock modular forms of weight 1/2 whose shadows are linear combinations of weight 3/2 theta series. Zwegers’ discovery connecting mock theta functions to harmonic Maass forms [33, 34] has inspired many works (see [5, 7, 32] and the references therein for examples). Recently Bruinier and Ono [11] considered harmonic Maass forms whose shadows are cusp forms orthogonal to the weight 3/2 theta series. They showed that the coefficients of these forms are related to central critical values and derivatives of weight 2 modular L-functions. Later Bruinier [8] connected these coefficients to periods of algebraic differentials of the third kind on modular and elliptic curves.

In this paper, we investigate the arithmetic nature of the coefficients of mock modular forms of weight 5/2 whose shadows are weakly holomorphic modular forms. This relates to work of Duke, Imamoḡlu, and Tóth [14] regarding a particularly interesting family of weight 1/2 mock modular forms which are related to Zagier’s proof [31] of Borcherds’ theorem [4] on infinite product formulas for modular forms. Let J denote the normalized Hauptmodul for SL2() given by

J(τ):=j(τ)-744=1q+196884q+,q:=exp(2πiτ).

For each nonzero discriminant d0,1(mod4), let 𝒬d denote the set of integral binary quadratic forms of discriminant d which are positive definite when d<0. The modular group Γ1:=PSL2() acts on these forms, and the set Γ1𝒬d is a finite abelian group.

If the quadratic form Q is positive definite then Q(τ,1) has exactly one zero τQ in . For fixed d, the values

J(τQ),QΓ1𝒬d

are conjugate algebraic integers known as singular moduli. Zagier [31] showed that the weighted sums

QΓ1𝒬dDχD(Q)wQJ(τQ)

appear as coefficients of a basis {gD}0<D0,1(4) for 3/2+, the plus-space of weakly holomorphic modular forms of weight 3/2 on Γ0(4). Here wQ denotes the order of the stabilizer of Q in Γ1, and χD:𝒬d{-1,0,1} is a generalized genus character (see [16, Section I.2]).

The coefficients of the forms in Zagier’s basis {gD} also appear as coefficients of forms in the basis {fD}0D0,1(4) given by Borcherds in [4, Section 14]. Borcherds showed that the coefficients of the fD are the exponents in the infinite product expansions of certain meromorphic modular forms. In [14], Duke, Imamoḡlu, and Tóth extended Borcherds’ basis to a basis {fD}D0,1(4) for 𝕄1/2, the space of mock modular forms of weight 1/2 on Γ0(4) satisfying Kohnen’s plus-space condition. When D>0, the shadow of the mock modular form fD is proportional to the weakly holomorphic form gD. Whereas the forms fD for D<0 encode imaginary quadratic information in the form of traces of singular moduli, the forms fD for D>0 encode real quadratic information in the form of traces of cycle integrals of J(τ). To explain this requires some notation.

For an indefinite quadratic form Q with non-square discriminant, let CQ denote the geodesic in connecting the two (necessarily irrational) roots of Q, modulo the stabilizer of Q. Then CQ defines a closed geodesic on the modular curve Γ1{}, and the cycle integral

CQJ(τ)dτQ(τ,1)

is a well-defined invariant of the equivalence class of Q. The beautiful result of Duke, Imamoḡlu, and Tóth [14] interprets the coefficient a(D,d) of qd in fD (D>0) in terms of the sums

12πQΓ1𝒬dDχD(Q)CQJ(τ)dτQ(τ,1),

whenever dD is not a square. However, their theorem does not include the coefficients a(D,d) when dD is a square since, in that case, the integral (1.3) diverges. There are two approaches to studying these square-indexed coefficients. The first method, by Bruinier, Funke and Imamoḡlu in [10], involves regularizing the divergent integral (1.3); their method generalizes the results of [14] to a large class of non-holomorphic modular functions. The second method, by the present author in [2], involves replacing J(τ) in (1.3) by JQ(τ), a dampened version of J(τ) which depends on Q (and for which the resulting integral converges).

Here we study H5/2!(ε), the space of harmonic Maass forms of weight 5/2 which transform as

f(γτ)=ε(γ)(cτ+d)5/2f(τ)for allγ=(abcd)SL2().

Here ε is the multiplier system defined by

ε(γ):=η(γτ)cτ+dη(τ),

where η is the Dedekind eta-function η(τ):=q1/24k1(1-qk). We will show that the coefficients of forms in H5/2!(ε) are given by traces of singular invariants; that is, sums of CM values of certain weight 0 weak Maass forms on Γ0(6) in the imaginary quadratic case, or cycle integrals of those forms over geodesics on the modular curve

X0(6):=Γ0(6)^

in the real quadratic case.

In [1], the author and Ahlgren constructed an infinite basis {hm}m1(24) for H5/2!(ε). For m<0 the hm are holomorphic, while for m>0 they are non-holomorphic harmonic Maass forms whose shadows are weakly holomorphic modular forms of weight -1/2. The first element, 𝐏:=h1, has Fourier expansion1

𝐏(τ)=iq1/24+0<n1(24)np(1,n)qn/24-iβ(-y)q1/24+0>n1(24)|n|p(1,n)β(|n|y)qn/24,

where β(y) is the normalized incomplete gamma function

β(y):=Γ(-32,πy6)Γ(-32)=34ππy/6e-tt-5/2𝑑t.

The function 𝐏(τ) is of particular interest since ξ5/2(𝐏) is proportional to

η-1(τ)=q-1/24+0>n1(24)p(1-n24)q|n|/24,

where p(1-n24) is the ordinary partition function. Corollary 2 of [1] shows that for negative n1(mod24) we have p(1,n)=|n|p(1-n24).

Building on work of Hardy and Ramanujan [17], Rademacher [25, 26] proved the exact formula

p(k)=2π(24k-1)-3/4c>0Ac(k)cI3/2(π24k-16c),

where I3/2(x) is the I-Bessel function and Ac(k) is a Kloosterman sum. Using Rademacher’s formula, Bringmann and Ono [6] showed that p(k) can be written as a sum of CM values of a certain non-holomorphic Maass–Poincaré series on Γ0(6). Later Bruinier and Ono [12] refined the results of [6]; by applying a theta lift to a certain weight -2 modular form, they obtained a new formula for p(k) as a finite sum of algebraic numbers.

The formula of Bruinier and Ono involves a certain non-holomorphic, Γ0(6)-invariant function P(τ), which the authors define in the following way. Let

F(τ):=12E2(τ)-2E2(2τ)-3E2(3τ)+6E2(6τ)(η(τ)η(2τ)η(3τ)η(6τ))2=q-1-10-29q-,

where E2 denotes the weight 2 quasi-modular Eisenstein series

E2(τ):=1-24n=1d|ndqn.

The function F(τ) is in M-2!(Γ0(6),1,-1), the space of weakly holomorphic modular forms of weight -2 on Γ0(6), having eigenvalues 1 and -1 under the Atkin–Lehner involutions W6 and W2, respectively (see Section 2 for definitions). The function P(τ) is the weak Maass form given by

P(τ):=14πR-2F(τ)=-(qddq+12πy)F(τ),

where R-2 is the Maass raising operator in weight -2 (see Section 2 for details).

For each n1(mod24) and r{1,5,7,11}, let

𝒬n(r):={[a,b,c]𝒬n:6a and br(mod12)}.

Here [a,b,c] denotes the quadratic form ax2+bxy+cy2. The group Γ:=Γ0(6)/{±1} acts on 𝒬n(r), and for each r we have the canonical isomorphism (see [16, Section I])

Γ𝒬n(r)Γ1𝒬n.

Bruinier and Ono [12, Theorem 1.1] proved the finite algebraic formula

p(1-n24)=1|n|QΓ𝒬n(1)P(τQ),

for each 0>n1(mod24). Therefore the coefficients of the non-holomorphic part of 𝐏(τ) are given in terms of singular invariants (note the similarity with the expression (1.2)).

For the holomorphic part of 𝐏(τ) one might suspect by analogy with (1.4) that the traces

QΓ𝒬n(1)CQP(τ)dτQ(τ,1)

give the coefficients p(1,n). However, as we show in the remarks following Proposition 5.1 below, the traces (1.6) are identically zero whenever n is not a square. In order to describe the arithmetic nature of the coefficients p(1,n), we must introduce the function P(τ,s), of which P(τ) is the specialization at s=2 (see Section 2 for the definition of P(τ,s)). Similarly, for each Q𝒬n(1) whose discriminant is a square, we will define PQ(τ,s), a dampened version of P(τ,s) (see Section 3). These are analogues of the functions JQ(τ) and are defined in (3.13) below. Then, for each n1(mod24) we define the trace

Tr(n):={1|n|QΓ𝒬n(1)P(τQ)if n<0,12πQΓ𝒬n(1)CQ[sP(τ,s)|s=2]dτQ(τ,1)if n>0 is not a square,12πQΓ𝒬n(1)CQ[sPQ(τ,s)|s=2]dτQ(τ,1)if n>0 is a square.

We have the following theorem which provides an arithmetic interpretation for the coefficients of 𝐏(τ) by relating them to the traces (1.7). It is a special case of Theorem 1.2 below.

For each n1(mod24) we have p(1,n)=Tr(n).

Since p(1,n)=|n|p(1-n24) for 0>n1(mod24), Theorem 1.1 recovers the algebraic formula (1.5) of Bruinier and Ono. However, our proof is quite different than the proof given in [12].

Recall that 𝐏=h1 is the first element of an infinite basis {hm} for H5/2(ε). We turn to the other harmonic Maass forms hm for general m1(mod24). For negative m, we have the Fourier expansion (see [1, Theorem 1])

hm(τ)=|m|3/2qm/24-0<n1(24)|mn|p(m,n)qn/24.

These forms are holomorphic on and can be constructed using η(τ) and j(τ):=-qddqj(τ). We list a few examples here:

23-32h-23=ηj
=q-23/24-q1/24-196 885q25/24-42 790 636q49/24-,
47-32h-47=ηj(j-743)
=q-47/24-2q1/24-21 690 645q25/24-40 513 206 272q49/24-,
71-32h-71=ηj(j2-1487j+355 910)
=q-71/24-3q1/24-886 187 500q25/24-8 543 738 297 129q49/24-.

For positive m, we have the Fourier expansion (see [1, Theorem 1])

hm=im3/2qm/24+0<n1(24)mnp(m,n)qn/24-im3/2β(-my)qm/24+0>n1(24)|mn|p(m,n)β(|n|y)qn/24.

For these m, the shadow of hm is proportional to the weight -1/2 form

gm(τ):=m-3/2q-m/24+0>n1(24)|mn|-1/2p(n,m)q|n|/24,

with p(n,m) as in (1.8). When mn<0, we have the relation p(n,m)=-p(m,n). The first few examples of these forms are

g1=η-1
=q-124+q2324+2q4724+3q7124+5q9524+7q11924+,
53g25=η-1(j-745)
=q-2524+196 885q23/24+21 690 645q47/24+,
73g49=η-1(j2-1489j+160 511)
=q-4924+42 790 636q23/24+40 513 206 272q47/24+.

We also have the relation p(m,n)=p(n,m) when m,n>0 (see [1, Corollary 2]).

In order to give an arithmetic interpretation for the coefficients p(m,n) for general m, we require a family of functions {Pv(τ,s)}v whose first member is P1(τ,s)=P(τ,s). We construct these functions in Section 2 using non-holomorphic Maass–Poincaré series. The specializations Pv(τ):=Pv(τ,2) can be obtained by raising the elements of a certain basis {Fv}v of M-2(Γ0(6),1,-1) to weight 0. The functions Fv are uniquely determined by having Fourier expansion Fv=q-v+O(1). They are easily constructed from F and the Γ0(6)-Hauptmodul

J6(τ):=(η(τ)η(2τ)η(3τ)η(6τ))4+(3η(3τ)η(6τ)η(τ)η(2τ))4=q-1-4+79q+352q2+.

For example, F1=F and

F2=F(J6+14)=q-2-50-832q-5693q2-,
F3=F(J62+18J6+27)=q-3-190-7371q-108 216q2-,
F4=F(J63+22J62+20J6-1160)=q-4-370-48 640q-1 100 352q2-.

We also require functions Pv,Q(τ,s) for each quadratic form with square discriminant. These are dampened versions of the functions Pv(τ,s), and are constructed in Section 3.

With χm:𝒬mn(1){-1,0,1} as in (3.7) below, we define the general twisted traces for m,n1(mod24) by

Trv(m,n):={1|mn|QΓ𝒬mn(1)χm(Q)Pv(τQ)if mn<0,12πQΓ𝒬mn(1)χm(Q)CQ[sPv(τ,s)|s=2]dτQ(τ,1)if mn>0 is not a square,12πQΓ𝒬mn(1)χm(Q)CQ[sPv,Q(τ,s)|s=2]dτQ(τ,1)if mn>0 is a square.

The main theorem relates the coefficients p(m,n) to these twisted traces, giving an arithmetic interpretation of the coefficients p(m,n) for all m,n.

Suppose that m,n1(mod24) and that m is squarefree. For each v1 coprime to 6 we have

Trv(m,n)=d|vd(mv/d)(12d)p(d2m,n).

The construction of the harmonic Maass forms hm for m>0 relies heavily on the fact that the space of cusp forms of weight 5/2 with multiplier system ε is trivial (see [1, §3.3]). It would be interesting to generalize Theorem 1.2 to other weights; however, the presence of cusp forms in higher weights adds significant complications.

In the proof of Theorem 1.2 we will encounter the Kloosterman sum

K(a,b;c):=dmodc(d,c)=1eπis(d,c)e(d¯a+dbc),

where d¯ denotes the inverse of d modulo c and e(x):=exp(2πix). Here s(d,c) is the Dedekind sum

s(d,c):=r=1c-1(rc-rc-12)(drc-drc-12)

which appears in the transformation formula for η(z) (see [19, Section 2.8]). The presence of the factor eπis(d,c) makes the Kloosterman sum quite difficult to evaluate. The following formula, proved by Whiteman in [30], is attributed to Selberg and gives an evaluation in the special case a=0:

K(0,b;c)=c3mod2c(32+)/2b(c)(-1)cos(6+16cπ).

Theorem 1.3 below is a generalization of (1.12) which is of independent interest. For each v with (v,6)=1 and for each m,n1(mod24) with m squarefree, define

Sv(m,n;24c):=bmod24cb2mn(24c)(12b)χm([6c,b,b2-mn24c])e(bv12c).

It is not difficult to show (see (4.3) below) that the right-hand side of (1.12) is equal to

14S1(1,24b+1;24c).

Suppose that n=24n+1 and that M=v2m=24M+1, where m is squarefree and (v,6)=1. Then

K(M,n;c)=4c3(12v)u|(v,c)μ(u)(mu)Sv/u(m,n;24c/u).

When (mn,c)=1 we clearly have Sv(m,n;24c)ϵcϵ for any ϵ>0, which yields the bound (for any fixed integers m, n)

K(m,n;c)ϵc12+ϵ,(mn,c)=1.

This is reminiscent of Weil’s bound (see [29] and [18, Lemma 2]) for the ordinary Kloosterman sum

k(m,n;c):=dmodc(d,c)=1e(d¯m+dnc)ϵ(m,n,c)12c12+ϵ.

Our proof of Theorem 1.2 follows the method of Duke, Imamoḡlu, and Tóth [14] in the case when mn is not a square, and the author [2] in the case when mn is a square. We construct the functions Pv(τ,s) and Pv,Q(τ,s) as Poincaré series and then evaluate the traces of these Poincaré series directly. We match these evaluations to the formulas given in [1] for the coefficients p(m,n).

In Section 2 we review some facts about weak Maass forms and construct the functions Pv(τ,s). In Section 3 we discuss binary quadratic forms and establish some facts which we will need for the proof of the main theorem. We construct the functions Pv,Q(τ,s) at the end of Section 3. In Section 4 we prove a proposition which is equivalent to Theorem 1.3 and is a crucial ingredient in the proof of Theorem 1.2. In the final section, we prove Theorem 1.2.

2 Poincaré series and Pv(τ,s)

In this section we construct the weak Maass forms Pv(τ,s) in terms of Poincaré series. We first recall the definition and basic properties of weak Maass forms, and show how such forms can be constructed using Poincaré series associated with Whittaker functions. We then discuss the Atkin–Lehner involutions Wd which are used to build the functions Pv(τ,s).

2.1 Weak Maass forms and Poincaré series

For γ=(abcd)GL2+() and k2, we define the weight k slash operator |k by

(f|kγ)(τ):=(detγ)k/2(cτ+d)-kf(aτ+bcτ+d).

Let ΓΓ1 be a congruence subgroup. A weak Maass form of weight k and Laplace eigenvalue λ is a smooth function f: satisfying

  1. f|kγ=f for all γΓ,
  2. Δkf=λf, where Δk is defined in (1.1), and
  3. f has at most linear exponential growth at each cusp of Γ.

If λ=0, we say that f is a harmonic Maass form. The differential operator

ξk:=2iyk¯τ¯

plays an important role in the theory of harmonic Maass forms. It commutes with the slash operator; that is,

ξk(f|kγ)=(ξkf)|2-kγ.

Thus, if f has weight k then ξkf has weight 2-k. The Laplacian Δk decomposes as

Δk=-ξ2-kξk,

which shows that ξk maps harmonic Maass forms to weakly holomorphic modular forms.

Define

Γ:=Γ0(6)/{±1}.

We follow [12, Section 2.6] in constructing Poincaré series for Γ attached to special values of the M-Whittaker function Mμ,ν(y) (see [27, Chapter 13] for the definition and relevant properties). Let v be a positive integer, and for s and y>0, define

s,k(y):=y-k/2M-k/2,s-1/2(y)

and

φv(τ,s,k):=s,k(4πvy)e(-vx).

Then

φv(τ,s,k)yRe(s)-k/2asy0.

Letting Γ:={(1*01)}Γ denote the stabilizer of , we define the Poincaré series

𝔽v(τ,s,k):=1Γ(2s)γΓΓ(φv|kγ)(τ,s,k).

On compact subsets of , we have (by (2.2)) the bound

|𝔽v(τ,s,k)|yRe(s)-k/2(abcd)ΓΓ|cτ+d|-2Re(s),

so 𝔽v(τ,s,k) converges normally for Re(s)>1. A computation involving [27, (13.1.31)] shows that

Δkφv(τ,s,k)=(s-k/2)(1-k/2-s)φv(τ,s,k).

Since 𝔽v clearly satisfies 𝔽v|kγ=𝔽v for all γΓ, we see that for fixed s with Re(s)>1, the function 𝔽v(τ,s,k) is a weak Maass form of weight k and Laplace eigenvalue (s-k/2)(1-k/2-s).

We are primarily interested in the case when k is negative. In this case the special value 𝔽v(τ,1-k/2,k) is a harmonic Maass form. Its principal part at is given by q-v+c0 for some c0, while its principal parts at the other cusps are constant. Thus, ξk𝔽v(τ,1-k/2,k) is a cusp form of weight 2-k on Γ.

2.2 Atkin–Lehner involutions

We recall some basic facts on Atkin–Lehner involutions (see, for example, [21, Section IX.7] or [23, Section 2.4]). Suppose that N is a positive squarefree integer and that dN. Let Wd=WdN denote any matrix with determinant d of the form

Wd=(dαβNγdδ)

with α,β,γ,δ. The relation

WdΓ0(N)Wd-1=Γ0(N)

shows that the map ff|kWd (called the Atkin–Lehner involution Wd) is independent of the choices of α,β,γ,δ and defines an involution on the space of weight k forms on Γ0(N). If d and d are divisors of N, then

f|kWd|kWd=f|kWd*d,

where d*d=dd/(d,d)2. When d=N it is convenient to take WN=(0-1N0). Finally, the Atkin–Lehner involutions act transitively on the cusps of Γ0(N); that is, for each cusp 𝔞Γ0(N)1(), there exists a unique dN such that Wd=𝔞.

Recall that Fv(τ)=q-v+O(1) is a weakly holomorphic modular form of weight -2 on Γ0(6) with eigenvalues 1 and -1 under W6 and W2, respectively. We claim that

Fv(τ)=d|6μ(d)(𝔽v|-2Wd)(τ,2,-2).

To prove this, let F~v(τ) denote the right-hand side of (2.4). Since ξ-2𝔽v(τ,2,-2) lies in the one-dimensional space of cusp forms of weight 4 on Γ0(6), it must be proportional to

g(τ):=(η(τ)η(2τ)η(3τ)η(6τ))2.

Since ξk commutes with the slash operator and g(τ) is invariant under Wd for each d6, we have, for some α, the relation

ξ-2F~v(τ)=αd6μ(d)g(τ)=0.

Thus Fv(τ)-F~v(τ) is holomorphic on and vanishes at every cusp. Hence Fv(τ)=F~v(τ).

2.3 The functions Pv(τ),Pv(τ,s)

To construct the functions Pv(τ) and Pv(τ,s), we require the Maass raising operator

Rk:=2iτ+ky

which raises the weight of a weak Maass form by 2. For each v1, we define

Pv(τ):=14πvR-2Fv(τ).

Then Pv(τ) is a weak Maass form of weight 0 and Laplace eigenvalue -2. By (2.4) and [12, Proposition 2.2] this is equivalent to defining

Pv(τ):=d|6μ(d)𝔽v(Wdτ,2,0)=16d6μ(d)γΓΓφv(γWdτ,2,0).

Similarly, we define Pv(τ,s) as

Pv(τ,s):=C(s)d|6μ(d)𝔽v(Wdτ,s,0)=C(s)Γ(2s)d6μ(d)γΓΓφv(γWdτ,s,0),

where

C(s):=2sπΓ(s+12)2.

We have chosen the non-standard normalizing factor C(s) so that later results are cleaner to state. Note that C(2)=1, so

Pv(τ,2)=Pv(τ).

In the next section we will define the dampened functions Pv,Q(τ,s).

3 Binary quadratic forms and Pv,Q(τ,s)

In this section we recall some basic facts about binary quadratic forms and the genus characters χm. A good reference for this material is [16, Section I]. Throughout this section, we assume that m,n1(mod24) and that m is squarefree. The latter condition ensures that m is a fundamental discriminant.

Suppose that r{1,5,7,11}. We recall that

𝒬n(r):={ax2+bxy+cy2:b2-4ac=n, 6a,br(mod12), and a>0 if n<0}.

Let Γ* denote the group generated by Γ=Γ0(6)/{±1} and the Atkin–Lehner involutions Wd for d6. Matrices γ=(ABCD)Γ* act on such forms on the left by

γQ(x,y):=1detγQ(Dx-By,-Cx+Ay).

It is easy to check that this action is compatible with the action γτ:=Aτ+BCτ+D on the roots of Q: for all γΓ*, we have

γτQ=τγQ.

The set Γ𝒬n(r) forms a finite group under Gaussian composition which is isomorphic to the narrow class group of (n)/ when n is a fundamental discriminant. Let 𝒬n denote the union

𝒬n:=r{1,5,7,11}𝒬n(r).

For d6, the Atkin–Lehner involution Wd=(dαβ6γdδ) acts on quadratic forms by

WdQ(x,y):=1dQ(dδx-βy,-6γx+dαy).

A computation involving (3.2) and the relation dαδ-6dβγ=1 shows that

Wd[6a,b,c]=[6*,b(1+12dβγ)+12*,*].

It is convenient to choose W2=(2-16-2) and W3=(3163). Then (3.3) shows that

Wd:𝒬n(r)𝒬n(r)

is a bijection, where

rr×{1if d=1,7if d=2,5if d=3,11if d=6,(mod12).

Moreover, we have

𝒬n=d|6Wd𝒬n(r)

for any r{1,5,7,11}.

We turn now to the extended genus character χm. For Q𝒬mn, define

χm(Q):={(mr)if (a,b,c,m)=1 and Q represents r with (r,m)=1,0if (a,b,c,m)>1.

The following lemma lists some properties of χm.

Suppose that m,n1(mod24) and that m is squarefree.

  1. P1)The map χm:Γ𝒬mn{-1,0,1} is well-defined; i.e. χm(γQ)=χm(Q) for all γΓ.
  2. P2)If (a,a)=1 then
    χm([6aa,b,c])=χm([6a,b,ac])χm([6a,b,ac]).
  3. P3)For each d6 we have
    χm(Q)=χm(WdQ).
  4. P4)Suppose that [6a,b,c]𝒬mn, and let g:=±(a,m), where the sign is chosen so that g1(mod4). Then
    χm([6a,b,c])=(m/g6a)(gc).
  5. P5)We have χm(-Q)=sgn(m)χm(Q).

Proof.

Property P3 for d=1,6 and P1, P2, and P4 are special cases of [16, Proposition 1]. Property P5 follows easily from P4. A generalization of P3 is stated without proof in [16], so we provide a proof here for our special case.

We want to show that P3 holds for d=2,3. Suppose that

Q=[6a,b,c]with (a,b,c,m)=1.

Choosing W2=(2-16-2) and W3=(3163), we find that

W2[6a,b,c]=[6(2a+b+3c),*,3a+b+2c],W3[6a,b,c]=[6(3a-b+2c),*,2a-b+3c].

We will use P4. For W2, we want to show that

(m/g6(2a+b+3c))(g3a+b+2c)=(m/g6a)(gc).

Since g divides mn+4ac=b2 and g is squarefree, we see that gb, so

(g3a+b+2c)=(gc)(g2).

If a=0 then g=m, and (3.8) follows from (3.9) and the fact that m1(mod8). If a0, then the relation

8a(2a+b+3c)=(4a+b)2-m

shows that

(m/g6(2a+b+3c))(m/g6a)=(m/g2).

Together with (3.9), this completes the proof for W2. The proof for W3 is similar. ∎

The remainder of this section follows [14, Sections 3 and 4] and [2, Section 3]. Let ΓQ denote the stabilizer of Q in Γ=Γ0(6)/{±1}. When the discriminant of Q is negative or a positive square, the group ΓQ is trivial. However, when the discriminant n>0 is not a square, the group ΓQ is infinite cyclic. If Q=[a,b,c]𝒬n with (a,b,c)=1, we have ΓQ=gQ, where

gQ:=(t+bu2cu-aut-bu2)

and t,u are the smallest positive integral solutions to Pell’s equation t2-nu2=4. When (a,b,c)=δ>1, we have ΓQ=gQ/δ.

For Q=[a,b,c]𝒬n with n>0, let SQ denote the geodesic in connecting the two roots of Q(τ,1). Explicitly, SQ is the curve in defined by

a|τ|2+bRe(τ)+c=0.

When a0, SQ is a semicircle, which we orient counter-clockwise if a>0 and clockwise if a<0. When a=0, SQ is the vertical line Re(τ)=-c/b, which we orient upward. If γΓ then we have

γSQ=SγQ.

Fix any zSQ and define the cycle CQ as the directed arc on SQ from z to gQz. We define

dτQ:=ndτQ(τ,1),

so that if τ=γτ for some γΓ*, we have

dτγQ=dτQ.

Suppose that Q has positive non-square discriminant and that f is a Γ-invariant function that is continuous on SQ. A straightforward generalization of [14, Lemma 6] shows that the integral

CQf(τ)𝑑τQ

is a well-defined (i.e. independent of the choice of zSQ) invariant of the equivalence class of Q.

We now define the functions Pv,Q(τ,s). Let Q=[a,b,c] be a binary quadratic form with square discriminant. Then the equation Q(x,y)=0 has two inequivalent solutions [r1:s1] and [r2:s2] in 1(), which we write as fractions 𝔞i:=ri/si, with (ri,si)=1 and possibly si=0. For each i, there is a unique di such that

Wdi𝔞iΓ.

Thus, up to translation, there is a unique γiΓ such that

γiWdi𝔞i=.

The function Pv,Q(τ,s) is defined by deleting the two terms of Pv(τ,s) in (2.5) corresponding to the pairs (γi,Wdi); that is,

Pv,Q(τ,s):=C(s)Γ(2s)d|6μ(d)γΓΓγWd𝔞iφv(γWdτ,s,0).

Suppose that σΓ. Then by (3.1), the roots of σQ are σ𝔞1 and σ𝔞2, and we find (using (2.3)) that

Pv,σQ(στ,s)=Pv,Q(τ,s).

Together with (3.10) and (3.12), this shows that the integral

CQPv,Q(τ,s)dτQ(τ,1)

is invariant under QσQ.

4 Kloosterman sums and the proof of Theorem 1.3

In this section we prove an identity (Proposition 4.2 below) connecting the Kloosterman sum (1.11) with the twisted quadratic Weyl sum (1.13). This is an essential ingredient in the proof of Theorem 1.2, and is equivalent to the evaluation of the Kloosterman sum in Theorem 1.3.

Throughout this section, v is a positive integer coprime to 6 and m,n1(mod24) with m squarefree. We will use the notation

a:=a-124

whenever a1(mod24). The Kloosterman sum is defined as

K(a,b;c):=d(c)*eπis(d,c)e(d¯a+dbc),

where d(c)* indicates that the sum is taken over residue classes coprime to c, and d¯ denotes the inverse of d modulo c. The factor eπis(d,c) makes the Kloosterman sum very difficult to evaluate. The following lemma shows that eπis(d,c) is related to the Gauss-type sums

Hd,c(δ):=12j(2c)e(d(6j+δ)224c)

which were introduced by Fischer in [15].

Suppose that (v,6)=(c,d)=1 and define

α:=1-d¯c-d¯v,β:=1-d¯c+d¯v,

with d¯ chosen such that

dd¯{1(modc)if c is odd,1(mod2c)if c is even.

Then we have

3c(12v)e(d¯(v2-1)24c)eπis(d,c)=e(2v+dα224c)H-d,c(α)+e(-2v+dβ224c)H-d,c(β).

Proof.

Define

f(v):=(12v)e(2v+dα2-d¯(v2-1)24c)H-d,c(α).

We will prove (4.2) by showing that f(v)+f(-v)=3ceπis(d,c).

We first show that for fixed d,c, the function f(v) depends only on vmod6. By [30, (3.8)] we find that H-d,c(α) depends only on αmod6. Define ε{-1,1} by vε(mod6). Then (12v)=e(v-ε12), and we have

(12v)e(2v+dα2-d¯(v2-1)24c)=e(v2d¯(dd¯-1)/c24-v(dd¯-1)/c-dd¯2-112-ε12+d(d¯c-1)2+d¯24c).

This depends only on vmod6 since v21(mod24) and, by definition,

dd¯-1c-dd¯2-12.

Now f(v)+f(-v)=f(ε)+f(-ε) is independent of v, so to prove (4.2) it suffices to show that

f(1)+f(-1)=3ceπis(d,c).

This is proved in [30, Section 4]. ∎

The quadratic Weyl sum Sv(m,n;24c) is defined as

Sv(m,n;24c):=b(24c)b2mn(24c)χ12(b)χm([6c,b,b2-mn24c])e(bv12c),

where χm is defined in (3.7). We clearly have Sv(m,n;24c)=Sv(m,n;24c)¯, so the exponential e(bv12c) may be replaced by cos(bvπ6c). When m=1, we obtain a simpler expression for S1(1,n;24c) as follows. The summands of Re(Sv(1,n;24c)) are invariant under both bb+12c and b-b, so we may sum over those b modulo 12c for which b1(mod6), and multiply the sum by 4. Writing b=6+1, we obtain (cf. formula (1.12))

S1(1,n;24c)=4mod2c(32+)/2n(c)(-1)cos((6+1)vπ6c).

The following proposition gives an expression for Sv(m,n;24c) in terms of Kloosterman sums. Its proof occupies the remainder of the section. Theorem 1.3 follows from (4.4) by Möbius inversion in two variables.

Suppose that m,n1(mod24) and that m is squarefree. Suppose that c,v>0 and that (v,6)=1. Then

Sv(m,n;24c)=43u|(v,c)(12v/u)(mu)ucK((v2u2m),n;cu).

Proposition 4.2 resembles [14, Proposition 3], which is proved using a slight modification of Kohnen’s argument in [22, Proposition 5]. Using an elegant idea of Tóth [28], Duke [13] greatly simplified Kohnen’s proof for the case m=D=1 (in the notation of [14]). Jenkins [20] later extended this argument to the case of general m. However, Kohnen’s argument remains the only proof of the general case.

Although special cases of (4.4) are amenable to the methods of Duke and Jenkins, we prove Proposition 4.2 in full generality by adapting Kohnen’s argument. The proof is quite technical.

In the proof of Proposition 4.2 we will encounter the quadratic Gauss sum

G(a,b,c):=x(c)e(ax2+bxc),c>0.

For any d(a,c), we see by replacing x by x+c/d that

G(a,b,c)=e(bd)G(a,b,c).

This implies that G(a,b,c)=0 unless db. In that case,

G(a,b,c)=dG(ad,bd,cd).

If (a,c)=1, we have the well-known evaluations (see [3, Theorems 1.5.1, 1.5.2, and 1.5.4])

G(a,0,c)={0if 2c,(1+i)εa-1c(ca)if 4c,εcc(ac)if c is odd,

where

εa:={1if a1(mod4),iif a3(mod4).

If 4c and (a,c)=1 then, by replacing x by x+c/2, we find that G(a,b,c)=0 if b is odd. If b is even and 4c, or if c is odd, then by completing the square and using (4.7), we find that

G(a,b,c)={e(-a¯b24c)(1+i)εa-1c(ca)if b is even and 4c,e(-4a¯b2c)εcc(ac)if c is odd.

Finally, these Gauss sums satisfy the multiplicative property

G(a,b,qr)=G(ar,b,q)G(aq,b,r),(q,r)=1

which is a straightforward generalization of [3, Lemma 1.2.5].

We will need an explicit formula for χm([6c,b,b2-mn24c]), which follows from P4 of Lemma 3.1 (see also [22, Proposition 6]). For each odd prime p, let

p*:=(-1)p-12p

so that (ap)=(p*a). If m is squarefree, then

χm([6c,b,b2-mn24c])=pλ||cpm(mpλ)pλ||cp|m(m/p*pλ)(p*(b2-mn)/pλ).

Proof of Proposition 4.2.

Both sides of (4.4) are periodic in v with period 12c, so it suffices to show that their Fourier transforms are equal. For each h we will show that

112cv(12c)e(-hv12c)Sv(m,n;24c)=4312cv(12c)e(-hv12c)u|(c,v)(12v/u)(mu)ucK((v2u2m),n;cu).

Let L(h) and R(h) denote the left- and right-hand sides of (4.11), respectively. Then we have

L(h)=b2mn(24c)χ12(b)χm([6c,b,b2-mn24c])×112cv(12c)e((b-h)v12c)
={2χ12(h)χm([6c,h,h2-mn24c])if h2mn(mod24c),0otherwise.

For the right-hand side, we have

R(h)=13cu|c(mu)ucv(12c/u)(12v)e(-hv12c/u)d(c/u)*eπis(d,c/u)e(d¯(v2m)+dnc/u)
=13cu|c(mc/u)1ud(u)*e(dnu)v(12u)eπis(d,u)e(-d¯24u)(12v)e(d¯mv2-2hv24u).

Using Lemma 4.1, definition (4.1), and the fact that u+v is even, we find that

eπis(d,u)e(-d¯24u)(12v)=123ue(-d¯v224u)j(2u)e(-d(3j2+j)/2u+j2)(e(v(6j+1)12u)+e(-v(6j+1)12u)).

Thus we obtain

R(h)=16cu|c(mc/u)u-1d(u)*e(dnu)j(2u)e(-d(3j2+j)/2u+j2)
×(G(d¯(m-1)/2,6j+1-h,12u)+G(d¯(m-1)/2,-6j-1-h,12u)),

where G(a,b,c) is the quadratic Gauss sum defined in (4.5). Since 12(m-1)/2, we see by (4.6) that

G(d(m-1)/2,±(6j+1)-h,12u)={12G(d¯m,±(6j+1)-h12,u)if h±(6j+1)(mod12),0otherwise.

In particular, R(h)=0 unless h±1(mod6).

For the remainder of the proof, we assume that h1(mod6) (the other case is analogous). Then in (4.13) the second Gauss sum is zero, and we take only those j for which jh-16(mod2). We write j=2k+h-16; then

j(2u)e(-d(3j2+j)/2u+j2)G(d¯(m-1)/2,6j+1-h,12u)
=12e(h-112-d(h2-1)/24u)k(u)e(-dk(6k+h)u)G(d¯m,k,u).

Since e(h-112)=χ12(h), we have

R(h)=2cχ12(h)u|c(mc/u)u-1Fh(u),

where

Fh(u):=d(u)*e(du(n-h2-124))k(u)e(-dk(6k+h)u)G(d¯m,k,u).

We show that Fh(u) is multiplicative as a function of u. To prove this, suppose that (q,r)=1, and choose r¯ and q¯ such that rr¯+qq¯=1. Let α:=n-(h2-1)/24, and write d=rr¯x+qq¯y and k=k1r+k2q. Using (4.5) and (4.9), we find that

Fh(qr)=x(q)*e(r¯xαq)k1(q)e(-k1x(6k1r+h)q)G(rx¯m,k1r,q)
×y(r)*e(q¯yαr)k2(r)e(-k2y(6k2q+h)r)G(qy¯m,k2q,r).

Replacing k1, k2, x, and y by k1r¯, k2q¯, xr, and yq, respectively, we conclude that

Fh(qr)=Fh(q)Fh(r).

Clearly 1cu|c(mc/u)u-1Fh(u) is multiplicative as a function of c. Thus, by (4.10), (4.12), and (4.14), to show that L(h)=R(h) it suffices to show that for each prime power pλc we have

p-λj=0λ(mpλ-j)p-jFh(pj)={(mpλ)if pm,(m/p*pλ)(p*(h2-mn)/pλ)if pm,

when h2mn(mod24pλ), and 0 otherwise.

Suppose first that p is an odd prime. Set

pμ:=(m,pj).

Then G(d¯m,k,pj)=0 unless pμk. In the latter case, using (4.6) and (4.8), we find that

G(d¯m,k,pj)=εpj-μpj+μ2(d¯m/pμpj-μ)e(-d(4m/pμ)¯(k/pμ)2pj-μ).

Writing k=pμ, we find that

Fh(pj)=εpj-μ(m/pμpj-μ)pj+μ2d(pj)*(dpj-μ)e(dpj(n-h2-124))(pj-μ)e(-d(6pμ+(4m/pμ)¯)2-dhpj-μ).

Since

(4m/pμ)¯m=(4m/pμ)¯(24pμm/pμ+1)6pμ+(4m/pμ)¯(modpj-μ),

the inner sum is equal to

G(-dm(4m/pμ)¯,-dh,pj-μ).

We first consider the case where pm, and we choose m¯ such that m¯m1(mod24pλ). Using (4.8) again, we find that

Fh(pj)=pj(-mpj-μ)εpj-μ2d(pj)*e(dpj(n-h2-124+m¯mh2)).

Note that

n-h2-124+m¯mh2n-m¯h224(modpλ).

Since εpj-μ2=(-1pj-μ) and m1(modpμ), we conclude that the quantity in (4.15) is

p-λj=0λ(mpλ-j)p-jFh(pj)=p-λ(mpλ)j=0λd(pj)*e(dpλ-jpλ(n-m¯h224))
=p-λ(mpλ)d(pλ)e(dpλ(n-m¯h224))
={(mpλ)if pλn-m¯h224,0otherwise.

The condition pλ(n-m¯h2)/24 is equivalent to h2mn(mod24pλ), so (4.15) is true in the case where p is an odd prime not dividing m.

We turn now to the case where pm. Then pm, so μ=0 in (4.16), and since m is squarefree, (m/p,p)=1. Furthermore, all of the terms in the sum on the left-hand side of (4.15) vanish except for the term j=λ. From (4.16) we have

Fh(pλ)=εpλ(mpλ)pλ/2d(pλ)*(dpλ)e(dpλ(n-h2-124))G(-dm(4m)¯,-dh,pλ),

which is zero unless ph. Assume that ph; then, using (4.8), we obtain

Fh(pλ)=pλ+12εp(mp)(-m/ppλ-1)d(pλ)*(dp)e(dpλ(n-h2-124+(m/p)¯mh2p)).

Set

α:=n-h2-124+(m/p)¯mh2p.

Replacing d by d+p, we see that the sum

d(pλ)*(dp)e(dαpλ)

is zero unless pλ-1α. Assume that pλ-1α. Then

d(pλ)*(dp)e(dαpλ)=pλ-1d(p)(dp)e(dα/pλ-1p)=εppλ-12(α/pλ-1p),

where the last equality uses [3, Theorem 1.1.5] and evaluations (4.7). We have p3 since m1(mod24), so (mp)=(-24p). Therefore

Fh(pλ)=p2λεp2(-m/ppλ-1)(-24α/pλ-1p).

We have

24αpλ-1=p(n-h2)+(m/p)¯24mh2pλ

which, together with the fact that (m/p)¯mp(modpλ+1), yields

24αpλ-1(m/p)¯[mn-mh2+24mh2]pλ(m/p)¯[mn-h2]pλ(modp).

Therefore

Fh(pλ)=p2λ(-m/ppλ)((h2-mn)/pλp)=p2λ(m/p*pλ)(p*(h2-mn)/pλ),

under the assumption that pλ-1α. This assumption is equivalent to h2mn(mod24pλ) and implies that ph, which justifies our previous assumption. Thus we conclude that

Fh(pλ)={p2λ(m/p*pλ)(p*(h2-mn)/pλ)if h2mn(mod24pλ),0otherwise,

which verifies (4.15) in the case where p is an odd prime dividing m.

Now suppose that p=2. Since 2m and (m2)=1, we want to show that

j=0λ2-jFh(2j)={2λif h2mn(mod242λ),0otherwise.

We recall the definition

Fh(u):=d(u)*e(du(n-h2-124))k(u)e(-dk(6k+h)u)G(d¯m,k,u).

Define μ by

2μ=(m,2j).

Then

G(d¯m,k,2j)={2μG(d¯m/2μ,k/2μ,2j-μ)if 2μk,0otherwise.

Let

β:=n-h2-124+h2m¯m,with m¯m1(mod242λ).

We claim that

Fh(2j)=2jd(2j)*e(dβ2j).

If μ=j then G(d¯m/2μ,k/2μ,2j-μ)=1 and 2jm, so

Fh(2j)=2jd(2j)*e(dβ2j).

If μ=j-1 then

G(d¯m/2μ,k/2μ,2j-μ)={2if k/2μ is odd,0if k/2μ is even.

Since 2j-1m and m¯ is odd, we have βn-(h2-1)/24-2j-1h(mod2j), which yields

Fh(2j)=2jd(2j)*e(d2j(n-h2-124)-d2j-1(62j-1+h)2j)=2jd(2j)*e(dβ2j).

If μj-2 then by (4.8) we have

G(d¯m/2μ,k/2μ,2j-μ)={(1+i)εd¯m/2μ-1 2j-μ2(2j-μd¯m/2μ)e(-d(m/2μ¯)(k/2μ)22j-μ+2)if k/2μ is even,0if k/2μ is odd.

Writing k=2μ+1, we have

Fh(2j)=(1+i) 2j+μ2d(2j)*e(d2j(n-h2-124))εd¯m/2μ-1(2j-μd¯m/2μ)
×(2j-μ-1)e(-d2μ+1(62μ+1+h)2j-d(m/2μ¯)22j-μ).

Since (m/2μ¯)+242μ(m/2μ¯)m(mod2j-μ), the inner sum equals

12G(-dm(m/2μ¯),-2dh,2j-μ)=(1+i) 2j-μ2-1ε-dm(m/2μ¯)-1(2j-μ-dm(m/2μ¯))e(dh2m¯m2j).

Since

εd¯m/2μ-1ε-dm(m/2μ)¯-1=-i,

we have

Fh(2j)=2jd(2j)*e(dβ2j).

We conclude in every case that

j=0λ2-jFh(2j)=j=0λd(2j)*e(dβ2j)=d(2λ)e(dβ2λ)={2λif h2mn(mod242λ),0otherwise,

which completes the proof. ∎

5 Proof of Theorem 1.2

We begin by recording exact formulas for the coefficients p(m,n) in terms of Kloosterman sums and the I- and J-Bessel functions Iα(x) and Jα(x). The following formulas are found in [1, Proposition 11]. Let h~m denote the functions in that paper; then our functions hm described in (1.9) and (1.8) are normalized as

hm={-3m3/24πh~mif m>0,|m|3/2h~mif m<0.

Suppose that m,n1(mod24) are not both negative. By [1, Propositions 8 and 11] we have

p(m,n)={2π|mn|14c>0K(m,n;c)cI32(π|mn|6c)if mn<0,4(mn)14c>0K(m,n;c)c[sJs-12(πmn6c)|s=2]if mn>0.

To prove Theorem 1.2 we will show that the traces Trv(m,n) can also be expressed as infinite series involving Kloosterman sums and Bessel functions. This is essentially accomplished in the following proposition. To simplify the statement of the proposition for square and non-square positive discriminants, we set Pv,Q(τ,s):=Pv(τ,s) whenever Q has positive non-square discriminant. The functions Pv(τ), Pv(τ,s), and Pv,Q(τ,s) are defined in (2.7), (2.5), and (3.13), respectively.

Suppose that m,n1(mod24) and that m is squarefree. If mn<0 then

1|mn|12QΓ𝒬mn(1)χm(Q)Pv(τQ)=2π|mn|14dvd(12d)(mv/d)c>0K((d2m),n;c)cI32(π|d2mn|6c).

If Re(s)>1 and m,n>0 then

12πQΓ𝒬mn(1)χm(Q)CQPv,Q(τ,s)dτQ(τ,1)
=4(mn)14dvd(12d)(mv/d)c>0K((d2m),n;c)cJs-12(πd2mn6c).

Before proving Proposition 5.1, we remark that when s=2, the right-hand side of (5.4) is often identically zero. This follows from [1, (3.15)], which states that

c>0K(m,n;c)cJ3/2(πmn6c)={0if mn,12πif m=n.

The only situation in which the right-hand side of (5.4) does not vanish is when n=mt2 for some integer t and v=t, for some integer with (,m)=1. In that case (5.4) becomes

Q𝒬(mt)2(1)χm(Q)CQPv,Q(τ,2)dτQ(τ,1)=4m(12t)(m).

Proof of Proposition 5.1.

Suppose that mn<0, and let Lv-(m,n) denote the left-hand side of (5.3). Using the definition of Pv(τ)=Pv(τ,2) in (2.5), we find that

Lv-(m,n)=16|mn|-1/2d6QΓ𝒬mn(1)γΓΓμ(d)χm(Q)φv(γWdτQ,2,0).

Using P1 and P3 of Lemma 3.1 and equation (3.1), this becomes

Lv-(m,n)=16|mn|-1/2d6QΓ𝒬mn(1)γΓΓμ(d)χm(γWdQ)φv(τγWdQ,2,0).

By (3.4) and (3.6) the map (γ,d,Q)γWdQ gives a bijection

ΓΓ×{1,2,3,6}×Γ𝒬mn(1)Γ𝒬mn.

If Q𝒬mn(1) and Q=WdQ=[a,b,c] then μ(d)=(12b) by (3.4) and (3.5). With s,k(y) as in (2.1), we have

Lv-(m,n)=16|mn|-1/2QΓ𝒬mnQ=[a,b,c](12b)χm(Q)2,0(4πvImτQ)e(-vReτQ).

Since 𝒬mn contains only positive definite forms (those with a>0) we have

τQ=-b2a+i|mn|2a.

By [27, (13.1.32) and (13.6.6)], we have

2,0(4πvy)=M0,3/2(4πvy)=12πvyI3/2(2πvy),

which gives

Lv-(m,n)=π2v|mn|14QΓ𝒬mnQ=[a,b,c](12b)χm(Q)aI32(πv|mn|a)e(bv2a).

Now suppose that m and n are both positive, and let Lv+(m,n) denote the left-hand side of (5.4). As in (3.11), we write

dτQ=mndτQ(τ,1).

If mn is not a square, then by (2.5) we have

Lv+(m,n)=C(s)2πΓ(2s)mnd6μ(d)QΓ𝒬mn(1)χm(Q)γΓΓCQφv(γWdτ,s,0)𝑑τQ.

For each Q, let ΓQΓ denote the stabilizer of Q. We rewrite the sum over ΓΓ as a sum over γΓΓ/ΓQ and a sum over gΓQ. Since SQ=gΓQCQ, the inner sum in (5.7) becomes

γΓΓ/ΓQSQφv(γWdτ,s,0)𝑑τQ.

In each integral, we replace τ by Wd-1γ-1τ. Using (3.12) and P1 and P3 of Lemma 3.1, we obtain

Lv+(m,n)=C(s)2πΓ(2s)mnd6QΓ𝒬mn(1)γΓΓ/ΓQμ(d)χm(γWdQ)SγWdQφv(τ,s,0)𝑑τγWdQ.

As in (5.5), we have the bijection

ΓΓ/ΓQ×{1,2,3,6}×Γ𝒬mn(1)Γ𝒬mn

given by (γ,d,Q)γWdQ. Thus

Lv+(m,n)=C(s)2πΓ(2s)mnQΓ𝒬mnQ=[a,b,c](12b)χm(Q)SQs,0(4πvImτ)e(-vReτ)𝑑τQ.

In order to treat the square case together with the non-square case, we will show that, with the added condition a0 in the sum, (5.9) holds when mn is a square. Of course, a0 is implied in (5.9) when mn is not a square.

Suppose that mn>0 is a square. For each Q𝒬mn(1) define 𝔞i,Q:=ri,Q/si,Q as in (3.13). The stabilizer ΓQ is trivial for all Q𝒬mn and, using (3.13), we have (5.8) with the added condition γWd𝔞i,Q on the third sum; that is,

Lv+(m,n)=C(s)2πΓ(2s)mnd6QΓ𝒬mn(1)γΓΓγWd𝔞i,Qμ(d)χm(γWdQ)SγWdQφv(τ,s,0)𝑑τγWdQ.

The quadratic forms Q having =[1,0] as a root are of the form Q=[0,±b,*], where b=mn. Thus the condition γWd𝔞i,Q is equivalent to γWdQ[0,±b,*]. Applying the bijection (5.5), which holds in this case since ΓQ is trivial, we obtain (5.9) with the restriction a0 on the sum.

Treating the square and non-square case together, we assume that m and n are arbitrary positive integers satisfying m,n1(mod24). Suppose that Q=[a,b,c]. The apex of the semicircle SQ is

-b2a+imn2|a|,

so we parametrize SQ by

τ=-b2a+mn2aeisgn(a)θ=-b2a+mn2acosθ+imn2|a|sinθ,0θπ.

Then we have

Q(τ,1)=mn4a(e2isgn(a)θ-1),

which gives

dτQ=mndτQ(τ,1)=dθsinθ.

Combining (5.9), (5.10), and (5.11), we obtain

Lv+(m,n)=C(s)2πΓ(2s)mnQΓ𝒬mnQ=[a,b,c],a0RQ(m,n),

where

RQ(m,n):=(12b)χm(Q)e(bv2a)0πs,0(2πvmn|a|sinθ)e(-vmn2acosθ)dθsinθ.

For each Q=[a,b,c]Γ𝒬mn with a>0, we have (using P5 of Lemma 3.1)

RQ(m,n)+R-Q(m,n)=2(12b)χm(Q)e(bv2a)0πs,0(2πvmnasinθ)cos(πvmnacosθ)dθsinθ.

By [27, (13.6.6)] we have

s,0(y)=M0,s-1/2(y)=22s-1Γ(s+1/2)yIs-1/2(y/2),

hence

Lv+(m,n)=22s-1/2C(s)Γ(s+1/2)πΓ(2s)(mn)1/4QΓ𝒬mn+(12b)χm(Q)vae(bv2a)
×0πIs-1/2(πvmnasinθ)cos(πvmnacosθ)dθsinθ,

where 𝒬mn+ consists of those Q=[a,b,c] with a>0. Lemma 9 of [14] asserts that for Res>0 we have

0πcos(tcosθ)Is-1/2(tsinθ)dθsinθ=2s-1Γ(s/2)2Γ(s)Js-1/2(t).

Since

23s-3/2C(s)Γ(s+1/2)Γ(s/2)2πΓ(2s)Γ(s)=22,

we obtain

Lv+(m,n)=22v(mn)14QΓ𝒬mn+(12b)χm(Q)ae(bv2a)Js-12(πvmna).

Summarizing (5.6) and (5.12), we have

Lv±(m,n)=2v|mn|14QΓ𝒬mn+Q=[a,b,c](12b)χm(Q)ae(bv2a)ϕ±(πv|mn|a),

where

ϕ-(x)=πI3/2(x),ϕ+(x)=2Js-1/2(x).

Since (1k01)[a,b,c]=[a,b-2ka,*], we have a bijection

Γ𝒬mn+{(a,b):a>0, 6a, 0b<2a},

which gives

Lv±(m,n)=2v|mn|14a>06aa-1/2ϕ±(πv|mn|a)bmod2ab2-mn4a(12b)χm([a,b,b2-mn4a])e(bv2a).

We write a=6c and find that the inner sum in (5.13) equals 12Sv(m,n;24c) (see (1.13)), so

Lv±(m,n)=v23|mn|14c>0Sv(m,n;24c)cϕ±(πv|mn|6c).

Applying Proposition 4.2, we obtain

Lv±(m,n)=2v|mn|14c>0c-1/2ϕ±(πv|mn|6c)u(v,c)(12v/u)(mu)ucK((v2u2m),u;cu).

We replace c by cu and switch the order of summation to obtain

Lv±(m,n)=2|mn|14uv(12v/u)(mu)vuc>01cK((v2u2m),u;c)ϕ±(πv/u|mn|6c).

Finally, letting d=v/u, we conclude that

Lv±(m,n)=2|mn|14dvd(12d)(mv/d)c>0K((d2m),n;c)cϕ±(π|d2mn|6c),

from which Proposition 5.1 follows. ∎

Theorem 1.2 now follows easily from Proposition 5.1.

Proof of Theorem 1.2.

As above, we let Pv,Q(τ,s):=Pv(τ,s) when Q has positive non-square discriminant. Then the definition of the traces in (1.7) becomes

Trv(m,n)={|mn|-1/2QΓ𝒬mn(1)χm(Q)Pv(τQ)if mn<0,12πQΓ𝒬mn(1)χm(Q)CQ[sPv,Q(τ,s)|s=2]dτQ(τ,1)if mn>0.

When mn<0, (1.10) follows immediately from (5.2) and (5.3). When m,n>0, we take the derivative of each side with respect to s, then set s=2. Comparing the resulting equation with (5.2) gives (1.10). ∎

Acknowledgements

The author is grateful to Jan Bruinier for encouraging him to investigate the arithmetic nature of the coefficients p(m,n).

References

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    Ahlgren S. and Andersen N., Weak harmonic Maass forms of weight 5/2 and a mock modular form for the partition function, Res. Number Theory 1 (2015), Article ID 10.

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    Andersen N., Periods of the j-function along infinite geodesics and mock modular forms, Bull. Lond. Math. Soc. 47 (2015), 407–415.

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    Berndt B. C., Evans R. J. and Williams K. S., Gauss and Jacobi Sums, Canad. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, New York, 1998.

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    Borcherds R. E., Automorphic forms on O s + 2 , 2 ( R ) ${{\rm O}_{s+2,2}({R})}$ and infinite products, Invent. Math. 120 (1995), no. 1, 161–213.

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    Bringmann K. and Ono K., The f ( q ) ${f(q)}$ mock theta function conjecture and partition ranks, Invent. Math. 165 (2006), no. 2, 243–266.

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    • Export Citation
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    Bringmann K. and Ono K., An arithmetic formula for the partition function, Proc. Amer. Math. Soc. 135 (2007), no. 11, 3507–3514.

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    • Export Citation
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    Bringmann K. and Ono K., Dyson’s ranks and Maass forms, Ann. of Math. (2) 171 (2010), no. 1, 419–449.

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    Bruinier J. H., Harmonic Maass forms and periods, Math. Ann. 357 (2013), no. 4, 1363–1387.

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    Bruinier J. H. and Funke J., On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45–90.

    • Crossref
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    Bruinier J. H., Funke J. and Imamoglu O., Regularized theta liftings and periods of modular functions, J. Reine Angew. Math. 703 (2015), 43–93.

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    Bruinier J. H. and Ono K., Heegner divisors, L-functions and harmonic weak Maass forms, Ann. of Math. (2) 172 (2010), no. 3, 2135–2181.

    • Crossref
    • Export Citation
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    Bruinier J. H. and Ono K., Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms, Adv. Math. 246 (2013), 198–219.

    • Crossref
    • Export Citation
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    Duke W., Modular functions and the uniform distribution of CM points, Math. Ann. 334 (2006), no. 2, 241–252.

    • Crossref
    • Export Citation
  • [14]

    Duke W., Imamoḡlu Ö. and Tóth Á., Cycle integrals of the j-function and mock modular forms, Ann. of Math. (2) 173 (2011), no. 2, 947–981.

    • Crossref
    • Export Citation
  • [15]

    Fischer W., On Dedekind’s function η ( τ ) ${\eta(\tau)}$, Pacific J. Math. 1 (1951), 83–95.

    • Crossref
    • Export Citation
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    Gross B., Kohnen W. and Zagier D., Heegner points and derivatives of L-series. II, Math. Ann. 278 (1987), no. 1–4, 497–562.

    • Crossref
    • Export Citation
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    Hardy G. H. and Ramanujan S., Asymptotic formulae in combinatory analysis, Proc. Lond. Math. Soc. (2) 17 (1918), no. 1, 75–115.

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    Hooley C., An asymptotic formula in the theory of numbers, Proc. Lond. Math. Soc. (3) 7 (1957), 396–413.

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    Iwaniec H., Topics in Classical Automorphic Forms, Grad. Stud. Math. 17, American Mathematical Society, Providence, 1997.

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    Jenkins P., Kloosterman sums and traces of singular moduli, J. Number Theory 117 (2006), no. 2, 301–314.

    • Crossref
    • Export Citation
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    Knapp A. W., Elliptic Curves, Math. Notes 40, Princeton University Press, Princeton, 1992.

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    Kohnen W., Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), no. 2, 237–268.

    • Crossref
    • Export Citation
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    Ono K., The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS Reg. Conf. Ser. Math. 102, American Mathematical Society, Providence, 2004.

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    Ono K., Unearthing the visions of a master: Harmonic Maass forms and number theory, Current Developments in Mathematics, International Press, Somerville (2009), 347–454.

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    Rademacher H., On the partition function p ( n ) ${p(n)}$, Proc. Lond. Math. Soc. (2) 43 (1937), 241–254.

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    Rademacher H., On the expansion of the partition function in a series, Ann. of Math. (2) 44 (1943), 416–422.

    • Crossref
    • Export Citation
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    Stegun I. A., Pocketbook of Mathematical Functions, Harri Deutsch, Thun, 1984.

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    Tóth A., On the evaluation of Salié sums, Proc. Amer. Math. Soc. 133 (2005), no. 3, 643–645.

    • Crossref
    • Export Citation
  • [29]

    Weil A., On some exponential sums, Proc. Natl. Acad. Sci. USA 34 (1948), 204–207.

    • Crossref
    • Export Citation
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    Whiteman A. L., A sum connected with the series for the partition function, Pacific J. Math. 6 (1956), 159–176.

    • Crossref
    • Export Citation
  • [31]

    Zagier D., Traces of singular moduli, Motives, Polylogarithms and Hodge Theory. Part I: Motives and Polylogarithms, Int. Press Lect. Ser. 3, International Press, Somerville (2002), 211–244.

  • [32]

    Zagier D., Ramanujan’s mock theta functions and their applications (after Zwegers and Ono–Bringmann), Séminaire Bourbaki 2007/2008. Exposés 982–996, Astérisque 326, Société Mathématique de France, Paris (2009), Exposé 986, 143–164.

  • [33]

    Zwegers S. P., Mock θ-functions and real analytic modular forms, q-Series with Applications to Combinatorics, Number Theory, and Physics, Contemp. Math. 291, American Mathematical Society, Providence (2001), 269–277.

  • [34]

    Zwegers S. P., Mock theta functions, Ph.D. thesis, Universiteit Utrecht, Utrecht, 2002.

Footnotes

1

Note that here, and in (1.8) and (1.9), we have renormalized the functions 𝐏 and hm from [1]. See (5.1) below.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Ahlgren S. and Andersen N., Weak harmonic Maass forms of weight 5/2 and a mock modular form for the partition function, Res. Number Theory 1 (2015), Article ID 10.

  • [2]

    Andersen N., Periods of the j-function along infinite geodesics and mock modular forms, Bull. Lond. Math. Soc. 47 (2015), 407–415.

    • Crossref
    • Export Citation
  • [3]

    Berndt B. C., Evans R. J. and Williams K. S., Gauss and Jacobi Sums, Canad. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, New York, 1998.

  • [4]

    Borcherds R. E., Automorphic forms on O s + 2 , 2 ( R ) ${{\rm O}_{s+2,2}({R})}$ and infinite products, Invent. Math. 120 (1995), no. 1, 161–213.

    • Crossref
    • Export Citation
  • [5]

    Bringmann K. and Ono K., The f ( q ) ${f(q)}$ mock theta function conjecture and partition ranks, Invent. Math. 165 (2006), no. 2, 243–266.

    • Crossref
    • Export Citation
  • [6]

    Bringmann K. and Ono K., An arithmetic formula for the partition function, Proc. Amer. Math. Soc. 135 (2007), no. 11, 3507–3514.

    • Crossref
    • Export Citation
  • [7]

    Bringmann K. and Ono K., Dyson’s ranks and Maass forms, Ann. of Math. (2) 171 (2010), no. 1, 419–449.

    • Crossref
    • Export Citation
  • [8]

    Bruinier J. H., Harmonic Maass forms and periods, Math. Ann. 357 (2013), no. 4, 1363–1387.

    • Crossref
    • Export Citation
  • [9]

    Bruinier J. H. and Funke J., On two geometric theta lifts, Duke Math. J. 125 (2004), no. 1, 45–90.

    • Crossref
    • Export Citation
  • [10]

    Bruinier J. H., Funke J. and Imamoglu O., Regularized theta liftings and periods of modular functions, J. Reine Angew. Math. 703 (2015), 43–93.

  • [11]

    Bruinier J. H. and Ono K., Heegner divisors, L-functions and harmonic weak Maass forms, Ann. of Math. (2) 172 (2010), no. 3, 2135–2181.

    • Crossref
    • Export Citation
  • [12]

    Bruinier J. H. and Ono K., Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms, Adv. Math. 246 (2013), 198–219.

    • Crossref
    • Export Citation
  • [13]

    Duke W., Modular functions and the uniform distribution of CM points, Math. Ann. 334 (2006), no. 2, 241–252.

    • Crossref
    • Export Citation
  • [14]

    Duke W., Imamoḡlu Ö. and Tóth Á., Cycle integrals of the j-function and mock modular forms, Ann. of Math. (2) 173 (2011), no. 2, 947–981.

    • Crossref
    • Export Citation
  • [15]

    Fischer W., On Dedekind’s function η ( τ ) ${\eta(\tau)}$, Pacific J. Math. 1 (1951), 83–95.

    • Crossref
    • Export Citation
  • [16]

    Gross B., Kohnen W. and Zagier D., Heegner points and derivatives of L-series. II, Math. Ann. 278 (1987), no. 1–4, 497–562.

    • Crossref
    • Export Citation
  • [17]

    Hardy G. H. and Ramanujan S., Asymptotic formulae in combinatory analysis, Proc. Lond. Math. Soc. (2) 17 (1918), no. 1, 75–115.

  • [18]

    Hooley C., An asymptotic formula in the theory of numbers, Proc. Lond. Math. Soc. (3) 7 (1957), 396–413.

  • [19]

    Iwaniec H., Topics in Classical Automorphic Forms, Grad. Stud. Math. 17, American Mathematical Society, Providence, 1997.

  • [20]

    Jenkins P., Kloosterman sums and traces of singular moduli, J. Number Theory 117 (2006), no. 2, 301–314.

    • Crossref
    • Export Citation
  • [21]

    Knapp A. W., Elliptic Curves, Math. Notes 40, Princeton University Press, Princeton, 1992.

  • [22]

    Kohnen W., Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), no. 2, 237–268.

    • Crossref
    • Export Citation
  • [23]

    Ono K., The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS Reg. Conf. Ser. Math. 102, American Mathematical Society, Providence, 2004.

  • [24]

    Ono K., Unearthing the visions of a master: Harmonic Maass forms and number theory, Current Developments in Mathematics, International Press, Somerville (2009), 347–454.

  • [25]

    Rademacher H., On the partition function p ( n ) ${p(n)}$, Proc. Lond. Math. Soc. (2) 43 (1937), 241–254.

  • [26]

    Rademacher H., On the expansion of the partition function in a series, Ann. of Math. (2) 44 (1943), 416–422.

    • Crossref
    • Export Citation
  • [27]

    Stegun I. A., Pocketbook of Mathematical Functions, Harri Deutsch, Thun, 1984.

  • [28]

    Tóth A., On the evaluation of Salié sums, Proc. Amer. Math. Soc. 133 (2005), no. 3, 643–645.

    • Crossref
    • Export Citation
  • [29]

    Weil A., On some exponential sums, Proc. Natl. Acad. Sci. USA 34 (1948), 204–207.

    • Crossref
    • Export Citation
  • [30]

    Whiteman A. L., A sum connected with the series for the partition function, Pacific J. Math. 6 (1956), 159–176.

    • Crossref
    • Export Citation
  • [31]

    Zagier D., Traces of singular moduli, Motives, Polylogarithms and Hodge Theory. Part I: Motives and Polylogarithms, Int. Press Lect. Ser. 3, International Press, Somerville (2002), 211–244.

  • [32]

    Zagier D., Ramanujan’s mock theta functions and their applications (after Zwegers and Ono–Bringmann), Séminaire Bourbaki 2007/2008. Exposés 982–996, Astérisque 326, Société Mathématique de France, Paris (2009), Exposé 986, 143–164.

  • [33]

    Zwegers S. P., Mock θ-functions and real analytic modular forms, q-Series with Applications to Combinatorics, Number Theory, and Physics, Contemp. Math. 291, American Mathematical Society, Providence (2001), 269–277.

  • [34]

    Zwegers S. P., Mock theta functions, Ph.D. thesis, Universiteit Utrecht, Utrecht, 2002.

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