Lifting cusp forms to Maass forms with an application to partitions

Lifting cusp forms to Maass forms with an application to partitions Kathrin Bringmann and Ken Ono* Department of Mathematics, University of Wisconsin, Madison, WI 53706 Communicated by George E. Andrews, Pennsylvania State University, University Park, PA, December 22, 2006 (received for review August 19, 2006)

For 2 < k 僆 21 ⺪, we define lifts of cuspidal Poincare´ series in Sk(⌫0(N)) to weight 2 ⴚ k harmonic weak Maass forms. This construction answers a question of Dyson by providing the general framework ‘‘explaining’’ Ramanujan’s mock theta functions. As an application, we show that the number of partitions of a positive integer n is the ‘‘trace’’ of singular moduli of a Maass form arising from the lift of a weight 4 cusp form corresponding to a Calabi–Yau threefold.

1. Introduction and Statement of Results amanujan proved that the partition function p(n) satisfies the congruences

R

p共5n ⫹ 4兲 ⬅ 0

共mod 5兲,

p共7n ⫹ 5兲 ⬅ 0

共mod 7兲,

p共11n ⫹ 6兲 ⬅ 0

共mod 11兲.

Although these congruences are not difficult to prove, the generic theory (1–3) of partition congruences is quite complicated and depends critically on the interplay between deeper structures in the theory of modular forms. Congruences such as p共48037937n ⫹ 1122838兲 ⬅ 0

共mod 17兲

depend on the Deligne–Serre theory of ᐉ-adic Galois representations and Shimura’s theory of half-integral weight modular forms. Shimura’s theory is built around lifts that map half-integral weight cusp forms to integer weight cusp forms. We describe another lift, one which maps cuspidal Poincare´ series to harmonic weak Maass forms. Using these maps, we obtain an arithmetic formula exhibiting p(n) as the ‘‘trace’’ of singular moduli of a Maass form arising from a Calabi–Yau threefold. 1 1 First we describe these lifts. Suppose that 2 ⬍ k 僆 2 ⺪ and that N is a positive integer (with 4兩N if k 僆 2 ⺪⶿⺪). Let Maass2⫺k(⌫ 0(N), 0) be the space of weight 2 ⫺ k harmonic weak Maass forms on ⌫0(N) (see Section 2), and let Weakk(⌫ 0(N)) be the space of weight k weakly holomorphic modular forms on ⌫ 0(N), where a weakly holomorphic modular form is any meromorphic modular form whose poles (if any) are supported at cusps. The differential operator

␰w :⫽ 2iyw

⭸ ⭸z៮

defines a map

Let Maass*2⫺k(⌫0(N)) be the subspace of those f(z) 僆 Maass2⫺k(⌫0(N), 0) for which ␰2⫺k(f(z)) 僆 Sk(⌫0(N)), the weight k elliptic cusp forms on ⌫0(N). It turns out that ker(␰2⫺k) ⫽ Weak2⫺k(⌫0(N)). For every positive integer m, let H(m, k, N; z) 僆 S k(⌫ 0(N)) be the classical holomorphic Poincare´ series (see Section 3.2). These forms generate S k(⌫ 0(N)). Similarly, for every positive integer m we construct (see Section 3.3) Maass–Poincare´ series F共m, 2 ⫺ k, N; z兲 僆 Maass*2⫺k共⌫ 0共N兲兲. Using these series, we let Lk,N共H共m, k, N; z兲兲 :⫽ F共m, 2 ⫺ k, N; z兲.

[1.1]

This defines the lifting of cuspidal Poincare´ series in S k(⌫ 0(N)) to Maass*2⫺k(⌫ 0(N)) which is dual to the differential operator ␰ 2⫺k. Theorem 1.1. Assume the notation and hypotheses above. The following are true:

(1) We have that

Author contributions: K.B. and K.O. wrote the paper. The authors declare no conflict of interest. *To whom correspondence should be addressed. E-mail: [email protected]. © 2007 by The National Academy of Sciences of the USA

www.pnas.org兾cgi兾doi兾10.1073兾pnas.0611414104

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MATHEMATICS

␰2⫺k : Maass2⫺k共⌫ 0共N兲, 0兲 3 Weakk共⌫ 0共N兲兲.

fH共m,k,N;z兲共z兲 :⫽ Lk,N共H共m, k, N; z兲兲 ⫺ i共k ⫺ 1兲共2 ␲ m兲 k⫺1

冕

i⬁

⫺z៮

H共m, k, N; ⫺␶៮ 兲 d␶ 共⫺i共 ␶ ⫹ z兲兲 2⫺k

is a holomorphic function on the complex upper half-plane ⺘. (2) We have that

␰2⫺k共Lk,N共H共m, k, N; z兲兲 ⫽ 共k ⫺ 1兲䡠共4␲m兲k⫺1䡠H共m, k, N; z兲. Remark: Since Poincare ´ series in S k(⌫ 0(N)) are dependent, we stress that these lifting maps are defined on Poincare´ series, not the space S k(⌫ 0(N)). Remark: Theorem 1.1 (1) is typical of results in the theory of automorphic integrals [for example, see works by Knopp and Niebur (4, 5)], where automorphic forms arise from period integrals of cusp forms. Thanks to explicit formulas for our Poincare´ series, Theorem 1.1 (1) follows from an elementary integral identity (see Proposition 4.1). Now we turn to the motivating problem of providing an arithmetic formula for p(n). To this end, let gC (z) 僆 S4(⌫0(6)) be the eta-product

冘 ⬁

gC共z兲 :⫽ ␩2共z兲␩2共2z兲␩2共3z兲␩2共6z兲 ⫽

a共n兲qn ⫽ q ⫺ 2q2 ⫺ 3q3 ⫹ 4q4 ⫹ · · ·,

[1.2]

n⫽1

where q :⫽ e 2␲iz and where ␩ (z) is Dedekind’s eta-function. This form corresponds [see Mortenson’s thesis (6) and Verrill’s paper (7)] to the Calabi–Yau threefold Y:

1 1 1 1 1 1 1 ⫽ t ⫹ ⫺ 2. x ⫹ ⫹ y ⫹ ⫹ z ⫹ ⫹ xy ⫹ ⫹ yz ⫹ ⫹ xyz ⫹ x y z xy yz xyz t

For odd primes p, this implies that a(p) ⫽ p 3 ⫺ p 2 ⫺ 13 ⫺ N(p), where N(p) :⫽ #Y(⺖ p). If 储䡠储 denotes the usual Petersson norm, then define C(z) 僆 Maass0(⌫0(6), ⫺2) by C共z兲 :⫽ ⫺

1 䡠 96 ␲ 3储g C 储 2

冉

冊

1 1 ⭸ ⫹ 共L4,6共g C 共z兲兲兲. 2 ␲ i ⭸z 2 ␲ Imz

[1.3]

Because the lifting maps are defined on Poincare´ series, C(z) is not well defined as given above. Later in the paper we resolve this issue by describing g C (z) in terms of the first Poincare´ series in S 4(⌫ 0(6)). Remark: In terms of symmetric-square L-functions, 1.3 implies C共z兲 ⫽ ⫺

64 䡠 27 ␲ 䡠L共Sym2共g C 兲, 1兲

冉

冊

1 1 ⭸ ⫹ 共L4,6共g C共z兲兲兲. 2 ␲ i ⭸z 2 ␲ Imz

For positive d ⬅ 0, 3 (mod 4), let Q (p) d be the set of positive definite integral binary quadratic forms (including imprimitive forms) Q(x, y) ⫽ [a, b, c] ⫽ ax 2 ⫹ bxy ⫹ cy 2 of discriminant ⫺d ⫽ b 2 ⫺ 4ac, where 6兩a. The group ⌫0(6) acts on Q (p) d in the usual way. For each Q, let ␶ Q be the unique root of Q(x, 1) ⫽ 0 in ⺘, and let ⌫ ␶Q 債 ⌫ 0(6) be its isotropy subgroup. As in the theory of complex multiplication, we refer to each C( ␶ Q) as a singular modulus. For positive integers n, let Tr共p兲共n兲 :⫽

冘

共p兲 Q僆Q 24n⫺1 兾⌫ 0共 6 兲

where ␹ 12(Q)⫽ ␹ 12([a, b, c]) :⫽

␹ 12共Q兲C共 ␶ Q兲 , #⌫ ␶Q

[1.4]

冉 12x 冊 . The following gives the formula for p(n).

Theorem 1.2. If n is a positive integer, then

p共n兲 ⫽

Tr共 p兲共n兲 . 24n ⫺ 1

Remark: This phenomenon, where coefficients of half-integral weight forms are ‘‘traces’’ of singular moduli was observed by Zagier (8). Recent papers by the authors (9) and Bruinier and Funke (10) give generalizations. In Section 2 we recall facts about weak Maass forms. In Section 3 we construct the Poincare´ series H(m, k, N; z) and F(m, 2 ⫺ k, N; z), and we compute their Fourier expansions. In Sections 4 and 5 we prove Theorems 1.1 and 1.2. In Section 6 we explain how Theorem 1.1 is related to Ramanujan’s mock theta functions.

2. Weak Maass Forms We recall the notion of a weak Maass form of weight k 僆 21 ⺪. If z ⫽ x ⫹ iy 僆 ⺘ with x, y 僆 ⺢, then the weight k hyperbolic Laplacian is given by ⌬k :⫽ ⫺y 2

3726 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0611414104

冉

冊

冉

冊

⭸ ⭸2 ⭸2 ⭸ ⫹i ⫹ ik y . 2⫹ ⭸x ⭸y 2 ⭸x ⭸y

[2.1]

Bringmann and Ono

For odd integers d, define ␧ d by

再

␧d :⫽

if d ⬅ 1 共mod 4兲, if d ⬅ 3 共mod 4兲.

1 i

[2.2]

1

If N is a positive integer (with 4兩N if k 僆 2 ⺪⶿⺪), then a weak Maass form of weight k on ⌫ 0(N) is any smooth function M:⺘ 3 ⺓ satisfying the following: (1) For all A ⫽

冉 冊 ab cd

僆 ⌫0(N) and all z 僆 ⺘, we have M共Az兲 ⫽

再

共cz ⫹ d兲kM共z兲

if k 僆 ⺪,

共 dc 兲 2k␧ ⫺2k 共cz ⫹ d兲 k M共z兲 d

if k 僆 21 ⺪⶿⺪.

Here ( dc ) denotes the extended Legendre symbol, and 公z is the principal branch of the holomorphic square root. (2) There is a complex number ␭ for which ⌬ kM ⫽ ␭ M. (3) The function M(z) has at most linear exponential growth at cusps. Remark: These transformation laws occur in Shimura’s theory of half-integral weight modular forms (11).

Let Maassk(⌫ 0(N), ␭ ) denote the space of weight k weak Maass forms on ⌫ 0(N) with eigenvalue ␭ with respect to ⌬ k. Those forms with ␭ ⫽ 0 are called harmonic, and they are relevant for Theorem 1.1. Here we recall some facts due to Bruinier and Funke (see Proposition 3.2 of ref. 12). Lemma 2.1. The differential operator

␰k : ⫽ 2iyk

⭸ ⭸z៮

maps

␰k : Maassk共⌫ 0共N兲, 0兲 3 Weak2⫺k共⌫ 0共N兲兲, and ker( ␰ k) ⫽ Weakk(⌫ 0(N)). 3. Poincare´ Series To prove Theorem 1.1, we rely on explicit Fourier expansions (one could also argue directly with the defining series). Throughout, we rely on classical special functions whose properties and definitions may be found in ref. 13. We give them since they are useful in applications (for example, see ref. 14). In Section 3.1 we first recall the classical construction of Poincare´ series (see refs. 15–17). In Sections 3.2 and 3.3 we give explicit Fourier expansions in terms of classical special functions (see ref. 13 for more on these special functions).

冉 冊

ab 僆 ⌫0(N), define j(A, z) by cd j共A, z兲 :⫽

1 2

再冑

⺪ and that N is a positive integer (with 4兩N if k 僆

cz ⫹ d

共 dc 兲␧ ⫺1 d 冑 cz ⫹ d

1 2

⺪⶿⺪). For A ⫽

if k 僆 ⺪, if k 僆 21 ⺪⶿⺪,

[3.1]

where ␧ d is defined by 2.2 and where 公z is the principal branch of the holomorphic square root as before. As usual, for A 僆 ⌫ 0(N) and functions f:⺘ 3 ⺓, we let 共 f兩k A兲共z兲 :⫽ j共A, z兲 ⫺2kf共Az兲.

[3.2]

Let m be an integer, and let ␸ m : ⺢ ⫹ 3 ⺓ be a function which satisfies ␸ m(y) ⫽ O(y ␣), as y 3 0, for some ␣ 僆 ⺢. If e( ␣ ) :⫽ e 2␲i␣ as usual, then

␸*m共z兲 :⫽ ␸ m共y兲e共mx兲 is fixed by the group of translations ⌫ ⬁ :⫽

再 冉 冊 ⫾

冎

[3.3]

1n :n 僆 ⺪ . Given this data we let 01

P共m, k, N, ␸m ; z兲 :⫽

冘

共 ␸ *m兩 k A兲共z兲.

[3.4]

A僆⌫ ⬁⶿⌫ 0共 N 兲

The explicit Fourier expansions are given in terms of the Kloosterman sums Bringmann and Ono

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MATHEMATICS

3.1. The Fundamental Poincare´ Series. Suppose that k 僆

Kk共m, n, c兲 :⫽

冦

冘冉 冘

mv៮ ⫹ nv c

e

v共c兲⫻

v共c兲⫻

共 vc 兲 2k␧ v2ke

冉

冊

if k 僆 ⺪,

mv៮ ⫹ nv c

冊

[3.5] 1 2

if k 僆 ⺪⶿⺪.

In the sums above, v runs through the primitive residue classes modulo c, and v៮ denotes the multiplicative inverse of v modulo c. The following lemma gives the fundamental properties of such Poincare´ series (for example, see Proposition 3.1 of ref. 18, where N ⫽ 4). Lemma 3.1. If k ⬎ 2 ⫺ 2 ␣ , then the following are true.

(1) Each Poincare´ series P(m, k, N, ␸ m; z) is a weight k ⌫ 0(N)-invariant function. (2) Near the cusp at ⬁, the function P(m, k, N, ␸ m; z) ⫺ ␸ *m(z) has moderate growth. Near the other cusps, P(m, k, N, ␸ m; z) has moderate growth. (3) If P(m, k, N, ␸ m; z) is twice continuously differentiable, then it has the locally uniformly absolutely convergent Fourier expansion P共m, k, N, ␸m ; z兲 ⫽ ␸*m共z兲 ⫹

冘

a共n, y兲e共nx兲,

n僆⺪

where

冘 ⬁

a共n, y兲 :⫽

c⬎0 c⬅0 共mod N 兲

c ⫺kK k共m, n, c兲

冕

⬁

z ⫺k␸ m

⫺⬁

3.2. The Holomorphic Poincare´ Series H(m, k, N; z). Suppose that 2 ⬍ k 僆

For positive integers m, let

1 2

冉 冊冉

冊

mx y e ⫺ 2 2 ⫺ nx dx. c 2兩z兩 2 c 兩z兩

⺪, and that N is a positive integer (with 4兩N if k 僆 21 ⺪⶿⺪).

H共m, k, N; z兲 :⫽ P共m, k, N, 1; z兲.

[3.6]

Lemma 3.1, combined with facts about Petersson norms, implies the following well known proposition (for example, see Chapter 3 of ref. 17). Proposition 3.2. The set of Poincare ´ series {H(m, k, N; z) : m ⱖ 1} spans S k(⌫ 0(N)). Moreover, if ␦ (m, n) is the Kronecker delta-function ⬁ b m(n)q n, where and J k⫺1 is the usual J-Bessel function, then H(m, k, N; z) ⫽ 兺 n⫽1

冉冊 冉

n bm共n兲 ⫽ m

k⫺1 2

␦共m, n兲 ⫹ 2␲i⫺k

冘

c⬎0 c⬅0 共mod N 兲

冉

K k共m, n, c兲 4 ␲ 冑nm 䡠 J k⫺1 c c

冊冊

.

3.3. Maass–Poincare´ Series F(m, 2 ⴚ k, N; z). Although the series F(m, 2 ⫺ k, N; z) are less well known, they have appeared in earlier 1 1 works (9, 15, 16, 18, 19). To define them, again suppose that 2 ⬍ k 僆 2 ⺪, and that N is a positive integer (with 4兩N if k 僆 2 ⺪⶿⺪). To employ Lemma 3.1, we first select an appropriate function ␸. Let M ␯,␮(z) be the usual M-Whittaker function. For complex s, let

Ms共y兲 :⫽ 兩y兩 ⫺k/2 M 共k/2兲 sgn共y兲, s⫺1/2共兩y兩兲, and for positive m let ␸ ⫺m(z) :⫽ M1⫺2k (⫺4 ␲ my). We now let F共m, 2 ⫺ k, N; z兲 :⫽ P共⫺m, 2 ⫺ k, N, ␸ ⫺m ; z兲.

[3.7]

Lemma 3.1 leads to the following proposition (for a proof in the N ⫽ 4 case see ref. 9). Proposition 3.3. Each F(m, 2 ⫺ k, N; z) is in Maass2⫺k(⌫ 0(N), 0). Moreover, if I k⫺1 is the usual I-Bessel function, and ⌫(a, x) is the incomplete ⌫-function, then

F共m, 2 ⫺ k, N; z兲 ⫽ 共1 ⫺ k兲共⌫共k ⫺ 1, 4␲my兲 ⫺ ⌫共k ⫺ 1兲兲q⫺m ⫹

冘

c共n, y兲qn.

n僆⺪

(1) If n ⬍ 0, then

冏冏

n c共n, y兲 ⫽ 2␲i 共1 ⫺ k兲⌫共k ⫺ 1, 4␲兩n兩y兲 m k

1⫺k 2

⫻

冘

c⬎0 c⬅0 共mod N 兲

冉

冊

K 2⫺k共⫺m, n, c兲 4 ␲ 冑兩mn兩 䡠 J k⫺1 . c c

(2) If n ⬎ 0, then 3728 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0611414104

Bringmann and Ono

冉冊

n c共n, y兲 ⫽ ⫺2 ␲ i k⌫共k兲 m

冘

1⫺k 2

c⬎0 c⬅0 共 mod N 兲

冉

冊

K 2⫺k共⫺m, n, c兲 4 ␲ 冑兩mn兩 䡠 I k⫺1 . c c

(3) If n ⫽ 0, then c共0, y兲 ⫽ ⫺2 k␲ ki km k⫺1

冘

c⬎0 c⬅0 共 mod N 兲

K 2⫺k共⫺m, 0, c兲 . ck

4. Proof of Theorem 1.1 Throughout, suppose that 2 ⬍ k 僆 21 ⺪, and that N is a positive integer (with 4兩N if k 僆 21 ⺪⶿⺪). We begin with an elementary integral identity. Proposition 4.1. If n is a positive integer, then

冕

i⬁

⫺z៮

e2␲in␶ d ␶ ⫽ i共2 ␲ n兲 1⫺k䡠⌫共k ⫺ 1, 4 ␲ ny兲q ⫺n. 共⫺i共 ␶ ⫹ z兲兲 2⫺k

Proof: This identity follows by the direct calculation

冕

i⬁

⫺z៮

e2␲in␶ d␶ ⫽ 共⫺i共 ␶ ⫹ z兲兲 2⫺k

冕

e 2␲in共␶⫺z兲 d ␶ ⫽ i共2 ␲ n兲 1⫺k䡠⌫共k ⫺ 1, 4 ␲ ny兲q ⫺n. 共⫺i ␶ 兲 2⫺k

i⬁

2iy

e

Proof of Theorem 1.1: We first prove Theorem 1.1 (2). For convenience, let

冘 ⬁

F共m, 2 ⫺ k, N; z兲 ⫽ 共k ⫺ 1兲⌫共k ⫺ 1兲q⫺m ⫹

冘 ⬁

a共n兲qn ⫹

n⫽0

b共n兲⌫共k ⫺ 1, 4␲ny兲q⫺n.

n⫽1

The operator ␰ 2⫺k is antilinear, and it has the property that ␰ 2⫺k(f) ⫽ 0 for holomorphic functions f. We also have the identity

␰2⫺k共⌫共k ⫺ 1, 4␲ny兲兲 ⫽ ⫺共4 ␲ n兲 k⫺1e ⫺4␲ny. These facts imply that

冘 ⬁

␰2⫺k共F共m, 2 ⫺ k, N; z兲兲 ⫽ ⫺共4 ␲ 兲 k⫺1

n k⫺1b共n兲q n.

n⫽1

By the definition of Lk,N, Theorem 1.1 (2) follows from the identity K2⫺k共⫺m, ⫺n, c兲 ⫽ K k共m, n, c兲.

冕

i⬁

⫺z៮

MATHEMATICS

To prove Theorem 1.1 (1), it suffices to compare the Fourier expansion of H共m, k, N; ⫺␶៮ 兲 d␶ 共⫺i共 ␶ ⫹ z兲兲 2⫺k

⬁ with the nonholomorphic part of F(m, 2 ⫺ k, N; z). If H(m, k, N; z) ⫽ 兺 n⫽1 b(n)q n, then by Proposition 4.1 we find that

冕

i⬁

⫺z៮

H共m, k, N; ⫺␶៮ 兲 d ␶ ⫽ i共2 ␲ 兲 1⫺k 共⫺i共 ␶ ⫹ z兲兲 2⫺k

冘 ⬁

n⫽1

b共n兲 ⌫共k ⫺ 1, 4 ␲ ny兲q ⫺n n k⫺1

The claim now follows from the formulas in Propositions 3.2 and 3.3. 5. Proof of Theorem 1.2 In earlier work (20), we proved that if n is a positive integer, then p共n兲 ⫽

1 24n ⫺ 1

冘

共p兲 Q僆Q 24n⫺1 兾⌫0共6兲

␹12共Q兲P共␶Q兲 , #⌫␶Q

where Bringmann and Ono

PNAS 兩 March 6, 2007 兩 vol. 104 兩 no. 10 兩 3729

冘

P共z兲 :⫽ 4 ␲

1

Im共Az兲 2 I 3共2 ␲ Im共Az兲兲e共⫺Re共Az兲兲.

[5.1]

2

A僆⌫ ⬁⶿⌫ 0共 6 兲

This was obtained by reformulating Rademacher’s exact formula using Salie´ sums. Therefore, it suffices to show that P共z兲 ⫽ ⫺

冉

冊

1 ⭸ 1 1 ⫹ 䡠 共L4,6共g C共z兲兲兲. 96␲3储gC 储2 2␲i ⭸z 2␲Imz

Since dim⺓(S 4(⌫ 0(6))) ⫽ 1, it follows (for example, see Chapter 3 of ref. 17) that gC共z兲 ⫽ 32␲3储gC 储2䡠H共1, 4, 6; z兲. Therefore, it suffices to show that P共z兲 ⫽ ⫺

冉

冊

1 1 ⭸ 1 ⫹ F共1, ⫺2, 6; z兲. 3 2␲i ⭸z 2␲Imz

For this we need the Fourier expansion of P(z) that was computed by Niebur (5). Correcting some typographical errors in his paper, we find that 1

P共z兲 ⫽ 4␲I 3共2␲y兲y2e⫺2␲ix ⫹ 2

冘

b共n, y兲qn,

n僆⺪

where

冦

8␲3 3y

冘

c⬎0 c⬅0 共mod 6 兲

K ⫺2共⫺1, 0, c兲 c4

冘 冘

1

2 ␲ ny 3 K 共2 ␲ ny兲 b共n, y兲 :⫽ 8 ␲ y 2 e 2

c⬎0 c⬅0 共 mod 6 兲

1

8 ␲ y 2 e 2␲nyK 3共2 ␲ 兩n兩y兲 2

if n ⫽ 0,

冉

K ⫺2共⫺1, n, c兲 4 ␲ 冑n 䡠 I3 c c

c⬎0 c⬅0 共 mod 6 兲

冉

冊

K ⫺2共⫺1, n, c兲 4 ␲ 冑 兩n兩 䡠 J3 c c

if n ⬎ 0,

冊

[5.2]

if n ⬍ 0.

Thanks to Proposition 3.3 and 5.2, the theorem is obtained (after some computation) by using the identities: ⌫共3, y兲 ⫽ e⫺y共y2 ⫹ 2y ⫹ 2兲, ⭸ ⌫共3, 4␲兩n兩y兲 ⫽ 32␲3i兩n兩3y2e⫺4␲兩n兩y, ⭸z I3共z兲 ⫽ 2

冉 冊冉 z 2␲

1/2

K3共z兲 ⫽ 2

冊

1 1 ⫺z 共e ⫺ ez兲 ⫹ 共ez ⫹ e⫺z兲 , z2 z

冉冊 冉 冊 ␲ 2z

1/2

e⫺z 1 ⫹

1 . z

6. Relationship with Ramanujan’s Mock Theta Functions Ramanujan’s mock theta functions are a collection of 22 ‘‘strange’’ q-series such as

冘 ⬁

f共q兲 :⫽ 1 ⫹

n⫽1

2

qn . 2 共1 ⫹ q兲 共1 ⫹ q 2兲 2· · ·共1 ⫹ q n兲 2

[6.1]

They do not arise as the minor modification of a modular form. Nevertheless, a wealth of evidence, such as identities involving mock theta functions and modular forms, suggested a strong connection between these objects (for example, see ref. 21 and the references therein). Determining their place in the theory of automorphic forms was a puzzle for many decades, a quandary nicely described by Freeman Dyson in 1987 (22): The mock theta-functions give us tantalizing hints of a grand synthesis still to be discovered. Somehow it should be possible to build them into a coherent group-theoretical structure, analogous to the structure of modular forms which Hecke built around the old theta-functions of Jacobi. This remains a challenge for the future. In his 2002 Ph.D. thesis (23, 24) Zwegers made an important breakthrough. Loosely speaking, he ‘‘completed’’ the mock theta functions to obtain weight 1兾2 weak Maass forms. In the case of f(q), it turns out that (see Corollary 2.3 of ref. 14) Mf共z兲 :⫽ q ⫺1f共q 24兲 ⫺ 2i 冑3䡠N f 共z兲 3730 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0611414104

Bringmann and Ono

is a weight 1兾2 weak Maass form on ⌫0(144) with Nebentypus ␹12(䡠) :⫽ Nf 共z兲 :⫽ ⫺

冕

i⬁

⫺24z៮

冘

⬁ n⫽⫺⬁ 共n

⫹

冉 12䡠 冊 , where 1 6

1 2 兲e 3 ␲ i 共n⫹6兲 ␶

冑⫺i共 ␶ ⫹ 24z兲

d␶.

In the context of Theorem 1.1 (1), N f (z) plays the role of the period integral, and the mock theta function f(q) plays the role of the holomorphic function f H(m,k,N;z)(z). Zwegers’ work, combined with recent work by the authors (refs. 14 and 25 and unpublished work), establishes that the mock theta functions (resp. certain q-series arising from the Rogers–Fine basic hypergeometric series) are the holomorphic parts of weight 1兾2 (resp. 3兾2) harmonic weak Maass forms. In these cases, the nonholomorphic parts are indeed period integrals of weight 3兾2 (resp. 1兾2) theta functions. Theorem 1.1 illustrates this phenomenon for all other possible half-integral weights. Despite this beautiful picture, many questions remain. For example, we ask the following. Question. Can any of the f H(m,k,N;z)(z) be represented as a combinatorial q-series such as those appearing in the theory of basic hypergeometric series? We thank Scott Ahlgren, Sharon Garthwaite, Karl Mahlburg, and the referees for their remarks on earlier versions of this paper. We also thank the National Science Foundation for its generous support. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

MATHEMATICS

24. 25.

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PNAS 兩 March 6, 2007 兩 vol. 104 兩 no. 10 兩 3731