Mock theta functions, weak Maass forms, and applications

MOCK THETA FUNCTIONS, WEAK MAASS FORMS, AND APPLICATIONS KATHRIN BRINGMANN

1. Introduction The goal of this article is to provide an overview on mock theta functions and their connection to weak Maass forms. The theory of modular forms has important applications to many areas of mathematics, e.g. quadratic forms, elliptic curves, partitions as well as other areas throughout mathematics. Let me explain this with the example of partitions. If p(n) denotes the number of partitions of an integer n, then by Euler, we have ∞ X 1 , P (q) := p(n) q 24n−1 = η(24z) n=0

where η(z) is Dedekind’s η-functions, a weight 12 cusp form (q = e2πiz throughout). The theory of modular forms can be employed to show many important properties of p(n). For example Rademacher used the circle method to prove that if n is a positive integer, then √ ∞ X π 24n − 1 2π Ak (n) . · I (1.1) p(n) = 3 2 k 6k (24n − 1)3/4 k=1

Here Is (x) is the usual I-Bessel function of order s. Furthermore, if k ≥ 1 and n are integers and e(x) := e2πix , then define X 2πihn Ak (n) := ωh,k e− k , h

(mod k)∗

where h runs through all primitive elements modulo k, and where ωh,k := exp (πis(h, k)) . Here X

s(h, k) := µ

µ hµ k

(mod k)

k

with ((x)) :=

x − bxc − 0

1 2

if x ∈ R \ Z, if x ∈ Z.

Moreover p(n) satisfies some nice congruence properties. The most famous ones are the Ramanujan congruences: (1.2) (1.3) (1.4)

p(5n + 4) ≡ 0 (mod 5), p(7n + 5) ≡ 0 (mod 7), p(11n + 6) ≡ 0 (mod 11).

Date: October 31, 2007. 1

2

KATHRIN BRINGMANN

In a celebrated paper Ono [32] treated these kinds of congruences systematically. Combining Shimura’s theory of modular forms of half-integral weight with results of Serre on modular forms modulo ` he showed that for any prime ` ≥ 5 there exist infinitely many non-nested arithmetic progressions of the form An + B such that p(An + B) ≡ 0

(mod `).

Moreover partitions are related to Eulerian series. For example we have ∞ X

(1.5)

p(n)q n = 1 +

n=0

∞ X n=1

2

qn . (1 − q)2 (1 − q 2 )2 · · · (1 − q n )2

Other examples that relate Eulerian series to modular forms are the Rogers-Ramanujan identities 1+ (1.6) 1+

∞ X n=1 ∞ X n=1

2

qn 1 , = Q∞ 5n−1 )(1 − q 5n−4 ) (1 − q (1 − q)(1 − q 2 ) · · · (1 − q n ) n=1 2

1 q n +n . = Q∞ 5n−2 )(1 − q 5n−3 ) (1 − q (1 − q)(1 − q 2 ) · · · (1 − q n ) n=1

Mock theta functions, which can also be defined as Eulerian series, stand out of this context. For example the mock theta function f (q), defined by Ramanujan [34] in his last letter to Hardy, is given by (1.7)

f (q) := 1 +

∞ X n=1

2

qn . (1 + q)2 (1 + q 2 )2 · · · (1 + q n )2

Even if (1.5) and (1.7) have a similar shape P (q) is modular whereas f (q) is not. The mock theta functions were mysterious objects for a long time, for example there was even a discussion how to rigorously define them. Despite those problems they have applications in vast areas of mathematics (e.g. [2, 3, 4, 18, 20, 28, 39] just to mention a few). In this context one has to understand Dyson’s quote given 1987 at the Ramanujan’s Centary Conference: “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.” Zwegers [39, 40] made a first step towards solving this challenge. He observed that Ramanujan’s mock theta functions can be interpreted as part of a real analytic vector valued modular form. The author and Ono build on those results to relate functions like f (q) to weak Maass forms (see Section 2 for the definition of a weak Maass form). It turns out that f (q) is the “holomorphic part” of a weak Maass form, the “non-holomorphic part” is a Mordell type integral involving weight 23 theta functions. This is the special case of an infinite family of weak Maass forms that arise from Dyson’s rank generating functions (see Section 3). This new theory has a wide range of applications. For example we obtain exact formulas for Ramanujan’s mock theta function f (q) (see Section 4), congruences for Dyson’s ranks (see Section 5), asymptotics and inequalities for ranks (see Section 6) and identities for rank differences that involve modular forms (see Section 8). In Section 7 we show furthermore a correspondence between weight 23 weak Maass forms and weight 12 theta functions. This paper is an extended version of a talk given at the conference “Modular forms” held in October 2006 in Schiermonnikoog. The author thanks the organizers B. Edixhoven, G. van der Geer, and B. Moonen for a stimulating athmosphere.

MOCK THETA FUNCTIONS, WEAK MAASS FORMS, AND APPLICATIONS

3

Acknowledgements The author thanks the referee for many helpful comments. 2. General facts on weak Maass forms Here we recall basic facts on weak Maass forms, first studied by Bruinier and Funke [17]. For k ∈ 21 Z \ Z and z = x + iy with x, y ∈ R, the weight k hyperbolic Laplacian is given by 2 ∂ ∂ ∂2 ∂ 2 ∆k := −y + iky . + +i ∂x2 ∂y 2 ∂x ∂y If v is odd, then define v by ( 1 if v ≡ 1 (mod 4), v := i if v ≡ 3 (mod 4). A (harmonic) weak Maass form of weight k on a subgroup Γ ⊂ Γ0 (4) is any smooth function f : H → C satisfying the following: (1) For all A = ac db ∈ Γ and all z ∈ H, we have 2k c d−2k (cz + d)k f (z). f (Az) = d (2) We have that ∆k f = 0. (3) The function f (z) has at most linear exponential growth at all the cusps of Γ. In a similar manner one defines weak Maass forms on Γ0 (4N ) (N a positive integer) with Nebentypus χ (a Dirichlet character) by requiring 2k c k −2k f (Az) = χ(d) d (cz + d) f (z) d instead of (1). Harmonic weak Maass forms have Fourier expansions of the form ∞ ∞ X X f (z) = γy (n)q −n + a(n)q n , n=n0

n=n1

P −n with n0 , n1 ∈ Z. The γy (n) are functions in y, the imaginary part of z. We refer to ∞ n=n0 γy (n)q P∞ n as the non-holomorphic part of f (z), and we refer to n=n1 a(n)q as its holomorphic part. Moreover we need the anti-linear differential operator ξk defined by ξk (g)(z) := 2iy k ∂∂z¯ g(z). If g is a harmonic weak Maass form of weight k for the group Γ, then ξk (g) is a weakly holomorphic modular form (i.e, a modular form with poles at most at the cusps of Γ) of weight 2 − k on Γ. Furthermore, ξk has the property that its kernel consists of those weight k weak Maass forms which are weakly holomorphic modular forms. 3. Dyson’s ranks and weak Maass forms In order to explain the Ramanujan congruences Dyson introduced the so-called rank of a partition [23]. The rank of a partition is defined to be its largest part minus the number of its parts. In his famous paper Dyson conjectured that ranks could be used to “explain” the congruences (1.2) and (1.3) with modulus 5 and 7. More precisely, he conjectured that the partitions of 5n + 4 (resp. 7n + 5) form 5 (resp. 7) groups of equal size when sorted by their ranks modulo 5 (resp. 7). In 1954, Atkin and Swinnerton-Dyer proved Dyson’s rank conjecture [7].

4

KATHRIN BRINGMANN

To study ranks, it is natural to investigate a generating function. If N (m, n) denotes the number of partitions of n with rank m, then it is well known that R(w; q) := 1 +

∞ ∞ X X

N (m, n)wm q n = 1 +

n=1 m=−∞

∞ X n=1

2

qn , (wq; q)n (w−1 q; q)n

where (a; q)n = (a)n := (1 − a)(1 − aq) · · · (1 − aq n−1 ). If we let w = 1 we recover P (q) in its Eulerian form (1.5), i.e., (up to a q-power) a weight − 21 modular form. Moreover R(−1; q) is the generating function for the number of partitions with even rank minus the number of partitions with odd rank and equals R(−1; q) = 1 +

∞ X n=1

2

qn = f (q) (1 + q)2 (1 + q 2 )2 · · · (1 + q n )2

with f (q) as in (1.7). The author and Ono [12] showed that the functions R(w; q) for w 6= 1 a root of unity are the holomorphic parts of weak Maass forms. To make this statement more precise suppose 2πi 2c that 0 < a < c are integers, and let ζc := e c . If fc := gcd(c,6) , then define for τ ∈ H the weight 32 cuspidal theta function Θ ac ; τ by a X aπ(6m + 1) τ m ; τ := (−1) sin · θ 6m + 1, 6fc ; , Θ c c 24 m

(mod fc )

where X

θ(α, β; τ ) := n≡α

2

ne2πiτ n .

(mod β)

Moreover let `c := lcm(2c2 , 24) and define a a a `c ;q = D ; z := −S1 ; z + q − 24 R(ζca ; q `c ), (3.1) D c c c where the period integral S1 ac ; z is given by 1 Z i∞ a `c 2 Θ ac ; `c τ −i sin πa c √ p dτ. (3.2) S1 ; z := c 3 −i(τ + z) −¯ z Theorem 3.1. (1) < a < c, then D ac ; z is a weak Maass form of weight 12 on D If 0E Γc := ( 10 11 ) , `12c 01 . (2) If c is odd, then D ac ; z is a weak Maass form of weight 12 on Γ1 6fc2 `c . If ac = 12 , it turns out, usingresults of Zwegers [39], that D 21 ; z is a weak Maass form on Γ0 (144) with Nebentypus χ12 (·) := 12· . A similar phenomenon as in Theorem 3.1 occurs in the case of overpartitions. Recall that an overpartition is a partition, where the first occurance of a summand may be overlined. For a nonnegative integer n we denote by p(n) the number of overpartions of n. We have the generating function [21] X η(2z) (3.3) P (q) := p(n) q n = , η(z)2 n≥0


5

which is a weight − 21 modular form. Moreover the generating function for N (m, n), the number of overpartitions of n with rank m, is given by ∞ X

1 ∞ X (−1)n q 2 n(n+1) . O(w; q) := 1 + N (m, n)w q = (wq; q)n wq ; q n n=1 n=0

m n

In particular the case w = 1 gives by (3.3) a modular form. Moreover it turns out [10] that for w 6∈ {−1, 1} a root of unity O(w; q) is the holomorphic part of a weight 21 weak Maass form. In contrast to the case of usual partitions one obtains in the case w = −1 the holomorphic part of a weight 23 weak Maass form. Sketch of proof of Theorem 3.1. We first determine the transformation law of R (ζca ; q). For this we define certain related functions. For q := e2πiz let a ∞ a X 3 (−1)n q n+ c 1 M ; q := · q 2 n(n+1) , a n+ c (q; q)∞ n=−∞ 1 − q c a ∞ X 3 1 (−1)n+1 q n+ c ; q := · q 2 n(n+1) , M1 a n+ c (q; q)∞ n=−∞ 1 + q c n a (1 − ζca ) X (−1)n q 2 (3n+1) N ; q := , c (q; q)∞ 1 − ζca q n

a

n∈Z

N1

a

; q :=

(n+1) X (−1)n q 3n 2

. 1 1 − ζca q n+ 2 As an abuse of notation we also write M ac ; z instead of M other functions. One can show [27] that a R (ζca ; q) = N ;q . c Moreover for 0 ≤ b < c, define M (a, b, c; z) by

In addition, if

(3.4)

b c

c

n∈Z

a c;q

and in the same way we treat the

a ∞ X 3 1 (−1)n q n+ c n(n+1) . M (a, b, c; q) := a · q2 n+ (q; q)∞ n=−∞ 1 − ζcb q c 6∈ 0, 12 , 61 , 65 , then define the integer k(b, c) by  if 0 < cb < 16 ,  0  1 if 16 < cb < 12 , k(b, c) :=  2 if 12 < cb < 56 ,    3 if 56 < cb < 1,

and let b n ∞ a q − 2c X iζ2c (−1)n q 2 (3n+1)−k(b,c)n N (a, b, c; q) := − . b 2 (q; q)∞ n=−∞ 1 − ζca q n− c

Remark. The above defined functions can also be rewritten in terms of the functions X (−1)n q n2 (3n+2k+1) 1 Tk (x; q) := (q; q)∞ 1 − xq n n∈Z

6

KATHRIN BRINGMANN

with k ∈ Z. These functions can all be expressed in terms of T0 : m

Tm (x; q) − xTm+1 (x; q) = (−1)m χ3 (1 − m)q − 6 (m+1) with

 if m ≡ 1 (mod 3),  1 −1 if m ≡ −1 (mod 3), χ3 (m) :=  0 if m ≡ 0 (mod 3).

Also Tm x−1 ; q = −xT−m (x; q). Moreover we need the following Mordell type integrals. Z ∞ 3a 3a a − 23 αx2 cosh c − 2 αx + cosh c − 1 αx J e ; α := dx, c cosh(3αx/2) 0 Z ∞ 3a 3a a − 23 αx2 sinh c − 2 αx − sinh c − 1 αx e ; α := dx, J1 c sinh(3αx/2) 0 Z ∞ b −αx + ζ 2b e−2αx c − 32 αx2 +3αx ac ζc e dx. J(a, b, c; α) := e cosh 3αx/2 − 3πi cb −∞ Modifying an argument of Watson [38] one can show using contour integration. Lemma 3.2. Suppose that 0 < a < c are coprime integers, and that α and β have the property that αβ = π 2 . If q := e−α and q1 := e−β , then we have r a r 3α aπ − 1 a a 3a a 1 π 1− − q 2c ( c ) 24 · M ;q = csc q1 6 · N ; q14 − ·J ;α , c 2α c c 2π c r r a a a 3a a 1 2π 43 3α q 2c (1− c )− 24 · M1 ;q = − q1 · N1 ; q12 − · J1 ;α . c α c 2π c If moreover cb 6∈ 12 , 61 , 56 , then 3a

a

1

q 2c (1− c )− 24 · M (a, b, c; q) = r r 2 8π −2πi a k(b,c)+3πi b ( 2a −1) −b 4bc k(b,c)− 6bc2 − 16 3α −5b 4 c c c · N (a, b, c; q1 ) − · J(a, b, c; α). e ζ c q1 ζ α 8π 2c Two remarks. 1) The case b = 0 is contained in [27]. 2) It is nowadays more common to write modular transformation laws in terms of τ and − τ1 than in q and q1 . The above transformation laws allow us to construct an infinite family of a vector valued weight 12 weak Maass forms (see [12] for the definition of vector valued weak Maass form). For simplicity we assume for the remainder of this section that c is odd. Using the functions a a aπ a 1 N ;q = N ;z := csc · q − 24 · N ;q , c c ac ac 3a a 1 a ;q = M ;z := 2q 2c ·(1− c )− 24 · M ;q , M c c c we define the vector valued (holomorphic) function F ac ; z by a a a T πa a πa a T F ; z := F1 ; z , F2 ;z = sin N ; `c z , sin M ; `c z . c c c c c c c


7

Similarly, define the vector valued (non-holomorphic) function G ac ; z by a T a a ; z = G1 ; z , G2 ;z G c c c !T πa p a 2√3 sin πa √ a 2πi c := 2 3 sin −i`c z · J ; −2πi`c z , ·J ; . c c i`c z c `c z Following a method of Zwegers[39], which uses the Mittag-Leffler partial fraction decomposition, we can realize the function G ac ; z as a vector valued theta integral. Lemma 3.3. For z ∈ H, we have 1

i`c2 sin √ G ;z = c 3 a

πa c

Z

i∞

0

3 (−i`c τ )− 2 Θ ac ; − `c1τ , Θ p −i(τ + z)

a c ; `c τ

T dτ.

We next determine the necessary modular transformation properties of the vector a a a S ; z = S1 ; z , S2 ;z c c c T 1 Z i∞ − 32 a a 1 πa Θ Θ ; ` τ , (−i` τ ) ; − c −i sin c `c 2 c c c `c τ √ p dτ. := 3 −i(τ + z) −¯ z Lemma 3.4. We have a a S ;z + 1 = S ;z , c c a a a 1 1 0 1 √ ·S ;− 2 = ·S ;z + G ;z . 1 0 c `c z c c −i`c z a Combining the above and using the transformation law for Θ ; τ [35], one can now conclude c that D ac ; z satisfies the correct transformation law under the stated group. To see that D ac ; z is annihilated by ∆ 1 , we write 2 3

∆ 1 = −4y 2

(3.5)

2

∂ √ ∂ y . ∂z ∂ z¯

`c

Since q − 24 R(ζba ; q `c ) is a holomorphic function in z, it is thus clearly annihilated by ∆ 1 . Moreover 2

πa

1 2

a sin c `c ∂ a √ S1 ;z =− ·Θ ; −`c z¯ . ∂ z¯ c c 6y √ ∂ Hence, we find that y ∂ z¯ D ac ; z is anti-holomorphic, and therefore by (3.5) annihilated by ∆ 1 . 2 Using that Θ ac ; τ is a weight 32 cusp form it is not hard to conclude that D ac ; z has at most linear exponential growth at the cusps. 4. The Andrews-Dragonette-Conjecture One can use the theory of weak Maass form to obtain exact formulas for the coefficients of the mock theta function f (q) which we denote by α(n) [11]. Recall that f (q) = R(−1; q) = 1 +

∞ X n=1

n

(Ne (n) − No (n)) q = 1 +

∞ X n=1

2

qn , (1 + q)2 (1 + q 2 )2 · · · (1 + q n )2

8

KATHRIN BRINGMANN

where Ne (n) (resp. No (n)) denotes the number of partions of n with even (resp. odd) rank. It is a classical problem to find exact formulas for Ne (n) and No (n). Since by (1.1) we have an exact formula for the partition function p(n) this is equivalent to the problem of determining exact formulas for α(n). Ramanujan’s last letter to Hardy includes the claim that  q q  1 n n 1 1 exp π 6 − 144  exp 2 π 6 − 144  q q α(n) = (−1)n−1 +O (4.1) . 1 1 2 n − 24 n − 24 Typical of his writings, Ramanujan did not give a proof. Dragonette finally showed (4.1) in her 1951 Ph.D. thesis [22] written under the direction of Rademacher. In his 1964 Ph.D. thesis, also written under Rademacher, Andrews [2] improved upon Dragonette’s work, and showed that √ k+1 k(1+(−1)k ) √ [ n ] (−1)b 2 c A n − X 2k 4 π 24n − 1 − 41 + O(n ). α(n) = π(24n − 1) · I1 2 k 12k k=1

Moreover they made the Conjecture. (Andrews-Dragonette) If n is a positive integer, then (4.2)

− 41

α(n) = π(24n − 1)

∞ (−1)b X

k+1 c 2

A2k n −

k(1+(−1)k ) 4

k

k=1

√ π 24n − 1 · I1 . 2 12k

The author and Ono [11] used the theory of weak Maass forms to prove this conjecture. Theorem 4.1. The Andrews-Dragonette Conjecture is true. Sketch of Proof. From Section 3 we know that D 21 ; z is a weight 12 weak Maass form on Γ0 (144) with Nebentypus character χ12 . We will construct a Maass-Poincaré series which we will show equals 1 D 2 ; z . The Andrews-Dragonette Conjecture can be concluded by computing the coefficients of the Poincaré series. For s ∈ C, k ∈ 21 + Z, and y ∈ R \ {0}, let k

Ms (y) := |y|− 2 M k sgn(y), s− 1 (|y|), 2

Ws (y) := |y|

− 14

2

W 1 sgn(y), s− 1 (|y|), 4

2

where Mν,µ (z) and Wν,µ (z) are the standard Whittaker function. Furthermore, let πy x e − . ϕs,k (z) := Ms − 6 24 It is straightforward to confirm that ϕs,k (z) is an eigenfunction of ∆k . Moreover for matrices Γ0 (2), with c ≥ 0, let ( b e − if c = 0, a b 24 χ := 1 −1 (c+ad+1) a+d a 3dc c d i−1/2 (−1) 2 e − 24c − 4 + 8 · ω−d,c if c > 0. Define the Poincaré series Pk (s; z) by (4.3)

2 Pk (s; z) := √ π

X M ∈Γ∞ \Γ0 (2)

χ(M )−1 (cz + d)−k ϕs,k (M z),

a b c d

∈


9

were Γ∞ := {± ( 10 n1 ) : n ∈ Z} . The series P 1 1 − k2 ; z is absolute convergent for k < 12 and anni2 hilated by ∆ k . The function P 1 34 ; z can be analytically continued by its Fourier expansion, which 2 2 bases on a modification of an argument of Hooley involving the interplay between solutions of quadratic congruences and the representation of integers by quadratic forms. This calculation is lengthy, and is carried out in detail in [11]. We compute the Fourier expansion of P 1 34 ; z as 2

P1 2

3 ;z 4

0 ∞ X X 1 1 1 1 1 πy · q − 24 + = 1 − π− 2 · Γ , γy (n)q n− 24 + β(n)q n− 24 , 2 6 n=−∞ n=1

where for positive integers n we have − 14

β(n) = π(24n − 1)

∞ (−1)b X

k+1 c 2

A2k n −

k(1+(−1)k ) 4

k

k=1

√ π 24n − 1 , · I1 2 12k

and for non-positive integers n we have 1 1 1 π|24n − 1| · y − γy (n) = π 2 |24n − 1| 4 · Γ , 2 6 k(1+(−1)k ) c b k+1 ∞ (−1) 2 A2k n − 4 X · J1 × 2 k

π

k=1

! p |24n − 1| . 12k

Here the incomplete gamma function Γ(a; x) is defined by Z ∞ (4.4) Γ(a; x) := e−t ta−1 dt. x

To finish the proof, we have to show that α(n) = β(n). For this we let 3 P (z) := P 1 ; 24z = Pnh (z) + Ph (z), 2 4 1 M (z) := D ; z = Mnh (z) + Mh (z) 2 canonically decomposed into a non-holomorphic and a holomorphic part. In particular ∞ X −1 Ph (z) = q + β(n) q 24n−1 , n=1

Mh (z) = q

−1

24

f (q ) = q

−1

+

∞ X

α(n) q 24n−1 .

n=1

The function P (z) and M (z) are weak Maass forms of weight 12 for Γ0 (144) with Nebentypus χ12 . We first prove that Pnh (z) = Mnh (z). For this compute that ξ 1 (P (z)) and ξ 1 (M (z)) are holomorphic 2

2

modular forms of weight 23 with Nebentypus χ12 with the property that their non-zero Fourier coefficients are supported on arithmetic progression congruent to 1 (mod 24). Choose a constant c such that 24 the coefficients up to q of ξ 1 (P (z)) and cξ 1 (M (z)) agree and since dimC M 3 (Γ0 (144), χ12 ) = 24, 2 2 2 ξ 1 (P (z)) = cξ 1 (M (z)), which implies that Pnh (z) = cMnh (z). Thus the function H(z) := P (z) − 2 2 cM (z) is a weakly holomorphic modular form. We have to show that c = 1. For this we apply the inversion z 7→ − z1 . By work of Zwegers [39] this produces a nonholomorphic part unless c = 1. Since H(z) is weakly holomorphic we conclude that c = 1. To be more precise, the Poincaré series considered

10

KATHRIN BRINGMANN

here is a component of a vector valued weak Maass form whose transformation law is known by work of Zwegers (see also [26], where such a vector valued Poincaré series is construced). Estimating the 3 coefficients of P (z) and M (z) against n 4 + , one obtains that H(z) is a holomorphic modular form of weight 12 on Γ0 (144) with Nebentypus χ12 . Since this space is trivial we obtain H(z) = 0 which employs the claim. 5. Congruences for Dyson’s rank generating functions In this section we prove an infinite family of congruences for Dyson’s ranks which generalizes partitions congruences [12]. Theorem 5.1. Let t be a positive odd integer, and let Q - 6t be prime. If j is a positive integer, then there are infinitely many non-nested arithmetic progressions An + B such that for every 0 ≤ r < t we have N (r, t; An + B) ≡ 0 (mod Qj ). Two remarks. 1) The congruences in Theorem 5.1 may be viewed as a combinatorial decomposition of the partition function congruence p(An + B) ≡ 0 (mod Qj ). 2) Congruences for t = Qj were shown in [9]. Sketch of proof. First observe that ∞ X

(5.1)

N (r, t; n)q n =

n=0

∞

t−1

n=0

j=1

1X 1 X −rj p(n)q n + ζt · R(ζtj ; q). t t

Using the results from Section 3 we can conclude that ∞ X p(n) `t n− `t 24 q N (r, t; n) − t n=0

is the holomorphic part of a weak Maass form of weight 12 on Γ1 6ft2 `t . We wish to apply certain quatratic twists which “kill” the non-holomorphic part of D ac ; q . For this we compute on which arithmetic progressions it is supported. This will enable us to use results on congruences for half integer weight modular forms. We obtain ∞ ∞ a X X `c `c ; z = q − 24 + D N (m, n)ζcam q `c n− 24 c n=1 m=−∞ πa X X 2 sin c ` c n2 aπ(6m + 1) 1 `c n 2 y m √ Γ ; q − 24 . − (−1) sin c 2 6 π m (mod fc ) n≡6m+1 (mod 6fc ) In particular the non-holomorphic part of D ac ; z is supported on certain fixed arithmetic progression. Generalizing the theory of twists of modular forms to twists of weak Maass forms, one can show. Proposition 5.2. If 0 ≤ r < t are integers, where t is odd, and P - 6t is prime, then X p(n) `t n− `t 24 N (r, t; n) − q t n≥1

t (24`tPn−`t )=−(−` P )

is a weight

1 2

weakly holomorphic modular form on Γ1 6ft2 `t P 4 .


11

To prove Theorem 5.1, we shall employ a recent general result of Treneer [36]. We use the following fact, which generalized Serre’s results on p-adic modular forms. Proposition 5.3. Suppose that f1 (z), f2 (z), . . . , fs (z) are half-integral weight cusp forms where fi (z) ∈ Sλi + 1 (Γ1 (4Ni )) ∩ OK [[q]], 2

and where OK is the ring of integers of a fixed number field K. If Q is prime and j ≥ 1 is an integer, then the set of primes L for which fi (z) | Tλi (L2 ) ≡ 0

(mod Qj ),

for each 1 ≤ i ≤ s, has positive Frobenius density. Here Tλi (L2 ) denotes the usual L2 index Hecke operator of weight λi + 21 . Now suppose that P - 6tQ is prime. By Proposition 5.2, for every 0 ≤ r < t ∞ X X p(n) `t n− `t n 24 (5.2) F (r, t, P; z) = a(r, t, P; n)q := N (r, t; n) − q t n=1 t (24`tPn−`t )=−(−` P ) is a weakly holomorphic modular form of weight 21 on Γ1 6ft2 `t P 4 . Furthermore, by the work of Ahlgren and Ono [1], it is known that P (t, P; z) =

(5.3)

∞ X

X

p(t, P; n)q n :=

`t

p(n)q `t n− 24

n=1

t (24`tPn−`t )=−(−` P ) is a weakly holomorphic modular form of weight − 12 on Γ1 24`t P 4 . Now since Q - 24ft2 `t P 4 , a generalization of a result of Treneer (see Theorem 3.1 of [36]), implies that there is a sufficiently large integer m for which X a(r, t, P; Qm n)q n ,

Q-n

for all 0 ≤ r < t, and X

p(t, P; Qm n)q n

Q-n

Qj

are all congruent modulo to forms in the graded ring of half-integral weight cusp forms with algebraic integer coefficients on Γ1 24ft2 `t . Applying Proposition 5.3 to these t + 1 forms gives that a positive proportion of primes L have the property that these t + 1 half-integral weight cusp forms modulo Qj are annihilated by the index L2 half-integral weight Hecke operators. Theorem 5.1 now follows mutatis mutandis as in the proof of Theorem 1 of [32]. 6. Asymptotics for Dyson’s rank partition functions We obtain asymptotic formulas for Dyson’s rank generating functions [8]. As an application, we solve a conjecture of Andrews and Lewis on inequalities between ranks. We write R(ζca ; q)

=: 1 +

∞ X n=1

A

a c

; n qn.

12

KATHRIN BRINGMANN

Let k and h be coprime integers, h0 defined by hh0 ≡ −1 (mod k) if k is odd and by hh0 ≡ −1 k c (mod 2k) if k is even, k1 := gcd(k,c) , c1 := gcd(k,c) , and 0 < l < c1 is defined by the congruence l ≡ ak1 (mod c1 ). Furthermore we define, for n, m ∈ Z, the following sums of Kloosterman type X 2πi 0 ωh,k · e k (nh+mh ) , Da,c,k (n, m) := (−1)ak+l h

(mod k)∗

Ba,c,k (n, m) := (−1)ak+1 sin

πa c

X h

(mod k)∗

3πia2 k1 h0 2πi ωh,k 0 − c · e · e k (nh+mh ) , 0 πah sin c

where for Ba,c,k (n, m) we require that c|k. Moreover, for c - k, let  2 l 3 l 1 1  + + r + 24 if 0 < cl1 < 16 , −   2 c1 2 c1 2 (6.1) δc,k,r := 3 5l l 25 l + − + − r 1 − if 56 < cl1 < 1,  24 c1   2c1 2 c1 0 otherwise, and for 0
N (1, 3; n) if n ≡ 1 (mod 3),

(6.2)

A careful analysis of the asymptotics in Corollary 6.2 gives the following theorem. Theorem 6.3. The Andrews-Lewis Conjecture is true for all n 6∈ {3, 9, 21} in which case we have equality in (6.2). Sketch of proof of Theorem 6.1. We use the Hardy Littlewood method. By Cauchy’s Theorem we have for n > 0 a

1 A ;n = c 2πi

Z C

N ac ; q dq, q n+1

where C is an arbitrary path inside the unit circle surrounding 0 counterclockwise. Now let 1 , k(k1 + k)

ϑ0h,k := where

h1 k1

0

P P P Using Lemma 6.4 we estimate 3 in a similar way. In 1 and 2 we next write Z 1 Z − 1 Z 1 Z ϑ00 h,k kN k(k+k1 ) kN = − − −ϑ0h,k

1 − kN

1 − kN

1 k(k+k2 )

and denote the associated sums by S11 , S12 , and S13 , respectively. The sums S12 and S13 contribute to the error terms which can be bounded as before. To finish the proof, we have to estimate integrals of the shape Z 1 kN 2π 1 r 1 z − 2 · e k (z (n− 24 )+ z ) dΦ. Ik,r := 1 − kN

One can show that Ik,r

1 = ki

Z

3 1 r 2π 1 z − 2 · e k (z (n− 24 )+ z ) dz + O n− 4 ,

Γ

where Γ denotes the circle through nk ± Ni and tangent to the imaginary axis at 0. Making the substitution t = 2πr kz and using the Hankel formula, we obtain ! r √ 3 2r(24n − 1) π 4 3 Ik,r = p sinh + O n− 4 3 k k (24n − 1) from which we can easily conclude the theorem.

16

KATHRIN BRINGMANN

7. Correspondences for weight

3 2

weak Maass forms

In [13] we classify those weak Maass forms whose holomorphic part arise from basic hypergeometric series, and we obtain a one-to-one correspondence {Θ(χ; z)} ←→ {Mχ (z)} between holomorphic weight 12 theta functions Θ(χ; z) and certain weight To state our result, define the basic hypergeometric-type series F (α, β, γ, δ, , ζ) :=

3 2

weak Maass forms Mχ (z).

2 ∞ X (α; ζ)n δ n n (β; ζ)n (γ; ζ)n

n=0

and by differentiation of a special case of the Rogers and Fine identity [25] X ∞ α ∂ 1 2 FRF (α, β) := · F (α, −αβ, 0, 1, α, β) = (−1)n nα2n β n . 2 ∂α 1 + α n=1

P

n2

Moreover let Θ0 (z) := n∈Z q be the classical Jacobi theta function. For a non-trivial even Dirichlet character χ with conductor b define X 2 Θ(χ; z) := χ(n)q n , n∈Z

which is a weight be given by

1 2

modular form on Γ0 (4b2 ). Moreover let the non-holomorphic theta integral Nχ (z) ib2 Nχ (z) := − π

Z

i∞

Θ(χ; τ ) 3

−z

dτ.

(−i(τ + z)) 2

Lastly, define the function Qχ by √ 4b 2 Qχ (z) := Θ0 (b2 z)

(7.1)

X a

(mod b)

X

χ(a) j

ζbja L q 2a ζbj , 2b; q ,

(mod b)

where (7.2)

L(w, d; q) :=

X n∈Z

2 1 (k+1)d 1 kd X q n wn −2 2 q 2 , −q 2 n + F w q , −q . = F w RF RF 1 − q dn

k≥0

In [13], we show: Theorem 7.1. If χ is a non-trivial even Dirichlet character with conductor b, then Mχ (z) := Qχ (z) − Nχ (z) is a weight

3 2

weak Maass form on Γ0 (4b2 ) with Nebentypus χ.

Corollary 7.2. We have that y 3/2 ·

∂ ∂ z¯ Mχ (z)

ib2 = − √ · Θ(χ; z). 2 2π

Since the Serre-Stark Basis Theorem asserts that the spaces of holomorphic weight 12 modular forms have explicit bases of theta functions, Corollary 7.2 gives a bijection between weight 32 weak Maass forms and weight 21 modular forms.


17

Sketch of proof of Theorem 7.1. We only give a sketch of proof here; details can be found in [13]. For 0 < a < b, we let X X 2 2 2 ψa,b (z) := 2 (d + bm) e2πi(b m −d )z . m≥1

−bm≤d 1 is an odd integer. If 0 ≤ r1 , r2 < t are integers, and P - 6t is prime, then X lt (N (r1 , t; n) − N (r2 , t; n)) q lt n− 24 n≥1

(24ltPn−lt )=−(−lPt ) is a weight

1 2

weakly holomorphic modular form on Γ1 (6ft2 lt P 4 ).

Since Theorem 8.2 follows easily from Section 3, we only consider Theorem 8.1 Proof of Theorem 8.1. Using the results from Sections 3 and 5, one can reduce the claim to the identity t−1 X πj πjα −r1 j −r2 j ζt − ζt sin =0 sin t t j=1

which can be easily verified using that sin(x) =

1 2i

eix − e−ix .

20

KATHRIN BRINGMANN

References [1] S. Ahlgren and K. Ono, Congruence properties for the partition function, Proc. Natl. Acad. Sci., USA 98, No. 23 (2001), pages 12882-12884. [2] G. E. Andrews, On the theorems of Watson and Dragonette for Ramanujan’s mock theta functions, Amer. J. Math. 88 No. 2 (1966), pages 454-490. [3] G. E. Andrews, Partitions with short sequences and mock theta functions, Proc. Natl. Acad. Sci. USA, 102 No. 13 (2005), pages 4666-4671. [4] G. E. Andrews, F. Dyson, and D. Hickerson, Partitions and indefinite quadratic forms, Invent. Math. 91 No. 3 (1988), pages 391-407. [5] G. E. Andrews and F. Garvan, Ramanujan’s “lost notebook”. VI: The mock theta conjectures, Adv. in Math. 73 (1989), pages 242-255. [6] G. E. Andrews and R. P. Lewis, The ranks and cranks of partitions moduli 2, 3 and 4, J. Number Th. 85 (2000), pages 74-84. [7] A. O. L. Atkin and H. P. F. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. 66 No. 4 (1954), pages 84-106. [8] K. Bringmann, Asymptotics for rank partition functions, Trans. Amer. Math. Soc., accepted for publication. [9] K. Bringmann, On certain congruences for Dyson’s ranks, submitted for publication. [10] K. Bringmann and J. Lovejoy, Dyson’s rank, overpartitions, and weak Maass forms, Int. Math. Res. Not., Article ID rnm063, 34 pages. [11] K. Bringmann and K. Ono, The f (q) mock theta function conjecture and partition ranks, Invent. Math. 165 (2006), pages 243-266. [12] K. Bringmann and K. Ono, Dysons ranks and Maass forms, Ann. of Math., accepted for publication. [13] K. Bringmann, A. Folsom, and K. Ono, q-series and weight 3/2 Maass forms, in preparation. [14] K. Bringmann and K. Ono, Arithmetic properties of coefficients of half-integral weight Maass-Poincaré series, Math. Ann. 337 (2007), pages 591-612. [15] K. Bringmann and K. Ono, Lifting elliptic cusp forms to Maass forms with an application to partitions, Proc. Nat. Acad. Sc. 104 (2007), pages 3725-3731. [16] K. Bringmann, K. Ono, and R. Rhoades, Eulerian series as modular forms, J. Amer. Math. Soc., accepted for publication. [17] J. H. Bruinier and J. Funke, On two geometric theta lifts, Duke Math. J. 125 (2004), pages 45-90. [18] Y.-S. Choi, Tenth order mock theta functions in Ramanujan’s lost notebook, Invent. Math. 136 (1999), pages 497-569. [19] Y.-S. Choi, Tenth order mock theta functions in Ramanujan’s lost notebook, II, Adv. in Math. 156 (2000), pages 180-285. [20] H. Cohen, q-identities for Maass waveforms, Invent. Math. 91 No. 3 (1988), pages 409-422. [21] S. Corteel and J. Lovejoy, Overpartitions, Trans. Amer. Math. Soc. 356 (2004), pages 1623-1635. [22] L. Dragonette, Some asymptotic formulae for the mock theta series of Ramanujan, Trans. Amer. Math. Soc. 72 No. 3 (1952), pages 474-500. [23] F. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), pages 10-15. [24] M. Eichler, On the class number of imaginary quadratic fields and the sums of divisors of natural numbers, J. Indian Math. Soc. (N.S.) 19 (1955), pages 153-180. [25] N. J. Fine, Basic hypergeometric series and applications, Math. Surveys and Monographs, Vol. 27, Amer. Math. Soc., Providence, 1988. [26] S. Garthwaite, The coefficients of the ω(q) mock theta function, Int. J. Number Theory, accepted for publication. [27] B. Gordon and R. McIntosh, Modular transformations of Ramanujan’s fifth and seventh order mock theta functions, Ramanujan J. 7 (2003), pages 193-222. [28] D. Hickerson, A proof of the mock theta conjectures, Invent. Math. 94 (1988), pages 639-660. [29] D. Hickerson, On the seventh order mock theta functions, Invent. Math. 94 (1988), pages 661-677. [30] F. Hirzebruch and D. Zagier, Intersection numbers of curves on Hilbert modular surfaces and modular forms with Nebentypus, Invent. Math. 36 (1976), pages 57-113. [31] R. P. Lewis, The ranks of partitions modulo 2, Discuss. Math. 167/168 (1997), pages 445-449. [32] K. Ono, Distribution of the partition function modulo m, Ann. of Math. 151 (2000), pages 293-307. [33] K. Ono, The web of modularity: Arithmetic of the coefficients of modular forms and q-series, CBMS Regional Conference, 102, Amer. Math. Soc., Providence, R. I., 2004. [34] S. Ramanujan, The lost notebook and other unpublished papers, Narosa, New Delhi, 1988. [35] G. Shimura, On modular forms of half integral weight, Ann. of Math. 97 (1973), pages 440-481.


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[36] S. Treneer, Congruences for the coefficients of weakly holomorphic modular forms, Proc. London Math. Soc. 93 (2006), pages 304–324. [37] H. Yesilyurt, Four identities related to third order mock theta functions in Ramanujan’s lost notebook, Adv. in Math. 190 (2005), pages 278-299. [38] G. N. Watson, The final problem: An account of the mock theta functions, J. London Math. Soc. 2 (2) (1936), pages 55-80. [39] S. P. Zwegers, Mock ϑ-functions and real analytic modular forms, q-series with applications to combinatorics, number theory, and physics (Ed. B. C. Berndt and K. Ono), Contemp. Math. 291, Amer. Math. Soc., (2001), pages 269-277. [40] S. P. Zwegers, Mock theta functions, Ph.D. Thesis, Universiteit Utrecht, 2002. School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A. E-mail address: [email protected]

Mock theta functions, weak Maass forms, and applications

Mock theta functions, weak Maass forms, and applications

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