Dyson's Ranks and Maass Forms

ANNALS OF MATHEMATICS

Dyson’s ranks and Maass forms

By Kathrin Bringmann and Ken Ono

SECOND SERIES, VOL. 171, NO. 1 January, 2010

anmaah Annals of Mathematics, 171 (2010), 419–449

Dyson’s ranks and Maass forms

By KATHRIN BRINGMANN and KEN ONO

For Jean-Pierre Serre in celebration of his 80th birthday.

Abstract

Motivated by work of Ramanujan, Freeman Dyson deﬁned the rank of an integer partition to be its largest part minus its number of parts. If N.m; n/ denotes the number of partitions of n with rank m, then it turns out that n2 X1 X1 X1 q R.w q/ 1 N.m; n/wmqn 1 : I WD C D C Qn .1 .w w 1/qj q2j / n 1 m n 1 j 1 D D1 D D C C We show that if 1 is a root of unity, then R. q/ is essentially the holomorphic ¤ I part of a weight 1=2 weak Maass form on a subgroup of SL2.ޚ/. For integers 0 Ä r < t, we use this result to determine the modularity of the generating function for N.r; t n/, the number of partitions of n whose rank is congruent to r .mod t/. We I extend the modularity above to construct an inﬁnite family of vector valued weight 1=2 forms for the full modular group SL2.ޚ/, a result which is of independent interest.

1. Introduction and statement of results

The mock theta-functions give us tantalizing hints of a grand synthe- sis 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. Freeman Dyson, 1987 Ramanujan Centenary Conference

The authors thank the National Science Foundation for their generous support. The second author is grateful for the support of a Packard and a Romnes Fellowship, and the support of the Manasse family. This work was completed when the ﬁrst author was a Van Vleck Assistant Professor at the University of Wisconsin.

419 420 KATHRIN BRINGMANN and KEN ONO

Dyson’s quote (see page 20 of[15]) refers to 22 peculiar q-series, such as

n2 X1 q (1.1) f .q/ 1 ; WD C .1 q/2.1 q2/2 .1 qn/2 n 1 D C C C which were defined by Ramanujan and Watson decades ago. In his last letter to Hardy dated January 1920 (see pp. 127–131 of[26]), Ramanujan lists 17 such functions, and he gives 2 more in his “Lost Notebook”[26]. In his paper “The final problem: An account of the mock theta functions”[31], Watson defines three further functions. Surprisingly, much remains unknown about these enigmatic series. Ramanu- jan’s claims about their analytic properties remain open, and there is even debate concerning the rigorous definition of such a function. Despite these seemingly prob- lematic issues, Ramanujan’s mock theta functions indeed possess many striking properties, and they have been the subject of an astonishing number of important works (for example, see[4],[5],[6],[7],[11],[12],[13],[17],[18],[19],[22], [26],[27],[31],[32],[34],[35] to name a few). Watson predicted this high level of activity in his 1936 Presidential Address to the London Mathematical Society with his prophetic words (see page 80 of[31]): Ramanujan’s discovery of the mock theta functions makes it obvious that his skill and ingenuity did not desert him at the oncoming of his untimely end. As much as any of his earlier work, the mock theta functions are an achievement sufficient to cause his name to be held in lasting remembrance. To his students such discoveries will be a source of delight and wonder until the time shall come when we too shall make our journey to that Garden of Proserpine (a.k.a. Persephone). . . G. N. Watson, 1936. In his 2002 Ph.D. thesis[35], written under the direction of Zagier, Zwegers made an important step in the direction of Dyson’s “challenge for the future”. He related many of Ramanujan’s mock theta functions to real analytic vector valued modular forms. We make another step by establishing that Dyson’s own rank generating function can be used to construct the desired “coherent group-theoretical structure, analogous to the structure of modular forms which Hecke built around old theta functions of Jacobi”. We show that the specializations of his partition rank generating function R. q/, where 1 is a root of unity, are “holomorphic I ¤ parts” of weak Maass forms. Moreover, we show that the “nonholomorphic parts” of these forms are period integrals of theta functions, thereby realizing Dyson’s speculation that such a picture should involve theta functions. We shall use these results to systematically obtain Ramanujan-type congruences for Dyson’s rank partition functions. DYSON’SRANKSANDMAASSFORMS 421

To describe the historical context of these results, we begin by recalling classical facts about partitions and modular forms which inspired Ramanujan to originally deﬁne the mock theta functions. A partition of a nonnegative integer n is any nonincreasing sequence of positive integers whose sum is n. As usual, let p.n/ denote the number of partitions of n. The partition function p.n/ has the well known inﬁnite product generating function

X1 Y1 1 (1.2) p.n/qn ; D 1 qn n 0 n 1 D D 1 which coincides with q 24 =.z/, where

Y1 .z/ q1=24 .1 qn/ .q e2iz/ WD WD n 1 D is Dedekind’s weight 1=2 modular form. Modular forms have played a central role in the theory of partitions, largely due to the fact that many generating functions in the subject, such as (1.2), are related to inﬁnite product modular forms such as Dedekind’s eta-function and the Siegel-Klein forms. On the other hand, many partition generating functions are “Eulerian” forms, also known as q-series, which do not naturally appear in modular form theory. However there are famous examples, such as the Rogers-Ramanujan identities

n2 X1 q 1 1 ; C .1 q/.1 q2/ .1 qn/ D Q .1 q5n 1/.1 q5n 4/ n 1 n1 1 D D n2 n X1 q 1 1 C ; C .1 q/.1 q2/ .1 qn/ D Q .1 q5n 2/.1 q5n 3/ n 1 n1 1 D D where Eulerian forms are essentially modular forms. As another example, we note that n2 X1 X1 q (1.3) p.n/qn 1 : D C .1 q/2.1 q2/2 .1 qn/2 n 0 n 1 D D The mock theta functions stand out in this context. Although they are not modular, they possess striking properties which prompted Dyson to set forth his challenge of 1987. In this regard, the focus of our attention is a particularly excep- tional family of such series, the specializations of Dyson’s own rank generating function. In an effort to provide a combinatorial explanation of Ramanujan’s congruences for p.n/, Dyson introduced[14] the so-called “rank” of a partition, a delightfully simple statistic. The rank of a partition is deﬁned to be its largest part minus the number of its parts. More precisely, he conjectured that the partitions of 422 KATHRIN BRINGMANN and KEN ONO

5n 4 (resp. 7n 5) form 5 (resp. 7) groups of equal size when sorted by their ranks C C modulo 5 (resp. 7)1. He further postulated the existence of another statistic, the so- called “crank”2, which allegedly would explain all three Ramanujan congruences p.5n 4/ 0 .mod 5/; C Á p.7n 5/ 0 .mod 7/; C Á p.11n 6/ 0 .mod 11/: C Á In 1954, Atkin and Swinnerton-Dyer proved[9] Dyson’s rank conjectures. If N.m; n/ denotes the number of partitions of n with rank m, then it is well known that n2 X1 X1 X1 q (1.4) R.w q/ 1 N.m; n/wmqn 1 ; I WD C D C .wq q/ .w 1q q/ n 1 m n 1 n n D D1 D I I where n 1 .a q/n .1 a/.1 aq/ .1 aq /; I WD Y1 .a q/ .1 aqm/: I 1 WD m 0 D Obviously, by letting w 1, we obtain (1.3). D Letting w 1, we obtain the series D n2 X1 q R. 1 q/ 1 : I D C .1 q/2.1 q2/2 .1 qn/2 n 1 D C C C This series is the mock theta function f .q/ given in (1.1). In earlier work[10], the present authors proved that q 1R. 1 q24/ is the “holomorphic part” of a weak I Maass form. This is a special case of our ﬁrst result. To make this precise, we begin by recalling the notion of a weak Maass form of half-integral weight k 1 ޚ ޚ. If z x iy with x; y ޒ, then the weight k 2 2 n D C 2 hyperbolic Laplacian is given by Â @2 @2 Ã Â @ @ Ã (1.5) y2 iky i : k WD @x2 C @y2 C @x C @y

1A short calculation reveals that this phenomenon cannot hold modulo 11. 2In 1988, Andrews and Garvan[8] found the crank, and they indeed conﬁrmed Dyson’s speculation that it explains the three Ramanujan congruences above. Recent work of Mahlburg[23] establishes that the Andrews-Dyson-Garvan crank plays an even more central role in the theory partition congruences. His work concerns partition congruences modulo arbitrary powers of all primes 5. Other work by Garvan, Kim and Stanton[16] gives a different “crank” for several other Ramanujan congruences. DYSON’SRANKSANDMAASSFORMS 423

If v is odd, then define v by 1 if v 1 .mod 4/; v Á WD i if v 3 .mod 4/: Á A weak Maass form of weight k on a subgroup 0.4/ is any smooth function f ވ ރ satisfying the following: W ! (1) For all A a b and all z ވ, we have3 D c d 2 2 !2k c f .Az/ 2k.cz d/k f .z/: D d d C (2) We have that f 0. k D (3) The function f .z/ has at most linear exponential growth at all the cusps of . 2i=c 2c Suppose that 0 < a < c are integers, and let c e . If fc , WD WD gcd.c;6/ then define the theta function ‚ a by c I Â Ã a Á X m a.6m 1/ Á (1.6) ‚ . 1/ sin C 6m 1; 6fc ; c I WD c C I 24 m .mod fc / where X 2 (1.7) .˛; ˇ / ne2i n : I WD n ˛ .mod ˇ/ Á 2 Throughout, let `c lcm.2c ; 24/, and let `c `c=24. It is well known that a WD z WD ‚ c `c is a cusp form of weight 3=2. Using this cuspidal theta function, we I a define the function S1 z by the period integral c I 1 a 2 Z i a a Á i sin c `c 1 ‚ c `c S1 z p I d: c I WD p3 z i. z/ N C Using this notation, define D a z by c I Á Á `c a a a `c D z S1 z q 24 R. q /: c I WD c I C c I Moreover, define the group c by Â1 1Ã Â 1 0Ã c ; 2 : WD 0 1 `c 1 a THEOREM 1.1. If 0 < a < c, then D z is a weak Maass form of weight c I 1=2 on c.

3This transformation law agrees with Shimura’s notion of a half-integral weight modular form [29]. 424 KATHRIN BRINGMANN and KEN ONO

1 When a=c 1=2, it turns out that D 2 z is a weak Maass form on 0.144/ D 12 I with Nebentypus character 12. / . This fact was established by the authors D in[10], and it plays a central role in the proof of the Andrews-Dragonette Conjec- ture on the coefﬁcients of f .q/. In view of this fact, it is natural to suspect that a D z is often a weak Maass form on a group larger than c. For odd c, we c I establish the following.

a THEOREM 1.2. If 0 < a < c, where c is odd, then D z is a weak Maass c I form of weight on 2 . 1=2 1.144fc `zc/ Theorem 1.2 is implied by a general result about vector valued weight 1/2 weak Maass forms for the modular group SL2.ޚ/ (see Theorem 3.4), a result which is of independent interest.

`c a 24 a `c Remark. We refer to S1 c z (resp. q R.c q /) as the nonholomorphic I a I (resp. holomorphic) part of the Maass form D c z . To justify this, one notes that a I S1 z is nonholomorphic in z, and that c I @ `c a ` Á q 24 R. q c / 0: @z c I D Here @ 1 @ i @ . In particular, q `c =24R.a q`c / is the part of the Fourier @z WD 2 @x C @y c I expansion of D a z which is given as a series expansion in q e2iz (see c I D Proposition 4.1).

Theorems 1.1 and 1.2 provide a new perspective on the role that modular forms play in the theory of partitions. They imply that the generating functions for Dyson’s rank partition functions are related to Maass forms and modular forms. If r and t are integers, then let N.r; t n/ be the number of partitions of n whose rank I is r .mod t/.

THEOREM 1.3. If 0 r < t are integers, then Ä Â Ã X1 p.n/ ` n `t N.r; t n/ q t 24 I t n 0 D is the holomorphic part of a weak Maass form of weight 1/2 on t . Moreover, if t is odd then it is on 2 . , 1.144ft `zt / This result allows us to relate many “sieved” generating functions to weakly holomorphic modular forms, those forms whose poles (if there are any) are supported at cusps. DYSON’SRANKSANDMAASSFORMS 425

THEOREM 1.4. If 0 r < t are integers, where t is odd, and ᏼ − 6t is prime, Ä then X Â p.n/Ã N.r; t n/ q`t n `t =24 I t n 1 24`t n `t 24`t . / . z / ᏼ D ᏼ is a weight weakly holomorphic modular form on 2 4 . 1=2 1.144ft `zt ᏼ / These results are useful for studying Dyson’s rank partition generating functions. Atkin and Swinnerton-Dyer[9] confirmed Dyson’s conjecture that for every integer n and every r we have p.5n 4/ (1.8) N.r; 5 5n 4/ C ; I C D 5 p.7n 5/ (1.9) N.r; 7 7n 5/ C ; I C D 7 thereby providing a combinatorial “explanation” of Ramanujan partition congruences with modulus 5 and 7. It is not difficult to use our results to give alternative proofs of these rank identities, as well as others of similar type. Armed with Theorems 1.2, 1.4 and 3.4, one can obtain deeper results about ranks. They can be used to obtain asymptotic formulas for the N.r; t n/ partition I functions. Indeed, the present authors have already successfully employed the theory of weak Maass forms to solve the more difficult problem of obtaining exact formulas in the case of the functions N.0; 2 n/ and N.1; 2 n/ (see Theorem 1.1 I I of[10]). For odd t, one can use Theorem 3.4 and the “circle method” to obtain asymptotics. Since the details are messy and lengthy, for brevity we have chosen to address asymptotics in a later paper. Here we turn to the question of congruences, the subject which originally motivated Dyson to define partition ranks. In this direction, we shall employ a method first used by the second author in[24] in his work on p.n/. We show that Dyson’s rank partition functions satisfy congruences of Ramanujan type, a result which nicely complements the recent paper[23] by Mahlburg on the Andrews- Garvan-Dyson crank.

THEOREM 1.5. Let t be a positive odd integer, and let Q − 6t be prime. If j is a positive integer, then there are inﬁnitely many nonnested arithmetic progressions An B such that for every 0 r < t we have C Ä N.r; t An B/ 0 .mod Qj /: I C Á Three remarks. (1) The congruences in Theorem 1.5 may be viewed as a combinatorial decomposition of the partition function congruence p.An B/ 0 .mod Qj /: C Á 426 KATHRIN BRINGMANN and KEN ONO

(2) By “nonnested”, we mean that there are inﬁnitely many arithmetic progressions An B, with 0 B < A, with the property that there are no progressions which C Ä contain another progression. (3) Theorem 1.5 is in sharp contrast to Mahlburg’s recent result on the Andrews- Garvan-Dyson crank (see[23]). For example, his results imply that congruences modulo Qj exist for all the crank partition functions with modulus t Q. On the D other hand, Theorem 1.5 proves congruences for powers of those primes Q 5 which do not divide the rank modulus t.

CONJECTURE. Theorem 1.5 holds for those primes Q 5 which divide t. To prove these theorems, we require a number of new results. First of all, the proof of Theorem 1.2 requires transformation laws for some new classes of mock theta functions. In Section 2, we derive these transformation formulas, and we recall recent work of Gordon and McIntosh[18]. In Section 3, we use the results of Section 2 to construct the vector valued Maass forms whose properties are the content of Theorem 3.4. We conclude Section 3 with proofs of Theorems 1.1, 1.2, and 1.3. In Section 4, we prove Theorem 1.4, and then give the proof of Theorem 1.5. The proof of Theorem 1.5 relies on Q-adic properties of weakly holomorphic half-integral weight modular forms, and Q-adic Galois representa- tions associated to modular forms.

Acknowledgements. The authors would like to thank Scott Ahlgren, George Andrews, Matthew Boylan, Michael Dewar, Jan H. Bruinier, Freeman Dyson, Sharon Garthwaite, Frank Garvan, Karl Mahlburg, Jean-Pierre Serre, and the referee for their helpful comments.

2. Modular transformation formulas

Here we derive modular transformation formulas for R. q/ and allied func- I tions. In Section 2.1, we ﬁrst recall transformation laws obtained recently by Gor- don and McIntosh[18], and in Section 2.2 we derive transformation formulas for closely allied functions. In Section 2.3 we combine these results to produce an inﬁnite family of vector valued modular forms under SL2.ޚ/.

2.1. Transformation laws of Gordon and McIntosh. To state the transformation formulas of Gordon and McIntosh, we require the following series. If 0 < a < c are integers and q e2iz, then we let WD a n n c a Á a Á 1 X1 . 1/ q C 3 n.n 1/ (2.1a) M z M q q 2 ; n a C c I D c I WD .q q/ n 1 q C c I 1 D1 DYSON’SRANKSANDMAASSFORMS 427 a Á a Á (2.1b) M1 z M1 q c I D c I a n 1 n c 1 X1 . 1/ C q C 3 n.n 1/ q 2 ; n a C WD .q q/ 1 q C c I 1 n C a Á a Á D1 (2.1c) N z N q c I D c I n n 2a ! 1 X1 . 1/ .1 q / 2 2 cos c n.3n 1/ 1 C q 2C ; WD .q q/ C 1 2qn cos 2a q2n I 1 n 1 c C a Á aD Á (2.1d) N1 z N1 q c I D c I n 2n 1 1 X1 . 1/ 1 q C 3n.n 1/ 2C : 1 q : D .q q/ n 2 2a 2n 1 n 0 1 2q C cos c q C I 1 D C Two remarks. (1) Gordon and McIntosh show the q-series identities

n.n 1/ a Á X1 q (2.2) M q a a ; c I D c 1 c n 1 .q q/n .q q/n D I I n2 a Á X1 q (2.3) N q 1 : c I D C Qn 2a j 2j n 1 j 1 1 2 cos c q q D D C (2) If 0 < a < c are integers, then (1.4) and (2.3) imply the important fact that a Á (2.4) R.a q/ N q : c I D c I To state their transformation laws, we require the following Mordell integrals: Z 3a 3a a Á 1 3 ˛x2 cosh c 2 ˛x cosh c 1 ˛x J ˛ e 2 C dx; c I WD cosh.3˛x=2/ (2.5) 0 Z 3a 3a a Á 1 3 ˛x2 sinh c 2 ˛x sinh c 1 ˛x J1 ˛ e 2 dx: c I WD 0 sinh.3˛x=2/ By modifying the seminal arguments of Watson[31], Gordon and McIntosh (see page 199 of[18]) proved the following theorem.

THEOREM 2.1. Suppose that 0 < a < c are integers, and that ˛ and ˇ have 2 ˛ ˇ the property that ˛ˇ . If q e and q1 e , then we have D WD WD r 3a a 1 r 1 1 a Á a Á 6 a 4Á 3˛ a Á q 2c . c / 24 M q csc q N q J ˛ ; c I D 2˛ c 1 c I 1 2 c I r r 3a a 1 Á 4 Á Á .1 / a 2 3 a 2 3˛ a q 2c c 24 M1 q q N1 q J1 ˛ : c I D ˛ 1 c I 1 2 c I 428 KATHRIN BRINGMANN and KEN ONO

2.2. Modular transformation formulas for allied series. Theorem 2.1 is insuf- a ficient for fully understanding the modularity properties of the functions N. c q/ a I R. q/ under the Mobius¨ transformations arising from SL2.ޚ/. Indeed, under D c I translations the functions M and M1 transform to allied functions whose modularity properties must be deduced. To this end, it is necessary to define further series which will allow us to view the functions in the previous subsection as pieces of the components of a vector valued function whose transformations we shall determine under the generators of SL2.ޚ/. Suppose that c is a positive integer, and suppose that a and b are integers for which 0 a; b < c. Using this notation, define Ä M.a; b; c z/ by I n n a 1 X1 . 1/ q c 3 C 2 n.n 1/ (2.6) M.a; b; c z/ M.a; b; c q/ a q C : b n I D I WD .q q/ n 1 c q C c I 1 D1 In addition, if b 0; 1 ; 1 ; 5 , then define the integer k.b; c/ by c 62 f 2 6 6 g 8 0 if 0 < b < 1 ; ˆ c 6 ˆ 1 b 1 <1 if 6 < c < 2 ; (2.7) k.b; c/ 1 b 5 WD ˆ2 if < < ; ˆ 2 c 6 :ˆ 5 b 3 if 6 < c < 1: Furthermore, throughout we let e.˛/ e2i˛. Using this notation, then define the WD series N.a; b; c z/ by I N.a; b; c z/ N.a; b; c q/ I D I 0 b 1 a 2c (2.8) 1 ie 2c q X1 m.3m 1/ K.a; b; c; m z/ q 2 C ; @ b Á A WD .q q/ a c C I 2 1 e c q m 1 I 1 D where K.a; b; c; m z/ I Á Á Á Á sin a b 2k.b; c/m z sin a b 2k.b; c/m z qm m c c C C c c . 1/ Á : WD 1 2 cos 2a 2bz qm q2m c c C Moreover, define the Mordell integral

b ˛x 2b 2˛xÁ Z c e c e 1 3 ˛x2 3˛x a C J.a; b; c ˛/ e 2 C c Á dx: I WD cosh 3˛x=2 3i b 1 c Using this notation, we obtain the following transformation laws. DYSON’SRANKSANDMAASSFORMS 429

THEOREM 2.2. Suppose that c is a positive integer, and that a and b are b ˚ 1 1 5 « integers for which 0 a < c, 0 < b < c and c 2 ; 6 ; 6 . Furthermore, suppose Ä 2 62 ˛ ˇ that ˛ and ˇ have the property that ˛ˇ . If q e and q1 e , then D WD WD 3a 1 a 1 q 2c . c / 24 M.a; b; c q/ I r 4b 6b2 1 a b 2a k.b;c/ 8 2i k.b;c/ 3i 1 b c c2 6 4 e c c . c / q N.a; b; c q / ˛ C c 1 I 1 r 3˛ 5b J.a; b; c ˛/: 8 2c I Three remarks. (1) Although b is nonzero in Theorem 2.2, note that we have M.a; 0; c q/ M a q. Therefore, Theorem 2.1, combined with Theorem 2.2, I D c I gives the appropriate transformation laws of every M.a; b; c q/, where 0 < a < c I and 0 b < c, provided that b ˚ 1 ; 1 ; 5 «. Ä c 62 2 6 6 1 (2) Observe that M.1; 1; 2 q/ M1 q . Therefore, the case where 2a c I D 2 I D and b 1 is also covered by Theorem 2.1. c D 2 (3) One could also prove Theorem 2.2 by using arguments which are analogous to those developed by Zwegers in[35].

Proof of Theorem 2.2. To prove this theorem, we argue with contour integration in a manner which is very similar to earlier work of Watson[31]. We consider a contour integral which is basically the function M.a; b; c q/. Deﬁne this integral I I by a Z i" ˛. c / 1 1 e C 3 ˛. 1/ I I1 I2 e 2 C d ˛ a WD C WD 2i i" sin./ 1 be . c / 1 c C (2.9) a Z i" ˛. c / 1 1C e C 3 ˛. 1/ e 2 C d: ˛ a 2i i" sin./ 1 be . c / 1C c C b ˛. a / Here " > 0 is sufﬁciently small enough so that 1 c e C c is nonzero for b ˛ a " Im./ ". This is indeed possible since 1 e . c / 0 if and only if Ä Ä c C D a 2i .b=c n/ C n D c C ˛ DW (here we need the condition that b 0). 6D By construction, we have that the poles of the integrand only arise from the roots of sin./, and they are the points ޚ. The residue of the integrand in 2 n ޚ equals D 2 n n a . 1/ q C c 3n.n 1/ q 2C : n a 1 b q c c C 430 KATHRIN BRINGMANN and KEN ONO

For Re./ , the integrand is of rapid decay, and so the Residue Theorem ! 1 implies that

n n a X1 . 1/ q c 3n.n 1/ C 2C (2.10) I a q .q q/ M.a; b; c q/: b n c 1 D n 1 c q C D I I D1

We now compute the integrals I1 and I2. We ﬁrst consider I2. Using (2.9) and the identity

1 X1 2i e.2n 1/i ; sin./ D C n 0 D which holds for ވ, we ﬁnd that 2 Z i" .2n 1/i ˛. a / 3 ˛. 1/ 1 X1 1C e C C c 2 C 1 X1 I2 2i a d Jn: D 2i b ˛. c / DW 2i n 0 i" 1 c e C n 0 D 1C D

We now reformulate I2 in a useful way by shifting the paths of integration through the points !n, the saddle points of

exp.2n 1/i 3 ˛ 2: C 2 These are the points given by

.2n 1/i !n C : D 3˛ By the Residue Theorem, we have

Z i Z !n 1C 1C X Jn f f ResŒf .m;n/; WD i D !n C 1C 1C m;n where m;n are those poles m of the integrand f lying between the original contour and the new contour (i.e. those m for which m 0 and Im.m/ < Im.!n//. These turn out to be those points m for which m 0 and that satisfy 2n 1 b Á (2.11) C > 2 m : 3 c C

b (That no poles lie on the path of integration follows from the condition that c ˚ 1 1 5 « 62 2 ; 6 ; 6 .) Using deﬁnition (2.7), we have that (2.11) is equivalent to

n 3m k.b; c/: C DYSON’SRANKSANDMAASSFORMS 431

At the points m, the integrand has the residue

2i a 3 .2n 1/im ˛.m / ˛m.m 1/ n;m e c 2 WD ˛ C C C 2i a b a b .2n 1/i c 3i.m c /.2 c 1/ c e C C C D ˛ 2 3a a b b 1 2.2n 1/. c m/ 6. c m/ q 2c . c / q C C C : 1 Hence the Residue Theorem, combined with a reordering of summation, implies that X X X (2.12) I2 n;m J ; D C n0 m 0 n 3m k.b;c/ n 0 C where a 3 Z !n .2n 1/i ˛. / ˛. 1/ 1C e C C c 2 C Jn0 a d: b ˛. / WD !n 1 e c 1C c C Using the fact that

a 4.m b / 2i c c n 1;m e q1 C n;m; C D we ﬁnd that X X X 3m k.b;c/;m (2.13) n;m C a b D 2i 4.m c / m 0 n 3m k.b;c/ m 0 1 e c q C C 1 2i .2k.b;c/ 1/ a i 3i 1 2 a b b e c . c / c D ˛ C c b 6b2 3a a 2.2k.b;c/ 1/ 1 c c2 q 2c . c / q C 1 2 m 6m 2.2k.b;c/ 1/m X1 . 1/ q C C 1 : a b 2i 4.m c / m 0 1 e c q C D 1 Now we compute the integral I1 by arguing as above using the identity

1 X1 2i e .2n 1/i ; sin./ D C n 0 D which holds for ވ. Again by the Residue Theorem, we ﬁnd that 2 X X X (2.14) I1 n;m K ; D C n0 m 1 n 3m k.b;c/ n 0 where a 3 Z !n .2n 1/iz ˛.z / ˛z.z 1/ 1C z e C C c 2 C Kn0 a dz: b ˛.z / WD !n 1 e c 1C z c C 432 KATHRIN BRINGMANN and KEN ONO .2n 1/i Here the points !n are given by !n C ; and z z WD 3˛ 2i a 3 .2n 1/i m ˛. m c / 2 ˛ m. m 1/ n;m e WD ˛ C C C 2i a b a 3a a b .2n 1/i c 3i. m c /.2 c 1/ 2c .1 c / c e C C C q D ˛ b b 2 2.2n 1/. c m/ 6. c m/ q1 C C : As in the case of I2, we obtain

X X 2i . 2k.b;c/ 1/ a i 3i.1 2 a / b b n;m e c c c D ˛ C c m 1 n 3m k.b;c/ b 6b2 3a a 2.2k.b;c/ 1/ 1 c c2 q 2c . c / q 1 2 m 6m 2. 2k.b;c/ 1/m X1 . 1/ q C C 1 : a b 2i 4.m c / m 1 1 e c q D 1 This fact, combined with (2.12), (2.13), and (2.14), implies that

b 6b2 a a b 3a a 4k.b;c/ 4 2k.b;c/ i 3i. 1 2 / b .1 / c c2 I1 I2 e c c c q 2c c q C D ˛ C C c 1 a 2b ! i c c ie q1 X1 6m2 2m K.a; b; c; m q1/ q C a 4b 1 2i c C z I 2 1 e c q m 1 1 D X J K ; C n0 C n0 n 0 where

K.a; b; c; m q1/ z I sin a iˇ 2b 4k.b; c/m sin a iˇ 2b 4k.b; c/mq4m . 1/m c c C C c c 1 : WD 1 2 cos2 a 4i b ˇ q4m q8m c c 1 C 1 By (2.8), we then ﬁnd that

4 a a b 3a a 2k.b;c/ c i 3i. 1 2 c / c b 2c .1 c / (2.15) I1 I2 e C C c q C D ˛ 2 4k.b;c/ b 6b q c c2 q4 q4 N a; b; c q4 1 1I 1 I 1 X 1 J K : C n0 C n0 n 0 Hence the proof of the theorem essentially boils down to the computation of X J K : n0 C n0 n 0 DYSON’SRANKSANDMAASSFORMS 433

We ﬁrst compute the Jn0 integrals. For this we need the identity 1 1 3 t t 2 t 2 t 2 C3 C3 ; 1 t D t 2 t 2 b ˛ a which we apply when t e . c /. This identity implies that the integrand D c C in Jn0 equals b ˛. a / 2b 2˛. a / 3b 3˛. a / 3 a 3 2 c e C c c e C c c e C c 5b .2n 1/i 2 ˛ c 2 ˛ 2c e C C C C : 3b 3 ˛ a 3b 3 ˛ a Á e 2 . c / e 2 . c / 2c C 2c C In the integrand we now put a p x, where D c C C .2n 1/i p C ; WD 3˛ and where x is a real variable running from to . This easily gives 1 1 n 1 2 .2n 1/ 3a a . 1/ C i 5b C6 1 J q q 2c . c / n0 D 2 2c 1 .2n 1/i 2.2n 1/i Z b C3 ˛x 2b C3 2˛x 3b 3˛x c e e c e e c e 3 ˛x2 3˛ a x C Á e 2 C c dx: ޒ cosh 3 ˛x 3i b 2 c In the same way, we obtain

n 2 .2n 1/ 3a a . 1/ i 5b C6 1 K q q 2c . c / n0 D 2 2c 1 .2n 1/i 2.2n 1/i Á b C ˛x 2b C 2˛x 3b 3˛x Z c e 3 e c e 3 e c e C 3 ˛x2 3˛ a x Á e 2 C c dx ޒ cosh 3 ˛x 3i b 2 c since we have that Â.2n 1/ Ã Â2.2n 1/ Ã sin C sin C ; 3 D 3 for every integer n we obtain the expression

2 .2n 1/ 3a a Â Ã n 1 5b C6 1 .2n 1/ J K . 1/ q q 2c . c / sin C n0 C n0 D C 2c 1 3 b ˛x 2b 2˛xÁ Z c e c e C 3 ˛x2 3˛ a x Á e 2 C c dx ޒ cosh 3 ˛x 3i b 2 c 2 .2n 1/ 3a a Â Ã n 1 5b C6 1 .2n 1/ . 1/ q q 2c . c / sin C J.a; b; c ˛/: D C 2c 1 3 I 434 KATHRIN BRINGMANN and KEN ONO

Now by Euler’s identity (see page 464 of[33])

Â Ã .2n 1/2 1 X1 .2n 1/ C 2 . 1/n sin C q 6 p3 q 6 q4 q4 ; 3 1 D 1 1I 1 n 0 1 D we ﬁnd that

p 1 3 a a X1 3 5b 6 1 4 4 J K q q 2 c . c / q q J.a; b; c ˛/: n0 C n0 D 2 2c 1 1I 1 I n 0 1 D This fact, combined with (2.9), (2.10), and (2.15) then gives

.q q/ M.a; b; c q/ I 1 I 4 a a b 3a a 2k.b;c/ c i 3i. 1 2 c / c b 2c .1 c / e C C c q D ˛ 2 4k.b;c/ b 6b c c2 4 4 4 q1 q1 q1 N a; b; c q1 I 1 I p3 1 3 a a 5b 6 2 c .1 c / 4 4 2c q1 q q1 q1 J.a; b; c ˛/: 2 I 1 I By the transformation law for Dedekind’s eta-function, it is straightforward to de- duce that r 2 1 1 24 6 4 4 .q q/ q q1 q1 q1 ; I 1 D ˛ I 1 from which the statement of the theorem follows easily.

2.3. An infinite family of vector valued Maass forms. It turns out that the transformations in Theorems 2.1 and 2.2 allow us to produce an infinite family of vector valued weight 1/2 weak Maass forms, one for every positive odd integer c. To this end, it suffices to determine the images of the components of these forms under the generators of SL2.ޚ/: 1 z z 1 and z : 7! C 7! z If c is a positive odd integer, then for every pair of integers 0 a; b < c define Ä the functions a Á a Á a Á 1 a Á (2.16) ᏺ q ᏺ z csc q 24 N q ; c I D c I WD c c I a Á a Á 3a 1 a 1 a Á (2.17) ᏹ q ᏹ z 2q 2c . c / 24 M q ; c I D c I WD c I 3a 1 a 1 (2.18) ᏹ.a; b; c q/ ᏹ.a; b; c z/ 2q 2c . c / 24 M.a; b; c q/; I D I WD I 2i a k.b;c/ 3i b . 2a 1/ (2.19) ᏺ.a; b; c q/ ᏺ.a; b; c z/ 4e c C c c I D I WD b 3b2 1 b k.b;c/ q c 2c2 24 N.a; b; c q/: c I DYSON’SRANKSANDMAASSFORMS 435

Remark. Notice that a must be nonzero for the function ᏺ a q. c I THEOREM 2.3. Suppose that c is a positive odd integer, and that a and b are integers for which 0 a < c and 0 < b < c. Ä (1) For all z ވ we have 2 a Á a Á ᏺ z 1 1 ᏺ z ; c I C D 24 c I 2 ᏺ.a; b; c z 1/ 3b 1 ᏺ.a b; b; c z/; I C D 2c2 24 I a Á 2 ᏹ z 1 5a 3a 1 ᏹ.a; a; c z/; c I C D 2c 2c 2 24 I 2 ᏹ.a; b; c z 1/ 5a 3a 1 ᏹ.a; a b; c z/; I C D 2c 2c 2 24 C I where a is required to be nonzero in the first and third formula. (2) For all z ވ we have 2 1 Âa 1Ã a Á a Á ᏺ ᏹ z 2p3p iz J 2iz ; p iz c I z D c I C c I Â Ã 1 1 5b ᏺ a; b; c ᏹ.a; b; c z/ 2c p3p iz J.a; b; c 2iz/; p iz I z D I C I 1 Âa 1Ã a Á 2p3i Âa 2i Ã ᏹ ᏺ z J ; p iz c I z D c I z c I z Â Ã Â Ã 1 1 5b p3i 2i ᏹ a; b; c ᏺ.a; b; c z/ 2c J a; b; c ; p iz I z D I z I z where a is required to be nonzero in the first and third formula. Remark. Strictly speaking, the functions in Theorem 2.3 do not always have the property that their defining parameters lie in the interval Œ0; c/. For example, this occurs whenever a b (resp. a b) is not in the interval Œ0; c/. In such cases, C one defines the corresponding functions in the obvious way, and then observes that the resulting functions equal, up to a precise root of unity, the corresponding functions where a b (resp. a b) are replaced by their reduced residue classes C modulo c. Lastly, the reader should recall the first remark after Theorem 2.2. Proof of Theorem 2.3. The first claim follows from the definitions of the series. The second claim follows from Theorems 2.1 and 2.2 by letting ˛ 2iz and D 2i=z.

3. Weak Maass forms

Here we prove Theorems 1.1 and 3.4 using the results from the previous section. In Section 3.1, we explicitly construct the nonholomorphic and holomorphic 436 KATHRIN BRINGMANN and KEN ONO

a parts of the functions D z , we derive their images under the generators of c, c I and we prove Theorem 1.1. 3.1. The nonholomorphic and holomorphic parts of D a z. Using Theo- c I rem 2.1, here we construct a weak Maass form of weight 1/2 using N a q. The c I arguments we employ are analogous to those employed by Zwegers in his work on Ramanujan’s mock theta functions (for example, see 3 of[34], or[35]). We begin with the transformation formulas for the relevant series. As in the introduction, suppose that 0 < a < c are integers. Define the vector valued function F a z by c I a Á a Á a ÁÁT F z F1 z ;F2 z (3.1) c I WD c I c I aÁ a Á aÁ a ÁÁT sin ᏺ `cz ; sin ᏹ `cz ; D c c I c c I 2 where `c lcm.2c ; 24/, as defined in the introduction. Similarly, define the WD vector valued (nonholomorphic) function G a z by c I (3.2) a Á a Á a ÁÁT G z G1 z ;G2 z c I D c I c I Â p a ÃT aÁp a Á 2 3 sin c a 2i Á 2p3 sin i`cz J 2i`cz ; J : WD c c I i`cz c I `cz The transformations in Theorem 2.1 imply that these two vector valued functions are intertwined by the generators of c. LEMMA 3.1. Assume the notation and hypotheses above. For z ވ, we have 2 a Á a Á F z 1 F z ; c I C D c I 1 Âa 1 Ã Â0 1Ã a Á a Á F 2 F z G z : p i`cz c I ` z D 1 0 c I C c I c Proof. The first transformation law follows from the simple fact that both components of F a z are given as series in q with integer exponents. The second c I transformation follows from Theorem 2.3. The Mordell vector G a z appearing in Lemma 3.1 may be interpreted in c I terms of period integrals of the theta function ‚ a . The next lemma makes c I this precise. LEMMA 3.2. Assume the notation and hypotheses above. For z ވ, we have 2 T 1 3 a 1 Á a Á 2 a Z i . i`c/ 2 ‚ ;‚ `c ; a Á i`c sin c 1 c I `c c I G z p d: c I D p3 0 i. z/ C DYSON’SRANKSANDMAASSFORMS 437

Proof. For brevity, we only prove the asserted formula for the first component of G a z. The proof of the second component follows in the same way. c I By analytic continuation, we may assume that z it with t > 0. By a change D of variables, using (2.5), we find that 3a 3a Á Z 2 cosh 2 2x cosh 1 2x a 2 1 3`c tx c c J `ct e C dx: c I `ct D 0 cosh.3x/ Using the Mittag-Leffler theory of partial fraction decompositions (see e.g.[33, pp. 134–136]), a direct calculation shows that cosh 3a 2 2x cosh 3a 1 2x c C c cosh.3x/ Á Á . 1/n sin a.6n 1/ . 1/n sin a.6n 1/ i X c C i X c C : D p 1 p 1 3 n ޚ x i n 6 3 n ޚ x i n 6 2 C 2 C By introducing the extra term 1 , we just have to consider i.n 1 / C 6 Z Â Ã ! 2 X a.6n 1/ 1 1 1 e 3`c tx . 1/n sin C dx: c 1 C 1 n ޚ x i n 6 i n 6 1 2 C C Since this expression is absolutely convergent, thanks to Lebesgue’s theorem of dominated convergence, we may interchange summation and integration to obtain

2 Â Ã Â Ã Z 3`c tx a 2 `cit X n a.6n 1/ 1 e J . 1/ sin C 1 dx: c I `ct D p c 3 n ޚ x i n 6 2 1 C For all s ޒ 0 , we have the identity 2 n f g 2 2 Z e tx Z e us 1 dx is 1 du x is D 0 pu t 1 C (this follows since both sides are solutions of @ s2f .t/ is and have @t C D pt the same limit 0 as t and hence are equal). Hence we may conclude that 7! 1 Â Ã Z .n 1=6/2u a 2 Á `ct X a.6n 1/ e J . 1/n.6n 1/ sin C 1 C du: c I `ct D p C c pu 3` t 6 3 n ޚ 0 c 2 C Substituting u 3`ci, and interchanging summation and integration (which is D allowed by Lebesgue’s theorem of dominated convergence) gives

2 a.6n 1/Á 1 3 P n 3i`c .n 6 / 2 Z i . 1/ .6n 1/ sin C e a 2 Á it`c 1 n ޚ C c C J 2 p d: c I `ct D 6 0 i. it/ C 438 KATHRIN BRINGMANN and KEN ONO

Now the claim follows since one can easily see that the sum over n coincides with deﬁnition (1.6).

To prove Theorem 1.1, we must determine the necessary modular transformation properties of the vector a Á a Á a ÁÁ S z S1 z ;S2 z c I D c I c I T 1 a 3 a 1 ÁÁ a 2 Z i ‚ `c ;. i`c/ 2 ‚ i sin c `c 1 c I c I `c p d: WD p3 z i. z/ N C a Since ‚ `cz is a cusp form, the integral above is absolutely convergent. The c I next lemma shows that S a z satisﬁes the same transformations as F a z. c I c I LEMMA 3.3. Assume the notation and hypotheses above. For z ވ, we have 2 a Á a Á S z 1 S z ; c I C D c I 1 Âa 1 Ã Â0 1Ã a Á a Á S 2 S z G z : p i`cz c I ` z D 1 0 c I C c I c Proof. Using the Fourier expansion of ‚ a z, one easily sees that c I a Á a Á S1 z 1 S1 z : c I C D c I Using classical facts about theta functions (for example, see equations (2.4) and (2.5) of[29]), we also have that a Á a Á S2 z 1 S2 z : c I C D c I Hence, it sufﬁces to prove the second transformation law. We directly compute 1 a 1 Á S 2 p i`cz c I `c z T 1 a 3 a 1 ÁÁ a 2 Z i ‚ `c ;. i`c/ 2 ‚ i sin `c c c `c c 1 I I d: 1 q 2 D p3p i`cz 2 `c z i 1=.`c z/ N 2 By making the change of variable 1=.`c /, we obtain 7! 1 a 1 Á S 2 p i`cz c I `c z T 1 3 a 1 Á a Á a 2 Z z . i`c/ 2 ‚ ;‚ ; `c i sin c `c N c I `c c p d: D p3 0 i . z/ C DYSON’SRANKSANDMAASSFORMS 439

Consequently, we obtain the desired conclusion:

1 Âa 1 Ã Â0 1Ã a Á S 2 S z p i`cz c I `c z 1 0 c I T 1 3 a 1 Á a Á a 2 Z i . i`c/ 2 ‚ ;‚ `c i sin c `c 1 c I `c c I p d D p3 0 i . z/ a Á C G z : D c I Proof of Theorem 1.1. Using (2.1a), (2.4), (2.16), and (3.1), we find that we have already determined the transformation laws satisfied by D a z. Since c I Â Ã Â ÃÂ ÃÂ 2Ã 1 0 0 1 1 1 0 1=`c 2 2 ; `c 1 D `c 0 0 1 1 0 where the first and third matrices on the right provide the same Mobius¨ transformation on ވ, the transformation laws for D a z follow from Lemma 3.1 and c I Lemma 3.3. Now we show that D a z is annihilated by c I Â 2 2 Ã Â Ã 2 @ @ iy @ @ 3 @ @ 1 y i 4y 2 py : 2 D @x2 C @y2 C 2 @x C @y D @z @z N Since q `c =24R.a q`c / is a holomorphic function in z, we get b I a @ a ÁÁ @ a ÁÁ sin c a Á D z S1 z ‚ `cz : @z c I D @z c I D p6y c I N N N @ a Hence, we find that py @z D c z is anti-holomorphic, and so N I @ @ a ÁÁ py D z 0: @z @z c I D N a To complete the proof, it suffices to show that D c z has at most linear a I exponential growth at cusps. The period integral S1 c z is convergent since a I ‚ `c is a weight 3=2 cusp form (for example, see Section 2 of[29]). This c I fact, combined with the transformation laws in Theorems 1.1 and 1.2, allow us to conclude that D a z has at most linear exponential growth at cusps. c I 3.2. Vector valued weak Maass forms of weight 1=2. Theorem 1.1 is a hint of a more general modular transformation law which holds for larger groups than c. Using Theorem 2.3, here we produce an infinite family of vector valued weak Maass forms for SL2.ޚ/. 440 KATHRIN BRINGMANN and KEN ONO

Suppose that c is a positive odd integer. For integers 0 a < c and 0 < b < c, Ä deﬁne the functions Z i a a Á i 1 ‚ c T1 z p I d; c I WD p3 z i. z/ N 3C Z i 2 a 1 a Á i 1 . i/ ‚ c T2 z p I d; c I WD p3 z i. z/ N C 5b Z i 2c 1 ‚ .a; b; c / T1 .a; b; c z/ p I d; I WD 2p3 z i. z/ N 3C 5b Z i 2 1 2c 1 . i/ ‚ a; b; c T2 .a; b; c z/ p I d: I WD 2p3 z i. z/ N C If we let tc lcm.c; 6/, then deﬁne ‚.a; b; c / by WD I X m Á 2im a Á ‚.a; b; c / . 1/ sin .2m 1/ e c 2cm 6b c; 2ctc : I WD 3 C C C I 24c2 m .mod tc/

Recall that the theta functions .˛; ˇ / are defined by (1.7). Using this notation, I define the following functions a Á a Á a Á (3.3) Ᏻ1 z ᏺ z T1 z ; c I WD c I c I a Á a Á a Á (3.4) Ᏻ2 z ᏹ z T2 z ; c I WD c I c I (3.5) Ᏻ1 .a; b; c z/ ᏺ .a; b; c z/ T1 .a; b; c z/ ; I WD I I (3.6) Ᏻ2 .a; b; c z/ ᏹ .a; b; c z/ T2 .a; b; c z/ : I WD I I These functions constitute a vector valued weak Maass form of weight 1=2. Here we recall this notion more precisely. A vector valued weak Maass form of weight k for SL2.ޚ/ is any finite set of smooth functions, say v1.z/; : : : ; vm.z/ W ވ ރ, which satisfy the following: ! Âa bÃ (1) If 1 n1 m and A SL2.ޚ/, then there is a root of unity .A; n1/ Ä Ä D c d 2 and an index 1 n2 m for which Ä Ä k vn .Az/ .A; n1/.cz d/ vn .z/ 1 D C 2 for all z ވ. 2 (2) For each 1 n m we have that vn 0. Ä Ä k D DYSON’SRANKSANDMAASSFORMS 441

If c is a positive odd integer, let Vc be the “vector” of functions deﬁned by n a Á a Á o Vc Ᏻ1 z ; Ᏻ2 z with 0 < a < c WD c I c I W S Ᏻ1.a; b; c z/; Ᏻ2.a; b; c z/ .a; b/ with 0 a < c and 0 < b < c : f I I W Ä g THEOREM 3.4. Assume the notation above. If c is a positive odd integer, then Vc is a vector valued weak Maass form of weight 1=2 for the full modular group SL2.ޚ/.

Sketch of the proof. The proof of Theorem 3.4 follows along the lines of the proof of Theorem 1.1. Therefore, for brevity here we simply provide a sketch of the proof and make key observations. As in the proof of Lemma 3.2, one first shows that Â Ã Z i a 2p3 a 2i i 1 ‚ c J p I d; iz c I z D p3 0 i. z/ C 3 a 1 a Á i Z i . i/ 2 ‚ p p 1 c 2 3 iz J 2iz p I d; c I D p3 0 i. z/ C 5b Â Ã 5b Z i 2c p3 2i 2c 1 ‚ .a; b; c / J a; b; c p I d; iz I z D 6c 0 i. z/ C 3 1 5b Z i . i/ 2 ‚ a; b; c 5bp p 2c 1 2c 3 iz J .a; b; c 2iz/ p I d: I D 6c 0 i. z/ C Arguing as in the proof of Lemma 3.3, one then establishes that the functions Ti satisfy the same transformation laws under the generators of SL2.ޚ/ as the corresponding functions ᏺ and ᏹ appearing in (3.3)–(3.6). That the functions Ᏻ1 and Ᏻ2 satisfy suitable transformation laws under SL2.ޚ/ follows easily from the “closure” of the formulas in Theorem 2.3. To complete the proof, it suffices to show that each component is annihilated by the weight 1=2 hyperbolic Laplacian 1 , and satisfies the required growth 2 conditions at the cusps. These facts follow mutatis mutandis as in the proof of Theorem 1.1.

Sketch of the proof of Theorem 1.2. By Theorem 3.4, the transformation laws of the components of the given vector valued weak Maass forms are completely a a determined under all of SL2.ޚ/. Observe that D c z is the image of Ᏻ1 c z I a I by letting z `cz. Therefore, the modular transformation properties of D c z ! a I are inherited by the modularity properties of ‚ c `c when applied to the def- a I a inition of S1 c z . By Proposition 2.1 of[29], it is known that ‚ c `c is on 2 I I 1.144fc `zc/, and the result follows. 442 KATHRIN BRINGMANN and KEN ONO

Remark. The phenomenon above where the modularity properties of a theta function imply the modular transformation laws of a Maass form was ﬁrst observed by Hirzebruch and Zagier[20]. In their work, the period integral of the classical P 2in2 Jacobi theta function ./ n ޚ e is the nonholomorphic part of their D 2 0.4/ weight 3=2 Maass form Ᏺ.z/. The modularity in Theorem 1.2 follows mutatis mutandis (see page 92 of[20]).

3.3. Proof of Theorem 1.3. Now we use Theorems 1.1 and 1.2 to prove The- orem 1.3. If 0 r < t are integers, then we begin by claiming that Ä t 1 X1 n 1 X1 n 1 X rj j (3.7) N.r; t n/q p.n/q R. q/: I D t C t t t I n 0 n 0 j 1 D D D There is just one partition of 0, the empty partition. We define its rank to be 0. Since we have X1 p.n/qn R.1 q/; D I n 0 D it follows that the right-hand side of (3.7) is t 1 1 X rj j R. q/: t t t I j 0 D Therefore the nth coefficient of this series, say a.n/, is given by t 1 t 1 1 X rj X1 mj 1 X1 X .m r/j a.n/ N.m; n/ N.m; n/ : D t t t D t t j 0 m m j 0 D D1 D1 D Equation (3.7) follows since the inner sum is t if m r .mod t/, and is 0 otherwise. Á By Theorems 1.1, 1.2, and (3.7), we obtain t 1 t 1 1 Á `t Á Á X p.n/ `t n 1 X rj j 1 X rj j N.r; t n/ q 24 S1 z D z : I t D t t t I C t t t I n 0 j 1 j 1 D D D Theorem 1.3 follows since each S1.j=t z/ is nonholomorphic. I 4. Ramanujan congruences for ranks Here we use Theorem 1.2 to prove that many of Dyson’s partition functions satisfy Ramanujan-type congruences. To prove this, we first show that “sieved” generating functions are indeed already weakly holomorphic modular forms. This observation is the content of Theorem 1.4. Armed with this observation, it is not difficult to prove Theorem 1.5. The proof is a generalization of an argument employed by the second author which proved the existence of infinitely many Ramanujan- type congruences for the partition function p.n/ [24]. DYSON’SRANKSANDMAASSFORMS 443

4.1. Sieved generating functions. To prove Theorem 1.4, we ﬁrst explicitly calculate the Fourier expansions of the Maass forms D a z. To give these ex- c I pansions, we require the incomplete Gamma-function Z 1 t a 1 (4.1) .a x/ e t dt: I WD x PROPOSITION 4.1. For integers 0 < a < c, we have

a Á `c X1 X1 am ` n `c D z q 24 N.m; n/ q c 24 c I D C c n 1 m D1 1 D a 2 i sin c `c C p3 Â Ã X m a.6m 1/ X ` n2 . 1/ sin C .c; y n/q zc ; c I m .mod fc / n 6m 1 .mod 6fc / Á C where i sign.n/ 1 2 Á .c; y n/ 4`cn y : I WD q 2I z 2`zc Proof. It suffices to compute the Fourier expansion of the period integral a S1 z . By definition, we find that c I 1 a 2 Â Ã a Á i sin c `c X m a.6m 1/ S1 z . 1/ sin C c I D p3 c m .mod fc / 2 Z i 2in `c X 1 ne z p d: z i. z/ n 6m 1 .mod 6fc / Á C C To complete the proof, one observes that

2 i 2in `c Z z 2 1 ne `c n p d .c; y n/ qz : z i. z/ D I C This integral identity follows by the following changes of variable

2 2 2 Z i 2in `c Z i 2in `c n . z/ 1 ne z 1 ne z p d d z i. z/ D 2iy p i C 2in2` .iu z/ Z ne zc i 1 du D 2y pu 2 2n `c u 2 Z z `c n 1 e inqz du: D 2y pu 444 KATHRIN BRINGMANN and KEN ONO

Proof of Theorem 1.4. If f .z/ is a function on the upper half-plane, 1 ޚ, 2 2 Âa bÃ and GL .ޒ/, then we deﬁne the usual slash operator by c d 2 2C Â Ã Â Ã ˇ a b az b (4.2) f .z/ˇ .ad bc/ 2 .cz d/ f C : c d WD C cz d C a Suppose that 0 < a < c are integers, where c is odd. Since S1 z is the c I period integral of a cusp form, and since R.a q/ has no poles in the upper half of c I the complex plane (which is easily seen by comparing with (1.3), a function with no poles in the upper half plane), it follows that D a z has no poles on the upper c I half of the complex plane. Furthermore, suppose that ᏼ − 6c is prime. For this prime ᏼ, let

ᏼ 1 ! X v 2iv g e ᏼ WD ᏼ v 1 D be the usual Gauss sum with respect to ᏼ. Define the function D a z by c I ᏼ ᏼ 1 a Á g X a Á v Á v 1 ᏼ (4.3) D z ᏼ D z 1 : c I ᏼ WD ᏼ c I j 2 0 1 v 1 D By construction, D a z is the ᏼ quadratic twist of D a z. In other words, c I ᏼ c I the nth coefficient in the q-expansion of D a z is n times the nth coefficient c I ᏼ ᏼ of D a z. That this holds for the nonholomorphic part follows from the fact that c I the factors .c; y n/ appearing in Proposition 4.1 are fixed by the transformations I in (4.3). Generalizing the classical argument on twists of modular forms in the obvious a way (for example, see Proposition 17 of[21]), D c z ᏼ is a weak Maass form of 2 2 I weight 1=2 on 1.144fc `zcᏼ /. By Proposition 4.1, it follows that ! a Á `c a Á (4.4) D z z D z c I ᏼ c I ᏼ

2 2 is a weak Maass form of weight 1=2 on 1.144fc `zcᏼ / with the property that ` ᏼ2n2 its nonholomorphic part is supported on summands of the form q zc . These terms are annihilated by taking the ᏼ-quadratic twist of this Maass form. Conse- quently, by the discussion above, we obtain a weakly holomorphic modular form 2 4 of weight 1=2 on 1.144fc `zcᏼ /. Thanks to (4.4), the conclusion of Theorem 1.4 follows easily by arguing as in the proof of Theorem 1.3.

4.2. Ramanujan-type Congruences. Here we use Theorem 1.4 and facts about eigenvalues of Hecke operators to prove Theorem 1.5. Basically, the result follows DYSON’SRANKSANDMAASSFORMS 445 from the general phenomenon that coefficients of weakly holomorphic modular forms satisfy Ramanujan-type congruences. This phenomenon was first observed by the second author in his first work on Ramanujan congruences for p.n/ [24]. Subsequent generalizations of this argument appear in[1],[2],[23],[30]. Since this strategy is now quite well known, for brevity we only offer sketches of proofs. To prove Theorem 1.5, we shall employ a recent general result of Treneer [30], which generalizes earlier works by Ahlgren and the second author on weakly holomorphic modular forms. In short, Theorem 1.4, combined with her result, reduces the proof of Theorem 1.5 to the fact that any finite number of half-integral weight cusp forms with integer coefficients are annihilated modulo a fixed prime power by a positive proportion of half-integral weight Hecke operators. The following theorem is easily obtained by generalizing the proof of Theo- rem 2.2 of[25].

THEOREM 4.2. Suppose that f1.z/; f2.z/; : : : ; fs.z/ are half-integral weight cusp forms where

f .z/ S 1 . .4N // ᏻ ŒŒq; i i 1 i K 2 C 2 \ and where ᏻK 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 2 j fi .z/ T .L / 0 .mod Q /; j i Á 2 for each 1 i s, has positive Frobenius density. Here Ti .L / denotes the usual 2 Ä Ä 1 L index Hecke operator of weight i . C 2 Sketch of the proof. By the commutativity of the Hecke operators of integer and half-integral weight under the Shimura correspondence[29], it suffices to show that a positive proportion of primes L have the property that

j Sh.fi / T .L/ 0 .mod Q /; j 2i Á for each 1 i s. Here Sh.fi / denotes the image of fi .z/ under the Shimura Ä Ä correspondence, and T2i .L/ denotes the usual Lth weight 2i Hecke operator. Theorem 2.2 of[25] ensures that the set of such primes L has positive Frobenius density provided that a single such prime L − lcm.4; Q; N1;:::;Ns/ exists. That such primes L exist is essentially a classical observation of Serre (for example, see 6 of[28]).

Two remarks. (1) The primes L in Theorem 4.2 may be chosen to lie in the arithmetic progression L 1 .mod lcm.4; Q; N1;:::;Ns//. Á (2) Strictly speaking, Serre only states his observations for integer weight modular forms on a congruence subgroup 0.N / with ﬁxed Nebentypus and ﬁxed weight. 446 KATHRIN BRINGMANN and KEN ONO

To verify the claim, one examines the Q-adic Galois representation ; WD ˚f f where the indices f walk over all the weight 2i newforms with Nebentypus whose levels divide 4Ni . By the Chebotarev Density Theorem, the claim follows since the number of such f is ﬁnite, and the fact that each f is odd and has the property that their corresponding traces of Frobenius elements Q-adically interpolate the Hecke eigenvalues of f .

Sketch of the proof of Theorem 1.5. Suppose that ᏼ − 6tQ is prime. By Theorem 1.4, for every 0 r < t Ä X1 F .r; t; ᏼ z/ a.r; t; ᏼ n/qn I D I n 1 D X p.n/Á ` n `t N.r; t n/ q t 24 WD I t 24`t n `t 24`t . / . z / ᏼ D ᏼ 2 4 is a weakly holomorphic modular form of weight 1=2 on 1.144ft `zt ᏼ /. Fur- thermore, by the work of Ahlgren and the second author[2], it is known that

X1 n X ` n `t (4.5) P.t; ᏼ z/ p.t; ᏼ n/q p.n/q t 24 I D I WD n 1 24`t n `t 24`t D . / . z / ᏼ D ᏼ 4 is a weakly holomorphic modular form of weight 1=2 on 1.576`zt ᏼ /. In partic- 2 4 ular, observe that all of these forms are modular with respect to 1.576ft `zt ᏼ /. − 2 4 Now since Q 576ft `zt ᏼ , a recent result of Treneer (see Theorem 3.1 of [30]), generalizing earlier observations of Ahlgren and Ono[2],[3],[24], implies that there is a sufﬁciently large integer m for which X a.r; t; ᏼ Qmn/qn; I Q−n for all 0 r < t, and Ä X p.t; ᏼ Qmn/qn I Q−n are all congruent modulo Qj to forms in the graded ring of half-integral weight 2 4 2 cusp forms with algebraic integer coefﬁcients on 1.576ft `zt ᏼ Q /. Theorem 4.2 applies to these t 1 forms, and it guarantees that a positive C proportion of primes L have the property that these t 1 half-integral weight C cusp forms modulo Qj are annihilated by the index L2 half-integral weight Hecke DYSON’SRANKSANDMAASSFORMS 447 operators. Theorem 1.5 now follows mutatis mutandis as in the proof of Theorem 1 of[24] (see the top of page 301 of[24]).

Remark. Treneer states her result for weakly holomorphic modular forms on 0.4N / with Nebentypus. We are using a straightforward extension of her result to 1.4N / which is obtained by decomposing such forms into linear combinations of forms with Nebentypus. It is not difficult to produce such decompositions involving algebraic linear combinations of modular forms whose Fourier coefficients are algebraic integers (which is important when proving congruences). For example, one 12 can multiply each such form by a suitable odd power of .24z/ S 1 .0.576/; / 2 2 to obtain an integer weight cusp form with integer coefficients. One may rewrite such forms as an algebraic linear combination of cusp forms with algebraic integer coefficients using the theory of newforms with Nebentypus. Then divide each resulting summand by the original odd power of .24z/, which is nonvanishing on ވ, to obtain the desired decomposition into weakly holomorphic forms with Nebentypus.

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(Received October 10, 2005) (Revised February 22, 2007)

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