l-adic PROPERTIES OF THE PARTITION FUNCTION

Transcription

1 -ADIC PROPERTIES OF THE PARTITION FUNCTION AMANDA FOLSOM, ZACHARY A. KENT, AND KEN ONO (Appendix by Nick Ramsey) Ceebrating the ife of A. O. L. Atkin Abstract. Ramanujan s famous partition congruences moduo powers of 5, 7, and 11 impy that certain sequences of partition generating functions tend -adicay to 0. Athough these congruences have inspired research in many directions, itte is known about the -adic behavior of these sequences for primes 13. Using the cassica theory of moduar forms mod p, as deveoped by Serre in the 1970s, we show that these sequences are governed by fracta behavior. Moduo any power of a prime 5, these sequences of generating functions -adicay converge to inear combinations of at most many specia q-series. For {5, 7, 11} we have = 0, thereby giving a conceptua expanation of Ramanujan s congruences. We use the genera resut to revea the theory of mutipicative partition congruences that Atkin anticipated in the 1960s. His resuts and observations are exampes of systematic infinite famiies of congruences which exist for a powers of primes since = 1. Answering questions of Mazur, in the Appendix we give a new genera theorem which fits these resuts within the framework of overconvergent haf-integra weight p-adic moduar forms. This resut, which is based on recent works by N. Ramsey, is due to Frank Caegari. 1. Introduction and statement of resuts A partition of a positive integer n is any nonincreasing sequence of positive integers which sum to n. The partition function p(n), which counts the number of partitions of n, defines a provocative sequence of integers: 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002,...,..., p(100) = ,..., p(1000) = ,.... The study of p(n) has payed a centra roe in number theory. Indeed, Hardy and Ramanujan invented the circe method in anaytic number theory in their work on p(n) asymptotics. In terms of congruences, p(n) has served as a testing ground for fundamenta constructions in the theory of moduar forms. Indeed, some of the deepest resuts on partition congruences have been obtained by making use of moduar equations, Hecke operators, Shimura s correspondence, and the Deigne-Serre theory of -adic Gaois representations. The authors thank the American Institute of Mathematics in Pao Ato for their generous support. The first and third authors thank the NSF for its generous support. The third author is aso gratefu for the support of the Hidae Foundation, the Manasse famiy, and the Cander Fund. 1

2 2 AMANDA FOLSOM, ZACHARY A. KENT, AND KEN ONO Here we revisit the theory of Ramanujan s ceebrated congruences, which assert that p(5 m n + δ 5 (m)) 0 (mod 5 m ), p(7 m n + δ 7 (m)) 0 (mod 7 m/2 +1 ), p(11 m n + δ 11 (m)) 0 (mod 11 m ), where 0 < δ (m) < m satisfies the congruence 24δ (m) 1 (mod m ). To prove these congruences, Atkin, Ramanujan, and Watson [6, 29, 30, 35] made use of specia moduar equations to produce -adic expansions of the generating functions ( ) b n + 1 (1.1) P (b; z) := p q n n=0 (note that q := e 2πiz throughout, p(0) = 1, and p(α) = 0 if α < 0 or α Z). Litte is known about the -adic properties of the P (b; z), as b +, for primes 13. We address this topic, and we show, despite the absence of moduar equations, that these functions are nicey constrained -adicay. They are sef-simiar, with resoution that improves as one zooms in appropriatey. Throughout, if 5 is prime and m 1, then we et m 2 if 5 and 7, (1.2) b (m) := m 2 if = 7 and m is even, m(m + 1) if = 7 and m is odd. To iustrate the genera theorem (see Theorem 1.2), we first highight the phenomenon for powers of the primes Theorem 1.1. Suppose that 5 31 is prime, and that m 1. If b 1 b 2 (mod 2) are integers for which b 2 > b 1 b (m), then there is an integer A (b 1, b 2, m) such that for every non-negative integer n we have ( b 2 ) ( n + 1 b 1 ) n + 1 p A (b 1, b 2, m) p (mod m ) If {5, 7, 11}, then A (b 1, b 2, m) = 0. Remark. Boyan and Webb [12] have recenty engthened the range on b in Theorem 1.1. They show that one can et b (m) := 2m 1. Numerics show that this is best possibe bound. Exampe. Here we iustrate Theorem 1.1 with = 13. For m = 1, Theorem 1.1 appies for every pair of positive integers b 1 < b 2 with the same parity. We et b 1 := 1 and b 2 := 3. It turns out that A 13 (1, 3, 1) = 6, and so we have that p(13 3 n ) 6p(13n + 6) (mod 13). By direct cacuation, we find that 6 p(13n + 6)q n = q q q q q n= q + 5q q 3 + 7q q (mod 13),

3 and -ADIC PROPERTIES OF THE PARTITION FUNCTION 3 p(13 3 n )q n = n= q q + 5q q 3 + 7q q (mod 13). We zoom in and consider m = 2. It turns out that b 1 := 2 and b 2 := 4 satisfy the concusion of Theorem 1.1 with A 13 (2, 4, 2) = 45, which in turn impies that p(13 4 n ) 45p(13 2 n + 162) (mod 13 2 ). For n = 0, 1, and 2, we find that the smaer partition numbers give 45p( ) = (mod 13 2 ), 45p( ) = (mod 13 2 ), 45p( ) = (mod 13 2 ). Athough the other partition numbers are way too arge to give here, we find p( ) = (mod 13 2 ), p( ) = (mod 13 2 ), p( ) = (mod 13 2 ). Theorem 1.1 foows from our genera theorem. To make this precise, we reca Dedekind s eta-function (1.3) η(z) := q 1 24 q n=1(1 n ) = ( 1) k q 3k2 +k k Z If 5 is prime and m 1, then we et k (m) := m 1 ( 1). In 3.3 we consider the action of a specia aternating sequence of operators appied to S k (m) Z[[q]], the space of weight k (m) cusp forms on SL 2 (Z) with integer coefficients. We define Ω (m) to be the Z/ m Z-modue of the reductions moduo m of those forms which arise as images after appying at east the first b (m) operators. We bound the dimension of Ω (m) independenty of m, and we reate the partition generating functions to the forms in this space. Theorem 1.2. If 5 is prime and m 1, then Ω (m) is a Z/ m Z-modue with rank Moreover, if b b (m), then we have that where F (b; z) Ω (m). P (b; z) F (b;z) η(z) (mod m ) if b is even, F (b;z) η(z) (mod m ) if b is odd, Five remarks. (1) As the proof wi show, each form F (b; z) Ω (m) is congruent moduo to a cusp form in S 1 Z[[q]]. Since these spaces are trivia for {5, 7, 11}, Theorem 1.2 for these foows

4 4 AMANDA FOLSOM, ZACHARY A. KENT, AND KEN ONO immediatey from the Ramanujan congruences. Conversey, if {5, 7, 11} and m 1, then the proof of Theorem 1.2 wi show that for b b (m) that p( b n + δ (b)) 0 (mod m ). We do not see how to obtain the fu strength of the Ramanujan congruences using ony the ideas in the proof of Theorem 1.2. The extra information provided by the moduar equations empoyed by Atkin, Ramanujan, and Watson seem to be necessary for this task. (2) As mentioned earier, Boyan and Webb [12] have shown that one can et b (m) := 2m 1. (3) Theorem 1.2 appies to every partition number with the exception of p(0) = 1. If n is positive, then the partition number p(n) occurs (at east with b = 1 and m = 1) for every prime 5 which divides 24n 1. Obviousy, there are such for every positive n. (4) Theorem 1.2 shows that the partition numbers are sef-simiar -adicay with resoutions that improve as one zooms in propery using the stochastic process which defines the P (b; z). Indeed, the P (b; z) (mod m ), for b b (m), form periodic orbits. Theorem 1.2 bounds the corresponding Hausdorff dimensions, and these dimensions ony depend on. For {5, 7, 11}, the dimension is 0, a fact that is beautifuy iustrated by Ramanujan s congruences, and for 13 23, the dimension is 1. Theorem 1.1 summarizes these observations for 5 23 and the proof wi show how to incude the primes = 29 and 31. (5) In October 2010 Mazur [24] asked the third author questions about the modues Ω (m) (see Section 7). Caegari has answered some of these questions by fitting Theorem 1.2 into the theory of overconvergent haf-integra weight p-adic moduar forms as deveoped in the recent works of Ramsey. The Appendix to this paper by Ramsey incudes a detaied discussion of this resut. Theorem 1.2 is inspired by the famous work of Atkin and O Brien [7, 8, 10] from the 1960s. Their papers suggested the existence of a richer theory of partition congruences than was known at the time. Athough Ramanujan s congruences had aready been the subject of many works (for exampe, see [4, 6, 7, 10, 21, 25, 26, 35, 36] to name a few), 1 mathematicians had itte uck in finding any further partition congruences. Then Atkin and O Brien [7, 10] surprisingy produced congruences moduo the primes For exampe, Atkin proved that p( n ) 0 (mod 19). In the ate 1990s, the third author revisited their work using -adic Gaois representations and the theory of haf-integra weight moduar forms [27], and he proved that there are such congruences moduo every prime 5. Ahgren and the third author [1, 3] ater extended this to incude a modui coprime to 6. Other recent works by the third author and Lovejoy, Garvan, Weaver 2, and Yang [16, 22, 36, 37] give more resuts aong these ines, further removing much of the mystery behind the wid congruences of Atkin and O Brien. Despite this new knowedge, one important enigmatic probem about p(n) in Atkin s program on congruence Hecke operators has remained open. In [7] he writes: The theory of Hecke operators for moduar forms of negative dimension [i.e. positive weight] shows that under suitabe conditions their Fourier coefficients possess mutipicative properties... I 1 Ramanujan s congruences have continued to inspire research. Indeed, the subject of ranks and cranks represents a different thread in number theory which has grown out of the probem of trying to better understand partition congruences (for exampe, see [5, 11, 13, 14, 15, 17, 20, 23] to name a few). 2 Weaver found many new congruences using ideas simiar to those in [7].

5 -ADIC PROPERTIES OF THE PARTITION FUNCTION 5 have overwheming numerica evidence, and some theoretica support, for the view that a simiar theory exists for forms of positive dimension [i.e.negative weight] and functions... ; the mutipicative properties being now congruentia and not identica. Remark. Guerzhoy [18, 19] has confirmed this specuation for eve 1 moduar functions using the theory of integer weight p-adic moduar forms as deveoped by Hida, and refined by Wan. For negative haf-integra weights, Atkin offered p(n) as evidence of this theory. Contrary to conventiona thinking, he suspected that the P (b; 24z) (mod m ), where the b, m +, converge to Hecke eigenforms for = 13 and 17. Since the P (b; 24z), as m +, ie in spaces whose dimensions grow exponentiay in m, Atkin beieved in the existence of a theory of congruence Hecke operators, one which depends on but is independent of m. To be precise, Atkin considered the weight 1 Hecke operator with Nebentypus χ 2 12( ) = ( ) 12. Reca that if λ is an integer and c is prime, then the Hecke operator T (c 2 ) on the space of forms of weight λ + 1 with Nebentypus χ is defined by 2 (1.4) ( ) a(n)q n T (c 2 ) := ( ( ) ) ( 1) a(c 2 n) + c λ 1 λ n χ(c) a(n) + c 2λ 1 χ(c 2 ) a(n/c 2 ) q n, c n n where a(n/c 2 ) = 0 if c 2 n. Atkin and O Brien found instances in which these series, as b varies, behave ike Hecke eigenforms moduo increasing powers of 13 and 17. For 13 (see Theorem 5 of [10])) they prove this observation moduo 13 and 13 2, and for 17 Atkin caims (see 6.3 of [7]) to have a proof moduo 17, 17 2, and Here we confirm Atkin s specuation for the primes 31 by decorating Theorem 1.2 with the theory of Hecke operators. Theorem 1.3. If 5 31 and m 1, then for b b (m) we have that P (b; 24z) (mod m ) is an eigenform of a of the weight k (m) 1 2 Hecke operators on Γ 0(576). As an immediate coroary, we have the foowing congruences for p(n). Coroary 1.4. Suppose that 5 31 and that m 1. If b b (m), then for every prime c 5 there is an integer λ (m, c) such that for a n coprime to c we have ( ) ( ) b nc b nc + 1 p λ (m, c)p (mod m ) Remark. Atkin [7] found such congruences moduo 13 2, 17 3, 19 2, 23 6, 29, and 31. To obtain the resuts in this paper, we begin in 2 by defining a sequence of distinguished moduar functions. By construction, these functions contain the P (b; z) as canonica factors. In 3 we briefy reca some important facts from the theory of moduar forms moduo as deveoped by Serre, and we study a specia sequence of operators to define a specia space Ω (m). In Section 4 we reate the P (b; z) to forms in Ω (m), which aows us to prove Theorem 1.2. In 5 we prove Theorem 1.3 using recent work of Ahgren and Boyan [2], Garvan [16] and Yang [37], and in 6 we concude with a discussion of some exampes.

6 6 AMANDA FOLSOM, ZACHARY A. KENT, AND KEN ONO Acknowedgements The authors thank Scott Ahgren, Matt Boyan, Kathrin Bringmann, Frank Caegari, Frank Garvan, Marie Jameson, Kar Mahburg, Barry Mazur, Robert Lemke Oiver, Christee Vincent, and John Webb for their hepfu comments on earier versions of this paper. The authors are gratefu to Frank Caegari for aowing us to incude a detaied account of his observations as an Appendix to this paper. 2. Partition generating functions For every prime 5 we define a sequence of q-series that naturay contain the generating functions P (b; z) as factors. Throughout, suppose that 5 is prime, and et (2.1) Φ (z) := η(2 z) η(z). We reca Atkin s U()-operator ( (2.2) a(n)q n) U() := a(n)q n, and we define D() by (2.3) f(z) D() := (Φ (z) f(z)) U(). This paper depends on a specia sequence of moduar functions. We begin by etting (2.4) L (0; z) := 1. If b 1, we then et { (2.5) L (b; z) := L (b 1; z) U() L (b 1; z) D() We have the foowing eementary emma. Lemma 2.1. If b is a nonnegative integer, then { η(z) P (b; z) L (b; z) = η(z) P (b; z) if b is even, if b is odd. if b is even, if b is odd. Remark. Sequences ike {L (b; z)} have payed a centra roe in the papers [6, 29, 30, 35]. Proof. If F (q) and G(q) are forma power series with integer exponents, then ( F (q ) G(q) ) U() = F (q) (G(q) U()). The emma now foows iterativey by combining the definition of Φ (z) with the fact that p(n)q n 1 = 1 q. n n=0 As usua, et M k (Γ 0 (N)) denote the space of weight k hoomorphic moduar forms on Γ 0 (N). We et M! k (Γ 0(N)) denote the space of weight k weaky hoomorphic moduar forms on Γ 0 (N), those forms whose poes (if any) are supported at the cusps of Γ 0 (N). When N = 1 we use the notation M k and M! k. We have the foowing crucia emma about the q-series L (b; z). n=1

7 -ADIC PROPERTIES OF THE PARTITION FUNCTION 7 Lemma 2.2. If b is a nonnegative integer, then L (b; z) is in M! 0(Γ 0 ()) Z[[q]]. In particuar, if b 1, then L (b; z) vanishes at i, and its ony poe is at the cusp at 0. Proof. Using standard facts about Dedekind s eta-function (for exampe, see Theorem 1.64 and 1.65 of [28]), it foows that Φ (z) M! 0(Γ 0 ( 2 )). It is we known that the U()-operator satisfies U() : M! 0(Γ 0 ()) M! 0(Γ 0 ()). Moreover, Atkin and Lehner (see Lemma 7 of [9]) prove that U() : M! 0(Γ 0 ( 2 )) M! 0(Γ 0 ()). The emma now foows from the recursive definition of the L (b; z) and the observation that In other words, Φ (z) vanishes at i. Φ (z) = q The space Ω (m) We sha appy the theory of moduar forms mod to define and study a distinguished space of moduar forms moduo m, a space we denote by Ω (m). It wi turn out that Ω (m) contains arge ranges of the L (b; z) (mod m ) Moduar forms moduo. We begin by recaing and deriving severa important facts about eve 1 integer weight moduar forms moduo. We begin with the foowing we known fact about moduar form congruences (for exampe, see 1.1 of [33] or p. 6 of [28]). Lemma 3.1. [p. 198 of [33]] Suppose that f 1 M k1 Z[[q]] and f 2 M k2 Z[[q]]. If 5 is prime, f 1 0 (mod ), and f 1 f 2 (mod m ), then k 1 k 2 (mod m 1 ( 1)). Suppose that 5 is prime. If f is a moduar form with integer coefficients, then define ω (f), the fitration of f moduo, by (3.1) ω (f) := inf k 0 {k : f g (mod ) for some g M k Z[[q]]}. The foowing emma sha pay a centra roe in the proof of Theorem 1.2. Lemma 3.2. If 5 is prime and f M k Z[[q]], then the foowing are true: (1) We have that ω (f U()) + (ω (f) 1). (2) If ω (f) = 1, then ω (f U()) = 1. (3) If ω (f) = d( 1), where d 2, then ω (f U()) (d 1)( 1). (4) If ω (f) = 1 and (z) := η(z) 24 S 12, then ω ( ( (z) 2 1 ) 24 f) U() {0, 1}. Remark. Lemma 3.2 (2) says that U() is a bijection on weight 1 moduar forms moduo. Proof. Caims (1) and (2) constitute Lemme 2 on p. 213 of [33]. Caim (3) foows from Théorème 6 (i) on p. 212 of [33]. To prove (4), we note that for = 5, the hypothesis is vacuousy true. Indeed, the space S 4 is empty, so f must be congruent moduo 5 to an Eisenstein series of weight 4. But a

8 8 AMANDA FOLSOM, ZACHARY A. KENT, AND KEN ONO Eisenstein series of weight 4 with integer coefficients have fitration zero by the Causen-von Staudt congruences. To prove (4) for 7, we note that Lemma 3.2 (1) impies that ( ) ω ( f) U() < 2( 1). 2 By Lemma 3.1, the fitration must be a mutipe of 1, and so we obtain (4) Some consequences. Here we appy the facts from the previous subsection to study the fitrations of specia sequences of moduar forms. To this end, suppose that 5 is prime, and suppose that f(0; z) M k (m) Z[[q]]. In anaogy with (2.5), if b 1, then define f(b; z) by { f(b 1; z) T () if b is even, (3.2) f(b; z) := (Φ (z) f(b 1; z)) T () if b is odd. Here T () is the usua th Hecke operator of weight k (m). Remark. We aso note that since (z) := η(z) 24, we have the simpe congruence Φ (z) (z) (mod ). We obtain the foowing proposition concerning the fitrations of these forms in the specia case where each f(b; z) (mod ) is the reduction moduo of a form in M k (m) Z[[q]]. Proposition 3.3. Assume the notation above, and suppose that each f(b; z) (mod ) is the reduction moduo of a form in M k (m) Z[[q]]. If b b (m)/m, then ω (f(b; z)) 1. Proof. On moduar forms moduo, we have that T () = U(), and so we may use the resuts of the ast section. To ease notation, we et ω b := ω (f(b; z)). We begin by showing that for integers 0 s m 1, we have the inequaity (3.3) ω s ( m 1 s + 2 ) ( 1). The inequaity triviay hods for s = 0 since f(0; z) M k (m). Now suppose that (3.3) is true for 0 s < m 1. Then we appy Lemma 3.2 (1) to obtain ω s+1 + ω s ( m 1 (s+1) + 2 ) ( ) 7 ( 1) + ( 1), 2 where we have taken the right hand side to be the maximum with respect to both operators U() and D(). Lemma 3.1 tes us that the fitration must be an integer mutipe of 1, and so the inequaity (3.3) immediatey foows since (7 )/2 < 1 for 5. Appying inequaity (3.3) for s = m 1, we find that ω m 1 3( 1). If m is even, then 3( 1) 1 ω m + = ( 1) + 4 ( 1), which gives the resut for 5. On the other hand, if m is odd, then ω m + 3( 1) ( ) = ( 1) + ( 1), 2 which gives the resut for 11. For = 5, we have ω m 2( 1) and, anaogous to the proof of Lemma 3.2 (4), this impies that ω m = 0. For = 7, we have ω m 2( 1), and by Lemma 3.2 (3), it foows that ω m+1 1.

9 -ADIC PROPERTIES OF THE PARTITION FUNCTION 9 The argument thus far estabishes the truth of the proposition for b = b (m)/m, so by Lemma 3.2 (2) and (4), the proposition hods for a b b (m)/m A specia sequence of operators and Ω (m). We consider the aternating sequence of operators X := {D(), U(), D(), U(), D(), U(),... }. To ease notation, we et G (0; z) := G(z), and for b 1 we et { G (b 1; z) U() if b is even, (3.4) G (b; z) := G (b 1; z) D() if b is odd. We say that G(z) S k (m) Z[[q]] is good for (, m) if for each b b (m) we have that G (b; z) is the reduction moduo m of a cusp form in S k (m) Z[[q]]. It wi turn out that each L (b; z), for b b (m), is the reduction moduo m of a cusp form in S k (m) Z[[q]]. This fact then guarantees that Proposition 3.3 can be appied to prove Theorem 4.3, our main technica resut about the forms L (b; z). Remark. We stress again that U() T () (mod m ) on these spaces. We define the space Ω (m) to be the Z/ m Z-modue generated by the set (3.5) {G (b; z) (mod m ) : where b b (m) and G(z) is good for (, m)}. Theorem 3.4. If 5 and m 1, then Ω (m) is a Z/ m Z-modue with rank Proof. Let G(z) be an eement of Ω (m). The space S 1 Z[[q]] is we known to be a Z-modue of rank d := 1. By Proposition 3.3, it foows that the reduction of the forms in Ω 12 (m) moduo gives a subspace of S 1 Z[[q]] (mod ), which is a Z/Z-modue with rank d. Let B (1; z),..., B (d (1); z), where d (1) d, be eements of Ω (m) which form a basis of these reductions moduo. Therefore, G(z) (mod ) is in the Z/Z-span of these forms, and so it foows that G 1 (z) := G(z) d (1) i=1 α 1 (i)b (i; z) 0 (mod ) for certain α 1 (i) Z/Z. Now we consider 1 G 1(z) (mod ). If this function is not in the Z/Z-span of the B (1; z),..., B (d (1); z), then there are forms B (d (1) + 1; z),... B (d (2); z) S k (m) Z[[q]] for which is ineary independent over Z/ m Z, and where {B (1; z),..., B (d (2); z)} B (d (1) + 1; z) B (d (2); z) 0 (mod ). These forms a ie in Ω (m), and, of course, we have that d (2) d. We then have that d (1) G(z) (α 1 (i) + α 2 (i))b (i; z) + i=1 d (2) i=d (1)+1 α 2 (i)b (i; z) (mod 2 ), where the α j (i) are in Z/Z. The proof foows now in an obvious way.

10 10 AMANDA FOLSOM, ZACHARY A. KENT, AND KEN ONO 4. The forms L (b; z) and Ω (m) Here we appy the previous resuts to prove Theorem 1.2. First we require some preiminaries invoving the -adic properties of the functions L (b; z) Basic -adic properties of the L (b; z). To prove Theorem 1.2, we sha reate the L (b; z) to forms in Ω (m). We begin with some preiminary facts about these eve moduar forms. Lemma 4.1. If 5 is prime and A (z) := η(z) /η(z), then for every m 1 we have that A (z) 2m 1 M k (m)(γ 0 ()), and satisfies the congruence A (z) 2m 1 1 (mod m ). Proof. Using facts about Dedekind s eta-function (for exampe, see Theorem 1.64 and 1.65 of [28]), it foows that A (z) is a weight ( 1)/2 hoomorphic moduar form on Γ 0 () with Nebentypus ( ). To compete the proof, notice that the caimed congruence foows easiy from and the fact that (1 X) (1 X ) 1 (mod ), (1 + Ψ(q)) m 1 1 (mod m ), where Ψ(q) = n=1 a(n)qn is a power series with integer coefficients. For a fixed m 1, we define weight k (m) auxiiary forms L (b; z) and L (b; z) for b 0 by (4.1) L (b; z) := L (b; z) A (z) 2m 1, and (4.2) L (b; z) := L (b; z) U() + k (m) 2 1 L (b; z) k (m) W. Here we used the usua weight k sash operator, which is defined for γ = ( a b c d ) by (f k γ)(z) := (det γ) k 2 (cz + d) k f(γz), with the matrix W := ( ). To use (4.2), we must contro k (m) 1 2 L (b; z) k (m) W (mod m ). The next proposition, whose proof mirrors a cacuation of Ahgren and Boyan (see 5 of [2]), suffices for our work. In what foows, we define B (b, m) by B (b, m) := { 2( m m)/3 2/3 if b is even, 2( m m)/3 1 if b is odd. Proposition 4.2. If b B (b, m) is a nonnegative integer, then we have that Proof. It is we known that k (m) 2 1 L (b; z) k (m)w 0 (mod m ). (4.3) η( 1/z) = z/i η(z) and η(z + 1) = α η(z),

11 -ADIC PROPERTIES OF THE PARTITION FUNCTION 11 where α is a 24-th root of unity. Using the definition of W and appying (4.3), we find ( ( )) 2 m 1 A (z) 2m 1 k (m)w = (1 )m 1 η 1 ( ) 2 z (1 )m 1 z η ( ) = ( i) m m 1 m + m 1 η 2 m 1 (z) 2. 1 η(z) z Let ζ ν := exp(2πi/ν). In view of this identity, by Lemma 4.1 and (4.1), it suffices to show that (4.4) m 1 L (b; z) 0 W 0 (mod m ) in an appropriate power series ring. The case when b = 0 is trivia, so we first consider the case when b = 1. We have that ( ( ) m 1 L (1; z) 0 W = m 1 η( 2 z) U() ) ( 1 W = m 2 η( 2 z) ( 1 j ) ) 0 ( η(z) 0 η(z) ) 0 (4.5) 1 = m 2 j=0 where we have used (4.3), and that (4.6) η ( 1 z η ( 1 2 z + j j=0 + j) ) = 1 m 2 z/i η(z) 1 ( f U() = k 2 1 f 1 j ) k 0. j=0 j=0 α j η ( jz 1 2 z Let 1 j 1. Because gcd(, j) = 1, there exist integers d j and b j such that b j j + d j = 1. Using this fact we can write ( ) ( ) ( ) j 1 j dj bj 2 =, 0 b j 0 where the first matrix on the right hand side is in SL 2 (Z). Thus we deduce that ( ) ( jz 1 η = ɛ 2 j, (z) 12 η z + b ) j z where ɛ j, is a 24th root of unity, and we have used the genera moduar transformation aw for η(z). Thus, we re-write (4.5) as (4.7) m 5/2 i 1 η(z) ( ɛ j, η j=1 α j z + b j ) + m 3 η(z) η( 2 z), where again we have used (4.3) to deduce the contribution arising from j = 0. One then has that the term corresponding to j in (4.7), where 1 j 1, is m 5/2 i ζ b j 24 αj (1 q n ) ɛ j, n 1 (1 ζ 24b jn 24 q n ), and the term corresponding to j = 0 in (4.7) is m 3 q 1 2 (1 q n ) 24 (1 q 2 n ). n 1 ),

12 12 AMANDA FOLSOM, ZACHARY A. KENT, AND KEN ONO These observations combined with the fact that m 5/2 > m 3 m for 5 and m 1 indicate vanishing (mod m ) in q (Z[ζ 24 ])[[q]], proving (4.4) when b = 1. Athough the detais are messier, the case for 1 < b < B (b, m) foows simiary. Indeed, for even b, we see that L(b; z) is defined by a nested sequence invoving b/2 instances of the function Φ(z), with that same number of appications of U( 2 ) as foows: (4.8) L(b; z) = (((... ( }{{} Φ 0U( 2 )) Φ) 0 U( 2 )) Φ) 0 U( 2 )) Φ... ) 0 U( 2 ) b 3 (for b even). Just ike U() introduced an 1 in (4.5) when empoying (4.6), each iteration of U( 2 ) introduces a factor 2. Moreover, one can show (as with the case b = 1) that of any additiona negative powers of that occur after mutipying (4.8) by m 1 and appying 0 W, the smaest wi be produced for j = 0 in (4.6) simutaneousy for each of the b/2 U( 2 ) operators that appear, each decreasing the power of by 1. This impies for even b, the smaest negative power of appearing is (b/2) ( 2) + (b/2) ( 1) = 3b/2. For odd b, beginning simiary as in (4.8), we find a tota of (b 1)/2 instances of the U( 2 ) operator, and one instance of the U() operator, yieding a maxima negative power of 3b/2 1/2. Thus, for even b we require m 1 3b/2 m, and for odd b we require m 3/2 3b/2 m, which are equivaent to the hypotheses given in the statement of Proposition A key theorem and the proof of Theorem 1.2. Using the resuts in the previous subsection, we can now prove the foowing crucia fact about the forms L (b; z). Theorem 4.3. If 5 is prime, m 1, and b b (m), then L (b; z) is in Ω (m). Proof of Theorem 4.3. By Lemma 2.2, we have that L (b; z) is in M 0(Γ! 0 ()) Z[[q]]. Lemma 4.1 gives us the congruence L (b; z) L (b; z) (mod m ). It is cear that L (b; z) is a weight k (m) eve weaky hoomorphic moduar form. Lemma 7 of [9] asserts that ) (U() + k (m) 2 1 W : M k! (m)(γ 0 ()) M k! (m). From Proposition 4.2, for b (m) b B (b, m), we find that L (b; z) L (b; z) U() (mod m ), and an inspection of the q-series shows that L (b; z) must be congruent moduo m to a eve 1 cusp form since L (b; z) vanishes at i by Lemma 2.2. For b (m) b B (b, m), we empoy Lemma 3.2 (2), which shows that U() defines a bijection on eve 1 weight 1 moduar forms moduo. We aso use Proposition 3.3 which determines when the fitration of such forms is 1. By appying this proposition iterativey moduo increasing powers of (as in the proof of Theorem 3.4), up to m, one obtains the caimed concusion for these b. Since U() = T () on Ω (m), to compete the proof it suffices to prove the caim for odd b > B (b, m). We make use of the fact that D() defines a Z/ m Z-inear map to Ω (m) on the submodue of Ω (m) generated by the set of functions L (b; z), where b (m) b B (b, m) is even. If there is an even b (m) < b B (b, m) for which L (b ; z) is a Z/ m Z inear combination of the previous functions in this set, then the concusion for b, and hence a even b b, foows from the iterative definition of these functions. We have (B (b, m) b (m) + 1)/2 many functions in this set, and the maximum number of eements in Ω (m) one can order before there is such a b is m 1 12 since the dimension of Ω (m) is A short cacuation using

13 -ADIC PROPERTIES OF THE PARTITION FUNCTION 13 the definition of b (m) and B (b, m) reveas that (B (b, m) b (m) + 1)/2 > m 1 12 without exception. Therefore, each L (b; z) is in Ω (m). Proof of Theorem 1.2. The resut foows immediatey from Theorem 4.3, Theorem 3.4, and Lemma 2.1. Proof of Theorem 1.1. The theorem foows triviay from the Ramanujan congruences when {5, 7, 11}. More generay, we consider the two subspaces, Ω odd (m) and Ω even (m), of Ω (m) generated by L (b; z) for odd b and even b, respectivey. We observe that appying D() to a form gives q-expansions satisfying F D() = a(n)q n. n> Combining this observation with Theorem 1.2 and the fact that the fu space Ω (m) is generated by aternatey appying D() and U(), we have that the ranks of Ω odd (m) and Ω even (m) are If 13 31, then direction cacuation, when m = 1, shows that each of these subspaces has dimension 1. The theorem now foows immediatey from Lemma The proof of Theorem 1.3 and Coroary 1.4 To prove resuts on partition congruences, we made use of the -adic properties of the sequence of specia operators. To prove Theorem 1.3 we combine these ideas with the works of Ahgren and Boyan, Garvan, and Yang reated to Ramanujan-type congruences Theorems of Ahgren-Boyan and Garvan and Yang. Here we combine the work of Ahgren and Boyan, with works by Garvan and Yang to show that our partition generating functions P (b; 24z) are congruent moduo m to haf-integra weight moduar forms in a certain Hecke invariant subspace. Define the even integer k + (m) by { k + (m) := Aso define r (m) by m 1 ( 1) 12 if m is even, ( m 1 +1)( 1) 12( /24 + 1) 2 if m is odd. r (m) := 24δ (m) 1 m. We reca a resut of Ahgren and Boyan (Theorem 3 of [2]) which gives congruence reations between our partition generating functions P (b; 24z) and a product of a power of the η-function and a eve one moduar form. Theorem 5.1 (Ahgren and Boyan). For a prime 5 and an integer m 1, there exists a moduar form F (k + (m); z) M k + (m) Z[[q]] such that P (m; 24z) = p( m n + δ (m))q 24n+r(m) η(24z) r(m) F (k + (m); 24z) (mod m ) n=0 We aso state a resut of Garvan (Proposition 3.1 of [16]) which has recenty been extended by Yang (Theorem 2 of [37]). The theorem asserts that the moduar forms appearing in the work of Ahgren and Boyan actuay ive in a very nice Hecke invariant subspace.

14 14 AMANDA FOLSOM, ZACHARY A. KENT, AND KEN ONO Theorem 5.2 (Garvan and Yang). Let 0 < r < 24 be an odd integer and s be a non-negative even integer. Then S r,s := {η(24z) r f(24z) : f(z) M s } is a Hecke invariant subspace of S s+r/2 (Γ 0 (576), χ 12 ) (i.e. for a primes c 2, 3 and F S r,s, we have F r+s/2 T (c 2 ) S r,s ) Proof of Theorem 1.3. Combining Lemma 2.1 with Theorem 4.3, arguing in a manner simiar to the proof of Theorem 1.2 and using Theorem 5.1, we find that if 5 is prime and b (m) b, then the moduar form { η(z) η(z) r(b) F (k + (b); z) if b is even, H (b; z) := η(z) η(z) r(b) F (k + (b); z) if b is odd, is congruent moduo m to a cusp form in S k (m) Z[[q]]. In particuar, we have that { η(z) P (b; z) (mod m ) if b is even, H (b; z) L (b; z) η(z) P (b; z) (mod m ) if b is odd. If c 6 is prime, then we consider P (b; z) T (c 2 ). As shown in the proof of Theorem 1.1, we may consider the even/odd parity subspaces of Ω (m), both of which were determined to have dimension 1 for Therefore it suffices to show that { η(z) (P (b; z) T (c 2 )) if b is even, K (b, c; z) := η(z) (P (b; z) T (c 2 )) if b is odd is in Ω (m). Since 0 < r (b) < 24 is odd, Theorem 5.2 appies, and one obtains the desired concusion by arguing with the expicit formuas again invoving Dedekind s eta-function exacty as in the proof of Theorem 4.3. Proof of Coroary 1.4. After etting n nc in (1.4), the concusion foows from Theorem 1.3 because ( ) nc c = Exampes Here we give exampes of Theorem 1.2 for the prime = 13. We have that Φ 13 (z) := η(169z)/η(z) = q 7 + q 8 + 2q For m = 1, we have that k 13 (1) = 12, and so if b 1, then by Theorem 4.3 we have that L 13 (b; z) is congruent moduo 13 to a weight 12 cusp form of eve 1, which of course must be a mutipe of Ramanujan s (z) = η(z) 24. The first few terms of L 13 (1; z) are L 13 (1; z) = Φ 13 (z) U(13) = 11q + 490q q q q On the other hand, we have that 11q + 9q 2 + 3q 3 + 6q q 5 + 6q (mod 13). 11 (z) = 11q 264q q q q + 9q 2 + 3q 3 + 6q q 5 + 6q (mod 13). Therefore we have that L 13 (1; z) 11 (z) (mod 13), which, by Lemma 2.1, impies that P 13 (1; z) 11 η(z) 11 (mod 13).

15 -ADIC PROPERTIES OF THE PARTITION FUNCTION 15 More generay (for exampe, see 4 of [27]), for every non-negative integer k we have that P 13 (2k + 1; z) 11 6 k η(z) 11 (mod 13), P 13 (2k + 2; z) 10 6 k η(z) 23 (mod 13). These congruences iustrate Theorem 1.1 for m = 13. For m = 2, we have, for b 4, that L 13 (b; z) is congruent moduo 169 to a form in S 156 Z[[q]]. If E 4 (z) is the usua weight 4 Eisenstein series, then one directy computes and finds that L 13 (2; z) 36q + 150q q q q q q q (mod 169) 36 E E E E E E E E E E E E (mod 169). Using Lemma 2.1, we find that P 13 (2; z) 1 η(z) (36 E E E E E E E E E E E E ) (mod 13 2 ). Theorem 1.3 is iustrated by the fact that this is a Hecke eigenform moduo 169. Concerning Theorem 1.1, we have that whie we have that P 13 (2; 24z) = q q q q q q (mod 13 2 ), 154P 13 (4; 24z) 36q q q q (mod 13 2 ). Indeed, it turns out that P 13 (4; 24z) 45P 13 (2; 24z) (mod 13 2 ), which in turn impies that p(13 4 n ) 45p(13 2 n + 162) (mod 13 2 ). 7. Appendix (By N. Ramsey) Theorem 1.2 in the main text asserts that a certain space of moduar forms moduo m is bounded in rank by a constant independent of m. These forms are obtained by repeated appication of two operators on spaces of integra weight moduar forms of eve one and increasing (and increasingy congruent) weights. One of these operators is Atkin s U = U(), whie the other is a twisted version of U obtained by pre-composing by mutipication by a moduar function of weight zero. This is the operator D(). Utimatey, one is interested in the anaog of this sort of resut in the setting of haf-integra weight moduar forms. Indeed, the twisting referred to in the previous paragraph is exacty set up to transate a very natura Hecke action in haf-integra weight to the integra weight setting under an identification of the form F F/η between haf-integra and integra weight forms. This aowed the authors to empoy Serre s theory of integra weight moduar forms moduo. It is natura to ask if one can work more directy in the haf-integra weight setting to arrive at these resuts. Indeed, we can, and in doing so we provide a very natura answer to the foowing questions of Mazur [24]. Question (Mazur). Do the spaces Ω (m) compie we to produce a cean free Z modue? Do the Hecke operators work we on these spaces?

16 16 AMANDA FOLSOM, ZACHARY A. KENT, AND KEN ONO Congruences simiar to those discussed in the main text were observed by Atkin in the coefficients of the moduar j-invariant. Atkin specuated that such congruences were part of a coherent interpay between Hecke operators and congruences (see his quote in the main text, as we as his quote preceding Theorem 1 of [19]). Since that time, a systematic theory of -adic (generay referred to as p-adic in the iterature) moduar forms of integra weight has emerged through the work of Serre, Hida, Katz, Coeman, and Mazur, among others. This theory provides a natura framework for studying this interpay between Hecke operators and congruences. Indeed, using this machinery, Atkin s specuation has been proven in many cases (see [18, 19]). In ight of the success of these methods in the integra weight setting, it is natura to frame the haf-integra weight question in the context of a theory of overconvergent -adic moduar forms of haf-integra weight. Such a theory has been deveoped by the author in [31, 32]. The most reevant features of this work wi be summarized in the next subsection. With it, we wi answer the questions of Mazur, and we sha prove the foowing generaization of Theorem 1.2. Theorem 7.1. Fix a prime, an odd integer k, a positive integer N, and a Dirichet character χ moduo 4N. There exists a modue Ω Z [χ][q ] of finite rank over Z [χ] that is preserved by a haf-integra weight Hecke operators and has the foowing property. Let F = a(n)q n be a cassica moduar form that is hoomorphic away from the cusps, and suppose that L is a number fied whose ring of integers contains the coefficients of F and the vaues of χ. Choose an embedding of L into Q. Then for a integers m and s sufficienty arge, there is a congruence a( 2s n)q n ω s (mod m ) for some ω s Ω Z [χ] O L,p, where O L,p is the competion of the ring of integers of L at the prime determined by the embedding L Q above. Here, Z [χ] denotes the ring of integers in the finite extension of Q generated by the vaues of χ (thought of as Q -vaued). Three remarks. (1) We note that, given, k, N, and χ, the modue Ω can be expicity computed. (2) If this modue vanishes, then this resut impies that the coefficients a( 2s n) tend to zero as s uniformy in n. This is exacty the situation of Watson s and Atkin s generaization of Ramanujan s congruences. (3) If the rank of Ω is 1, then since Ω is Hecke-stabe, it is spanned by a singe nonzero Hecke eigenform. This theorem then impies that the images of F under sufficienty many iterates of U 2 are congruent moduo m to mutipes of this singe form. This is the phenomenon encapsuated by Theorem 1.1 of the main text. Acknowedgements. Frank Caegari observed that the resuts in the main text fit into the genera framework deveoped by the author in [31] and [32]. Caegari outined the proof of Theorem 7.1, and the author is gratefu that he has aowed us to pubish a detaied version of this proof in Subsection 7.2 beow.

17 -ADIC PROPERTIES OF THE PARTITION FUNCTION adic moduar forms of haf-integra weight. Let K denote a compete and discreteyvaued subfied of C such as, for exampe, a finite extension K/Q. Purey for the sake of exposition we wi assume that 2 in what foows. The resuts of the primary reference [32] hod for a, but the weight and eve book-keeping are sighty different for = 2. Fix integers k and N with k odd and N positive, and assume that 4N. One shoud think of 4N as the -adic tame eve. In [31], the author defined Banach spaces M k/2 (4N, K, r) of r-overconvergent moduar forms of weight k/2 and eve 4N over K, for any r Q [0, 1]. For r < r there is a natura incusion M k/2 (4N, K, r) M k/2 (4N, K, r ). The forms in spaces with r < 1 are caed overconvergent, and the entire space of overconvergent forms of this weight and eve is the direct imit M k/2 (4N, K) = im M k/2(4n, K, r). r 1 Stricty speaking, there were further restrictions on, k, and N in [31]. However, the subsequent paper [32] defined these spaces in the current generaity, and even greater generaity with respect to the weight. Indeed, for any p-adic weight κ, a Banach space M κ (4N, K, r) is defined for sufficienty arge r < 1, and thus an overconvergent space M κ(4n, K) is defined. The nature of these more genera weights is detaied in Section 2.4 of [32]. The cassicay-minded reader shoud think of these weights roughy as incuding a k/2 with k odd (positive or negative) and information about the -part of nebentypus character at (though they are in fact much more genera). This wi be expicated in sighty more detai beow when we discuss how the cassica forms sit among the -adic forms. Despite these restrictions, it is worth mentioning that the arguments in [31] go through for genera odd k (if not for the more genera κ of [32]), as that paper is rather ess technica than [32] and may provide a more accessibe reference for readers unfamiiar with rigid-anaytic spaces. These spaces of -adic moduar forms contain the cassica moduar forms of haf-integra weight in the foowing sense. Let F be a cassica hoomorphic moduar form of weight k/2 and eve 4N s (for any integer s 0) with coefficients in a finite extension L/Q. Suppose for the moment that F is fixed by the diamond operators at, so F is actuay a moduar form for the group Γ 1 (4N) Γ 0 ( s ). Given an embedding i : L Q, the cassica form F gives rise to an -adic form in i(f) M k/2 (4N, i(l), r) for a r < 1 sufficienty cose to 1. For a genera cassica form on Γ 1 (4N s ) that is an eigenform for the diamond action at (but perhaps with non-trivia character), F aso gives rise to an -adic form, but in one of the spaces M κ (4N, i(l), r) of more genera weight κ discussed in [32]. One can think of this form as ying in a space of -adic forms with nebentypus character at, but for technica reasons it is better to package that character as part of the -adic weight κ. The reader can refer to Section 6 of [32] for the detais. The upshot is that the space of overconvergent -adic forms of tame eve 4N sees the cassica forms with arbitrariy high powers of in the eve. To any cusp on X 1 (4N) that ies over the cusp on X 0 (), and any -adic moduar form F M k/2 (4N, K, r), one may associate a q-expansion an q n whose coefficients ie in a finite extension of K (see Definition 4.1 of [31] and Section 5.3 of [32]). These q-expansions recover the cassica ones if the form f is cassica in the above sense.

18 18 AMANDA FOLSOM, ZACHARY A. KENT, AND KEN ONO These spaces are equipped with a coection of commuting continuous Hecke operators, incuding an operator U 2 having the effect (7.1) an q n a 2 nq n on q-expansions at the cusps mentioned above. More generay, by the resuts of Section 8 of [31] and Section 5 of [32], a of these operators have the same effect on q-expansions as their cassica counterparts, and therefore recover the cassica operators on cassica forms. By Theorem 8.2 of [31] and the remarks foowing Proposition 5.1 of [32], the operator U 2 is compact on the Banach spaces of overconvergent forms. This operator aso has the effect of increasing the degree of overconvergence on such forms. It foows that, for any σ R, the sope σ subspace of M k/2 (4N, K) ies in M k/2(4n, K, r) for some r < 1, and hence that this subspace is finitedimensiona by the genera theory of compact operators on p-adic Banach spaces (see [34]). Taking σ = 0, we see that the space of ordinary overconvergent forms is finite-dimensiona. With an eye towards the appications in the main text, the forms in Theorem 7.1 are aowed to have poes at the cusps. The construction of the operator U 2 given in [31] and [32] appies directy to such forms, and this operator has the effect (7.1) on the q-expansions of such forms at cusps ying over the cusp on X 0 (). Indeed, the cacuations of Theorem 8.2 of [31] and Proposition 5.5 of [32] carry over easiy to this situation. Examining the effect (7.1) on q-expansions, we see that U 2 stricty reduces the order of any poes of a form at these cusps. In particuar, if F is an -adic moduar form that is reguar except perhaps with poes at these cusps, then U t F is a genuine -adic moduar form in the above sense 2 for sufficienty arge integers t. Moreover, if F is a cassica moduar form of haf-integra weight with poes at any cusps, then we arrive at this same concusion, since the associated -adic form ony sees the cusps ying above on X 0 () in the first pace. For a Dirichet character χ tm moduo 4N, the above spaces contain a Hecke-stabe subspace of forms with tame nebentypus χ tm (denoted by appending χ tm to the ist of arguments of the space). Let F be a cassica moduar form of weight k/2, eve 4N s, and nebentypus χ. The character χ factors as χ = χ χ tm for a character χ moduo s and a character χ tm moduo 4N, and the -adic form that F gives rise to has tame nebentypus χ tm Proof of Theorem 7.1. First note that the N beow is not the N in the statement of Theorem 7.1, but rather its prime-to- part, so that we may appy to the framework of the previous subsection. Accordingy, χ tm beow is the tame part of the χ in the statement (whie χ is packaged as part of the weight κ). Let K denote the competion of the number fied L in the statement at the prime above determined by the chosen incusion L Q. Then F gives rise to an eement of the Banach space M κ (4N, K, χ tm, r) for some r < 1. Genera facts about the spectra theory of compact operators impy that the map e : F im U s! 2F s is defined on M κ (4N, K, χ tm, r) (in that the imit exists) and is the projector onto the finitedimensiona ordinary subspace of this Banach space. Since U 2 increases overconvergence, these projectors form a compatibe famiy of projections onto the same finite-dimensiona space of ordinary forms for varying r, and hence provide a projector e : M κ(4n, K, χ tm ) M κ(4n, K, χ tm ) ord

19 -ADIC PROPERTIES OF THE PARTITION FUNCTION 19 from the fu space of overconvergent forms of this weight and eve onto this space of ordinary forms. We can extend the ordinary projector e to overconvergent moduar forms with poes at the cusps as foows. As observed above, for any such form F, the form U t 2 F (which is of the same weight, tame eve and nebentypus) is a genuine moduar form for sufficienty arge t. We set e(f ) = U t eu t 2 2(F ). First note that this makes sense as U 2 is invertibe on the space of ordinary forms. It is aso independent of t (sufficienty arge) as is easiy checked. Note that, whie this projector is a natura thing to consider, a we reay need in the present discussion is that eu t (F ) ies in the 2 finite-dimensiona space of ordinary forms. The upshot of a this is that, for this fixed weight κ, tame eve 4N, and tame nebentypus χ tm there is a finite-dimensiona vector space of moduar forms, namey M κ(4n, K, χ tm ) ord, with the foowing property: For any moduar form F of this weight, eve, and tame nebentypus with coefficients in L, perhaps with poes at the cusps, there exists a positive an integer t such that im U k!+t k (F ) M κ(4n, K, χ 2 tm ) ord. Let Q (χ tm, κ) be the extension of Q generated by the vaues of χ tm and κ, which is finite in the situation of Theorem 7.1. Repacing K by its Gaois cosure over Q (χ tm, κ) if necessary, the finite-dimensiona space M κ(4n, K, χ tm ) ord carries an action of Ga(K/Q (χ tm, κ)). By Gaois descent, this space has a basis of Gaois-invariants, which is to say that it is generated over K by M κ(4n, Q(χ tm, κ), χ tm ) ord. Passing to q-expansions, we arrive at a finite-dimensiona subspace of Q (χ tm, κ)[q ], and the submodue of this space consisting of series with integra coefficients is the Ω that we seek. References [1] S. Ahgren, The partition function moduo composite integers M, Math. Ann. 318 (2000), pages [2] S. Ahgren and M. Boyan, Arithmetic properties of the partition function, Invent. Math. 153 (2003), pages [3] S. Ahgren and K. Ono, Congruence properties for the partition function, Proc. Nat. Acad. Sci., USA 98 (2001), pages [4] G. E. Andrews, The theory of partitions, Cambridge Univ. Press, Cambridge, [5] G. E. Andrews and F. Garvan, Dyson s crank of a partition, Bu. Amer. Math. Soc. 18 (1988), pages [6] A. O. L. Atkin, Proof of a conjecture of Ramanujan, Gasgow Math. J. 8 (1967), pages [7] A. O. L. Atkin, Mutipicative congruence properties and density probems for p(n), Proc. London Math. Soc. (3) 18 (1968), pages [8] A. O. L. Atkin, Congruence Hecke operators, Proc. Symp. Pure Math., Amer. Math. Soc., 12, Amer. Math. Soc., Providence, 1969, pages [9] A. O. L. Atkin and J. Lehner, Hecke operators on Γ 0 (m), Math. Ann. 185 (1970), pages [10] A. O. L. Atkin and J. N. O Brien, Some properties of p(n) and c(n) moduo powers of 13, Trans. Amer. Math. Soc. 126 (1967), pages [11] A. O. L. Atkin and H. P. F. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. 66 (1954), pages [12] M. Boyan and J. Webb, preprint. [13] K. Bringmann and K. Ono, Dyson s ranks and Maass forms, Ann. Math. 171, (2010), pages [14] F. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), pages

20 20 AMANDA FOLSOM, ZACHARY A. KENT, AND KEN ONO [15] F. Garvan, New combinatoria interpretations of Ramanujan s partition congruences mod 5, 7, and 11, Trans. Amer. Math. Soc. 305 (1988), pages [16] F. Garvan, Congruences for Andrews s smaest parts partition function and new congruences for Dyson s rank, Int. J. Numb. Th. 6 (2010), pages [17] F. Garvan, D. Kim, and D. Stanton, Cranks and t-cores, Invent. Math. 101 (1990), pages [18] P. Guerzhoy, On a conjecture of Atkin for the primes 13, 17, 19, and 23, Math. Res. Lett. 13 (2006), pages [19] P. Guerzhoy, On a conjecture of Atkin, J. Reine Angew. Math. 639 (2010), pages [20] M. Hirschhorn and D.C. Hunt, A simpe proof of the Ramanujan conjecture for powers of 5, J. Reine Angew. Math. 326 (1981), pages [21] T. Hjee and T. Kove, Congruence properties and density probems for coefficients of moduar forms, Math. Scand. 23 (1968), pages [22] J. Lovejoy and K. Ono, Extension of Ramanujan s congruences for the partition function moduo powers of 5, J. Reine Angew. Math. 542 (2002), pages [23] K. Mahburg, Partition congruences and the Andrews-Garvan-Dyson crank, Proc. Nat. Acad. Sci. USA 102 (2005), pages [24] B. Mazur, private communication. [25] M. Newman, Periodicity moduo m and divisibiity properties of the partition function, Trans. Amer. Math. Soc. 97 (1960), pages [26] M. Newman, Congruences for the coefficients of moduar forms and some new congruences for the partition function, Canadian J. Math. 9 (1957), pages [27] K. Ono, The partition function moduo m, Ann. of Math. 151 (2000), pages [28] K. Ono, The web of moduarity: arithmetic of the coefficients of moduar forms and q-series, CBMS 102, Amer. Math. Soc., Providence, [29] S. Ramanujan, Congruence properties of partitions, Proc. London Math. Soc. (2) 19, pages [30] S. Ramanujan, Ramanujan s unpubished manuscript on the partition and tau-functions (Commentary by B. C. Berndt and K. Ono) The Andrews Festschrift (Ed. D. Foata and G.-N. Han), Springer-Verag, Berin, 2001, pages [31] N. Ramsey, Geometric and p-adic moduar forms of haf-integra weight, Ann. Inst. Fourier (Grenobe) 56 (3) (2006), pages [32] N. Ramsey, The haf-integra weight eigencurve, Agebra and Number Theory 2 (7) (2008), pages [33] J.-P. Serre, Formes moduaires et fonctions zêta p-adiques, Springer Lect. Notes in Math. 350 (1973), pages [34] J.-P. Serre, Endomorphisms compètement continus des espaces de Banach p-adiques, Inst. Hautes Études Sci. Pub. Math. 12 (1962), pages [35] G. N. Watson, Ramanujan s vermutung über zerfäungsanzahen, J. reine Angew. Math. 179 (1938), pages [36] R. Weaver, New congruences for the partition function, Ramanujan J. 5 (2001), pages [37] Y. Yang, Congruences of the partition function, Int. Math. Res. Notices, to appear. Department of Mathematics, Yae University, New Haven, Connecticut E-mai address: amanda.fosom@yae.edu Department of Mathematics and Computer Science, Emory University, Atanta, Georgia E-mai address: kent@mathcs.emory.edu E-mai address: ono@mathcs.emory.edu Department of Mathematics, University of Wisconsin, Madison, Wisconsin E-mai address: ono@math.wisc.edu Department of Mathematics, DePau University, Chicago, Iinois E-mai address: naramsey@gmai.com