l-adic PROPERTIES OF PARTITION FUNCTIONS

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1 -ADIC PROPERTIES OF PARTITION FUNCTIONS EVA BELMONT, HOLDEN LEE, ALEXANDRA MUSAT, SARAH TREBAT-LEDER Abstract. Fosom, Kent, and Ono used the theory of moduar forms moduo to estabish remarkabe sef-simiarity properties of the partition function and give an overarching expanation of many partition congruences. We generaize their work to anayze powers p r of the partition function as we as Andrews s spt-function. By showing that certain generating functions reside in a sma space made up of reductions of moduar forms, we set up a genera framework for congruences for p r and spt on arithmetic progressions of the form m n + δ moduo powers of. Our work gives a conceptua expanation of the exceptiona congruences of p r observed by Boyan, as we as striking congruences of spt moduo 5, 7, and 13 recenty discovered by Andrews and Garvan. Keywords: congruences, partitions, Andrews spt-function, moduar forms, Hecke operators 1. Introduction A partition of a nonnegative integer n is a non-increasing sequence of positive integers summing to n. Letting pn denote the number of partitions of n, we have that the generating function for p is 1 1 q = pnq n. 1.1 n Ramanujan congruences such as n=1 n=0 p 5 m n + δ 5 m 0 mod 5 m, where 4δ 5 m 1 mod 5 m, have been known for quite some time. Using the theory of -adic moduar forms, Fosom, Kent, and Ono estabished a genera framework for partition congruences in [8], which not ony expains this phenomenon but aso aows them to show additiona congruences such as p13 4 n p13 n + 16 mod 13. More precisey, they defined a specia sequence of power series L b, z reated to pn, whose -adic imit resides in a finite-dimensiona space. In particuar, when the dimension of this space is zero, which happens when = 5, 7, 11, one obtains the famous congruences of Ramanujan moduo powers of. Note that the congruences obtained in [8] are systematic ones which can be controed by the use of operators. They are not to be confused with congruences of the type p n mod 13, proven by Ono in [1], which are expained by a different phenomenon. We generaize their techniques, utiizing carefu refinements due to Boyan and Webb in [6], to estabish a framework for the reated functions p r and spt, to be defined beow. 1

2 EVA BELMONT, HOLDEN LEE, ALEXANDRA MUSAT, SARAH TREBAT-LEDER 1.1. Powers of the partition function. For positive vaues of r, we et p r n denote the coefficients of the r th power of the partition generating function: 1 1 q n = p r r nq n. 1. n=1 We can think of p r n as the number of r-coored partitions of n. Congruences for p r n have been we-studied; for exampe, in [4], Atkin showed congruences for p r n moduo powers of 5, 7, 11, and 13, whie in [5], Boyan cassified Ramanujan-type congruences for p r under mid assumptions. To state our resuts, we define a sequence of generating functions reated to p r n which are anaogous to those that appear in [8] and [6], P r, b; z := n=0 n=0 p r b n + r 4 q n 4, 1.3 where q denotes e πiz. Our main theorem is that these functions, reduced moduo powers of, reside in a reativey sma space. Fixing integers b 0, m 1, foowing [6] we define Λ odd r, b, m := Span Z/ Z{ηz r P m r, β; z mod m : β b odd} 1.4 Λ even r, b, m := Span Z/ Z{ηz r P m r, β; z mod m : β b even}. 1.5 Here, ηz := q q n is Dedekind s eta-function. In Section 4 we wi define d r and d r, which are reated to the dimension of the kerne of an important operator. The foowing is our main resut concerning the partition functions p r n. Theorem 1.1. Let r + 5 be prime and et m, r 1. Then there is an integer b r, m that satisfies the foowing. 1 The nested sequence of Z/ m Z modues Λ odd r, 1, m Λ odd r, 3, m... Λ odd r, b + 1, m is constant for a b with b + 1 b r, m. Moreover, if one denotes the stabiized Z/m Z modue by Ω odd r, m, then its rank, r r, is at most k odd r,1 r 1 k 1 4 odd r, 1 mod 1, R r := k odd 1 k odd r, 1 mod 1, where r,1 1 k odd r, 1 := { r 1 4 r+1 r+ 1 if r is odd, 1 if r is even. The nested sequence of even Z/ m Z modues {Λ even r, b, m : b b r, m} is constant for a b with b b r, m. If we denote the stabe modue by Ωeven r, m, then its rank is at most R r.

3 -ADIC PROPERTIES OF PARTITION FUNCTIONS 3 Moreover, if we define b r, m to be the east such integer, then Remarks. b r, 1 d r + 1 b r, m d r d r + 1m 1. 1 From the definition of d r and basic properties of inear maps, we get the trivia bound d r dims 1. The constant d r can aways be cacuated agorithmicay. For exampe, when 9 and r is such that r r = 1 we compute that d r = 0. In most of these cases d r = 0 as we; in genera d r d r+1. The question of resoving when d r is zero is an open probem. We sha make use of the theory of moduar forms moduo, which is coherent and we deveoped for 5. The restriction that r + 5, however, is necessary for the cacuations in some of our proofs. We make no caims for primes < r + 5. When the dimension r r is zero or one, we get the foowing congruences. Coroary 1.. Fix r, m 1, and et be a prime such that r r 1. If b 1, b b r, m and b 1 b mod, then there is an integer C r, b 1, b, m such that for a n, we have b 1 n + r b n + r p r C r, b 1, b, m p r mod m. 4 4 If r r = 0, then C r, b 1, b, m = 0. Remarks. 1 When r r 1, Coroary 1. gives rise to natura orbits. In fact, there exist C even r, m and C odd r, m such that if b 1, b are even, we have that C r, b 1, b, m = C even r, m b b 1 and if b 1, b are odd, we have that C r, b 1, b, m = C odd r, m b b 1. Hence there are at most two orbits, possiby depending on the parity of b. In fact, there is ony one orbit: if b is odd then Lemma.1, together with the notation of.7, shows that so C odd L r, b + 1, z L r, b, z U C odd r, b, b +, ml r, b +, z U = C even. C odd r, b, b +, ml r, b + 3, z, We note that theorems of this type, and more generay a of the theorems in this paper, can be impemented to get estimates of when the coefficients of the partition generating functions are in any of the residue casses moduo m, not just zero. Foowing [11] and [5], a pair r, is caed exceptiona if there is a congruence of the form p r n + a 0 mod, and superexceptiona if the congruence is not expained by any of the three criteria Boyan gives in Theorem.3 in [5]. We find that for a pairs r, which are exceptiona but not superexceptiona and satisfy r + 5, we have r r = R r = 0. For the first two superexceptiona pairs, 5, 3 and 7, 19, we find that r r = 1. In other words, for m = 1 we have C r, m = 0, but for m > 1, C r, m 0.

4 4 EVA BELMONT, HOLDEN LEE, ALEXANDRA MUSAT, SARAH TREBAT-LEDER Finay, as a direct coroary of Theorem 7.1 in [8], in the case r r = 1 we have that the forms P r, b; 4z mod m converge to Hecke eigenforms as b, m. This is the anaogue of Theorem 1.3 in [8]. For even r, P r, b, 4z has integra weight, whie for odd r, it has haf-integra weight, so we first review the definitions of Hecke operators. For odd r and c a prime not dividing the eve, reca that for λ Z and c prime, the Hecke operator with Nebentypus χ is defined by T c of weight λ + 1 anq n T c := n n 1 ac n + c λ 1 λ n χcan + c λ 1 an/c q n. c 1.6 For even r and c a prime not dividing the eve, reca that the Hecke operator T c on the space M k! Γ 0, χ is defined as anq n T c := n anc + p k 1 χca q n. 1.7 c Theorem 1.3. If r r = 1, then P r, b; 4z mod m is an eigenform of a the weight k r, m r Hecke operators on Γ 0576, for b b r, m. As an immediate coroary, we get the foowing congruences for p r n. Coroary 1.4. Suppose, r are such that r r = 1 and m 1. 1 If r is odd and b b r, m, then there is an integer λ m, c such that for a n coprime to c we have [ ] b nc 3 + r b nc + r p r λ m, c p r mod m. 4 4 If r is even and b b r, m, then there is an integer λ m, c such that for a n coprime to c we have [ ] b c n + r b cn + r p r λ m, c p r mod m The spt function. In the previous subsection, we obtained theorems anaogous to those proven in [8], where pn is repaced by p r n. These theorems rey on the fact that the generating functions for p r n are essentiay moduar forms. However, it turns out that the genera strategy for studying p r n aso appies to partition functions which do not directy reate to moduar forms, for exampe, the spt-function introduced by Andrews in []. The spt function counts the number of smaest parts among the partitions of n. For exampe, for n = 3, we have 3, + 1, The smaest parts are underined, giving us spt3 = 5. For convenience of notation, we define sn := sptn. Andrews [] proved the foowing Ramanujan-type congruences s5n mod 5, s7n mod 7, s13n mod 13.

5 -ADIC PROPERTIES OF PARTITION FUNCTIONS 5 Garvan recenty proved that these three congruences are the simpest exampes of eegant systematic congruences moduo arbitrary powers of 5, 7, and 13, namey s5 b n + δ 5 b 0 mod 5 b+1, s7 b n + δ 7 b 0 mod 7 b+1, 1.8 s13 b n + δ 13 b 0 mod 13 b+1, where δ b denotes the east nonnegative residue of 4 1 moduo b. For a primes 5, Ono [15] found a systematic modified type of congruence which Ahgren, Bringmann, and Lovejoy [1] have generaized to a powers of primes 5. We woud ike to appy simiar techniques as in the p r case, but the generating function for spt, snq n = pnq n q n n 1 m=1 1 qm, q n n 0 is not moduar. Instead, it is essentiay the hoomorphic part of a weight 3 harmonic Maass form. More precisey, if we work with the sequence n=1 an := 1sn + 4n 1pn, 1.10 and define αz := n 0 anq n 1 4, 1.11 then α4z is the hoomorphic part of a weight 3 weak Maass form, as shown by Bringmann [7]. Athough harmonic Maass forms are moduar, they have Fourier expansions that are not hoomorphic. Fortunatey, we can return to the theory of moduar forms by annihiating the non-hoomorphic part. Garvan [10] accompished this by Atkin s U operator, whie Ono [15] used the weight 3 Hecke operator T, and Ono and others have used the theory of twists for exampe, see Theorems 10.1 and 10. of [14]. We foow Garvan, and define α z := a n 1 n 4 1 χ 1 a q n 4, 1.1 n 0 where χ 1 := 1. As shown by Garvan in [10], for primes 5, α z is a weaky hoomorphic moduar form of weight 3 see the discussion before 1.0 of [10]. For the primes and 3, a fairy compete theory has been obtained by Fosom and Ono in [9], with an aternate approach for the prime by Andrew, Garvan, and Liang in [3]. We woud ike to appy simiar techniques as the p r case to obtain resuts for 5. Foowing Garvan, our main objects of study are the series P spt, b; z := b n + 1 b n + 1 a χ 1 a q n n Note that P spt, 1; z = α z.

6 6 EVA BELMONT, HOLDEN LEE, ALEXANDRA MUSAT, SARAH TREBAT-LEDER As in the p r case, fixing integers b 0 and m 1, we define Λ odd spt, b, m := Span Z/ Z{ηzP m spt, β; z mod m : β b, β b mod }, Λ even spt, b, m := Span Z/ Z{ηzP m spt, β; z mod m : β b, β b mod } for odd and even b, respectivey. The quantity d spt is anaogous to d r defined earier, and is reated to the kerne of an important operator we define in Section 4. Theorem 1.5. If 5 is prime and m 1, then there is an integer b spt, m such that the foowing are true. 1 The nested sequence of Z/ m Z modues Λ odd spt, 1, m Λ odd spt, 3, m... Λ odd spt, b + 1, m is constant for a b with b + 1 b spt, m. Moreover, if one denotes the stabiized Z/m Z- modue by Ω odd spt, m, then its rank, r spt, is at most if 1 mod 1, 4 R spt := 1 if 1 mod The nested sequence of even Z/ m Z-modues {Λ even spt, b, m : b b r, m} is constant spt, m, then its for a b with b b spt, m. If we denote the stabiized modue by Ω even rank is at most R spt. Moreover, if we define b spt, m to be the east such integer, then b spt, m d spt + 1m + 1. Remark. From the definition of d spt, we get the trivia bound d spt dims +1. For = 5, 7, or 13, we have that r spt = 0 and d spt = 0, and therefore our generaization of the theory of [8] reproduces Garvan s congruences but with the power of the moduus is one smaer. It is interesting to note that -adic theory can repace the theory of moduar equations. With sighty more extra input, we woud be abe to reproduce the fu congruences. Coroary 1.6. For the primes = 5, 7, and 13, we have s b n + δ b 0 mod b 1. For = 11, 17, 19, 9, 31, or 37, we have that r spt = 1, and therefore we get the foowing theorem. Coroary 1.7. Let be one of the primes above and m 1. If b 1, b b spt, m and b 1 b mod, then there is an integer C m, b 1, b such that for a n, we have s b 1 n + δ b 1 χ 1 s b 1 n + δ b 1 Remarks. C m, b 1, b [s b n + δ b χ 1 s b n + δ b ] mod m.

7 -ADIC PROPERTIES OF PARTITION FUNCTIONS 7 1 Aso note that when m = 1, the above formua dramaticay simpifies as s b 1 n + δ b 1 C 1, b 1, b s b n + δ b mod. Just as in the case of p r, when r spt 1 then the theorem above gives rise to natura orbits. Finay, we show that for sma, the forms P spt, b; z mod m converge to Hecke eigenforms as b, m, where the Hecke operators are as defined in 1.6. This is anaogous to Theorem 1.3 in [8]. Theorem 1.8. If 5 37 is prime with 3, then P spt, b; 4z mod m for b b m is an eigenform of a the weight k spt, m 1 Hecke operators on Γ As an immediate coroary, we get the foowing congruences for sn. Coroary 1.9. Suppose = 11, 17, 19, 9, 31, or 37, and m 1. If b b spt, m, then there is an integer λ m, c such that for a n coprime to c we have b nc b nc s χ 1 s 4 4 [ ] b nc + 1 b nc + 1 λ m, c s χ 1 s mod m. 4 4 Remark. In the case when m = 1, this simpifies to b nc b nc + 1 s λ m, cs 4 4 mod Exampes. Here, we give numerica exampes of the main theorems in this paper. In a cases, using the methods in 6 of [6], we find that d r and d spt are zero. Exampe 1. We iustrate Coroary 1. in the case that = 13, r =, and m = 1. We cacuate that P 13, 4; z 10 P 13,, z 1 + 4q + q + mod 13 and therefore p 13 4 n p 13 n mod 13. More generay, for every even b 1, b 1, we have 13 b 1 n + p 10 b 1 b 13 b n + p mod Exampe. We iustrate Coroary 1.7 in the case that m = 1 and = 11. We cacuate that P 11 spt, ; z P 11 spt, 4; z 4q + 7q + 7q 3 + mod 11 and therefore s11 n s11 4 n mod 11. More generay, for every even b 1, b, we have 11 b 1 n b n + 1 s s mod

8 8 EVA BELMONT, HOLDEN LEE, ALEXANDRA MUSAT, SARAH TREBAT-LEDER Exampe 3. We iustrate Theorem 1.8 for = 17 when b = and c = 5. We cacuate that P 17 spt, ; 4z T 5 P 17 spt, ; 4z 13q q q q mod 17 and therefore for n reativey prime to 5. s n s n mod Outine. In Section we introduce the sequence of functions L r, b; z and L spt, b; z, which are reated to the functions P r, b; z and P spt, b; z, and which are obtained by repeatedy appying certain operators U and D. We aso reca various facts about fitrations of moduar forms that we wi need. In Section 3, we show that U and D preserve spaces of moduar forms, and in Section 4, we show that iterating these operators resuts in spaces Ω r, m and Ω spt, m with sma rank. Since L spt, b; z and L r, b; z reside in these spaces, this proves the finiteness portions of Theorems 1.1 and 1.7 and their coroaries. In Section 5, we prove the bounds on b r, m and b spt, m given in these two theorems. In Section 6, we give proofs of the main theorems. Acknowedgements The authors woud ike to thank Ken Ono for hosting the Emory REU, where the research was conducted, and for his guidance and comments on this paper. We woud aso ike to thank Zachary Kent for usefu discussions, M. Boyan and J. Webb for providing a preprint of their paper which was usefu for our research, and the Nationa Science Foundation for funding the REU. Finay, we woud ike to thank the referee for heping to improve the bound for b r, m in Theorem Combinatoria and Moduar Properties of Important Generating Functions.1. Basic definitions. The proofs of the theorems in this paper rey on a detaied study of a pecuiar sequence of power series. To define them, we need combinatoria properties of some very specia generating functions. Throughout the paper, we wi consider 5 prime. In the case of p r, we wi aways be considering r + 5. Reca that the Atkin U-operator acts on q-series by anq n U := anq n..1 Foowing [8], we define and define the operator D r by Φ z := η z ηz,. f D r := f Φ z r U..3 To study p r we wi consider the sequence of operators U, D r, U, D r,.... To study spt we wi consider the same sequence, with r = 1. It wi occasionay be usefu for us to

9 -ADIC PROPERTIES OF PARTITION FUNCTIONS 9 think of this sequence as simpy repeated use of the operators U D r or D r U. To this end, we define f X r := f U D r and.4 f Y r := f D r U..5 We now define for b 1 two specia sequences of functions. Let L spt, 1; z := ηzα z, and for b, define L spt, b; z by { L spt, b 1; z U if b is even, L spt, b; z :=.6 L spt, b 1; z D 1 if b is odd. Let L r, 0; z := 1, and for b 1, define L r, b; z by { L r, b 1; z U L r, b; z := L r, b 1; z D r if b is even, if b is odd..7 The foowing emma wi reate L spt, b; z and L r, b; z to our main objects of interest, the functions P spt, b; z and P r, b; z. Lemma.1. For b 1, we have that { ηz P spt, b; z L spt, b; z = ηz P spt, b; z For b 0, we have that { η r z P r, b; z L r, b; z = η r z P r, b; z if b is even, if b is odd. if b is even, if b is odd..8.9 Proof. Note that by definition, we have L spt, 1; z = ηz P spt, 1; z, L r, 0; z = η r z P r, 0; z. To prove the emma in genera, we use induction on b and the foowing fact about the U-operator. If F q and Gq are forma power series with integer exponents, then The emma now foows by direct computation. F q Gq U = F q Gq U..10 We now prove a resut about the moduarity of the functions L spt, b; z and L r, b; z. Using standard notation, we denote by M k Γ 0 N the space of hoomorphic moduar forms of weight k on Γ 0 N, and we denote by M! k Γ 0N the space of weaky hoomorphic moduar forms on Γ 0 N of weight k, i.e. those forms whose poes if any are supported at the cusps of Γ 0. Lemma.. If b 1 is a positive integer, then L spt, b; z is in M! Γ 0 Z[[q]] and L r, b; z is in M! 0Γ 0 Z[[q]].

10 10 EVA BELMONT, HOLDEN LEE, ALEXANDRA MUSAT, SARAH TREBAT-LEDER Proof. For b = 1, we have that L spt, b; z = α zηz, which is in M! Γ 0 by a resut from [10]. By we-known facts about the Dedekind η function for exampe, see Theorem 1.64 and 1.65 in [13], it aso foows that Φ z M! 0Γ 0. Moreover, it is we-known see for exampe Lemma.1 in [6] that U : M! kγ 0 M! kγ 0, and that U : M kγ! 0 M kγ! 0. Combining the two facts above for k =, an inductive argument shows that L spt, b; z is in M Γ! 0 Z[[q]] for a b 1. The case b = 0 is obvious for L r, b; z, so the anaogous resut foows... Fitrations. The theory of fitrations has cassicay been used to understand moduar forms moduo. Foowing [16] and [17], we review some of the most pertinent facts. For f M k Z[[q]], define the fitration of f moduo by ω f := inf k 0 {k : f g mod for some g M k Z[[q]]}..11 If f 0 mod then define ω f =. Note that if f g mod and g M k, then we have ω f k mod 1. We use fitrations to understand how U acts on moduar forms moduo. The foowing is Lemme on p. 13 of [16]; it tes us how U decreases the fitration. Lemma.3. If f M k Z[[q]], then we have ω f U + ω f 1. Next we wi need the foowing facts from. Lemme 1 of [16], which shows how the θ operator acts on this graded ring. Lemma.4. Letting θf := q d f, the foowing hod. dq 1 We have ω θf ω f + + 1, with equaity if and ony if ω f 0 mod. We have ω f i = iω f for a i 1. In the case of spt, we wi use the U and D 1 operators to decrease the fitration to + 1, at which point it is stabe. In the case of p r, appying the operators decreases the fitration to 1, at which point we have { 1 if b is even, ω r, b; z = k odd r, 1 if b is odd. The foowing emma describes this process in both cases. Lemma.5. Let f Z[[q]] be a moduar form and be prime. 1 Suppose 5 and ω f mod 1. a. If ω f = + 1, then we have ω f U = + 1 and ω f X Hence U is a bijection on weight + 1 moduar forms moduo. b. If ω f = + 1, then we have ω f D c. If ω f > + 1, then we have ω f X 1 < ω f. Suppose r + 5 and ω f 0 mod 1. a. If ω f k odd r, 1, then we have ω f U 1 and ω f X r k odd r, 1.

11 -ADIC PROPERTIES OF PARTITION FUNCTIONS 11 b. If ω f = 1, then we have ω f D r k odd r, 1. c. If ω f > k odd r, 1, then we have ω f X r < ω f. Proof. To prove part 1a, use Lemma.41 repeatedy to see that ω θ i f = i for i 1. The identity f U = f θ 1 f, aso from [16], gives ω f U = maxf, θ 1 f = + 1. By Lemma.4, we get ω f U = + 1. From this and part 1b beow, we have that ω f X To prove part 1b, first note that By Lemma.3 we have ω z 1 4 f U + Φ z z 1 4 mod < for >. Since the fitration is congruent to + 1 moduo 1, the resut foows. For part 1c, write ω f = k 1 + with k > 1. Using Lemma.3, we compute that + k ω f X 1 + = k 1 +. Since for > and k > 1, we have that 3+ < k 1 +, this part foows. For a, using the definition of k odd r, 1, we get that r ω f U = r + 4. Since > r + 4, we get that ω f U 1. The second part of a foows from this and b beow. For b, we use Lemma.3 to see that ω f D r r 1 1 r = r + 4 < k odd r, when > r + 4. Lasty, for part c, using Lemma.3, we get that + ω f 1 + r 1 1 ω f X r +. Computation shows that the right-hand side is ess than ω f if and ony if ω f > r r+4. Since r + 5, 1 ω f, and ω f > k odd r, 1, we find that this ast condition is satisfied. 3. The action of U and D 3.1. U and D preserve moduar forms. We show in this section that the functions L r, b; z and L spt, b; z are reductions moduo of certain cusp forms. For the spt case, we need to show this for the base case b = 1. For both cases, we need to show the induction step, that U and D preserve these spaces of moduar forms. It is important to choose

12 1 EVA BELMONT, HOLDEN LEE, ALEXANDRA MUSAT, SARAH TREBAT-LEDER weights so that these operators preserve spaces of those weights moduo powers of. For the spt case we define k spt, m := m ; and in the case of p r we wi use the weights m 1 1 without specia notation. For ease of notation, we wi write f j S 3.1 if f is congruent moduo j to a moduar form in a space S. Foowing [8], we define A z := η z ηz. 3. Using standard facts about the Dedekind eta-function see for exampe Theorem 1.64 and 1.65 of [13], we see that A z is a hoomorphic moduar form of weight 1 on Γ 0 with Nebentypus. This function is usefu to us because it changes the weight of a form whie preserving the form moduo. Lemma 3.1. If 5 is prime, then A z m 1 M m 1 1Γ 0. Moreover, Proof. See the proof of Lemma 4.1 in [8]. Reca that the weight k sash operator is defined by A z m 1 1 mod m. 3.3 f k Az := det A k/ cz + d k faz, a b where A =. As in [16], we define the trace operator Tr : M c d k! Γ 0 M k! by Trf := f + 1 k 0 1 f k W U, W := Note that Tr takes M k Γ 0 to M k. Proposition 3.. If m 1, then L spt, 1; z m S k spt,m. Proof. Foowing [10], we define G z := α z η z ηz. 3.5 Note that G z M +1 Γ 0 by Theorem. in [10], hence we have and L spt, 1; za z m 1 = G za z m 1 1 M k spt,mγ 0 TrL spt, 1; za z m 1 = L spt, 1; za z m m 1 + G z +1 W A z m 1 1 m W U M k spt,m. The first term is congruent to L spt, 1; z moduo m by Lemma 3.1, so it suffices to show the second term has vauation at east m.

13 -ADIC PROPERTIES OF PARTITION FUNCTIONS 13 We use equation.19 on page 9 on [10], 1 G = iz +1 Eq z Eq 1 4n χan n= s 1 + a where Eq := 1 q n, and the transformation aw for η to cacuate 1 1m 1 + = 1 1m 1 + = 1 1m S1 q z m m 1 1 n + s G z +1 W A z m 1 1 m 1 1 1W U [ η 1 ] m 1 1 z [ ] m 1 +1 S q U = m 1 S 3 q, η 1 z m 1 1 q n+s, U 3.6 where S 1, S, S 3 are power series with agebraic integer coefficients. Since m 1 m, we get the desired concusion. Now we show D 1 preserves the spaces S k spt,m. Lemma 3.3. Suppose m 1 and Ψz Z[[q]]. 1 Let 5 be prime, and suppose that for a 1 j m, we have Ψz j M k spt,j. Then for a 1 j m, we have Ψz D 1 j S k spt,j. Let r+3 be prime, r, and suppose that we have Ψz j M j 1 1 for 1 j m. Then for a 1 j m, we have Ψz Y r j S j 1 1. Furthermore, defining r k odd + 1 1, j = 1 r, j = r 1, j = j 1 1, j 3, we have that Ψz D r j S k odd r,j for 1 j m. 1 Note that in, one appication of D r can increase the weight because r 4 Φ r mod has arge weight. However, if we use the operator Y r rather than D r, the weight is preserved. Proof. The proof is simiar to that of Lemma 3.1 in [6], so we wi omit some detais and instead note the differences. For 1, we et g j denote the forms in M k spt,j congruent to Ψ moduo j, and for, we et g j denote the forms in M j 1 1 congruent to Ψ moduo and moduo j, respectivey.

14 14 EVA BELMONT, HOLDEN LEE, ALEXANDRA MUSAT, SARAH TREBAT-LEDER 1 For the first part, the base case j = 1 foows by Lemma.51b. We write Ψz D 1 as gj 1 Ψz D 1 E 1 j 1 D A j + + g j 1 E j 1 1 g j z g j 1 ze j 1 1 g j 1 D A j D 1 mod j. 3.9 It suffices to show that for each summand S, we have S j S k spt,j. For 3.7 and 3.8 we wi show this using properties of the trace operator and for 3.9 we wi use the standard fitration argument Lemma.3. Note that we have the congruences Define fz := E j 1 1 A j 1 1 mod j g j 1z A z j D 1 and hz := E 1 z 1 E 1 z. By 3.10, the first summand 3.7 is congruent to f moduo j. We wi show that f Tr fh j 1 mod j Since Tr sends M k! spt,j Γ 0 to M k! spt,j, and f is cuspida, this woud show that f j S k spt,j. Because the coefficients of the terms with nonpositive exponents in Trfh j 1 are 0 moduo, we can subtract from Trfh j 1 a moduar function to cance out those terms, and this wi differ from fh j 1 by a mutipe of j. To show 3.11, we use Lemme 9 of [16], which gives ord Trfh j 1 f minj + ord f, j ord f W 1. The first argument is at east j; it suffices to show the second argument is at east j as we. Note that if F is a moduar form moduo of weight w, 1 1 k F U = 1 F w. 0 k=0 k 1 Letting γ k =, we rewrite h 0 W as 1 f W = 1 gj 1 1 k 0 1 Φ A j 0 0 k=0 1 = 1 k=0 g j 1 k spt,j 1γ k Φ 0 γ k. 3.1 A j j 1γ k Note that the extra factor of in 3.1, not present in [6], comes from combining the matrices after the sash operator. The rest of the proof is the same: for k 0 we decompose the

15 matrix γ as -ADIC PROPERTIES OF PARTITION FUNCTIONS 15 γ k = k bk k k, 0 where k is chosen so that kk 1 mod, b k = kk 1. For k = 0 we break up the matrix as γ 0 =. 0 0 Using transformation properties of g j 1 and η, we compute the foowing ower bounds for the vauation of the factors in the terms in 3.1. g j 1 k spt,j 1γ k A j j 1 γ k 1 Φ 0 γ k 1 k k = 0 k spt, j 1 j 1 1 The ony change from [6] is that in our case, k spt, j 1 repaces j 1. From the above tabe and 3.1 we get j 1 + ord f W j 1 + k spt, j 1 j 1 1 = j 1 j + 1 j, as needed. For the second summand 3.8, note that by 3.10, if we et B j, := g j 1 g j 1 A j E j 1, 3.13 we get that 3.8 is congruent to B j, D 1 M! k spt,j 1 Γ 0 moduo j. Furthermore, we have B j, 0 mod j Because 1 4 Φ mod, we can repace Φ with 1 4 without changing B j, moduo j. We wi show that B j, D 1 j S k spt,j 1, where for convenience we set k j := k spt, j + 1 = j Then, in ight of 3.14, mutipying by a suitabe power of E 1 wi give that B j, D 1 j S k j 1, finishing the proof of this part. First, note that if we define C j, := g j 1 A j 1 1 4, E j then by the definitions of B j, and D 1, we have B j, D 1 g j U C j, U g j U k j 1 1 TrC j, k j 1 W k j 1 1 C j, k j 1 W mod j.

16 16 EVA BELMONT, HOLDEN LEE, ALEXANDRA MUSAT, SARAH TREBAT-LEDER Note that the first term is the reduction of a cusp form in S k j 1. Since A M 1Γ 0, we have C j, M! k j 1Γ 0 and TrC j, k j 1 W M k j 1. It remains to show that k j 1 1 C j, k j 1 W 0 mod j. By using transformation properties of g j 1 and η we can compute ord k j 1 1 C j, k j 1 W k j 1 [ 1 + ord g j 1 E j ord A j 1 j 1 W = k j kspt, j 1 + j k spt,j 1+ j 1+ 1 j 1 + j ] W as needed. The ony difference in [6] is that k spt, j 1 in the ast ine above is repaced by j 1; this bound for the vauation is 1 greater than in [6]. Write the third summand 3.9 as F j, D 1, where F j, is defined by F j, := g j g j 1 E j 1 1. Since g j g j 1 mod j 1, we get the congruence We have that F j, 1 j 1 4 S k j, F j, 0 mod j j, so by a cacuation using Lemma.3 we obtain 1 j ω Fj, j 1 D 1 Mutipying by j 1 and an appropriate power of E 1 gives that F j, D 1 j This finishes the proof of the first part. S k spt,j. The second part is simiar except with the modified operator Y r and the weights j 1 1. Again we write Ψz D r moduo j as in The j = 1 case foows from Lemma.5, so et j. To study the first summand 3.7, et f = We have 1 f 0 W = 1 k=0 g j 1 D A j r and repace f W by f 0 W. g j 1 j 1γ k Φ r A j 0 γ k. j 1 γ k We have the foowing ower bounds on the vauations of the factors. g j 1 j 1γ k A j j 1 γ k 1 Φ r 0 γ k 1 k r k = 0 j 1 j 1 r

17 -ADIC PROPERTIES OF PARTITION FUNCTIONS 17 The change here is that the exponents coming from Φ are mutipied by r. Hence we get that From the inequaity Lemme 9 of [16] ord f 0 W 1 + j 1 j 1 r = 1 j r ord Trfh j 1 f min j + ord f, j ord f 0 W and from using r + 3 j ord f 0 W j 1 j r j we obtain as before that f j S j 1 1. Because U T mod m, we obtain f U j S j 1 1 as we. For the second summand 3.8, note that repacing D 1 by D r means repacing 1 4 by r 1 4. Thus the proof is simiar to that of Proposition 3.3 in [6], with 1 repaced by r 1. We get that B j, D r is the reduction of a cusp form of weight j 1 + r When j >, this is at most j 1 1, since the inequaity is equivaent to r 1 j 1, which is true since r 1. For j =, note that we get B +1 j, D r S r 1+ 1 and hence that B j, D r S r 1. By Lemma.3, ω Bj, r 1 1 D r U + 1, which shows, after mutipying by a suitabe power of E 1, that B j, Y r S 1. r 1 j 1 For the third summand 3.9, we cacuate the fitration of F j, 4 U using Lemma.3. Since the fitration of F j, is at most j 1 1, we get that j 1 Fj, r 1 ω 4 U + j r 1 1. j 1 We aso have that r + 3 and the fitration is a mutipe of 1, so we get that Fj, r 1 ω 4 U 1 j + r + j j 1 Another appication of U wi not increase the fitration. Lemma 3.4. We have the foowing. 1 If b 1 and m 1, then we have that L spt, b; z m M k spt,m. For b > 1, L spt, b; z is a cusp form.

18 18 EVA BELMONT, HOLDEN LEE, ALEXANDRA MUSAT, SARAH TREBAT-LEDER If b 0 is even and m 1, then we have that L r, b; z m M m 1 1. For b > 0, L r, b; z is a cusp form. If b 1 is odd and m 1, then L r, b; z m M k odd r,m. Proof. In both cases the proof is by induction on b. For 1, the base case b = 1 is Proposition 3.. For, the base case foows from 1 E 1 j 1 mod j for a j. In 1, the induction step for even b to b + 1 is given by Lemma 3.31, whie the induction step for odd b comes from the fact that U T mod j as ong as the weight is greater than j, and the atter operator preserves spaces of moduar forms. In, the induction steps for even b to b +, and even b to b + 1, are given by Lemma 3.3. Remark. Let M r, m and Mr odd, m denote the space of moduar forms in M m 1 1 Z [[q]] and M k odd r,m Z [[q]], respectivey, with coefficients reduced moduo m. The previous coroary shows that we have the foowing nesting of Z/ m Z-modues in the case p r : M r, m Λ even M odd r, m Λ odd r, 0, m Λ even r,, m Λ even r, b, m r, 1, m Λ odd r, 3, m Λ odd r, b + 1, m In the case of spt, et M spt, m denote the space of moduar forms in M k spt,m with coefficients reduced moduo m. Then we have the foowing incusions: M spt, m Λ even M spt, m Λ odd spt,, m Λ even spt, 4, m Λ even spt, b, m spt, 1, m Λ odd spt, 3, m Λ odd spt, b + 1, m Since each sequence is contained in a finite-dimensiona space, it must stabiize after a finite number of steps. 3.. Reducing the weight of moduar forms. The next emma shows that X 1 and Y r reduce the weight of those moduar forms moduo j in the Z/Z kerne of U or D. Think of the emma as an anaogue of Lemma.5 when we re considering q-series moduo higher powers of. Lemma 3.5. Let n 1 and Ψz Z[[q]]. 1 Let 5 be prime. Suppose that, for a 1 j n, we have Ψz j M k spt,j. If Ψz D 1 0 mod, then for a j n, we have Ψz Y 1 j S k spt,j 1. Let r + 5 be prime. Suppose that Ψz 1 M 1 and for a j n, we have Ψ j M j 1 1. If Ψz 0 mod, then for a j n, we have Ψz Y r j S j 1. Proof. This proof foows the same argument as Lemma 3.6 in [6]. We keep the notation from the proof of Lemma We break up Ψz Y 1 into three summands as foows: Ψz Y 1 gj A j E j 1 1 g j 1 E j 1 1 Y 1 3. g j 1 g j g j 1 E j 1 1 A j E j 1 1 Y Y 1 mod j. 3.4

19 -ADIC PROPERTIES OF PARTITION FUNCTIONS 19 We prove that for each summand S we have S j S k spt,j 1, and begin by assuming j 3. First summand 3.: By hypothesis, 0 fz D 1 D 1 mod. Hence g j 1 A j using E 1 z j 1 mod j 1, 3. is congruent to G j, z U moduo j, where gj 1 z G j, z := D 1 E 1 z j M! A z j j 1+ Γ 0. We have that j 1 Tr G j, k spt,j 1 W = G j, U + j 1 G j, k spt,j 1 W. We wi show that the vauation of the ast term is at east j. Since Tr sends M! k spt,j 1 Γ 0 to M! k spt,j 1, and G j,z U is cuspida, this woud show that G j, z U j S k spt,j 1, as needed. Note that ord g j 1 A j j 1 j 1 D 1 W is given by 3.1. Hence, we have G j, k spt,j 1 W gj 1 + ord E 1 j j 1 W + ord D 1 W A z j j 1 + j 1 + k spt, j 1 j 1 1 j. Second summand 3.3: Defining B j, as in 3.13, we see that 3.3 is congruent to B j, Y 1 moduo j. From 3.15 and Lemma.3 we get Bj, ω D 1 j , so j 1 Bj, ω D 1 U + j k j 1 spt, j 1. Mutipication by an appropriate power of E 1 then shows B j, z Y 1 j S k spt,j 1. Third summand 3.4: Defining F j, as in 3.17, we find that 3.4 is congruent to F j, Y r moduo j. Using 3.18, Lemma.3 gives ω Fj, j 1 D 1 U We finish with the same argument as before. + 1j A j k spt, j 1. First consider j >. As in the first part, we decompose Ψz Y r into three summands, each of which we wi show to be the reduction of a form in S j 1: gj 1 Ψ Y r E 1 j 1 Y r g j 1 E j 1 1 g j 1 g j g j 1 E j 1 1 A j E j 1 1 Y r 3.6 Y r. 3.7

20 0 EVA BELMONT, HOLDEN LEE, ALEXANDRA MUSAT, SARAH TREBAT-LEDER gj 1 z First summand 3.5: Letting G j, z = D A z j r E 1 z j M! j 1 Γ 0 and recaing E 1 j 1 mod j 1, we find that the first summand is congruent moduo j to G j, z U. As before, considering j 1 1 TrG j, z U = G j, z U + j 1 1 G j, z j 1 W, it suffices to show the vauation of the ast term is at east j. Using 3.19, we have that ord j 1 1 G j, z j 1 W j 1 gj 1 = 1 + ord D 1 0 W + ord E j A j 1 j 1 W j j r + j 1 j. The second summand 3.6 is B j, D r U, where we define B j, as in From 3.0 it foows that B j, D r j S j 1+ r 1. Lemma.3 and the fact that r + < then give Bj, ω D r U + j 1 + r 1 1 j 1. j 1 Since B j, D r U f j, U 0 mod j 1, we may mutipy by an appropriate power of E 1 to show B j, D r U j S j 1 moduo j. For the third summand, we define F j, as in From 3.1 we have that F j, z D r is congruent moduo to a form in S j + r Again using Lemma.3, we get Fj, ω D r U + j + r j 1. j 1 Exacty as before, mutipication by a power of E 1 S j 1. To check the j = case, note that by hypothesis, we have ω g Lemma.3 repeatedy and noting > r + 3 gives g D r r + 1 ω 1, g D r ω U 1. gives that F j, z D r U j 1. Appying Hence Ψ Y r g D r U mod is the reduction of a form in S 1 moduo. 4. The spaces Ω odd m and Ω even m In this section, we wi give injections from our stabiized spaces into spaces of cusp forms of sma weight. Foowing [6], we reca two eementary commutative agebra resuts. Lemma 4.1. Let A be a finite oca ring, M be a finitey generated A-modue, and T : M M be an A-isomorphism.

21 -ADIC PROPERTIES OF PARTITION FUNCTIONS 1 1 There exists an integer n > 0 such that T n is the identity map on M. For a µ M and n 0, we have µ A[T n µ, T n+1 µ,...]. Now, we wi state our main theorem, which we wi need a few emmas to prove. Theorem 4.. The foowing hod. 1 If 5 is prime and m 1, then there exists an injective Z/ m Z modue homomorphism Π odd spt : Ω odd spt, m S +1 Z [[q]] satisfying the foowing property: for a ν Ω odd spt, m with ord ν = j < m, we have Π odd sptν ν mod j+1. If r+5 is prime and m 1, then there exist injective Z/ m Z modue homomorphisms Π even r : Ω even r, m S 1 Z [[q]], Π odd r : Ω odd r, m S k oddr,1 Z [[q]] satisfying the foowing property: for a µ 1 Ω even r, m and µ Ω odd r, m with ord µ i = j i < m, we have Π even rµ 1 µ 1 mod j 1+1, Π odd rµ µ mod j +1. We wi work out the first case in detai, and state resuts for the second ony when they differ significanty from the r = 1 case in [6]. For the spt case, we consider the foowing two submodues of S k spt,m Z [[q]]: { } S 0 spt, m := fze 1 z m 1 1 : fz S +1 Z [[q]] and 4.1 { } S 1 spt, m := gz : gz = a j q j S k spt,m Z [[q]] with m 0 > dims +1. j=m 0 4. By the existence of diagona bases with integer Fourier coefficients for spaces of cusp forms, we have S k spt,m Z [[q]] = S 0 spt, m S 1 spt, m. We define S odd spt, m S k spt,m Z [[q]] to be the argest Z/ m Z submodue such that X 1 is an isomorphism on S odd spt, m moduo m. For p r, we consider the foowing two submodues of S m 1 1 Z [[q]]: { } S0 even r, m := fze 1 z m 1 1 : fz S 1 Z [[q]] and 4.3 { } S1 even r, m := gz : gz = a j q j S m 1 1 with m 0 > dims j=m 0 We simiary see that S m 1 1 Z [[q]] = S0 even r S1 even r, m, and define S even r to be the argest Z/ m Z submodue such that Y r is an isomorphism on S even r, m moduo m. Define S odd r, m simiary but use k odd r, m in pace of m 1 1 and with kodd in pace of m 1 1. r,m k odd r,1 1

22 EVA BELMONT, HOLDEN LEE, ALEXANDRA MUSAT, SARAH TREBAT-LEDER The foowing emma hods for both spt and p r, and for both S even and S odd. Therefore, for brevity, we sha et S = S odd spt, m, S even r, m, or S odd r, m, and et S 0, and S 1 denote the corresponding spaces. Lemma 4.3. Suppose that fz S has ord f = i < m, and that fz = f 0 z + f 1 z with f i z S i. Then we have ord f 1 > i and ord f 0 = i. Proof. Using Lemma.5, this proof proceeds as Lemma 4.4 of [6]. Now, we state a coroary of Lemma 4.3, whose proof foows Coroary 4.7 of [6] exacty. Coroary 4.4. Let fz, gz S, and suppose that fz = f 0 z + f 1 z and gz = g 0 z + g 1 z with f i, g i S i. Suppose further that f 0 z g 0 z mod m. Then we have fz gz mod m. We are now ready to construct our injection and prove Theorem 4.. Proof of Theorem 4.. We construct the injection Π odd spt as the composition of three Z/ m Z-modue homomorphisms Ψ 1 spt, Ψ spt, and Ψ 3 spt. By Lemma 3.4, we have that X 1 is an isomorphism on Ω odd spt, m, which impies that Ω odd spt, m S odd spt, m by the definition of S odd spt, m. Therefore, we et the map Ψ 1 spt be defined by Ψ 1 spt : Ω odd spt, m S odd spt, m. To define Ψ spt, we et fz = f 0 z + f 1 z S odd spt, m with f i S i spt, m, and we suppose that ord f < m. Lemma 4.3 impies that fz f 0 z mod ord f+1. Therefore, we can et Ψ spt : S odd spt, m S 0 spt, m be defined by Ψ spt : fz f 0 z mod ord f+1. This map is injective by Coroary 4.4. We next define the map Ψ 3 : S 0 spt, m S +1 Z [[q]]. Suppose that fz S 0 spt, m. By the definition of S 0 spt, m, there exists gz S +1 Z [[q]] such that fz = gze 1 z m 1 1. We therefore define Ψ 3 spt : fz gz. Since the first two are injections and the third is an isomorphism, the composition Π odd spt is an injection. Moreover, if we suppose that fz Ω odd spt, m with ord f < m, then we have that Π odd sptfz fz mod ord f+1. This proves the theorem for the spt case. The homomorphisms Ψ 1 r, Ψ r, and Ψ 3 r, whose composition gives us Π even r or Π odd r, are defined simiary. Remark. These injections preserve the order of vanishing. Since appying D r to a form gives q-expansions satisfying F D r = a n q n, this gives us that rank Z/ m Z n r 1 4 Ω odd spt, m dim S

23 -ADIC PROPERTIES OF PARTITION FUNCTIONS 3 For the p r case, note that we have isomorphisms D r : S even r, m = S odd r, m and U : S odd r, m = S even r, m. Hence rank Z/ m Z Ω even r, m = rank Z/ m Z Ω odd r, m r 1 dim S k odd r, Bounds on b r, m and b spt, m In this section, we prove that under certain assumptions, we can give bounds for b r, m and b spt, m of size roughy m. We wi first introduce some notation. Let Sspt := Sspt, 1 denote the argest subspace of S +1 Z [[q]] over Z/Z on which X 1 is an isomorphism. As in [6], we define d spt by d spt := min{t 0 : f M +1 Z [[q]], f D 1 X 1 t Sspt}. 5.1 Simiary define S even r := S even r, 1 to be the argest subspace of S 1 Z [[q]] over Z/Z on which X r is an isomorphism and S odd r := S odd r, 1 to be the argest subspace of r,1 on which Y r is an isomorphism. We define S k odd d r := min{t 0 : f M 1 Z [[q]], f D r X r t S odd r}. 5. We aso define a reated quantity d r := min{t 0 : f M 1 q r 1 4 Z [[q]], f Y r t S even r}. 5.3 By the fact that we have isomorphisms D r : S even r = S odd r, U : S odd r = S even r and the fact that Φ r U U ony has terms with exponents at east r 1, we have 4 that d r d r + 1 and d r d r + 1. We conjecture that d r = 0 for a r + 5. The same is not true of d r; however in many cases we sti have d r = 0. We wi prove the foowing theorem, which gives the ast component of our main theorems. Theorem 5.1. If m 1, then the foowing bounds hod. 1 For 5 prime, we have For r + 5 prime, we have b r, 1 d r + 1, b spt, m d spt + 1m + 1. b r, m d r d r + 1m 1, m Remark. Notice that if d spt = 0 and d r = d r = 0, then we get that b spt, m m + 1 and b r, m m. Our proof wi be simiar to the proof of Theorem 5.1 in [6]. We wi need the foowing emmas which we wi use in the proof of Theorem 5.1. Lemma 5.. The foowing hod.

24 4 EVA BELMONT, HOLDEN LEE, ALEXANDRA MUSAT, SARAH TREBAT-LEDER 1 Suppose that, for some odd b 0, we have fz Λ odd spt, b, m and 0 ord f < m. If, moreover, we assume that there exists gz M +1 Z [[q]] such that fz ord f gz mod ord f+1, then there exists hz Ω even spt, m such that fz D 1 X 1 d spt hz mod ord f+1. Suppose that, for some even b 1, we have fz Λ even r, b, m and 0 ord f < m. If, moreover, we assume that there exists gz M 1 Z [[q]] such that then there exists hz Ω odd r, m such that Proof. See the proof of Lemma 5. in [6]. fz ord f gz mod ord f+1, fz D r X r d r hz mod ord f+1. We wi aso need the foowing emma about the spaces Ω even spt, 1 and Ω odd r, 1. Lemma If 5 is prime, then we have that L spt, d spt + 3; z If r + 5 is prime, then we have that L r, d r + 1; z mod Ω odd spt, 1. mod Ω odd r, 1. Remark. Notice that the emma gives the case m = 1 of Theorem 5.1. Proof. 1 Let fz = L spt, 1; z Λ odd spt, 1, 1. Notice that if α z 0 mod, then L spt, b; z 0 mod, and the concusion hods triviay. Therefore, we may assume α z 0 mod, which impies that ord f = 0. By Théorème 11 on p. 8 of [16], we have that fz 1 M +1. By Lemma 5. we obtain hz Ω odd spt, m such that fz U D 1 Y 1 d = L spt, d spt + 3; z hz Since reduction moduo maps Ω odd mod. spt, m to Ω odd spt, 1, the concusion now foows. The proof is simiar to that of the first part of this emma. We start with L r, 0; z = 1, use Lemma 5., and argue as above. We use the next emma in the proof of Lemma 5.5. Lemma 5.4. The foowing hod. 1 If 5 is prime and fz S odd spt, m is such that 0 ord f < m, then for a 1 s m ord f, we have that fz ord f+s S k spt,s. If r + 5 is prime and fz S even r, m is such that 0 ord f < m, then for a 1 s m ord f, we have that fz ord f+s S s 1 1.

25 -ADIC PROPERTIES OF PARTITION FUNCTIONS 5 Proof. 1 See the proof of Lemma 5.3 in [6]. We use induction on s, the case s = 1 being Lemma 4.3. The proof goes through the same way because k spt, s + 1 k spt, s equas k s + 1 k s as defined in [6]. The proof foows the same outine as in [6], but for the even rather than the odd space. The foowing emma is the ast emma required in the proof of Theorem 5.1. Mutipe appications of U and D r to L spt, b; z and L r, b; z form a finite sequence that -adicay approaches eements of Ω odd spt, m and Ω even r, m, respectivey; we count the number of steps this process takes. Lemma 5.5. The foowing hod for m. 1 Suppose 5 is prime. If 1 s m 1, then there exist f 1,s z = f 1 d spt + 1s + 1; z Ω even spt, m and f,s z = f d spt + 1s + 1; z S k spt,m s Z [[q]] such that the foowing properties hod. a. We have that f,s z 0 mod s. b. For a k with s + 1 k m, we have that f,s z k S k spt,k s. c. We have that L spt, d spt + 1s + ; z f 1,s z + f,s z mod m. Let r be given and suppose r + 5 is prime. If 0 s m, then there exist f 1,s z Ω even r, m and f,s z S m s 1 1 Z [[q]] such that the foowing properties hod. a. We have that f,s z 0 mod s. b. For a k with s + 1 k m, we have that f,s z k S k s 1. c. We have that L r, d r d r + 1s 1; z f 1,sz + f,s z mod m. Proof. In both parts we foow Lemma 5.5 in [6] and proceed by induction on s. 1 We wi ony sketch the base case, since the induction step foows exacty as in [6]. Using Lemma 5.3, we get a form gz Ω odd spt, m with L spt, d spt + 3; z gz mod. Since U : Ω even spt, m Ω odd spt, m is a bijection, it foows that there exists f 1 d spt + ; z Ω even spt, m such that f 1 d spt + ; z U gz mod m. Now we wi appy Lemma 3.5 to L spt, d spt + ; z f 1 d spt + ; z. It is easy to check that the hypotheses of Lemma 3.5 are satisfied. Thus we get, for every k m, a form h k z S k spt,k 1 such that L spt, d spt + ; z f 1 d spt + ; z X r h k z mod k. If we define f d spt ; z := h m z, then f d spt ; z 0 mod, and f d spt ; z k S k s Moreover, we have L spt, d spt + 4; z =f 1 d spt + ; z Y 1 which gives the concusion. + L spt, d spt + ; z f 1 d spt + ; z Y 1 f 1 d spt + 4; z + f d spt + 4; z mod m,

26 6 EVA BELMONT, HOLDEN LEE, ALEXANDRA MUSAT, SARAH TREBAT-LEDER By Lemma 5.3 we get a form gz Ω odd r, m with L r, d r + 1; z gz mod. Let f 1,1 z = gz U Ω even r, m, f,1 z = L r, d r + 1; z gz U. This shows the base case. For the induction step, assume the emma true for s 1; we show it hods for s. By Lemma 3.5 appied to 1 f s 1,s 1, we have that one appication of Y r decreases the weight: f,s 1 Y r k S k s 1, s k m. By definition of d r we obtain that d r more appications of Y r brings f into the stabiized space: f,s 1 Y r d r+1 s s 1 Ω even r, 1. Suppose f,s 1 β mod s where β s 1 Ω even r, m and et This competes the induction step. f 1,s = f 1,s 1 Y r d r+1 + β f,s = f,s 1 Y r d r+1 β. We wi now return to the proof of Theorem 5.1. We use Lemma 5.5 aong with a simiar argument for going from m 1 to m. Proof of Theorem 5.1. The proof being simiar to that in [6], we wi give fu detais for the spt case, and for the second statement we wi ony sketch the detais. 1 We wi show that L spt, d spt + 1m + 1; z Ω odd spt, m. The case m = 1 is Lemma Now suppose m > 1. Using Lemma 5.5 with s = m 1, we get f 1 d spt + 1m 1 + ; z Ω even spt, m and f d spt + 1m 1 + ; z S +1 Z [[q]] satisfying the properties in Lemma 5.5. We have f d spt + 1m 1 + ; z 0 mod m 1. If f d spt + 1m 1 + ; z 0 mod m, then we get that L spt, d spt + 1m 1 + ; z f 1 spt, d spt + 1m 1 + ; z mod m, so L spt, d spt + 1m 1 + ; z Ω even spt, m. Since we have L spt, d spt + 1m + 1; z = L spt, d spt + 1m 1 + ; z D 1 X 1 d spt, it foows that L spt, d spt + 1m + 1; z Ω odd spt, m. If ord f d spt + 1m 1 + ; z = m 1, then by Lemma 5. we find hz Ω odd spt, m with f d spt + 1m 1 + ; z D 1 X 1 d spt hz mod m.

27 Hence -ADIC PROPERTIES OF PARTITION FUNCTIONS 7 L spt, d spt + 1m + 1; z = L spt, d spt + 1m 1 + ; z D 1 X 1 d spt f 1 d spt + 1m 1 + ; z D 1 X 1 d spt + hz + f d spt + 1m 1 + ; z D 1 X 1 d spt hz f 1 d spt + 1m 1 + ; z D 1 X 1 d spt + hz mod m, and therefore L spt, d spt + 1m + 1; z Ω even spt, m, proving the theorem. If m = 1, this foows directy from Lemma 5.3; if m this foows directy from the s = m case of Lemma Proofs of main theorems Proof of Theorems 1.1 and 1.5. By the remark after Lemma 3.4, the stabiized modues Ω odd spt, m and Ω even spt, m exist. By the remark foowing the proof of Theorem 4., we have that the ranks of these modues are at most R spt. Theorem 5.1 gives the bound on b spt, m that is stated in Theorem 1.1 and 1.5. The same reasoning hods for p r. Proof of Theorems 1.3 and 1.8. In the case of spt, one easiy checks that the -adic imit of the spaces Ω odd spt, m, Ω even spt, m is in the ordinary part of the space of -adic moduar forms. For these primes, the dimension of the space is 1. The theorem foows by Theorem 7.1 of [8]. For p r, the same argument appies when r is odd. When r is even, the theorem foows by cassica facts about p-adic moduar forms and the ordinary space, since P r, b; 4z has integra weight. Proof of Theorems 1.4 and 1.9. Let n nc in 1.6 and 1.7. The concusion foows from Theorems 1.3 and 1.8 because nc c = 0 and an/c = 0 when n is coprime to c. References [1] S. Ahgren, K. Bringmann, and J. Lovejoy. -adic properties of smaest parts functions. Advances in Math., 8:1, 011. [] G. Andrews. The number of smaest parts in the partitions of n. J. reine Angew. Math., 64:133 14, 008. [3] G. Andrews, F. Garvan, and J. Liang. Sef-conjugate vector partitions and the parity of the spt-function. preprint. [4] A. Atkin. Ramanujan congruences for p k n. Canad. J. Math., 0:67 78, [5] M. Boyan. Exceptiona congruences for powers of the partition function. Acta Arith., 111:187 03, 004. [6] M. Boyan and J. Webb. The partition function moduo prime powers. In submission, 011. [7] K. Bringmann. On the expicit construction of higher deformations of partition statistics. Duke Math J., 144: [8] A. Fosom, Z. Kent, and K. Ono. -adic properties of the partition function. Advances in Math., 9: , 01. [9] A Fosom and K Ono. The spt-function of Andrews. Proceedings of the Nationa Academy of Sciences, 10551: , 008. [10] F. Garvan. Congruences for Andrews s spt-function moduo powers of 5, 7, and 13. Trans. Amer. Math. Soc., accepted.

28 8 EVA BELMONT, HOLDEN LEE, ALEXANDRA MUSAT, SARAH TREBAT-LEDER [11] I. Kiming and J. Osson. Congruences ike Ramanujan s for powers of the partition function. Arch. Math, 59: , 199. [1] K. Ono. Distribution of the partition function moduo m. Ann. Math., 151:pg , 000. [13] K. Ono. The Web of Moduarity: Arithmetic of the Coefficients of Moduar Forms and q-series. AMS, 003. [14] K. Ono. Unearthing the visions of a master: Harmonic Maass forms and number theory. Current deveopments in mathematics, pages , 008. [15] K. Ono. Congruences for the Andrews spt function. PNAS, 108, no. : , 011. [16] J.-P. Serre. Formes moduaires et fonctions zêta p-adiques. Springer Lect. Notes in Math, 350:191 68, [17] H. P. F. Swinnerton-Dyer. On -adic representations and congruences for coefficients of moduar forms. Springer Lect. Notes in Math, 350:1 55, Department of Mathematics, Harvard University, Cambridge, Massachusetts 0138 E-mai address: ebemont@fas.harvard.edu Department of Mathematics, Massachusetts Institute of Technoogy, Cambridge, Massachusetts 0139 E-mai address: hoden1@mit.edu Department of Mathematics, Caifornia Institute of Technoogy, Pasadena, Caifornia 9115 E-mai address: amusat@catech.edu Department of Mathematics, Princeton University, Princeton, New Jersey E-mai address: strebat@math.princeton.edu