Congruences of the Partition Function

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Advance Access publication October 12, 2010 doi:10.1093/imrn/rnq194

Congruences of the Partition Function

Yifan Yang

Department of Applied Mathematics, National Chiao Tung University

and National Center for Theoretical Sciences, Hsinchu, Taiwan 300

Correspondence to be sent to: yfyang@math.nctu.edu.tw

Let p(n) denote the partition function. In this article, we will show that congruences of

the form

p(mk_n_{+ B) ≡ 0 mod m for all n≥ 0}

exist for all primes m and satisfying m ≥ 13 and = 2, 3, m, where B is a suitably

cho-sen integer depending on m and. Here, the integer k depends on the Hecke eigenvalues

of a certain invariant subspace of Sm/2−1(Γ0(576), χ12) and can be explicitly computed.

More generally, we will show that for each integer i> 0 there exists an integer k

such that with a properly chosen B the congruence

p(mikn+ B) ≡ 0 mod mi

holds for all integers n not divisible by.

1 Introduction

Let p(n) denote the number of ways to write a positive integer n as unordered sums of

positive integers. For convenience, we also set p(0) = 1, p(n) = 0 for n< 0, and p(α) = 0

ifα ∈ Z. A remarkable discovery of Ramanujan [14] is that the partition function p(n)

Received November 6, 2009; Revised July 2, 2010; Accepted September 6, 2010 c

at National Chiao Tung University Library on April 24, 2014

http://imrn.oxfordjournals.org/

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satisfies the congruences

p(An+ B) ≡ 0 mod m, (1)

for all nonnegative integers n for the triples

(A, B, m) = (5, 4, 5), (7, 5, 7), (11, 6, 11).

Ramanujan also conjectured that congruences (1) exist for the cases A= 5j_{, 7}j_{, or 11}j_.

This conjecture was proved by Watson [18] for the cases of powers of 5 and 7 and Atkin

[4] for the cases of powers of 11. (Apparently, Ramanujan actually found a proof of the

congruences modulo powers of 5 himself. The proof was contained in an unpublished manuscript, which was hidden from the public until 1988. It appeared that Ramanujan intended to prove congruences modulo powers of 7 along the same line of attack, but his ailing health prevented him from working out the details. See the commentary at the end

of [8] for more details.) Since then, the problem of finding more examples of such

con-gruences has attracted a great deal of attention. However, Ramanujan-type concon-gruences appear to be very sparse. Prior to the late twentieth century, there are only a handful of

such examples [5,7]. In those examples, the integer A is no longer a prime power.

It turns out that if we require the integer A to be a prime, then the congruences proved or conjectured by Ramanujan are the only ones. This was proved recently in a

remarkable paper of Ahlgren and Boylan [2]. On the other hand, if A is allowed to be a

nonprime power, a surprising result of Ono [13] shows that for each prime m≥ 5 and

each positive integer k, a positive proportion of prime has the property

p mk3_n_{+ 1} 24 ≡ 0 mod m (2)

for all nonnegative integers n relatively prime to . This result was later extended to

composite m,(m, 6) = 1, by Ahlgren [1]. The results of [1,13] were further extended by

Ahlgren and Ono [3].

Neither of [1,13] addressed the algorithmic aspect of finding congruences of the

form (2). For the cases m∈ {13, 17, 19, 23, 29, 31}, this was done by Weaver [19]. In effect,

she found 76,065 new congruences. (However, we should remark that some congruences

listed in [19, Theorem 2] were already discovered by Atkin [5]. Had Atkin had the

com-puting power of the present day, he would have undoubtedly discovered many more

congruences.) For primes m≥ 37, this was addressed by Chua [9], although no explicit

examples were given therein.

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Another remarkable discovery of Ono [13, Theorem 5] is that the partition func-tion possesses a certain periodic property modulo a prime m. Specifically, he showed

that for every prime m≥ 5, there exist integers 0 ≤ N(m) ≤ m48(m3_−2m+1)

and 1≤ P (m) ≤ m48(m3_−2m+1) such that p mi_n_{+ 1} 24 ≡ p mP(m)+i_n_{+ 1} 24 mod m (3)

for all nonnegative integers n and all i≥ N(m). Note that the bound m48(m3−2m+1)can be

improved greatly using a result of Garvan [10]. See Corollary 3.3 in Section 3 for details.

In this paper, we will obtain new congruences for the partition function and discuss related problems. In particular, we will show that there exist congruences of the form

p(mk_n_{+ B) ≡ 0 mod m}

for all primes m and such that m ≥ 13 and not equal to 2, 3, and m, where B is a

suitably chosen integer depending on m and.

Theorem 1.1. Let m and be primes such that m ≥ 13 and = 2, 3, m. Then there exists

an explicitly computable positive integer k≥ 2 such that

p m2k−1n+ 1 24 ≡ 0 mod m (4)

for all nonnegative integers n relatively prime to m.

For instance, in Section 5, we will find that for m= 37, congruences (4) hold with

5 7 11 13 17 19 23 29 31 41 43 47 53 59 61

k 228 57 18 684 38 38 684 684 228 171 18 333 18 12 684

As far as we know, this is the first example in literature where a congruence (1) modulo

a prime m≥ 37 is explicitly given.

Theorem 1.1 is in fact a simplified version of one of the main results. (See Theorem 3.6.) In the full version, we will see that the integer k in Theorem 1.1 can be determined quite explicitly in terms of the Hecke operators on a certain invariant

sub-space of the sub-space Sm/2−1(Γ0(576), χ12) of cusp forms of level 576 and weight m/2 − 1

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with characterχ12= (12_· ). This invariant subspace of Sm/2−1(Γ0(576), χ12) was first

dis-covered by Garvan [10] and rediscovered by the author of the present paper. To describe

this invariant subspace and to see how it comes into play with congruences of the

parti-tion funcparti-tion, perhaps we should first review the work of Ono [13] and other subsequent

papers [9,19]. Thus, we will postpone giving the statements of our main results until

Section 3.

Our method can be easily extended to obtain congruences of p(n) modulo a prime

power. In Section 6, we will see that for each prime power mi and a prime = 2, 3, m,

there always exists a positive integer k such that

p mi2k−1_n_{+ 1} 24 ≡ 0 mod mi

for all positive integers n not divisible by. One example worked out in Section 6 is

p 132_{· 5}56783_n_{+ 1} 24 ≡ 0 mod 132_.

In the same section, we will also discuss congruences of type p(5jk_n_{+ B) ≡ 0 mod 5}j+1_.

1.1 Notation

Throughout the paper, we let S_λ(Γ0(N), χ) denote the space of cusp forms of weight λ and

level N with characterχ. By an invariant subspace of S_λ(Γ0(N), χ) we mean a subspace

that is invariant under the action of the Hecke algebra on the space.

For a matrixγ =a b

c d

∈ GL(2, Q) and a modular form f(τ) of an even weight k, the slash operator is defined by

f(τ)|kγ := (det γ )k/2(cτ + d)−kf aτ + b cτ + d .

For a power series f(q) =af(n)qnand a positive integer N, we let UN and VN denote

the operators UN: f(q) −→ f(q)|UN:= ∞ n=0 af(Nn)qn, VN: f(q) −→ f(q)|VN:= ∞ n=0 af(n)qNn.

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Moreover, if ψ is a Dirichlet character, then f ⊗ ψ denotes the twist f ⊗ ψ :=

af(n)ψ(n)qn.

Finally, for a prime m≥ 5 and a positive integer j, we write

F_{m, j}= n≥0,mj_n_{≡−1 mod 24} p mj_n_{+ 1} 24 qn.

Note that we have

Fm, j|Um= Fm, j+1. (5)

2 Works of Ono [13], Weaver [19], and Chua [9]

In this section, we will review the ideas in [9,13,19].

First of all, by a classical identity of Euler, we know that the generating function of p(n) has an infinite product representation

∞ n=0 p(n)qn₌ ∞ n=1 1 1− qn.

If we set q= e2πiτ_{, then we have}

q−1/24 ∞ n=0

p(n)qn= η(τ)−1,

whereη(τ) is the Dedekind eta function. Now assume that m is a prime greater than 3.

Ono [13] considered the functionη(mk_τ)mk

/η(τ). On the one hand, one has η(mk_τ)mk η(τ) Umk= ∞ n=1 (1 − qn₎mk · _∞ n=0 p(n)qn+(m2k_−1)/24 Umk.

On the other hand, one has

η(mk_τ)mk

η(τ) ≡ η(τ)

m2k₋₁

= Δ(τ)(m2k_−1)/24

mod m,

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whereΔ(τ) = η(τ)24 is the normalized cusp form of weight 12 on SL(2, Z). From these,

Ono [13, Theorem 6] deduced that

F_m,k≡(Δ(τ)

(m2k_−1)/24

|Umk)|V₂₄

η(24τ)mk mod m.

Now it can be verified that for k= 1, the right-hand side of the above congruence is

con-tained in the space S_(m2_−m−1)/2(Γ₀(576m), χ₁₂) of cusp forms of level 576m and weight

(m2_{− m − 1)/2 with character χ}

12= (12_· ). Then by (5) and the fact that Umdefines a lin-ear map

Um: Sλ+1/2(Γ0(4Nm), ψ) → Sλ+1/2(Γ0(4Nm), ψχm),

whereχmis the Kronecker character attached toQ(

√

m), one sees that Fm,k≡ Gm,k=

am,k(n)qnmod m

for some G_m,k∈ S_(m2_−m−1)/2(Γ₀(576m), χ₁₂χ_mk−1).

Now the general Hecke theory for half-integral weight modular forms states that if f(τ) =∞_n₌₁af(n)qn∈ Sλ+1/2(Γ0(4N), ψ) and is a prime not dividing 4N, then the Hecke operator defined by T2: f(τ) → ∞ n=1 af(2n) + ψ() (−1)λ_n λ−1_a f(n) + ψ(2)2λ−1af n 2 qn

sends f(τ) to a cusp form in the same space. In the situation under consideration, if is

a prime not dividing 576m such that

Gm,k T2≡ 0 mod m, then we have 0≡ (Gm,k T2) Umod m =∞ n=1 am,k(3n) + ψ(2)m 2_−m−3 am,kn qn

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since(n) = 0. In particular, if n is not divisible by , then am,k(3n) ≡ 0 mod m, which implies p mk3_n_{+ 1} 24 ≡ 0 mod m.

Finally, to show that there is a positive proportion of primes such that

G_m,k|T2≡ 0 mod m, Ono invoked the Shimura correspondence between half-integral

weight modular forms and integral weight modular forms [16] and a result of Serre [15,

6.4].

As mentioned earlier, Ono [13] did not address the issue of finding explicit

con-gruences of the form (2). However, [13, Section 4] did give us some hints on how one might

proceed to discover new congruences, at least for small primes m. The key observation is the following.

The modular form Gm,kitself is in a vector space of big dimension, so to

deter-mine whether Gm,k|T2 vanishes modulo m, one needs to compute the Fourier

coeffi-cients of Gm,k for a huge number of terms. However, it turns out that Fm,kis congruent

to another half-integral weight modular form of a much smaller weight. For example,

using Sturm’s theorem [17] Ono verified that

F13,2k+1≡ G13,2k+1≡ 11 · 6kη(24τ)11mod 13,

F13,2k+2≡ G13,2k+2≡ 10 · 6kη(24τ)23mod 13

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for all nonnegative integers k. The modular formη(24τ)11 _{is in fact a Hecke eigenform.}

(The modular formη(24τ)23 _{is also a Hecke eigenform. It has been known since Morris}

Newman’s work in the 1950s that for odd r with 0< r < 24, the function η(24τ)ris a Hecke

eigenform.) More generally, for m∈ {13, 17, 19, 23, 29, 31}, it is shown in [13, Section 4],

[11, Proposition 6] and [19, Proposition 5] that Gm,1 is congruent to a Hecke eigenform

of weight m/2 − 1. Using this observation, Weaver [19] then devised an algorithm to find

explicit congruences of the form (2) for m∈ {13, 17, 19, 23, 29, 31}.

The proof of congruences (6) given in [11,19] is essentially “verification” in the

sense that they all used Sturm’s criterion [17]. That is, by Sturm’s theorem to show that

two modular forms on a congruence subgroupΓ are congruent to each other modulo a

prime m, it suffices to compare sufficiently many coefficients, depending on the weight

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and index(SL(2, Z) : Γ ). Naturally, this kind of argument will not be very useful in prov-ing general results.

In [9], instead of the congruence

η(mτ)m

η(τ) ≡ η(τ)m

2₋₁

mod m

used by Ono, Chua considered the congruence η(mτ)m

η(τ) ≡ η(mτ)m−1η(τ)m−1mod m

as the starting point. The function on the right is a modular form of weight m− 1 on

Γ0(m). Thus, by the level reduction lemma of Atkin and Lehner [6, Lemma 7], one has η(mτ)m−1_η(τ)m−1_|(U

m+ m(m−1)/2−1Wm) ∈ Sm−1(SL(2, Z)),

where for a modular form f(τ) of an even integral weight k on Γ0(m), the Atkin–Lehner

operator Wmis defined by Wm: f(τ) −→ f(τ)|k 0 −1 m 0 = (√mτ)−kf −_mτ1 . (7) It follows that F_m,1= 1 η(24τ) Um≡ fm(24τ) η(24τ)mmod m

for some cusp form fm(τ) ∈ Sm−1(SL(2, Z)). (Incidently, this also proves Ramanujan’s

con-gruences for m= 5, 7, and 11, since there are no nontrivial cusp forms of weight 4, 6, and

10 on SL(2, Z).) By examining the order of vanishing of fm(τ) at ∞, Chua [9, Theorem 1.1]

then concluded that if we let rm denote the integer in the range 0< rm< 24 such that

m≡ −rmmod 24, then

Fm,1≡ η(24τ)rmφ

m,1(24τ) mod m

for some modular formφ_m,1 on SL(2, Z) of weight (m − rm− 2)/2. More generally, one

has the following proposition.

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Proposition 2.1. Let m≥ 13 be a prime and rm be the integer in the range 0< rm< 24

such that m≡ −rmmod 24. Set

rm, j= ⎧ ⎨ ⎩ rm if j is odd, 23 if j is even. Then Fm, j≡ η(24τ)rm, jφm, j(24τ) mod m

for some modular formφm, j(τ) on SL(2, Z), where the weight of φm, j is(m − rm− 2)/2 if

j is odd and is m− 13 if j is even.

Proof. Consider the function fm, j(τ) = η(mjτ)mj/η(τ). It is a modular form of weight

(mj_{− 1)/2 on Γ}

0(mj) with character (_m·)j. Consider also the auxiliary function

hm, j(τ) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ η(τ)m η(mτ) if j is odd, _η(τ)m η(mτ) 2 if j is even.

It is a modular form onΓ0(m) with character (·/m)j and satisfies

hm, j(τ) ≡ 1 mod m. (8)

By the level reduction lemma of Atkin and Lehner [6, Lemma 7], if we apply Um to fm,i

j− 1 times and then multiply the resulting function by h_{m, j}, we get a modular form on

Γ0(m) with trivial character. That is,

f_{m, j}(τ)|U_mj−1· h_{m, j}(τ)

is a modular form onΓ0(m) with trivial character. The weight is

λm, j= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ (mj_{− 1)} 2 + (m − 1) 2 if j is odd, (mj_{− 1)} 2 + m − 1 if j is even.

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Then by the level reduction lemma again

( fm, j(τ)|Umj−1· hm, j(τ))|(Um+ mλm, j/2−1Wm)

is a modular form on SL(2, Z), where Wm is the Atkin–Lehner operator defined in (7).

Considering the order of vanishing at∞, we see that this modular form on SL(2, Z) is

Δ(τ)μm, j_φ m, j(τ), whereΔ(τ) = η(τ)24_, μm, j=m 2 j_{+ 24ν} m, j− 1 24mj

with νm, j being the unique integer satisfying 0< νm, j< mj and 24νm, j≡ 1 mod mj, and

φm, jis a modular form of weightλm, j− 12μm, jon SL(2, Z).

Now observe that hm, j|mλm, j/2−1Wmis congruent to 0 modulo a high power of m.

Then, by (8), we have

Δ(24τ)μm, j_φ

m, j(24τ) ≡ fm, j(τ) Umj V₂₄= η(24τ)m j

F_{m, j}mod m.

In other words, we have

F_{m, j}≡ η(24τ)24μm, j−mj_φ

m, j(24τ) = η(24τ)(24νm, j−1)/mjφm, j(24τ) mod m.

The integer(24ν_{m, j}− 1)/mj _{is in the range between 0 and 24. Also, it is congruent to}

−1/mj _{modulo 24. Thus, we have}

24νm, j− 1 mj = rm, j= ⎧ ⎪ ⎨ ⎪ ⎩ rm if j is odd, 23 if j is even.

From this, we get

λm, j− 12μm, j= ⎧ ⎪ ⎨ ⎪ ⎩ (m − rm− 2) 2 if j is odd, m− 13 if j is even.

This proves the proposition.

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Remark 2.2. Proposition 2.1 was stated as [9, Conjecture 1]. The proof sketched here

was suggested by one of the referees and was adapted from the proof of [2, Theorem 3].

Alternatively, one can combine Proposition 3.1 with an induction step proved in [9] to

get the same conclusion. See the arxiv version arXiv:0904.2530 of the present paper for

more details.

3 Main results

In this section, we will state our main results. Before doing so, let us first recall a prop-erty about the subspace

{η(24τ)r_{f(24τ) : f ∈ M}

s(SL(2, Z))}

of Ss+r/2(Γ0(576), χ12), in which the function η(24τ)rm, jφm, j(24τ) in Proposition 2.1 lies. Proposition 3.1 ([10, Proposition 3.1]). Let r be an odd integer with 0< r < 24. Let s be a nonnegative even integer. Then the space

Sr,s:= {η(24τ)rf(24τ) : f(τ) ∈ Ms(SL(2, Z))} (9)

is an invariant subspace of Ss+r/2(Γ0(576), χ12) under the action of the Hecke algebra.

That is, for all primes = 2, 3 and all f ∈ Sr,s, we have f|T2∈ S_r_,s.

Remark 3.2. This property ofSr,swas first discovered by Garvan [10], and later redis-covered by the author of the present paper. (See the arxiv version arXiv:0904.2530 of

the present paper.) Garvan stated the proposition under the assumption that (r, 6) = 1

instead of 2 r, but it can be easily checked that his proof also works for the cases

r= 3, 9, 15, and 21 as well. The author’s proof is more complicated, but can be applied

in other similar situations. However, at the hindsight, the invariance ofSr,s under the

action of Hecke algebra is best explained (and proved) as follows.

The usual definition of modular forms of half-integral weights, as per Shimura

[16], is given in terms of the theta functionθ(τ) =_n_∈Zqn2

. Specifically, we say a

holo-morphic function f :H → C is a modular form of half-integral weight λ +1₂ on Γ0(4N)

with characterχ, where χ is a Dirichlet character modulo 4N, if f(τ) is holomorphic at

each cusp and satisfies

f(γ τ)

f(τ) = χ(d)

θ(γ τ)2λ+1 θ(τ)2λ+1

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for allγ =a b c d

∈ Γ0(4N). It is in this sense we say η(24τ) is a modular form of weight 1/2 on Γ0(576) with character χ12.

Now the choice ofθ in the definition of half-integral modular forms is perhaps

the most natural and simplest from the view point of Weil representations, but one drawback of this choice is that the levels of the modular forms have to be a multiple of 4. On the other hand, if we define modular forms of half-integral weights in terms of η(τ), then the levels can be taken all the way down to 1. Explicitly, let Γ be a congruence

subgroup of SL(2, Z) for an odd integer r with 0 < r < 24 and a nonnegative even integer

s, we say a function f :H → C is a modular form of (ηr_{, s)-type on Γ if it is holomorphic}

inH and at each cusp such that

f(γ τ) f(τ) = (cτ + d) sη(γ τ)r η(τ)r for allγ =a b c d

∈ Γ . Let Sr,s(Γ ) be the space of all such modular forms on Γ .

Consider the case Γ = SL(2, Z). On the space S_r,s(SL(2, Z)), we can also define

Hecke operators T2 for primes = 2, 3 and show that their actions on f(τ) =

af(n)qn/24∈ Sr,s(SL(2, Z)) is T2: f(τ) → ∞ n=1 af(2n) + 12 (−1)λ_n λ−1_a f(n) + 2λ−1af n 2 qn/24

with λ = (r + 2s − 1)/2. Now observe that if g(τ) ∈ Sr,s(SL(2, Z)), then g(τ + 1) = e2πir/24_g_{(τ), which implies that g(τ) = q}r/24_(c

0+ c1q+ · · · ), ci∈ C. Therefore, f(τ) =

g(τ)/η(τ)r _{is a function holomorphic on} _{H and at each cusp and satisfies f(γ τ) =}

(cτ + d)s_{f(τ) for all γ =}a b c d ∈ SL(2, Z). In other words, Sr,s(SL(2, Z)) = {η(τ)rf(τ) : f ∈ Ms(SL(2, Z))} and Sr,s= {g(24τ) : g ∈ Sr,s(SL(2, Z))}.

This explains whySr,sis an invariant subspace of Sr/2+s(Γ0(576), χ12).

Using the pigeonhole principle, one can see that Propositions 2.1 and 3.1 yield

Ono’s periodicity result (3), with an improved bound.

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Corollary 3.3. Let m≥ 5 be a prime. Then there exist integers N(m) and P (m) with

0≤ (N(m) − 1)/2 ≤ mA(m)_{and 0}_{≤ P (m) ≤ m}A(m)_{such that}

p mi_n_{+ 1} 24 ≡ p m2P(m)+i_n_{+ 1} 24 mod m

for all nonnegative integers n and all i≥ N(m), where

A(m) = dim M_(m−r_m_−2)/2(SL(2, Z)) = m 12 − m 24 (10)

and rmis the integer satisfying 0< rm< 24 and m ≡ −rmmod 24.

From Proposition 3.1, we can deduce the following corollary, which will be proved in the next section.

Corollary 3.4. Let r be an odd integer satisfying 0< r < 24 and s be a nonnegative even

integer. LetS_r,sbe defined as (9) and{ f1, . . . , ft} be a Z-basis for the Z-module Z[[q]] ∩ Sr,s.

Given a prime ≥ 5, assume that A is the t × t matrix such that

⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ T2= A ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ . Then we have ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ U k 2= Ak ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ + Bk ⎛ ⎜ ⎜ ⎝ g1 ... gt ⎞ ⎟ ⎟ ⎠ + Ck ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ V2,

where gj= fj⊗ (·), and for nonnegative integers k, Ak, Bk, and Ckare t× t matrices

sat-isfying Ak Ak−1 = A −r+2s−2It It 0 k It 0

with Itbeing the t× t identity matrix, and

Bk= −s+(r−3)/2 ₍₋₁₎_(r−1)/2 12 Ak−1, Ck= −r+2s−2Ak−1.

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Remark 3.5. It is well known that for nonnegative even integer s, the space Ms(SL(2, Z))

has a basis consisting of g1, . . . , gd satisfying gi∈ Z[[q]] and gi= qi−1+ · · · , where d=

dim Ms(SL(2, Z)). (Usually, gi are chosen to be products of Δ(τ) and Eisenstein series.)

Then it can be easily verified that the functions fi(τ) = η(24τ)rgi(24τ) form a Z-basis of

theZ-module Z[[q]] ∩ Sr,s. In particular, the rank of theZ-module Z[[q]] ∩ Sr,s is the same

as the dimension ofSr,s.

Note also that if r+ 2s ≥ 3, then the Hecke operator T2 maps Z[[q]] ∩ S_r_,s into

Z[[q]] ∩ Sr,s. Therefore, the matrix A in the above corollary has entries inZ. This property

is crucial in our subsequent discussion when we need to take A modulo a prime.

Now we can state our main results. The first one is a more precise version of Theorem 1.1. The proof utilizes the corollary above and will be given in the next section.

Theorem 3.6. Let m≥ 13 be a prime. Set rmto be the integer satisfying 0< rm< 24 and

m≡ −rmmod 24. Let t= m 12 − m 24

be the dimension of Srm,(m−rm−2)/2 and assume that { f1, . . . , ft} is a basis for the

Z-module Z[[q]] ∩ Srm,(m−rm−2)/2. Let be a prime different from 2, 3, and m, and assume

that A is the t× t matrix such that

⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ T2= A ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ .

Assume that the order of the square matrix A −m−4_I t It 0 mod m (11) in PGL(2t, Fm) is K. Then we have p _m2uK−1_n_{+ 1} 24 ≡ 0 mod m (12)

for all positive integers u and all positive integers n not divisible by.

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Also, if the order of the matrix (11) in GL(2t, Fm) is M, then we have p _mi_n_{+ 1} 24 ≡ p _m2M+i_n_{+ 1} 24 mod m (13)

for all nonnegative integer i and all positive integers n.

Remark 3.7. Note that if the matrix A in the above theorem vanishes modulo m,

then the matrix in (11) has order 2 in PGL(2t, Fm), and the conclusion of the theorem

asserts that p mj3_n_{+ 1} 24 ≡ 0 mod m.

This is the congruence appearing in Ono’s theorem.

Remark 3.8. In general, the integer K in Theorem 3.6 may not be the smallest positive

integer such that congruence (4) holds. We choose to state the theorem in the current

form because of its simplicity. See the remark following the proof of Theorem 3.6.

4 Proof of Corollary 3.4 and Theorem 3.6

Proof of Corollary 3.4. By the definition of T2, we have

⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ U2= A1 ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ + B1 ⎛ ⎜ ⎜ ⎝ g1 ... gt ⎞ ⎟ ⎟ ⎠ + C1 ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ V2, where gj= fj⊗ (·) and A1= A, B1= −s+(r−3)/2 ₍₋₁₎_(r−1)/2 12 It, C1= −r+2s−2It.

Now we make the key observation

gj|U2= 0, f_j|V2|U2= f_j,

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from which we obtain ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ U 2 2= (A2₁+ C1) ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ + A1B1 ⎛ ⎜ ⎜ ⎝ g1 ... gt ⎞ ⎟ ⎟ ⎠ + A1C1 ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ V2.

Iterating, we see that in general if ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ U k 2= Ak ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ + Bk ⎛ ⎜ ⎜ ⎝ g1 ... gt ⎞ ⎟ ⎟ ⎠ + Ck ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ V2,

then the coefficients satisfy the recursive relation

Ak+1= AkA1+ Ck, Bk+1= AkB1, Ck+1= AkC1.

(Note that B1 and C1 are scalar matrices. Thus, all coefficients are polynomials in A.)

Finally, we note that the relation Ak+1= AkA1+ Ck= AkA1+ C1Ak−1can be written as

Ak+1 Ak = A C1 It 0 Ak Ak−1 , which yields Ak+1 Ak = A C1 It 0 k A It = A C1 It 0 k+1 It 0 .

This proves the corollary.

Proof of Theorem 3.6. Let m≥ 13 be a prime. Let r be the integer satisfying 0 < r < 24

and m≡ −r mod 24 and set s = (m − r − 2)/2. By Proposition 2.1, F_m,1 is congruent to a

modular form inS_r,s, where S_r,s is defined by (9). Now let { f1, . . . , ft} be a Z-basis for

Z[[q]] ∩ Sr,sand A be given as in the statement of the theorem. (Note that A has entries in

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Z. See Remark 3.5.) Then by Corollary 3.4, we know that ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ U k 2= Ak ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ + Bk ⎛ ⎜ ⎜ ⎝ g1 ... gt ⎞ ⎟ ⎟ ⎠ + Ck ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ V2,

where gj= fj⊗ (·), and Ak, Bk, and Ckare t× t matrices satisfying

Ak Ak−1 = Xk It 0 , (14) Bk= −(m−5)/2 ₍₋₁₎_(r−1)/2 12 Ak−1, Ck= −m−4Ak−1 (15) with X= A −m−4_I t It 0

for all k≥ 1. Now we have

X−1= −(m−4) 0 m−4_I t −It A .

Therefore, if the order of X mod m in PGL(2t, Fm) is K, then we have, for all positive

integers u, AuK−1 AuK−2 = XuK−1 It 0 ≡ 0 U mod m

for some t× t matrix U, that is, AuK−1≡ 0 mod m. The rest of proof follows Ono’s

argument. We have ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ U uK−1 2 ≡ BuK−1 ⎛ ⎜ ⎜ ⎝ g1 ... gt ⎞ ⎟ ⎟ ⎠ + CuK−1 ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ V2mod m

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and ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ U uK−1 2 U≡ CuK−1 ⎛ ⎜ ⎜ ⎝ f1 ... ft ⎞ ⎟ ⎟ ⎠ Vmod m.

This implies that the2uK−1_{nth Fourier coefficients of f}

jvanishes modulo m for all j and

all n not divisible by. Since F_m,1is a linear combination of fjmodulo m, the same thing

is true for the(2uK−1n)th Fourier coefficients of Fm,1. This translates to

p m2uK−1_n_{+ 1} 24 ≡ 0 mod m

for all n not divisible by. This proves (12).

Finally, if the matrix X has order M in GL(2t, Fm), then from the recursive

rela-tions (14) and (15), it is obvious that (13) holds. This completes the proof.

Remark 4.1. In general, the integer K in Theorem 3.6 may not be the smallest positive

integer such that congruence (4) holds. For example, consider the case whereSr,s has

dimension t≥ 2 and the reduction of Z[[q]] ∩ Sr,s modulo m has a basis consisting of

Hecke eigenforms f1, . . . , ft defined overFm. Suppose that the eigenvalues of T2 for f_i

modulo m are a(1) , . . . , a(t)∈ Fm. Let ki denote the order of

a(i)−m−4

1 0

in PGL(2, Fm). Let k

be the least common multiple of ki. Then we can show that

fi|U2k−1≡ cifi|Vmod m

for some ci∈ Fm and consequently congruence (4) holds. Of course, the least common

multiple of kimay be smaller than the integer K in Theorem 3.6 in general.

5 Examples

Example 5.1. Let m= 13. According to Proposition 2.1, we have

F13,1≡ cη(24τ)11mod 13

for some c∈ F13. (In fact, c= 11. See [13, page 303].) The eigenvalues amodulo 13 of T2

for the first few primes are

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5 7 11 17 19 23 29 31 37 41 43 47 53 59 61 67 73

a 10 8 5 1 8 8 4 4 5 9 12 6 10 0 2 4 0

9 ₅ ₈ ₈ ₁₂ ₅ ₁₂ ₁ ₅ ₈ ₅ ₁₂ ₈ ₁ ₈ ₁ ₅ ₅

For = 5, the matrix

X= a −9 1 0 ≡ 10 8 1 0 mod 13

has eigenvalues 5±√7 over F13. Now the order of(5 +

√

7)/(5 −√7) in F169 is 14. This

implies that 14 is the order of X in PGL(2, F13) and we have

p 13· 528u−1_n_{+ 1} 24 ≡ 0 mod 13

for all positive integers u and all positive integers n not divisible by 5. Likewise, we find

that congruence (4) holds with

5 7 11 17 19 23 29 31 37 41 43 47 53 59 61 67 73

k 14 14 14 7 14 3 6 12 14 12 7 12 7 2 13 12 2

Example 5.2. Let m= 37. By Proposition 2,1, we know that F37,1is congruent to a cusp

form inS11,12 modulo 37. In fact, according to [9, Table 3.1],

F37,1≡ η(24τ)11(E4(24τ)3+ 17Δ(24τ)) mod 37.

The two eigenforms ofS11,12are defined over a certain real quadratic number field, but

the reduction ofS11,12∩ Z[[q]] modulo 37 has eigenforms defined over F37. They are

f1= η(24τ)11(E4(24τ)3+ 24Δ(24τ)), f2= η(24τ)11Δ(24τ).

Let a(i) denote the eigenvalue of T2 associated to f_i. We have the following data.

5 7 11 13 17 19 23 29 31 41 43 47 53 59 61

a(1) 1 33 22 7 11 0 1 9 35 11 28 14 30 24 12

a(2) 32 10 0 6 7 8 31 36 9 10 1 35 9 3 16

33 ₈ ₂₆ ₃₆ ₈ ₂₃ ₈ ₆ ₃₁ ₃₁ ₁₁ ₆ ₁ ₁₀ ₂₃ ₂₉

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Let Xi= a(i) −33 1 0 .

For = 5, we find the orders of X1and X2in PGL(2, F37) are 38 and 12, respectively. The least common multiple of the orders is 228. Thus, we have

p 37· 5456u−1_n_{+ 1} 24 ≡ 0 mod 37

for all positive integers u and all positive integers n not divisible by 5. Note that this is an example showing that the integer K in the statement of Theorem 3.6 is not optimal.

(Here we have K= 456.)

For other small primes, we find that the congruence

p 372uk−1_n_{+ 1} 24 ≡ 0 mod 37

holds for all n not divisible by with

5 7 11 13 17 19 23 29 31 41 43 47 53 59 61

k 228 57 18 684 38 38 684 684 228 171 18 333 18 12 684

6 Generalizations

There are several directions in which one may generalize Theorem 3.6. Here, we only

consider congruences of the partition function modulo prime powers. The case m= 5

will be dealt with separately because in this case we have a very precise congruence result.

In his proof of Ramanujan’s conjecture for the cases m= 5 and 7, Watson [18,

page 111] established a formula

F5, j= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ i≥1 c_j,iη(120τ) 6i−1 η(24τ)6i if j is odd, i≥1 cj,iη(120τ) 6i η(24τ)6i+1 if j is even,

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where cj,i≡ ⎧ ⎨ ⎩ 3j−1₅j_{mod 5}j+1 _{if i}_{= 1,} 0 mod 5j+1 _{if i}_{≥ 2.}

From the identity, one deduces that

F5, j≡ 3j−15j ⎧ ⎨ ⎩ η(24τ)19_{mod 5}j+1 _{if j is odd}_, η(24τ)23_{mod 5}j+1 _{if j is even}_. (16)

Then Lovejoy and Ono [12] used this formula to study congruences of the partition

func-tion modulo higher powers of 5. One distinct feature of [12] is the following lemma.

Lemma 6.1 (Lovejoy and Ono [12, Theorem 2.2]). Let ≥ 5 be a prime. Let a and b be

the eigenvalues of η(24τ)19 _and_η(24τ)23 _{for the Hecke operator T}

2, respectively. Then we have a, b ≡ 15 (1 + ) mod 5.

With this lemma, Lovejoy and Ono obtained congruences of the form

p 5jk_n_{+ 1} 24 ≡ 0 mod 5j+1

for primes congruent to 3 or 4 modulo 5. Here, we shall deduce new congruences using

our method.

Theorem 6.2. Let ≥ 7 be a prime. Set

K= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 5 if ≡ 1 mod 5, 4 if ≡ 2, 3 mod 5, 2 if ≡ 4 mod 5. Then we have p 5j2uK−1_n+ 1 24 ≡ 0 mod 5j+1

for all positive integers j and u and all integers n not divisible by.

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Proof. In view of (16), We need to study when a Fourier coefficient ofη(24τ)19orη(24τ)23 vanishes modulo 5.

Let f= η(24τ)19_{. Let}_{≥ 7 be a prime and a be the eigenvalue of T}

2 associated to f . By Corollary 3.4 we have f|Uk2= akf+ bkf⊗ · + ckf|V2, (17) where a1= a, b1= −8(−12/), c1= −17, and ak= ak−1a1+ ck−1, bk= ak−1b1, ck= ak−1c1.

According to the proof of Theorem 3.6, if the order of a −17 1 0 mod 5 (18) in PGL(F5) is k, then f|U2uk−1≡ f|Vmod 5 (19)

for all positive integers u. Now by Lemma 6.1 the characteristic polynomial of (18) has a

factorization x− 15 x− 15

modulo 5. From this we see that the order of (18) in PGL(F5) is

K= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 5 if ≡ 1 mod 5, 4 if ≡ 2, 3 mod 5, 2 if ≡ 4 mod 5.

Thus, (19) holds with k= K. This yields the congruence

p 5j2uK−1_n+ 1 24 ≡ 0 mod 5j+1

for odd j, positive integer u, and all positive integers n not divisible by.

The proof of the case j even is exactly the same because21≡ 17mod 5.

Remark 6.3. Watson [18] also had an identity for F7, j, with which one can study con-gruences modulo higher powers of 7. However, because there does not seem to exist an

analog of Lemma 6.1 in this case, we do not have a result as precise as Theorem 6.2.

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The next congruence result is an analog of [19, Theorem 2], which in turn

origi-nates from the argument outlined in [13, page 301].

Theorem 6.4. Let ≥ 7 be a prime. Assuming one of the three situations below occurs,

we set kand mto be

(1) k= 2 and m= 5 if ≡ 1 mod 5, (−n/) = −1,

(2) k= 2 and m= 4 if ≡ 2 mod 5, (−n/) = −1, and

(3) k= 1 and m= 4 if ≡ 3 mod 5, (−n/) = −1. Then p 5i2(um+k)_n+ 1 24 ≡ 0 mod 5i+1

for all nonnegative integers u and all positive integers i.

Proof. Assume first that i is odd. Again, in view of (16), we need to study when the

Fourier coefficients of f(τ) = η(24τ)19vanish modulo 5.

Let ≥ 7 be a prime and a be the eigenvalue of T2 associated to . By (17), we

have

f|Uk2= akf+ bkf⊗ ·

+ ckf|V2, (20)

where ak, bk, and cksatisfy

ak ak−1 = a −17 1 0 k 1 0 , bk≡ − ₋₁₂ ak−1, ck≡ −ak−1mod 5.

From Lemma 6.1, we know that for ≡ 1 mod 5, we have a1≡ 2 and thus the values of ak

modulo 5 are

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 . . .

2 3 4 0 2 3 4 0 1 2 3 . . .

where  = (15/). Now assume that f(τ) =c(n)qn_{. Comparing the nth Fourier}

coeffi-cients of the two sides of (20) for integers n relatively prime to, we obtain

c(2k_n_{) =}_a k+ bk n c(n) ≡ ak− ak−1 −12n c(n) mod 5.

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When k= 5u+ 2 for a nonnegative integer u, we have c(2(5u+2)n) ≡ 3 15 u 1+ 15 −12n c(n) = 3 15 u 1+ −n c(n) mod 5. (21)

Thus, if(−n/) = −1, then c(2(5u+2)_n_{) ≡ 0 mod 5. This translates to the congruence}

p 5i2(5u+2)_n_{+ 1} 24 ≡ 0 mod 5i+1_.

This proves the first case of the theorem. The proof of the other cases is similar.

Remark 6.5. Note that the case ≡ 4 mod 5 is missing in Theorem 6.4. This is because

in this case, by Lemma 6.1, the Hecke eigenvalues of T2 for η(24τ)19 andη(24τ)23 are

both multiples of 5. Then the numbers akin (20) satisfy

ak ak−1 = 0 1 1 0 k 1 0 .

From this, we see that ak± ak−1can never vanish modulo 5.

Example 6.6. We now give some examples of congruences predicted in Theorem 6.4. (1) Let = 11, i = 1, and n= 67. Then the first situation occurs. We find

p 5· 114_{· 67 + 1} 24 = p(204364) = 28469 . . . 24450, which is a multiple of 25.

(2) Let = 11, i = 1, and n= 19. The condition in the theorem is not fulfilled, but

(21) implies that p 5· 114_{· 19 + 1} 24 ≡ p 5· 19 + 1 24 mod 25.

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Indeed, we have p(4) = 5,

p(57954) = 37834 . . . 45055,

and they are congruent to each other modulo 25.

(3) Let = 7, i = 2, and n= 23. Then the second situation occurs. We have

p 52_{· 7}4_{· 23 + 1} 24 = p(57524) = 38402 . . . 43875,

which is indeed a multiple of 53_.

Theorem 6.7. Let m≥ 13 be a prime and be a prime different from 2, 3, and m. For each positive integer i, there exists a positive integer K such that for all u≥ 1 and all

positive integers n not divisible by, the congruence

p mi2uK−1n+ 1 24 ≡ 0 mod mi

holds. There is also another positive integer M such that

p mirn+ 1 24 ≡ p miM+rn+ 1 24 mod mi

holds for all nonnegative integers n and r.

Proof. Letβ_m,ibe the integer satisfying 1≤ β_m,i≤ mi_{− 1 and 24β}

m,i≡ 1 mod mi. Define

km,i= ⎧ ⎪ ⎨ ⎪ ⎩ (mi−1_{+ 1)(m − 1)} 2 − 12  m 24 − 12 if i is odd, mi−1_{(m − 1) − 12} _{if i is even}_.

By [2, Theorem 3], for all i≥ 1, there is a modular form f ∈ Mkm,i(SL(2, Z)) such that

Fm,i≡ η(24τ)(24βm,i−1)/m

i

f(24τ) mod mi.

The rest of proof is parallel to that of Theorem 3.6.

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Example 6.8. Consider the case m= 13 and i = 2 of Theorem 6.7 and assume that is a

prime different from 2, 3, and 13. By [2, Theorem 3], F13,2is congruent to a modular form

in the spaceS23,144of dimension 13. Choose aZ-basis

fi= η(24τ)23E4(24τ)3(13−i)Δ(24τ)i−1, i = 1, . . . , 13,

forZ[[q]] ∩ S23,144and let A be the matrix of T2 with respect to this basis. If the order of

the matrix A −309_I 13 I13 0 mod 169 in PGL(26, Z/169) is K, then we have p 1692K−1_n_{+ 1} 24 ≡ 0 mod 169

for all integers n not divisible by. For instance, for = 5, we find

A= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 20 101 52 52 166 148 46 135 96 51 73 49 128 166 164 159 66 123 50 144 85 29 116 22 93 10 158 152 90 65 20 167 27 96 109 154 127 164 76 120 154 132 110 22 113 115 51 25 104 108 82 33 43 148 131 45 81 2 164 145 117 157 4 108 61 134 23 151 120 151 44 30 1 76 32 60 132 165 121 40 83 4 56 88 3 134 100 85 88 18 3 23 20 20 31 66 24 41 126 47 137 33 112 49 143 18 44 26 89 109 118 148 35 16 35 122 150 144 51 47 143 109 164 52 38 92 50 98 60 104 70 165 89 80 28 75 19 110 101 41 155 78 67 123 147 54 4 60 133 49 151 30 32 157 108 82 95 139 50 70 124 168 87 63 13 104 58 107 113 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

modulo 169, and the order K is 28, 392, which yields

p 132_{· 5}56,783_n_{+ 1} 24 ≡ 0 mod 132

for all n not divisible by 5.

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Acknowledgements

The author would like to thank the referees for thorough reading of the manuscript and providing many invaluable comments. In particular, the author is very grateful to one of the referees for giving a more accurate account of the history of the partition congruence problem and to another referee for bringing his attention to a very recent paper of Garvan [10]. Also, the proof of Proposi-tion 2.1 presented here was suggested by the second referee.

This paper was dedicated to Prof. B. C. Berndt on the occasion of his 70th birthday.

Funding

This work was partially supported by the National Science Council of Taiwan, grant no. 98-2115-M-009-001 (to Y.Y.). This work was also supported by the National Center for Theoretical Sciences.

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