CONGRUENCES FOR POWERS OF THE PARTITION FUNCTION MADELINE LOCUS AND IAN WAGNER Abstract. Let p tn denote the number of partitions of n into t colors. In analogy with Ramanujan s work on the partition function, Lin recently proved in [] that p n + 7 0 mod for every integer n. Such congruences, those of the form p tln + a 0 mod l, were previously studied by Kiming and Olsson. If l 5 is prime and t {l, l 3}, then such congruences satisfy a t mod l. Inspired by Lin s example, we obtain natural infinite families of such congruences. If l 2 mod 3 resp. l 3 mod 4 and l mod 2 is prime and r {4, 8, 4} resp. r {6, 0} and r 26, then for t ls r, where s 0, we have that p t ln + rl2 rl 2 0 mod l. l Moreover, we exhibit infinite families where such congruences cannot hold.. Introduction and Statement of Results A t-color partition, where t, of an integer n is a t-tuple λ λ,..., λ t of partitions such that λ +... + λ t n see [2]. The t-color partition function, which counts partitions of n into t colors, has the following generating function: p t nq n q n t. When t, this becomes the usual partition function pn. Ramanujan discovered the following congruences for the partition function: p5n + 4 0 mod 5, p7n + 5 0 mod 7, pn + 6 0 mod. Notice that the modulus of the progression matches the modulus of the congruence in these three examples. Motivated by this observation, it is natural to classify other congruences where these moduli agree. Kiming and Olsson proved in [3] that in order for a Ramanujan-type congruence modulo l to exist for the progression ln + a, it must be true that a t mod l. Along these lines, there have been other works which study specific cases in which congruences of the partition function exist. In [4], Harper gave several arithmetic progressions of primes and stated the possible values of b such that the coefficients an of certain color partitions satisfy similar congruences, namely aln + b 0 mod l. More recently, in [] Lin found and proved the specific congruence p n + 7 0 mod using elementary methods. In view of these recent works, it is natural to ask if there is a general phenomenon of when congruences can and cannot occur. Is there a more general form of Harper s and Lin s work? Kiming and Olsson raise the question of whether there are infinitely many primes for which a congruence of this type exists. We address both of these questions in the following theorem. Theorem. Assume the notation above, and let t ls r. Then the following are true.
2 LOCUS AND WAGNER If r {4, 8, 4} and l 2 mod 3 is prime, then for every integer n, we have p t ln + rl2 rl 2 0 mod l. l 2 If r {4, 8, 4} and l mod 3, then there are infinitely many n for every arithmetic progression mod l for which p t ln + rl2 rl 2 0 mod l l Theorem 2. Assume the notation above. Then the following are true. If r {6, 0} and l 3 mod 4 is prime, then for every integer n, we have p t ln + rl2 rl 2 0 mod l. l 2 If r {6, 0} and l mod 4, then there are infinitely many n for every arithmetic progression mod l for which p t ln + rl2 rl 2 0 mod l l Theorem 3. Assume the notation above. Then the following are true. If r 26 and l mod 2 is prime, then for every integer n, we have p t ln + rl2 rl 2 0 mod l. l 2 If r 26 and l, 5, 7 mod 2, then there are infinitely many n for every arithmetic progression mod l for which p t ln + rl2 rl 2 0 mod l. l To prove these statements, we use a famous work of Serre on powers of Dedekind s eta-function see [5], together with the theory of Hecke operators, modular forms with complex multiplication, and some combinatorial observations. 2. Nuts and Bolts In Section 2., we give basic definitions from the theory of modular forms and Hecke operators, and we introduce a theorem of Serre which gives the basis for our proofs of Theorems, 2, and 3. In Section 2.2, we define newforms with complex multiplication, which we reveal in Section 3 are closely connected with Ramanujan-type congruences. 2.. Basic Definitions. Recall the congruence subgroup Γ 0 N defined as { } a b Γ 0 N : SL c d 2 Z : c 0 mod N. Suppose that fz is a meromorphic function on H, the upper half of the complex plane, that k Z, and that Γ is a congruence subgroup of level N. Then fz is called a meromorphic modular form with integer weight k on Γ if the following hold: We have az + b f cz + d k fz cz + d a b for all z H and all Γ. c d
CONGRUENCES FOR POWERS OF THE PARTITION FUNCTION 3 2 If γ 0 SL 2 Z, then f k γ 0 z has a Fourier expansion of the form f k γ 0 z n n γ0 a γ0 nq n N, where q N : e 2πiz/N and a γ0 n γ0 0. If k 0, then fz is known as a modular function on Γ. Furthermore, if χ is a Dirichlet character modulo N, then we say that a form fz M k Γ N resp. S k Γ N has Nebentypus character χ if az + b f χdcz + d k fz cz + d a b for all z H and all Γ c d 0 N. The space of such modular forms resp. cusp forms is denoted by M k Γ 0 N, χ resp. S k Γ 0 N, χ. If fz is an integer weight meromorphic modular form on a congruence subgroup Γ, then we call fz a holomorphic modular resp. cusp form if fz is holomorphic on H and is holomorphic resp. vanishes at the cusps of Γ. We now define the Hecke operators, which are linear transformations that act on spaces of modular forms by transforming the coefficients of their Fourier expansions. Let k Z + and let p be prime. If fz Anqn is in M k Γ 0 N, χ resp. S k Γ 0 N, χ, then the action of the Hecke operator T p,k,χ on fz is defined by fz T p,k,χ : n Apn + χpp k A q n. p If p n, then we say that An/p 0. For our purposes, we shorten T p,k,χ to T p. It will later be valuable to note that fz T p,k,χ 0 mod p only when Apn 0 mod p for every n. We define the following function as a power of a Dedekind eta-product, with r, δ Z. We only consider integer weight modular forms, so we assume that r is even. * F r,δ z : η r δz q q δn r A r nq n S k Γ 0 N, χ. When r {2, 4, 6, 8, 0, 4, 26}, the level N of η r δz is the corresponding value of δ 2 in Theorem 4, and the Nebentypus χ depends on N. Several nice properties of this family of functions will become clear when we introduce newforms in the next section. We now give esult of Serre from [5], which follows from the theory of modular forms with complex multiplication. Theorem 4 Serre. Suppose that r {2, 4, 6, 8, 0, 4, 26}. Then the following are true: If l 2 mod 3 is prime, r {4, 8, 4}, and δ is the corresponding {6, 3, 2}, then F r,δ z T l 0 mod l. 2 If l 3 mod 4 is prime, r {6, 0}, and δ is the corresponding {4, 2}, then F r,δ z T l 0 mod l. 3 If l mod 2 is prime, r 26, and δ 2, then F r,δ z T l 0 mod l. 2.2. Modular forms with Complex Multiplication. For a generic number field K which is a finite field extension of Q, we denote the ring of algebraic integers in K by O K. This ring is a finite field extension of Z, and in the cases we will consider will be a principal ideal domain. Remark. In general, O K is rarely a unique factorization domain, but for the number fields we consider, its ideals always factor uniquely into products of prime ideals since it is a principal ideal domain.
4 LOCUS AND WAGNER It will be beneficial to consider p a + bia bi as ideals in Z[i], the ring of integers of O K where K Qi, and p x + y x y as ideals in Z [ + ] 2, the ring of integers of O where K Q, for those primes which factor i.e. the split primes. These are the unique factorizations of p into prime ideals in their respective fields. In order to connect this object with congruences for powers of the partition function we will make use of newforms. A newform is a normalized cusp form in Sk new Γ 0 N that is an eigenform of all the Hecke operators and all the Atkin-Lehner involutions k W Q p, for primes p N, and k W N. Newforms come with the following properties. Theorem. If fz A r nq n Sk new Γ 0 N is a newform, then the following are true. If p is prime with p 2 N, then A r p 0. 2 If p N is prime with ord p N, then A r p λ p p k 2, where λ p is oot of unity. If K Q D is an imaginary quadratic field with discriminant D and O K is its ring of algebraic integers, then a Hecke Grossencharacter φ of weight k 2 with modulus Λ can be defined in the following way. Let Λ be a nontrivial ideal in O K and let IΛ denote the group of fractional ideals prime to Λ. A Hecke Grossencharacter φ with modulus Λ is a homomorphism φ : IΛ C such that for each α K with α mod Λ we have φαo K α k. Using this notation the newform Ψz can be defined by Ψz α φαq Nα anq n. The sum is over the integral ideals α that are prime to Λ and Nα is the norm of the ideal α [6]. We can use this fact write our functions F r,δ z as q-series or linear combinations of q-series with complex units φα as coefficients over ideals α in Z[i] O Qi resp. Z [ + ] 2 OQ. This gives epresentation of A r p in terms of the factorization of primes p mod 4 as p a + bia bi a 2 + b 2 and primes p mod 3 as p x + y x y x 2 + 3y 2. The congruences for primes p 3 mod 4 resp. primes p 2 mod 3 come from Serre s result that A r np 0 mod p for all n Z. Showing that this does not happen for any prime p mod 4 resp. p mod 3 will show that all of the Ramanujan-type congruences have been found for every prime except p 2. Because of the way prime ideals factor in the principal ideal domains above, we will make use of the following theorem by Fermat. Theorem Fermat. The following are true. Every prime p mod 4 has exactly one representation of the form a 2 + b 2 p, where a, b > 0. 2 Every prime p mod 3 has exactly one representation of the form x 2 + 3y 2 p, where x, y > 0. 3. Proof of Theorems, 2, and 3 3.. Combinatorial Analysis of Modular Forms with CM. Let us define nq n qn r. Then, using the definition of t as t ls r, p t nq n q n t q n r q n ls q n ls nq n.
So then we have nq n CONGRUENCES FOR POWERS OF THE PARTITION FUNCTION 5 q n ls p t nq n q ln s p t nq n mod l. It can be seen that if n 0 mod l for some n, then p t n 0 mod l since qln s is supported only on multiples of l. To complete the proofs of Theorems, 2, and 3, we will show that ln + rl2 rl 2 l 0 mod l when l is in the correct residue class. Recall that F r,δ z A rnq n and from Theorem 4 that F r,δ z T l 0 mod l. It can be seen that A rnq n a rnq. Then the q-series nq n can be manipulated in the following way in order to take advantage of Theorem 4 of Serre. Let us define d : rl 2 l. Then we have ln d + rl2 q n δ ln d + rl2 δ ln d + rl2 δl n d + rl q δn q. If l and r are in the progressions given by Serre, then we have δl n d + rl q χll r 2 χll r 2 + q δ n d + rl l δn d l χll r δn d 2 l χll r 2 n d l χll r 2 nq ln+d χll r 2 q d q ln. Thus ln d + rl2 q n χll r 2 q d This proof gives a stronger result about ln + rl2 l rl 2 q n + q δn q ln. q q than is actually necessary. 3.2. Examples. Recall that Olsson proved that congruence relations of this type on progressions ln + a only occur when a is in exactly the residue classes given by Theorems, 2, and 3. We will now show for exactly which primes these congruence relations exist. It is noted above that η r δz is either a newform or a linear combination of newforms with CM. The representation using the sum
6 LOCUS AND WAGNER over ideals can then be used to gain insight about the coefficients of the eta-products. The cases when r 8 and r 26 are worked out as examples below. For r 8, we have A 8 nq n η 8 3z q q 3n 8. Using the ideal representation, A 8 nq n α The field is Z [ + ] 2 in this case, so prime ideals factor as p x + y x y. Then we have A 8 pq p p 3 q pp p 3 q x2 +3y 2 p p p φαq Nα α α 3 q αα. where p denotes the sum over primes. We have already proven congruences for primes p 2 mod 3 for r 8, so now we want to study primes p mod 3. From the theorem by Fermat, we know that primes p mod 3 have a unique representation as p x 2 + 3y 2, so selecting the x and y values that produce the prime we want should also give us the coefficient of that term. We see that p 3 q x2 +3y x 2 3 3 + y + x y q x2 +3y 2, p so the coefficients are of the form A 8 p x + y 3 + x y 3 2xx 2 9y 2, where p x 2 + 3y 2. Requiring that x mod 3 and y > 0 gives the correct coefficients as shown in the following table. p mod 3 x, y such that p x 2 + 3y 2 A 8 p 2xx 2 9y 2 7 2, 20 3, 2 70 9 4, 56 3 2, 3 308 37 5, 2 0 It can be seen that none of the coefficients in the table are divisible by the corresponding primes. In general, this can also be seen by reducing the general form of the coefficients mod p as follows. A 8 p 2xx 2 9y 2 2xx 2 + 3y 2 xy 2 xy 2 mod p x 2 + 3y 2. Since p x 2 + 3y 2, we have that p x and p y. Then, since 2 3 3 and p does not divide 2 or 3, we have that xy 2 0 mod p. This shows that there are no primes p mod 3 such that A 8 np 0 mod p for any n Z. Therefore there are no simple congruence relations for those primes for r 8. The same method can be used to show that A 4 p 2x 0 mod p x 2 + 3y 2 where Z [ + ] 2 is the ring, p mod 3, x mod 3, and y > 0, and that A 6 p 2a 2 b 2 4b 2 0 mod p a 2 + b 2 where Z[i] is the ring, p mod 4, a mod 2, and b > 0. For r 0, 4 and 26, the eta functions are no longer newforms but linear combinations of newforms. The same ideas can be used, but the process becomes more complicated. The case when r 26 is shown as an example below.
CONGRUENCES FOR POWERS OF THE PARTITION FUNCTION 7 For r 26, we have A 26 nq n η 26 2z q 3 q 2n 26. This is a linear combination of newforms, so the ideal representation is A 26 nq n [ φ 3267728 +αq Nα + φ αq Nα α α α φ +αq Nα α φ αq Nα]. The case when r 26 is unique even when compared to the cases when r 0 and r 4 because it makes use of two quadratic fields; K Q and K Qi. Prime ideals that factor, factor as p x + y x y in Z [ + ] 2 and as p a + bia bi in Z[i]. We define φ ±α : x y 2 α 2 and φ ±α : 3a ±i 3b α 2. Using these definitions, we can write A 26 pq p p [ φ 3267728 +pq pp + φ pq pp φ +pq pp φ pq pp]. p p p p We have already shown congruences for primes p mod 2 which corresponds to the primes that are both 2 mod 3 and 3 mod 4, so now we want to study primes p, 5, or 7 mod 2 which correspond to the primes that are mod 3 or mod 4. From the theorem above, we know that primes p mod 3 have a unique representation of the form x 2 + 3y 2 and primes p mod 4 have a unique representation of the form a 2 + b 2, so selecting x, y, a, and b such that x 2 + 3y 2 a 2 + b 2 p should give us the coefficients of the primes p mod 2. Primes which are 5 mod 2 resp. 7 mod 2 have a unique representation of the form a 2 + b 2 resp. x 2 + 3y 2 but cannot be represented by x 2 + 3y 2 resp. a 2 + b 2, so x and y resp. a and b are both 0 in those cases. Then we have the coefficients A 26 p [ x y 2 x + y 2 + x+y 2 x y 2 3267728 + x+y 2 x + y 2 + x y 2 x y 2 3a i 3b a + bi 2 3a i b a bi 2 3a i 3b a + bi 2 3a i b a bi 2]. So A 26 p [ x x y 2 + x+y 2 + y 2 + x y 2 3267728 3a i 3b + 3a i 3b a + bi 2 + a bi 2 ] The coefficients for the first few primes are given in the following table.
8 LOCUS AND WAGNER p, 5, 7 mod 2 a, b, x, y such that p a 2 + b 2 x 2 + 3y 2 A 26 p 5 2,, 0, 0 0 7 0, 0, 2, 0 3 3, 2,, 2 7 6,, 0, 0 0 9 0, 0, 4, 0 29 2, 5, 0, 0 0 3 0, 0, 2, 3 0 37, 6, 5, 2 299 It can be seen that for primes which are mod 2, none of the coefficients are divisible by the corresponding primes. This can also be seen by reducing the coefficients mod p as shown below. A 26 p 55296x 6 y 6 2048a 6 b 6 0 mod p a 2 + b 2 x 2 + 3y 2. Primes which are 5 mod 2 and 7 mod 2 are visibly zero, because A 26 n is only supported on the progression mod 2. But in order for there to be a congruence, it must be true that η r δz T p 0 mod p. This only occurs for r 26 when A 26 np 0 mod p for all n. It is easy to show that if A 26 p 0, then A 26 p 2 0 mod p, so there are no congruence relations for r 26 other than for primes p mod 2. Note that there is actually one prime, namely 3, which is not mod 2 but for which congruence relations exist. This is true because 3s 26 2 s, so then we have p 32 s nq n q n 32 s q 3n 2 s mod 3. All of the coefficients, except p 32 s 3n, are visibly 0 mod 3. The same method can be used to show that A 0 p a i b i b a + bi 4 a bi 4 96 i a i b i b a 3 b ab 3 2 0 mod p. This holds unless p mod 2, since A 0 n is only supported on the progression 5 mod 2. In the cases where A 0 p 0, it is easy to show A 0 5p 0 mod p. Thus there are no congruences for r 0 except for primes p 3 mod 4 and p 5 for the same reason that there are congruences for p 3 when r 26. Similarly, for r 4, we have A 4 p x y 2 x+y 2 720 x y 2 x+y 2 4x 3 y 3 5 0 mod p. x + y 6 x y 6 This holds unless p mod 2, since A 4 n is only supported on the progression 7 mod 2. In the cases where A 4 p 0, it is easy to show A 4 7p 0 mod p. Thus there are no congruences for r 4 except for primes p 2 mod 3 and p 7. References [] B. Lin. Ramanujan-style proof of p n + 7 0 mod. The Ramanujan Journal, 205. [2] G. Andrews. A Survey of Multipartitions: Congruences and Identities. Surveys in Number Theory, K. Alladi ed., Developments in Mathematics, vol. 7, pages -9, 2008. [3] I. Kiming and J. Olsson. Congruences like Ramanujan s for powers of the partition function. Arch. Math., 594:348-360, 992.
CONGRUENCES FOR POWERS OF THE PARTITION FUNCTION 9 [4] T. Harper. Partition Congruences Arising from Lacunary η-quotients. Submitted in partial fulfillment of the requirements for graduation with honors from The South Carolina Honors College. [5] J.P. Serre. Sur La Lacuranite Des Puissances De η. Glasgow Mathematical Journal, vol. 27, pages 203-22, 985. [6] K. Ono. The web of modularity: arithmetic of the coefficients of modular forms and q-series, volume 02 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004.