SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores. Frank G. Garvan

Transcription

1 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores Frank G. Garvan Department of Mathematics Statistics & Computing Science Dalhousie University Halifax, Nova Scotia Canada B3H 3J5 October 18, 1991 Abstract. A number of linear congruences modulo r are proved for the number of partitions that are p-cores where p is prime, 5 p 23, and r is any prime divisor of (p 1)/2. Analogous results are derived for the number of irreducible p-modular representations of the symmetric group S n. The congruences are proved using the theory of modular forms. 1. Introduction A partition of n is a finite nonincreasing sequence of positive integers whose sum is n. It is well-known that p(n), the number of partitions of n, is also the number of irreducible representations of the symmetric group S n. p-cores are important in the study of p- modular representations of S n. A p-core is a partition that has no hook numbers which are divisible by p [Ga-K-S]. We let a p (n) denote the number of partitions of n that are 1980 Mathematics Subject Classification (1991 Revision) Primary 11P83, Secondary 05A17, 11B65, 11F11, 11F20, 11F33, 11F60, 20C20, 20C30 Key words and phrases. Partitions, congruences, p-cores, p-modular representations, Dedekind s eta function, q-series, modular forms, Eisenstein series, Hecke operators. Research began at Macquarie University, NSW 2109, Australia, where the author was a Macquarie University Research Fellow. It was completed at the University of Florida, Gainesville FL 32611, and Dalhousie University where the author is an NSERC International Fellow. On leave (1991) from the University of Florida. frank@cs.dal.ca, frank@math.ufl.edu 1

2 2 FRANK G. GARVAN p-cores. In [Ga-K-S] we gave a combinatorial proof of the following generating function identity: (1.1) a p (n)q n = n 0 m=1 (1 q pm ) p (1 q m, ( q < 1). ) See [Ga-K-S, (2.1)]. See also [Kl] for a different combinatorial proof. Hence if p > 3 is prime then δ p := (p 2 1)/24 Z and (1.2) a p (n δ p )q n = ηp (pτ) =: X p (τ), η(τ) n δ p where q = e 2πiτ, τ H (the complex upper half-plane), and η(τ) is Dedekind s well-known η-function (1.3) η(τ) := e πi/12 (1 q 2mπiτ ). m=1 When p is a prime > 3, Hecke has observed that the function X p on the right side of (1.2) is a modular form of level p, weight (p 1)/2 and character the Legendre symbol ( p), for the congruence subgroup Γ 0 (p). See [O, p.28]. We may obtain good asymptotic estimates for a p (n) by using known estimates for cusp forms and by noting that we obtain a cusp form from X p by subtracting off a suitable multiple of an Eisenstein series. In this way it can be shown that for p prime > 3 then (1.4) a p (n δ p ) = α p u p (n) + O(n (p 3)/4+ɛ ) for any ɛ > 0, where u p (n) = n (p 3)/2 d n ( ) 1 d, d (p 3)/2 p and α p is some non-zero constant. This is a consequence of Deligne s proof of the Ramanujan-Petersson conjecture (see [Ka]). It follows that for two distinct primes 3 < p 1 < p 2 and n sufficiently large we have (1.5) a p1 (n) a p2 (n). Dennis Stanton has conjectured that this result holds for all 3 < p 1 < p 2 < n with p 1, p 2 not necessarily prime. It is also open whether a p (n) > 0 for all prime p > 3 and all n 0.

3 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 3 There is a general result of Serre s [Se2] that states that for a fixed integer m 1 almost all of the fourier coefficients of a modular form of positive integral weight on a congruence subgroup are divisible by m. Explicitly, let f = c n e 2πiτ/M, M 1, n=0 be such a modular form. We suppose the coefficients c n are contained in some ring of integers O K of some algebraic number field K. If m is an integer 1 we write a 0 (mod m) if a mo K. We define E f,m to be the set of integers n N such that c n 0 (mod m). Serre s result is that E f,m has density 1. Hence, in our case, we deduce that a p (n) 0 (mod m) for almost all n, where m is any integer 1 and p is any prime > 3. In this paper we give some explicit subsets of E f,m of positive density for certain f = X p and certain m. Ramanujan s [H-W, 19.12] congruences modulo 5, 7 and 11 for p(n) together with (1.1) easily yield the following congruences: (1.6) (1.7) (1.8) a 5 (5n 1) 0 (mod 5), a 7 (7n 2) 0 (mod 7), a 11 (11n 5) 0 (mod 11). In [Ga-K-S] we were able to explain and prove (1.6) (1.8) combinatorially. More general congruences hold (1.9) a p (p α n δ p ) 0 (mod p α ), for all α 1 and where p = 5, 7 or 11. The cases p = 5, 7 were proved in [Ga-K-S]. The case p = 11 is proved in 3. The congruences (1.6) (1.9) come from the level of the relevant space of modular forms. It should also be noted that the proof of (1.9) is much easier than its analog for the partition function p(n). In this paper we are mainly concerned with congruences of a different sort. This time the modulus comes from the weight of the space of modular forms instead of the level. The weight here is (p 1)/2 so we let r be any prime divisor of (p 1)/2. Then for each prime 5 p 23 there is an ɛ p (= 1 or 1) such that (1.10) a p (n δ p ) 0 (mod r), whenever ( ) n = ɛ p and n 0 (mod r). p

4 4 FRANK G. GARVAN In particular, ɛ p = 1 for p = 11, 13 and ɛ p = 1 for p = 5, 7, 17, 19, 23. For p = 17 stronger congruences hold: (1.11) and (1.12) a 17 (n 12) 0 (mod 8) if a 17 (n 12) 0 (mod 2) if ( ) n = 1 and n 0 (mod 2), 17 ( ) n = 1 and n 0 (mod 4). 17 These congruences are reminiscent of some congruences due to Kolberg [Kol2] for c n the coefficients of the modular invariant j(τ). However, in Kolberg s case, the weight is 0 and the level is 1. An easy consequence of (1.10) (1.12) are analogous congruences for b p (n), the number of irreducible p-modular representations of S n. It is well-known (see [An-O]) that (1.13) b p (n) = the number of partitions of n in which no multiple of p occurs as a part, so that (1.14) b p (n)q n = n 0 = m=1 (1 q pm ) (1 q m ) n 0 a p(n)q n m=1 (1 qpm ) p 1. Now as before let p be prime, 5 p 23, and let r be any prime divisor of (p 1)/2. Then n 0 b p (n)q n a p(n)q n m=1 (1 (mod r). qprm ) (p 1)/r n 0 It follows that b p (n) satisfies the same congruence as a p (n) given in (1.10). The results analogous to (1.11) and (1.12) also hold. These congruences have been observed independently by Gordon [Go]. The modular function on the right side of (1.14) and its powers has been previously studied by Newman [N5]. However, our congruences do not follow from Newman s identities. Our proof depends on the classical Hecke theory of modular forms. We need a certain twist operator (see 2.3), the existence of special bases for cusp forms (see 2.4) and the

5 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 5 existence of certain r-th powers of modular forms (see 2.5). For the case p = 5, 7, 19, 23 the problem is reduced to computing the constant term of a certain Eisenstein series (see Table I, 2.2). The remaining cases involve computing the first few terms of the q-expansion of certain modular forms. The necessary preliminary results are set up in 2. The majority of the congruences are proved in 3 with the special congruence (1.11) being left until 4. The q-expansions were computed using the computer algebra package MAPLE. It would be interesting to know if an elementary proof of the congruences could be found using the multidimensional theta function representation for the generating function for a p (n) given in [Ga-K-S, (2.2)]. Klyachko [Kl] has observed that this representation follows from Macdonald s [M] identity for the root system A p Preliminary Results 2.1 Definitions and Notation Let H denote the complex upper half plane. SL 2 (Z) acts transitively on H by linear fractional transformations (2.1.1) γτ = aτ + b cτ + d, where γ = Let N be a positive integer. We define {( ) a b (2.1.2) Γ(N) := c d ( ) a b SL c d 2 (Z). } SL 2 (Z) : a d 1 (mod N), b c 0 (mod N), so that Γ(1) = SL 2 (Z). A subgroup of Γ(1) is called a congruence subgroup of level N if it contains Γ(N). We will also be concerned with the following congruence subgroups: (2.1.3) {( ) } a b Γ 0 (N) := Γ(1) : c 0 (mod N), c d (2.1.4) {( ) } a b Γ 1 (N) := Γ c d 0 (N) : a 1 (mod N), Now we consider functions f : H C. The notation f [γ] k is used to denote the function whose value at τ is (cτ + d) k f(γτ). The value of f [γ] k at τ is denoted by f(τ) [γ] k so that ( ) (2.1.5) f(τ) [γ] k := (cτ + d) k a b f(γτ), for γ = Γ(1). c d

6 6 FRANK G. GARVAN We define modular forms and cusp forms for a congruence subgroup Γ Γ(1) of level N. Let q N denote e 2πiτ/N. Let k Z. A function f : H C is said to be a modular form of weight k for Γ if it satisfies the following conditions: (i) f is holomorphic on H, (ii) f [γ] k = f for all γ Γ, (iii) f is holomorphic at the cusps γ( ) for γ Γ(1); i.e. n 0 a nqn n. f(τ) [γ] k has the form We let M k (Γ ) denote the set of such modular forms. It turns out that M k (Γ ) is a finite dimensional C-vector space. If a 0 = 0 for all γ Γ(1) we say f is a cusp form. We let S k (Γ ) denote the subspace of cusp forms. Now we define what it means for a function to be a modular form of weight k and character χ for the group Γ 0 (p) when p is prime. Let χ be a Dirichlet character modulo p (2.1.6) χ : (Z / pz) C, with χ( 1) = ( 1) k. A function f : H C is said to be a modular form of weight k and character χ for Γ 0 (p) if it satisfies conditions (i) (iii) above except (ii) is replaced by (2.1.7) f [γ] k = χ(d) f for all γ Γ 0 (p). In this case, condition (iii) amounts to the following two conditions: (2.1.8a) and (2.1.8b) f(τ) = a n q n, q = e 2πiτ n=0 τ k f( 1/pτ) = b n q n, q = e 2πiτ. n=0 We call the Fourier series in (2.1.8a) the q-expansion of f at, and we call (2.1.8b) the q-expansion at 0. We let M k (p, χ) denote the space of such modular forms. If a 0 = b 0 = 0 in (2.1.8) above then f is a cusp form. We let S k (p, χ) denote the subspace of cusp forms. We have for k 2 M k (p, χ) = E k (p, χ) S k (p, χ), where E k (p, χ) is a space of Eisenstein series of dimension 2 (the number of inequivalent cusps in this case).

7 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 7 We will be concerned with a number of characters modulo p. We let χ 0 denote the trivial character. For p an odd prime and d 1 2 (p 1) we define (2.1.9) χ p,d : (Z /pz) C, by χ p,d (g) = exp(πi/d), where g is the smallest primitive root modulo p. We note that ) ( (2.1.10) χ p := χ p,1 = (the Legendre symbol), p and (2.1.11) χ d p,d = χ p = ) (. p 2.2 Eisenstein Series and η-products Kolberg [Kol3] has studied the Eisenstein series for Γ 0 (p) and has shown how to derive certain Ramanujan-type identities that involve Eisenstein series and certain η-products. Let χ be a Dirichlet character modulo p. We consider the Gauss sum p 1 (2.2.1) S χ (n) := χ(j)exp(2πijn/p). Then j=1 (2.2.2) S χ (n) = χ(n)s χ (1). We need the q-expansion of the two Eisenstein series for M k (p, χ). We define (2.2.3a) V χ,k := A k + where m=1 n=1 (2.2.3b) A k = A χ,k := 1 2k χ(n)n k 1 q mn for k 1, k 1 j=0 ( k p 1 )B j p j 1 j n=1 χ(n)n k j, and B 0 = 1, B 1 = 1/2, B 2 = 1/6, B 3 = 0, B 4 = 1/30,... are the Bernoulli numbers. We define (2.2.4) and U χ,k := m=1 n=1 U χ,1 := V χ,1. χ(m)n k 1 q mn for k 2,

8 8 FRANK G. GARVAN Proposition (2.2.5)([Kol3]). Let p be an integer > 1, let k be an integer 1, and let χ be a character modulo p. If χ( 1) = ( 1) k and χ χ 0 then (i) U χ,k, V χ,k M k (p, χ), (ii) V χ,k ( 1/pτ) = (pτ)k S χ (1) U χ,k(τ), (iii) U χ,k ( 1/pτ) = pτ k S χ (1) V χ,k(τ). We note that in the case that χ is real and p is an odd prime (i.e. χ = χ p ) then (2.2.6) S χ (1) = S χ (1) = i (p 1)2 /4 p. We define (2.2.7a) (2.2.7b) U p,k := U χp,k, V p,k := V χp,k, (2.2.7c) A p,k := A χp,k. We now determine which η-products belong to M k (p, χ). If we let of the square root having nonnegative real part, then denote the branch (2.2.8) η( 1/τ) = τ/i η(τ), for τ H. See [Kob, Prop.14 p.121]. Proposition (2.2.9). Let p be an odd prime and suppose k and m are integers satisfying (2.2.10a) m 1 (mod 2), (p + 1)k 0 (mod 12), (p 1)m 2k (mod 24), (2.2.10b) Then kp (p 1) 2 m k. Y k,m := η(τ)2k+m η(pτ) m M k(p, χ p ). Further, if there is strict inequality in (2.2.10b) then the η-product above is a cusp form. Remark: Some η-products that are cusp forms are given below in Table II. Proof. Let γ Γ 0 (p), and suppose that the conditions of the Proposition are satisfied. Kolberg [Kol3, Lemma 4 p.13] has shown that (2.2.11) Y k,m [γ] k = χ p (d)y k,m. It is clear from (1.3) that Y k,m is holomorphic on H and that it satisfies (2.1.8a). Condition (2.1.8b) follows by using (2.2.8). Hence Y k,m M k (p, χ p ).

9 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 9 Corollary (2.2.12). Let p > 3 be prime. Then (i) X p := ηp (pτ) η(τ) M 1 2 (p 1) (p, χ p ), (ii) c p X p U p, 1 2 (p 1) S 1 2 (p 1) (p, χ p ), where c p := pa p, 1 2 (p 1) i 1 4 (p 1)(p 3) 0. Remark: A table for some of the c p is given below. Proof. (i) follows immediately from Proposition (2.2.9). The q-expansion of X p at is X p = q rp +..., where r p = (p 2 1)/24 > 0. We choose c p so that the constant term of the q-expansion at 0 of c p X p U p, 1 2 (p 1) is zero. Using (2.2.5)(iii), (2.2.6) and (2.2.8) we find c p := pa p, 1 2 (p 1) i 1 4 (p 1)(p 3), which gives the desired result. Finally, c p 0 follows from the fact that A p, 1 2 (p 1) is a nonzero multiple ([Kol3, (2.26)]) of a Dirichlet L-series L( 1 2 (p 1), χ p) ([Se1, p.71]) which is nonzero. p c p = = = = = = Table I. c p of Corollary (2.2.12) (ii). We need to calculate which η-products are cusp forms for S 1 2 (p 1) (p, χ p ), 5 p 23. For k = 1 2 (p 1) we need m 1 (mod 2), (p 1)(m 1) 0 (mod 24) and (p 1) m 0. In this way we may calculate Table II below.

10 10 FRANK G. GARVAN p η-product cusp forms 5 7 η 3 (τ)η 3 (7τ) η 2n+1 (τ)η 11 2n (13τ) (0 n 5) 17 η 5 (τ)η 11 (17τ), η 5 (17τ)η 11 (τ) 19 η 4n+3 (τ)η 15 4n (19τ) (0 n 4) 23 η 11 (τ)η 11 (23τ) Table II. η-products in S 1 2 (p 1) (p, χ p ) for small p. Klyachko [Kl, p.1883] has the following explicit formula { [ (p 1) 2 /24 ], if p 1 (mod 12), dim S 1 2 (p 1) (p, χ p ) = [ (p 1) 2 /24 ] 1, if p 1 (mod 12). A table of these dimensions for small p is given below. We note that Proposition (2.4.13) below also gives these dimensions. For general k and χ, dim S k (p, χ) may be calculated from [C-O, Thm 1]. p dim S 1 2 (p 1) (p, χ p ) Table III. Dimension of certain spaces of cusp forms. From this table we observe that (2.2.13) dim S 1 2 (p 1) (p, χ p ) < p for p prime, 5 p 23. Our proof of the congruences depends in part on this inequality. This explains to some extent why the congruences seem to stop at p = 23.

11 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores Operators Let N > 1 be an odd integer. Let f M k (N, χ) where χ is a Dirichlet character modulo N. Suppose f(τ) = n 0 a nq n (q = e 2πiτ ). For p prime, the Hecke operator T p is given by T p (f(τ)) = n 0 b nq n where (2.3.1) b n = a pn + χ(p)p k 1 a n/p. Here we take χ(p) = 0 if p N and take a n/p = 0 if n is not divisible by p. See [Kob, Prop.37 p.161]. The Hecke operators T n may be defined for all n (see [Kob, (5.8) p.158]). The T n preserve the spaces M k (N, χ) and S k (N, χ) [Kob, Prop.35 p.160]. For f, g S k (Γ 1 (N)) S k (N, χ) we may define the Petersson product of f and g as (2.3.2) f, g := We need 1 k dx dy f(τ)g(τ)y [Γ(1) : Γ 1 (N)] Γ 1(N)\H y 2 (τ = x + iy). Theorem (2.3.3). (Petersson) Let T n be a Hecke operator on S k (N, χ) with gcd (n, N) = 1. If f, g S k (N, χ) then See [Kob, Prop.48 p.171] for a proof. T n f, g = χ(n) f, T n g. Corollary (2.3.4). ( [Kob, Prop.51 p.173]) There exists a basis of the C-vector space S k (N, χ) whose elements are eigenforms for all the Hecke operators T n for which gcd (n, N) = 1. Corollary (2.3.5). Suppose λ n is an eigenvalue of the Hecke operator T n for S k (p, χ p ). Then (i) λ n is real if ( n p) = χp (n) = 1, (ii) λ n is pure imaginary if ( n p) = χp (n) = 1. We now describe the needed twist-operator for S k (p, χ p ). We follow [Z, p.36]. There is a natural involution on S k (p, χ p ) due to Hecke: (2.3.6) ρ : S k (p, χ p ) S k (p, χ p ), f ρ (τ) = f( τ). If f(τ) = n 1 a nq n then f ρ (τ) = n 1 a nq n. An eigenform for the Hecke operators T n with gcd (n, p) = 1 is normalized if a 1 = 1 in its q-expansion at. In this case the

12 12 FRANK G. GARVAN eigenvalues are λ n = a n ([Kob, Prop.40 p.163]). By Corollary (2.3.5) we have (2.3.7) a ρ n = χ p (n)a n = ( ) n a n if p n. p Since, by Corollary (2.3.4), the normalized Hecke eigenforms form a basis for S k (p, χ p ) we may extend the action of ρ on the eigenforms by linearity to a linear operator on all of S k (p, χ p ) so that (2.3.8) a ρ n = ρ : S k (p, χ p ) S k (p, χ p ), ( ) n a n if p n, p where f(τ) = n 1 a nq n is any f S k (p, χ p ). We let (2.3.9) (2.3.10) S + k,p := f S k(p, χ p ) : f(τ) = a n q n, n 1 a n = 0 for S k,p := f S k(p, χ p ) : f(τ) = a n q n, n 1 a n = 0 for ( ) n p ( ) n p = 1, = +1. Clearly, (2.3.11) S k (p, χ p ) = S + k,p + S k,p, since we may write any f S k (p, χ p ) as (2.3.12) f = 1 2 (f + f ρ ) (f f ρ ). A well-known lemma of Hecke ([O, p.32]) says that a nonzero cusp form of level p and character χ χ 0 can not have a development of the form n 1 a nq pn. It follows that S + k,p S k,p = {0} so that (2.3.13) S k (p, χ p ) = S + k,p S k,p.

13 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 13 It is clear that the S ± k,p are the (±1)-eigenspaces of ρ. Let p be a fixed prime, 5 p 23. Let d := dim S 1 2 (p 1) (p, χ p ). Then, by (2.2.13), (2.3.14) d < p. In 2.4 we will show that we may a construct a basis B = {B 1, B 2,..., B d } for S 1 2 (p 1) (p, χ p ) such that each B j Z[[q]] and (2.3.15) B j = q j Proposition (2.3.16). Let ɛ = ±1 be fixed. Suppose p is prime with 7 p 23. Let d := dim S 1 2 (p 1) (p, χ p ). Suppose f S 1 2 (p 1) (p, χ p ) and f(τ) = n 1 a nq n. If then a n = 0 a n = 0 for ( ) n = ɛ and n d, p for all n with Remark: A table for the d s is given in Table III in 2.2. ( ) n = ɛ. p Proof. Let g := f + ɛf ρ = n 1 b nq n, so that g S 1 2 (p 1) (p, χ p ). Then, by (2.3.14), we have b n = 2a n = 0 for all n d. Hence, by (2.3.15), g = 0, f S ɛ 1 and the result follows. (p 1),p Example. For p = 7, d = 1 and a basis for S 3 (7, χ 7 ) is 2 η 3 (τ)η 3 (7τ) = q + = n 1 a n q n. Hence, a n = 0 for ( n 7) = 1. We note that there is an alternative approach to this problem. Instead of using the operator ρ we could have used the the Atkin-Lehner operator R p ([At-L, p.155]). See also [Kob, Prop.17 p.127]. However this would have required working in the space S 1 2 (p 1) (p 2, χ p ) rather than S 1 2 (p 1) (p, χ p ). Clearly the smaller space S 1 2 (p 1) (p, χ p ) and ρ are preferable. The action of Hecke operators on Eisenstein series is easily calculated.

14 14 FRANK G. GARVAN Lemma (2.3.17). Let p be an odd prime and let r be prime. For the Hecke operator T r on S k (p, χ p ) we have (i) (ii) T r (U p,k ) = (χ p (r) + p k 1 )U p,k, T r (V p,k ) = (1 + χ p (r)p k 1 )V p,k. We now introduce some elementary operators that act on formal power series in Z[[q]]. Let p be an odd prime. For ɛ = + or we define the operator Ψ ɛ = Ψ ɛ,p that acts on a series n 0 a nq n and picks out those terms in which χ p (n) = ɛ; i.e. (2.3.18) Ψ ɛ a n q n = n 0 n 1 χ p(n)=ɛ a n q n. We note that Ψ ɛ does not necessarily preserve the space M k (p, χ p ). However, the action of Ψ ɛ on Eisenstein series is nice. Lemma (2.3.19). Let p be an odd prime then for ɛ = ± we have (2.3.20) Ψ ɛ (U p,k ) = ɛψ ɛ (V p,k ). Proof. Let r be prime, r p. Let u(n) (v(n) resp.) be the n-th coefficient in the q- expansion of U p,k (V p,k resp.) at. An easy calculation gives So, by multiplicativity, we have and the result follows. u(r α ) = χ p (r α )v(r α ). u(n) = χ p (n)v(n) for p n, Finally, for r > 1 we define Φ r to be the operator that acts on a series n 0 a nq n and picks out those terms in which n 0 (mod r); i.e. (2.3.21) Φ r a n q n = n 0 n 1 n 0 (mod r) a n q n. We note that Φ r does not necessarily preserve the space M k (p, χ p ). Let O denote a ring of algebraic integers. For W = n 0 a n q n O[[q]] we write W 0 (mod r) if and only if a n 0 (mod r) in O for all n 0. In our applications we consider O = Z or Z[ζ], where ζ is some root of unity.

15 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 15 Lemma (2.3.22). Let p be an odd prime and let r and s be two distinct primes. Let O be a ring of algebraic integers. Suppose k > 1, r k and W M k/r (p, χ) O[[q]], where χ is a Dirichlet character modulo p such that χ r = χ p. Then (i) Φ r (W r ) 0 (mod r), (ii) Proof. (i) is trivial. We have Φ r (T s (W r )) 0 (mod r). W r = a n q n M k (p, χ p ), n 0 where a n Z, and a n 0 (mod r) for n 0 (mod r). Now T s (W r ) = n 0 b n q n, where b n = a sn + χ p (s)s k 1 a n/s. If n 0 (mod r) then sn 0 (mod r) and n/s 0 (mod r), so that b n 0 (mod r), which is (ii). There is an elementary congruence for Eisenstein series. Lemma (2.3.23). Let p be an odd prime and k 1. If d is prime and d 1 k 1 then Φ d Ψ (U p,k ) 0 (mod d). Proof. Let u(n) denote the n-th coefficient in the q-expansion of U p,k at τ =. If ( n ) ( p = 1 and n 0 (mod d) then there is a prime q, q d, such that q ) p = 1 and q α n with α odd. Now q d 1 1 (mod d) since q d. Since d 1 k 1 we have q k 1 1 (mod d) u(q α ) = q α(k 1) q (α 1)(k 1) (mod d),

16 16 FRANK G. GARVAN so that u(n) 0 (mod d) (by multiplicativity), which is the result. 2.4 Construction of Bases In this section we construct bases for S 1 2 (p 1) (p, χ p ) for p prime, 7 p 23. Let d := dim S 1 2 (p 1) (p, χ p ) (see Table III, 2.2). We want to construct a basis, B = {B 1, B 2,...,B d } with the following properties: For each j, (2.4.1a) (2.4.1b) B j Z[[q]], B j = q j We call such a basis good. To do this we first construct polynomial bases for certain sets of modular functions on Γ 0 (p). This problem has been studied previously by Kolberg [Kol3] and Newman [N3], [N4]. Let p be prime. A function f : H C is said to be a modular function on Γ 0 (p) if it satisfies the following conditions: (i) f is meromorphic on H, (ii) f [γ] 0 = f for all γ Γ 0 (p), (iii) f is meromorphic at the cusps γ( ) for γ Γ(1); i.e. f(τ) [γ] 0 has the form n ν a nq n p for some ν Z where q p = e 2πiτ/p. As in the case of modular forms, condition (iii) amounts to the following two conditions: (2.4.2a) f(τ) = a n q n, q = e 2πiτ, some ν Z n ν and (2.4.2b) f( 1/pτ) = n ν 0 b n q n, q = e 2πiτ, some ν 0 Z. We say ν is the valence of f at τ = and ν 0 is the valence at τ = 0, when a ν, b ν0 0. Let F p denote the set of functions f with the following properties: (i) f is a modular function on Γ 0 (p), (ii) f is holomorphic on H and the only pole of f in the fundamental region of Γ 0 (p) is at τ = (i.e. ν 0 0).

17 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 17 When p is a prime > 3 the genus of Γ 0 (p) is given by [p/12] 1, if p 1 (mod 12), (2.4.3) g = [p/12], if p 5, 7 (mod 12), [p/12] + 1, if p 11 (mod 12). This follows from [Sc, p.103]. For each prime 5 p 23 we will construct a polynomial basis for F p consisting of g + 1 functions f j, where f j has valence j for g + 1 j 2g + 1 at τ =. It is an open problem as to whether this can be done for general p. Once we have constructed such a basis it is an easy matter to construct a (linear) basis for S 1 2 (p 1) (p, χ p ). Following Kolberg [Kol1] we construct our polynomial bases out of η-products and Fine s W k -functions [F]. Kolberg has constructed such bases however we require our bases to have two additional properties: For each j, (2.4.4) (i) f j = q j +... with integral coefficients, and (2.4.4) (ii) We call such a basis good. f j is zero at τ = 0 (i.e. ν 0 1). Let p > 3 be prime. Following Fine [F] we define (2.4.5) W k (τ) = W k,p (τ) := q 6k2 p k (1 q pm 4k )(1 q pm+4k p ) (1 q pm 2k )(1 q pm+2k p ), m 1 for k 0 (mod p). We note that W k may be defined for all (p, 6) = 1 with p not necessarily prime. Atkin and Swinnerton-Dyer [At-S, Lemma 6] observed that the W k (pτ) occur in the p-dissection of the modular function η(τ)/η(p 2 τ). Proposition (2.4.6) ([F]). Let p > 3 be prime, and let k Z with k 0 (mod p). Then (i) (ii) (iii) W k = W k = W k+p, W k [γ] 0 = exp(12πik 2 ab/p)w ak, for ( ) a b γ = Γ c d 0 (p), W k ( 1/pτ) = 2 cos(2kπ/p) + O(q). From (i) there are 1 2 (p 1) distinct W k namely W 1, W 2,..., W 1 2 (p 1). We say that a polynomial in the W k s is cyclic if it is invariant under k ak for every a 0 (mod p). Further, a polynomial in the W k s is called (p, 0)-isobaric if each term has weight 0 (mod p) provided that W k is assigned weight k 2. Since each W k is holomorphic on H we have

18 18 FRANK G. GARVAN Corollary (2.4.7) ([F, p.153]). Let p > 3 be prime. Then (i) Every cyclic (p, 0)-isobaric polynomial in the W k (1 k 1 2 (p 1)) is a modular function on Γ 0 (p). In fact, each such function belongs to F p. (ii) The constant term (in the q-expansion at τ = 0) of any cyclic polynomial in the W k with integral coefficients is a rational integer. Proof. (i) is immediate from Proposition (2.4.6). From (2.4.6) (iii) the constant term of W k (in the q-expansion at τ = 0) is 2 cos(2kπ/p) which is an algebraic integer in the field Q(ζ p ) where ζ p = exp(2πi/p). It follows that the constant term of a cyclic Z-polynomial is a rational integer since it is invariant under all the Q-automorphisms ζ p ζ a p, a 0 (mod p). Let p > 3 be prime. Let ϱ be the least positive even integer such that (p 1)ϱ 0 (mod 24). We define (2.4.8) w p (τ) := Then ( ) ϱ η(τ). η(pτ) (2.4.9) w p (τ) = q (p 1)ϱ/ , and by (2.2.8) we have (2.4.10) w p ( 1/pτ) = p ϱ/2 q (p 1)ϱ/ Newman [N4] has determined which η-products are modular functions on Γ 0 (N). From [N4, Thm1 p.375] we may obtain Proposition (2.4.11). Let p > 3 be prime. Then w p F p. We obtain the following result which is analogous to [Kol1, Lemma 4 p.6]. Proposition (2.4.12). Let p > 3 be prime. If p 23 then there exists a good polynomial basis for the set of modular functions F p. Remarks. See (2.4.4) for the definition of good. This proposition is probably true for larger p. Proof. Let 5 p 23 be prime. By utilising Corollary (2.4.7), and Proposition (2.4.11) we have constructed functions f j (g + 1 j 2g + 1) that satisfy (2.4.4):

19 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 19 p = 5, g = 0 : f 1 = w 5 = p = 7, g = 0 : f 1 = w 7 = p = 11, g = 1 : ( ) 6 η(τ) = q q + 10 q 2 30 q q 4 25 q η(5τ) ( ) 4 η(τ) = q q + 8 q 2 5 q 3 4 q 4 10 q η(7τ) f 2 = W 3 1 W 4 W 5 + W 3 2 W 3 W 1 + W 3 3 W 1 W 4 + W 3 4 W 5 W 2 + W 3 5 W 2 W 3 17 = q q q + 8 q 2 + q q , f 3 = 2 ( W 6 1 W 4 + W 6 2 W 3 + W 6 3 W 1 + W 6 4 W 5 + W 6 5 W 2 ) = q 3 + q q + 2 q q q p = 13, g = 0 : ( ) 2 η(τ) f 1 = w 13 = = q 1 2 q + 2 q 2 + q q 4 2 q η(13τ) p = 17, g = 1 : f 2 = W 1 W 2 2 W 5 + W 2 W 2 4 W 7 + W 3 W 2 6 W 2 + W 4 W 2 8 W 3 + W 5 W 2 7 W 8 + W 6 W 2 5 W 4 + W 7 W 2 3 W 1 + W 8 W 2 1 W 6 9 = q 2 + q q + 2 q q q , f 3 = W 3 1 W 2 W 5 W 6 + W 3 2 W 4 W 7 W 5 + W 3 3 W 6 W 2 W 1 + W 3 4 W 8 W 3 W 7 + W 3 5 W 7 W 8 W 4 + W 3 6 W 5 W 4 W 2 + W 3 7 W 3 W 1 W 8 + W 3 8 W 1 W 6 W 3 36 = q q q q + 8 q q p = 19, g = 1 : f 2 = W 1 W 7 W 8 + W 2 W 5 W 3 + W 4 W 9 W 6 5 = q q + 2 q q 3 q , ( ) 4 η(τ) f 3 = w 19 = = q 3 4 q q q 4 q 2 10 q η(19τ)

20 20 FRANK G. GARVAN p = 23, g = 2 : f 3 = 8 ( W 2 1 W 6 W 10 + W 2 2 W 11 W 3 + W 2 3 W 5 W 7 + W 2 4 W 1 W 6 + W 2 5 W 7 W 4 + W 2 6 W 10 W 9 + W 2 7 W 4 W 1 + W 2 8 W 2 W 11 + W 2 9 W 8 W 2 + W 2 10W 9 W 8 +W 2 11W 3 W 5 ) = q q 2 + q q + 2 q q , f 4 = 7 ( W 3 1 W 5 W 8 + W 3 2 W 10 W 7 + W 3 3 W 8 W 1 + W 3 4 W 3 W 9 + W 3 5 W 2 W 6 + W 3 6 W 7 W 2 + W 3 7 W 11 W 10 + W 3 8 W 6 W 5 + W 3 9 W 1 W 3 + W 3 10W 4 W 11 +W 3 11W 9 W 4 ) = q 4 q 3 q q q 3 6 q , f 5 = 32 ( W 4 1 W 6 W 11 + W 4 2 W 11 W 1 + W 4 3 W 5 W 10 + W 4 4 W 1 W 2 + W 4 5 W 7 W 9 + W 4 6 W 10 W 3 + W 4 7 W 4 W 8 + W 4 8 W 2 W 4 + W 4 9 W 8 W 7 + W 4 10W 9 W 5 +W 4 11W 3 W 6 ) = q 5 + q 4 + q 3 + q q q The first few terms of each q-expansion above were obtained using the symbolic algebra package MAPLE. Now suppose f F p. Then it is clear that there is a polynomial P with integral coefficients such that ν (f P (f g+1,..., f 2g+1 )) g. It follows by the Weierstrass gap theorem [Sp, p.272] that f = c + P (f g+1,..., f 2g+1 )) for some constant c. Hence {f g+1,..., f 2g+1 } is a good polynomial basis for F p. It is now a simple matter to construct a good (linear) basis for S 1 2 (p 1) (p, χ p ) for p prime, 7 p 23. Proposition (2.4.13). Let p > 5 be prime. If p 23 then (i) d := dim S 1 2 (p 1) (p, χ p ) = r p g 1, where r p := (p 2 1)/24 and g is the genus of Γ 0 (p), (ii) S 1 2 (p 1) (p, χ p ) has a good linear basis B = {B 1, B 2,..., B d } where

21 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 21 B j = X p f rp j, if d g j d, B j = X p f (t 1) g+1 f s+g+1, if j < d g, t := [(r p j)/(g + 1)], s := r p t(g +1) j, and {f g+1,..., f 2g+1 } form a good polynomial basis for F p as given in the proof of Proposition (2.4.12). Remarks. See (2.4.1) for the definition of good. This proposition is probably true for larger p. Proof. Suppose there is an f M 1 2 (p 1) (p, χ p ) with ν (f) > r p g 1. Then f/x p F p since X p M 1 2 (p 1) (p, χ p ) and has no zeros in H. But now, ν (f/x p ) g which contradicts our construction in Proposition (2.4.12) unless f = cx p (for some constant c). Since X p is not a cusp form this means that if f S 1 2 (p 1) (p, χ p ) then ν (f) r p g 1. We will show that for each 1 j r p g 1 there is a B j S 1 2 (p 1) (p, χ p ) that satisfies (2.4.1). Then the B j form a basis and we have (i). From the proof of Proposition (2.4.12) there are modular functions f g+1,..., f 2g+1 on Γ 0 (p) that satisfy (2.4.4). For 1 j r p g 1 we let m = m j = r p j so that m g +1. Case (i). m 2g + 1. We take B j := X p f m S 1 2 (p 1) (p, χ p ), since X p M 1 2 (p 1) (p, χ p ), ν (B j ) = r p m = j 1 and ν 0 (B j ) = ν 0 (X p ) + ν 0 (f m ) = ν 0 (f m ) 1. Case (ii). m > 2g + 1. We find t 2 and s 0 such that where 0 s < g + 1, so that and In this case we take which has the desired properties. m = t(g + 1) + s, m = (t 1)(g + 1) + (s + g + 1) g + 1 s + g + 1 2g + 1. B j := X p f (t 1) g+1 f s+g+1, Hence we have (i) and (ii) is the construction given above. The following result follows easily from Propositions (2.3.16) and (2.4.13).

22 22 FRANK G. GARVAN Corollary (2.4.14). Let p be prime, 7 p 23, and d := dim S 1 2 (p 1) (p, χ p ). Then there exists a good basis C = {C 1, C 2,..., C d } for S 1 2 (p 1) (p, χ p ) such that (i) C i = q i + O(q d+1 ) (1 i d), (ii) C + = {C i : ( ) i p = 1} is a good basis for S + 1 (iii) C = {C i : ( ) i p = 1} is a good basis for S 2 (p 1),p, 1 (p 1),p. 2 Remark. This result is not true for p > 23 since for such p we have p > d. 2.5 Construction of r-th powers Let p be prime 11 p 23, and let r be a prime divisor of 1 2 (p 1). In this section we construct modular forms f of weight (p 1) 2r such that f r M 1 2 (p 1) (p, χ p ). Where possible we construct a cusp form f and where needed we calculate a superset of the coefficient set of f. If f = n 0 a nq n then the coefficient set of f is simply {a n : n 0}. If this superset is a ring of algebraic integers this construction yields a congruence modulo r. p = 11 In this case r = 1 2 (p 1) = 5. Since 11 3 (mod 4) we have, by Proposition (2.2.5), (2.5.1) W := 2V 11,1 = m,n 1 ( ) n q nm M 1 (11, χ 11 ). 11 Hence If we define (2.5.2) and (2.5.3) W 5 M 5 (11, χ 11 ). V := 11V 11,5 = U := U 11,5 = m,n 1 m,n 1 ( ) n m 4 q nm, 11 ( ) n n 4 q nm 11 then some combination of U, V and W 5 is a cusp form. We find (2.5.4) C := 11 3 U + V 1275W 5 S 5 (11, χ 11 ). p = 13

23 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 23 In this case 1 2 (p 1) = 6 so that r = 2 or 3. Case (i). r = 3. We consider the character χ 13,3, which is defined in (2.1.9) and has the property χ 3 13,3 = χ 13 = ( 13). Let ω = exp(πi/3). More explicitly we may define χ13,3 by χ 13,3 (2) = ω, since 2 is a primitive root modulo 13. Lemma (2.5.5). If χ = χ 13,3 then (i) dim S 2 (13, χ) = 1, (ii) C(τ) := U χ,2 [1 + (3 + 2ω)η 2 (13τ)/η 2 (τ)] 1 is a cusp form with coefficient set in Z[ω]. Proof. (i) follows from [C-O, Thm 1]. We provide a more elementary proof for fun. By Proposition (2.4.13), there is a good basis for S 6 (13, χ 13 ) given by (2.5.6) B j := η 13 2j (τ)η 2j 1 (13τ) (1 j 6). Now suppose dim S 2 (13, χ) > 1, then there is a cusp form F with q-expansion with ν 2. But F 3 S 6 (13, χ 13 ) so that for some constant c 0. By (2.2.8) we have F (τ) = q ν +... F 3 (τ) = cb 6 = c η(τ)η 11 (13τ) [ τ 2 F ( 1/13τ) ] 3 = c 13η(13τ)η 11 (τ) = c 13 q This is impossible since τ 2 F ( 1/13τ) S 2 (13, χ) by [Kob, Ex.15 p.145]. Hence, dim S 2 (13, χ) 1 so we need only construct a nonzero cusp form. By Proposition (2.2.5) and (2.2.8) we have (2.5.7) C(τ) := { ω A χ,2 η 2 } (τ) η 2 (13τ) U χ,2 + V χ,2 S 2 (13, χ). = q (1 + ω) q 2 + 2(ω 1) q 3 + ω q 4 + (1 2ω) q 5 + 2(2 ω) q Hence dim S 2 (13, χ) = 1 and C provides a basis. It appears that the coefficient set of C is in Z[ω]. To show this we find another representation for C. Clearly, C(τ)η 2 (13τ)/η 2 (τ) M 2 (13, χ) and since dim M 2 (13, χ) = 3 we find (3 + 2ω)C(τ) η2 (13τ) η 2 (τ) = U χ,2 C(τ).

24 24 FRANK G. GARVAN Hence [ ] 1 (2.5.8) C(τ) = U χ,2 1 + (3 + 2ω) η2 (13τ) η 2, (τ) which implies the coefficient set of C is in Z[ω]. Case (ii). r = 2. We consider the character χ modulo 13 defined by χ(2) = i, so that χ 2 = χ 13 = ( 13). In the notation of (2.1.9) χ = χ13,2. Lemma (2.5.9). If χ = χ 13,2 then (2.5.10) D(τ) := 1 + i 2 and the coefficient set of D is in Z[i]. { η 2 } (τ) ( i) η 2 (13τ) U χ,3 + 13V χ,3 S 3 (13, χ) Proof. Noting that ( ) i A χ,3 =, 13 the proof of (2.5.10) is analogous to that of (2.5.7) in the proof of Lemma (2.5.5). We show that the coefficient set is in Z[i]. We define D 1 (τ) := ( ) 2 D(τ), 1 + i so that the coefficient set of D 1 is in Z[i]. A MAPLE calculation gives (2.5.11) D 2 1 = (2058 i 21560) B 2 + (66374 i ) B 3 + ( i ) B 4 + ( i ) B 5. Hence in Z[i] we have D 2 1 n 1 d 2 nq 2n 0 (mod 2), where D 1 (τ) = n 1 d nq n. Now for each n, d n = a n + b n i for some a n, b n Z. We know d 2 n 0 (mod 2) which implies a n b n (mod 2) so that 1 2 (1 + i)d n Z[i] and the coefficient set of D is in Z[i]. p = 17

25 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 25 In this case 1 2 (p 1) = 8. We consider r = 8 in view of the congruences (1.11) and (1.12). We consider the character χ modulo 17 defined by χ(3) = ω 8 := exp(πi/8), so that χ 8 = χ 17 = ( 17). In the notation of (2.1.9) χ = χ17,8. It can be shown that dim S 1 (17, χ) = 0 so we can only consider Eisenstein series. By applying Proposition (2.2.5), we find (2.5.12) W := V χ,1 A χ,1 = 1 + (ω 2 8 ω ω 5 8 ω 7 8) m,n 1 χ(n)q nm M 1 (17, χ). Hence W 8 M 8 (17, χ 8 ). The coefficient set of W 8 is in Z[ω 8 ]. However, it seems that all coefficients except the coefficient of q 0 are even. Instead we consider (2.5.13) Y := W 8 (1 ω 8 ). Since (2.5.14) (ω 2 8 ω ω 5 8 ω 7 8) = (1 ω 8 )(ω ω ω 6 8) it follows that, except for the first term, the coefficient set of Y is in Z[ω 8 ]. To simplify calculations we will construct an element of M 8 (17, χ 17 ) from Y whose coefficient set lies in Z except for the first term. Let Frob := Gal (Q[ω 8 ]/ Q). Every element α of Q[ω 8 ] can be uniquely written α = a 0 + a 1 ω a 7 ω 7 8 (a i Q). We define a Q-linear map CT : Q[ω 8 ] Q by CT(α) = a 0. CT stands for constant term. Then CT(α) = 1 8 ρ Frob For ρ Frob we define W ρ := n 0 aρ nq n where W = n 0 a nq n. It is clear that W ρ M 1 (17, χ ρ ) and (χ ρ ) 8 = χ 17 = ( 17). Hence (2.5.15) Z := CT (Y ) = CT(W 8 /(1 ω 8 )) = 1 8 α ρ. ρ Frob (W 8 ) ρ (1 ω 8) ρ M 8 (17, χ 17 ).

26 26 FRANK G. GARVAN Via MAPLE, we have (2.5.16) Z = q q q q q q q q q q q q q q q q q q q Some combination of U 17,8, V 17,8 and Z is a cusp form. We find (2.5.17) C := U 17,8 / 2 17 V 17,8 / Z S 8 (17, χ 17 ). Since the coefficients of Z are difficult to compute we express C as a combination of our basis {B 1, B 2,..., B 10 } for S 8 (17, χ 17 ). A MAPLE calculation gives (2.5.18) C = B B B B B B B B B B 10. p = 19 In this case 1 2 (p 1) = 9 and r = 3. From Proposition (2.2.9) (or Table II, 2.2), we find two cubes in S 9 (19, χ 19 ): [ η(19τ)η 5 (τ) ] 3 (2.5.19) = q and (2.5.20) [ η(τ)η 5 (19τ) ] 3 = q p = 23 In this case r = 1 2 (p 1) = 11. From Proposition (2.2.9) (or Table II 2.2), we find an 11-th power in S 11 (23, χ 23 ): (2.5.21) [η(τ)η(23τ)] 11 = q The Congruences In this section we prove our main result (1.10) as well as (1.12). The proof of (1.11), the stronger congruence for p = 17 is delayed until 4. We distinguish three cases.

27 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 27 p = 5, 7, 19, 23 Let r be a prime divisor of 1 2 (p 1). In this first case ɛ p = 1 and our goal is to prove (3.1) a p (n δ p ) 0 (mod r) if ( ) n = 1 and n 0 (mod r), p i.e. Φ r Ψ (X p ) 0 (mod r). We recall from Corollary (2.2.12) that (3.2) C := c p X p U p, 1 2 (p 1) S 1 2 (p 1) (p, χ p ), where c p is some positive integer (given explicitly in Table I, 2.2). We observe that in this case we have (3.3a) (3.3b) gcd (c p, r) = 1, (r 1) ( 1 2 (p 1) 1). Hence, from Lemma (2.3.23), we have (3.4) Φ r Ψ (U p, 1 2 (p 1) ) 0 (mod r). We need the existence of a special basis for S 1 2 (p 1) (p, χ p ) which follows easily from Corollary (2.4.14) and the construction of our r-th powers from 2.5. Proposition (3.5). Suppose p = 7, 19, 23, d := dim S 1 2 (p 1) (p, χ p ), and r is a prime divisor of 1 2 (p 1). There is a good basis D = {D 1, D 2,..., D d } with the following properties: For 1 i d (i) D i = q i +..., (ii) If ( i p) = 1 then Di S (p 1),p, (iii) If ( i p) = 1 and i 0 (mod r) then Di S 1 2 (p 1),p, (iv) If ( i p) = 1 and i 0 (mod r) then Φr (D i ) 0 (mod r). Proof. Let C = {C 1, C 2,..., C d } be the good basis of Corollary (2.4.14). When p = 7, d = 1 and we take D 1 = C 1 = η 3 (τ)η 3 (7τ).

28 28 FRANK G. GARVAN For p = 19, d = 13 and r = 3. If 1 i 13, ( i 19) = 1 or i 0 (mod 3) we take D i := C i. Otherwise, i = 3 or 12. In this case we use our cubes (2.5.19), (2.5.20): D 3 := [ η(19τ)η 5 (τ) ] 3, D 12 := [ η(τ)η 5 (19τ) ] 3. These clearly satisfy (iv). The case p = 23 is similar. Here d = 19 and r = 11. If 1 i 19 and i 11 we take D i := C i. Otherwise, we use our 11-th power (2.5.21): D 11 := [η(τ)η(23τ)] 11, which satisfies (iv). We now complete the proof of (3.1). Now (3.6) C := c p X p U p, 1 2 (p 1) = d α i D i for some α i Z. We note that in the case p = 5 we interpret this to say C = 0 since d = 0. Since ν (X p ) = δ p > d and (3.4) holds we have (by Proposition (3.5)) i=1 (3.7) α i 0 (mod r) if ( ) i = 1 and i 0 (mod r). p Hence, since Ψ annihilates S +, we have Φ r Ψ (C) 0 (mod r). Now (3.3a) and (3.4) imply Φ r Ψ (X p ) 0 (mod r), which is our result (3.1). p = 13

29 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 29 In this case r = 2 or 3 and ɛ p = 1. Our goal is to prove (3.8a) (3.8b) i.e. a 13 (n 7) 0 (mod 2) if a 13 (n 7) 0 (mod 3) if Φ r Ψ + (X 13 ) 0 (mod r) for r = 2 or 3. ( ) n = 1 and n is odd, 13 ( ) n = 1 and n 0 (mod 3); 13 Recall, by Proposition (2.4.13), that the B j := η 13 2j (τ)η 2j 1 (13τ) (1 j 6) provide a good basis for S 6 (13, χ 13 ). Let C be as in Lemma (2.5.5). Then C 3 S 6 (13, χ 13 ) and a calculation shows that (3.9) C 3 = B 3 + (4 3ω)B 4. It follows that [ (3.10) X 13 + B 6 = η6 (13τ) η 2 ] 3 η 6 (τ) (B (13τ) 3 + B 4 ) η 2 (τ) C(τ) (mod 3). Hence, since the coefficient set of C is in Z[ω], we have (3.11) Φ 3 (X 13 ) + Φ 3 (B 6 ) 0 (mod 3). Since ν (B 6 ) = dim S 6 (13, χ 13 ) = 6 we have (3.12) Ψ + (B 6 ) = 0, (by Proposition (2.3.16)) which together with (3.11) and the fact that the operators Φ 3 and Ψ + commute yields the result (3.8b). We note that an elementary proof of (3.12) may be obtained using Jacobi s identity for η 3 (τ) ([H-W, Thm 357]). The proof of (3.8a) is analogous. Let D be as in Lemma (2.5.9). Then D 2 S 6 (13, χ 13 ). A calculation shows (3.13) D 2 = ( i) B 2 + ( i) B 3 + ( i) B 4 + ( i) B 5,

30 30 FRANK G. GARVAN so that (3.14) D 2 B 2 + B 3 (mod 2). But [ (3.15) X 13 + B 6 = η8 (13τ) η 4 ] 2 η 8 (τ) (B (13τ) 2 + B 3 ) η 4 (τ) D(τ) (mod 2), so that (3.16) Φ 2 (X 13 ) + Φ 2 (B 6 ) 0 (mod 2), and the result (3.8a) follows from (3.12). It is interesting to note that (3.11) and (3.16) combine to give the following result. Curious Result. If and r = 2 or 3 then b n q n = n 0 m=1 (1 q m ) (1 q 13m ) + q m=1 (1 q 13m ) (1 q m ), b n 0 (mod r) if n 0 (mod r). p = 11, 17 For p = 11, r = 5 and ɛ p = 1, and our goal is to prove (3.17) a 11 (n 5) 0 (mod 5) if ( ) n = 1 and n 0 (mod 5). 11 We first observed (3.17) in [Ga-K-S, Thm 8, p.16] where the proof was omitted. Here we provide the details. We also give a proof of the p = 11 case of (1.10) which was omitted from [Ga-K-S]. For the case p = 17 we have r = 4 and ɛ p = 1, and our goal is the proof of (1.12) which is a stronger version of (1.10) and which we restate below. (3.18) a 17 (n 12) 0 (mod 2) if ( ) n = 1 and n 0 (mod 4). 17

31 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 31 We delay the proof of (1.11) until 4. Our proofs of (3.17) and (3.18) are similar and involve Hecke operators. In both cases we consider an Eisenstein series W M 1 (p, χ) where χ (p 1)/2 = χ p = ( ) p for some Dirichlet character χ, so that W (p 1)/2 M 1 2 (p 1) (p, χ p ). For the case p = 11 we take Z := W 5. To simplify computations for the p = 17 case we take Z := CT (W 8 /(1 ω 8 )) instead of W 8. The details of this construction are given in 2.5. Now let (3.19) (3.20) d ɛ := dim S ɛp 1 2 (p 1),p, d := dim S 1 2 (p 1) (p, χ p ). We can easily calculate d ɛ using Corollary (2.4.14). p ɛ p d ɛ d We build a cusp form C out of Z (see (2.5.4) and (2.5.17)). Now we use Hecke operators T p (with gcd (p, r) = 1) to build a subspace S of S 1 2 (p 1) (p, χ p ) so that (3.21) S 1 2 (p 1) (p, χ p ) = S S ɛp 1 2 (p 1),p. We describe S in terms of a basis C. For p = 11 we take C := {C, T 2 (C)}. For p = 17 we take C := {C, T 3 (C), T 5 (C), T 7 (C), T 11 (C)}. In each case we need to show that C is linearly independent and that the S that C generates satisfies S S ɛ 1 = {0}. This (p 1),p can be done by showing the Ψ ɛp (C) is linearly independent. For p = 11 this could (in principle) be done by hand. We give some details. We find 2 (3.22) and (3.23) C = 16 ( 713 q 1950 q q q q q ), T 2 (C) = ( q + 6 q q q ). Hence (3.24) (3.25) Ψ + (C) = 16 (713 q q ), Ψ + (T 2 (C)) = ( q + 3 q ),

32 32 FRANK G. GARVAN which are clearly linearly independent. Since Ψ + annihilates S5,11, Ψ +(X 11 ) is some linear combination of Ψ + (U 11,5 ), Ψ + (Z), and Ψ + (T 2 (Z)), by (3.21) and Lemmas (2.3.17) and (2.3.19). With the help of MAPLE we find (3.26) Ψ + (X 11 ) = 26 Ψ + (Z) + 9 Ψ + (T 2 (Z)) 100 Ψ + (U 11,5 ), and our result (3.17) follows by Lemma (2.3.22). We have checked the identity (3.26) to order O(q 100 ). We now include the proof of the p = 11 case of (1.10) since its proof also involves Hecke operators. By Corollary (2.2.12), we have (3.27) F := 1275 X 11 U 11,5 S 5 (11, χ 11 ). Since dim S 5 (11, χ 11 ) = 3 it can be shown that {F, T 2 (F ), T 3 (F )} forms a basis of cusp forms. We find (3.28) T 11 (F ) = 11 (T 3 2 T 2 + 4)(F ). Since the Hecke operators commute we have (3.29) and (3.30) T α 11(F ) = 11 α (T 3 2 T 2 + 4) α (F ) T α 11(F ) 0 (mod 11 α ). From Lemma (2.3.17) we have (3.31) T α 11(U 11,5 ) 0 (mod 11 4α ) and the result follows. The proof of the case p = 17 with r = 4 is analogous. Now the problem of verifying that Ψ (C) is linearly independent is a tedious calculation but one that is easily performed by MAPLE. For example, in order to compute T 11 (C) up to q 10 it is necessary to compute C up to q 110. Here Z and C are defined in (2.5.15) and (2.5.17) respectively. We omit most details. The reader who wishes to carry out these calculations will find (2.5.18), the alternative expression for C, helpful. Using MAPLE we have shown that Ψ (C) is linearly independent so that, analogous to the p = 11 case, Ψ (X 11 ) is some linear combination of Ψ (U 17,8 ), Ψ (Z), Ψ (T 3 (Z)), Ψ (T 5 (Z)), Ψ (T 7 (Z)), and Ψ (T 11 (Z)). We have found

33 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 33 this linear combination in MAPLE but since the coefficients are huge we give a version that is reduced modulo 8. (3.32) 2 Ψ (X 17 ) 6 Ψ (Z) + 2 Ψ (T 3 (Z)) + 5 Ψ (T 5 (Z)) + 5 Ψ (T 11 (Z)) (mod 8). We have checked (3.32) directly to order O(q 15 ) and by an indirect method to order O(q 100 ). It follows that (3.33) Φ 4 Ψ (X 17 ) 0 (mod 2) (which is the desired result (3.18)), and (3.34) Φ 2 Ψ (X 17 ) 0 (mod 4). However a stronger result than (3.34) holds, namely (3.35) Φ 2 Ψ (X 17 ) 0 (mod 8). We attack this stronger result in the next section. inadequate. The methods of this section seem In this section we prove (4.1) a 17 (n 12) 0 (mod 8) if 4. The Proof of (1.11) ( ) n = 1 and n 0 (mod 2). 17 Our solution to this problem is mildly computational. It turns out that the problem can be reduced to verifying (4.1) for the first 7 values of n (i.e. n = 23, 27, 29, 31, 37, 39, 41) and computing the first 14 coefficients of a certain modular form in M 8 (17, χ 17 ). First we show how (4.1) is equivalent to a certain congruence (see (4.7) below) in S 20 (17, χ 17 ). We have (4.2) B(τ) = n 29 by Proposition (2.2.9). Now (4.3) B(τ) = η 24 (τ) η17 (17τ) η(τ) = q 17 b n q n := η41 (17τ) η(τ) m=1(1 q 17m ) 24 n 12 S 20 (17, χ 17 ) a 17 (n 12)q n q m=1(1 34m ) 12 a 17 (n 12)q n+17 (mod 8), n 12

34 34 FRANK G. GARVAN since (1 x) 24 (1 x 2 ) 12 (4.4) i.e. (4.5) b n 0 (mod 8) if b 2n 0 (mod 8) if (mod 8). It follows that (4.1) is equivalent to ( ) n = 1 and n 0 (mod 2), 17 ( ) n = 1, 17 since ( 2 17) = 1. Now we consider the Hecke operator T2 on S 20 (17, χ 17 ). Then (4.6) T 2 (B) = n 15 c n q n n 15 b 2n q n (mod 2 19 ), since c n = b 2n + ( n 17) 2 19 b n/2. Hence (4.1) is equivalent to (4.7) Ψ (T 2 (B)) 0 (mod 8). We now show the existence of certain special bases for S 20,17 + and S 20,17 (defined in (2.3.9) and (2.3.10)). We let {B 1, B 2,..., B 10 } be our good basis for S 8 (17, χ 17 ) constructed in Proposition (2.4.13). We may complete this is to a good basis {B 0, B 1,..., B 10, B 12 } for M 8 (17, χ 17 ) by defining (4.8) (4.9) B 0 := η17 (τ) η(17τ) = , B 12 := X 17 = η17 (17τ) η(τ) = q We note that there is no form in M 8 (17, χ 17 ) with a q-expansion of the form q , since dim M 8 (17, χ 17 ) = = 12. It is clear from this construction that there is a modular form D M 8 (17, χ 17 ) with integral coefficients and a q-expansion of the form (4.10) D = 1 + γ q 11 + O(q 13 ), for some γ Z. Using MAPLE we have computed (4.11) D = q q q

35 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 35 From (4.17) and Proposition (4.19) below it follows that Ψ (D) = 0. As a check on our taylor expansion we have confirmed this to order O(q 100 ). We show that there is a basis C = {C 1, C 2,..., C 27, C 29 } for S 20 (17, χ 17 ) such that (4.12a) C j Z[[q]] (for each j), (4.12b) (4.12c) C j = q j + β j q 28 + O(q 30 ) C 29 = q 29 + O(q 30 ). (for 1 j 27 and β j Z), We define (4.13) (τ) := η 24 (τ) = q As is well-known S 12 (Γ(1)). It follows, from [Kob, Prop. 17 p.127], that (4.14) 17 := (17τ) = q 17 + S 12 (17, χ 0 ). In addition 17 has no zeros in H. If C S 20 (17, χ 17 ) and ν (C) 17 then 1 17 C M 8 (17, χ 17 ). It follows that there is no cusp form in S 20 (17, χ 17 ) with ν (C) = = 28 and that for C S 20 (17, χ 17 ), C 0, we have (4.15) ν (C) 29. We take (4.16) (4.17) C 29 := B = η41 (17τ), η(τ) C 17 := 17 D, where D is the form in M 8 (17, χ 17 ) described before so that (4.18) C 17 = q q q q We next construct C i S 20(17, χ 17 ), 1 i 27 i 17, with as in Proposition (2.4.13). We simply set C i = q i + Z[[q]], C i = C 29 f k 2 f l 3,

36 36 FRANK G. GARVAN where f 2, f 3 are given in the proof of Proposition (2.4.12) with p = 17, and k 0, l 0 are chosen such that 29 2k 3l = i. Now the remaining C j may be easily constructed from the C i, C 17, C 29. We now use the twist operator ρ described in 2.3. We have for some γ C. Hence From (4.15) it follows that and C 17 S + 20,17. Propostion (4.19). (a) {C j : 1 j 26 and (b) For certain constants γ j, ρ(c 17 ) = γ q q , ρ(c 17 ) γ C 17 = O(q 30 ) S 20 (17, χ 17 ). ρ(c 17 ) = γ C 17 = C 17 {C j γ j C 17 : 3 j 29 j 28 and a basis for S 20,8. ( ) j = 1 or j = 17} is a basis for S 20, ( ) j = 1} is 17 Proof. Suppose 1 j 26 and ( j 17) = 1. We know for some β j Z, so that C j = q j + β j q 28 + O(q 30 ) for some γ j C. We have ρ(c j ) = q j + 2γ j q 17 β j q 28 + O(q 30 ) ρ(c j ) C j 2γ j C 17 = 2β j q 28 + O(q 30 ). It follows that β j = 0, and ρ(c j ) = C j + 2γ j C 17,

37 SOME CONGRUENCES FOR PARTITIONS THAT ARE p-cores 37 so that ρ 2 (C j ) = C j + 4γ j C 17, (since C 17 S + 20,17 ). But ρ 2 (C j ) = C j which means that γ j = 0 and C j S + 20,17. Now suppose 3 j 29, j 28 and ( j 17) = 1. Then ρ(c j ) = q j + 2γ j q 17 β j q 28 + O(q 30 ) for certain γ j, β j C. Arguing as before we deduce ρ(c j ) = C j + 2γ j C 17. Hence so that ρ(c j γ j C 17 ) = C j + γ j C 17, C j γ j C 17 S 20,17. The result follows from (2.3.13) since dim S 20 (17, χ 17 ) = 28. have We need to verify (4.1) for n 41 (i.e. n = 23, 27, 29, 31, 37, 39, 41).Via MAPLE we (4.20) X 17 := q 12 + q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 12 + q q q q q q q q q q q q q q q q q q q q q q q (mod 8). We complete the proof of (4.7). Since C is a basis that satisfies (4.12), there are α j Z such that (4.21) T 2 (B) = 29 j=15 j 28 α j C j.