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 pmodular 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. Email: frank@cs.dal.ca, frank@math.ufl.edu

2 FRANK G. GARVAN ^{∗}

p-cores. In [Ga-K-S] we gave a combinatorial proof of the following generating function identity:

(1.1)  ∑_{n≥0} a_{p}(n)q^{n} = ∏_{m=1}^{∞} \frac{(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)  ∑_{n≥δ_{p}} a_{p}(n − δ_{p})q^{n} = \frac{η^{p}(pτ)}{η(τ)} =: X_{p}(τ),

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} ∏_{m=1}^{∞}(1 − q^{2mπiτ}).

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 (\frac{·}{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+ϵ})  for any ϵ > 0,

where

u_{p}(n) = n^{(p−3)/2} ∑_{d|n} \frac{1}{d^{(p−3)/2}} (\frac{d}{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_{p_{1}}(n) ≤ a_{p_{2}}(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.

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 = ∑_{n=0}^{∞} c_{n}e^{2πiτ/M}, M ≥ 1,

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)  a_{5}(5n − 1) ≡ 0 (mod 5),
(1.7)  a_{7}(7n − 2) ≡ 0 (mod 7),
(1.8)  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 ϵ_{p} (= 1 or −1) such that

(1.10)  a_{p}(n − δ_{p}) ≡ 0 (mod r),  whenever (\frac{n}{p}) = ϵ_{p} and n ≢ 0 (mod r).

In particular, ϵ_{p} = 1 for p = 11, 13 and ϵ_{p} = −1 for p = 5, 7, 17, 19, 23. For p = 17 stronger congruences hold:

(1.11)  a_{17}(n − 12) ≡ 0 (mod 8) if (\frac{n}{17}) = −1 and n ≢ 0 (mod 2),

and

(1.12)  a_{17}(n − 12) ≡ 0 (mod 2) if (\frac{n}{17}) = −1 and n ≢ 0 (mod 4).

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)  ∑_{n≥0} b_{p}(n)q^{n} = ∏_{m=1}^{∞} \frac{(1 − q^{pm})}{(1 − q^{m})}

= \frac{∑_{n≥0} a_{p}(n)q^{n}}{∏_{m=1}^{∞}(1 − q^{pm})^{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} ≡ \frac{∑_{n≥0} a_{p}(n)q^{n}}{∏_{m=1}^{∞}(1 − q^{prm})^{(p−1)/r}} (mod r).

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 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−1}.

2. 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 = \frac{aτ + b}{cτ + d},  where γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(ℤ).

Let N be a positive integer. We define

(2.1.2) Γ(N) := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ 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)  Γ_{0}(N) := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ(1) : c ≡ 0 (mod N)\} ,
(2.1.4)  Γ_{1}(N) := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{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}f(γτ),  for γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ(1).

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. f(τ) | [γ]_{k} has the form

∑_{n≥0} a_{n}q_{N}^{n}. 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)  f(τ) = ∑_{n=0}^{∞} a_{n}q^{n},  q = e^{2πiτ}

and

(2.1.8b)  τ^{−k}f(−1/pτ) = ∑_{n=0}^{∞} b_{n}q^{n},  q = e^{2πiτ} .

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).

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 | \frac{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} = (\frac{·}{p})  (the Legendre symbol),

and

(2.1.11)  χ_{p,d}^{d} = χ_{p} = (\frac{·}{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

(2.2.1)  S_{χ}(n) := ∑_{j=1}^{p−1} χ(j)exp(2πijn/p).

Then

(2.2.2)  S_{χ}(n) = \overline{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} + ∑_{m=1}^{∞} ∑_{n=1}^{∞} χ(n)n^{k−1}q^{mn}  for k ≥ 1,

where

(2.2.3b)  A_{k} = A_{χ,k} := −\frac{1}{2k} ∑_{j=0}^{k−1} \binom{k}{j}B_{j}p^{j−1} ∑_{n=1}^{p−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)  U_{χ,k} := ∑_{m=1}^{∞} ∑_{n=1}^{∞} χ(m)n^{k−1}q^{mn}  for k ≥ 2,
and  U_{χ,1} := V_{χ,1}.

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) = \frac{(pτ)^{k}}{S\overline{χ}(1)}U_{\overline{χ},k}(τ),

(iii) U_{χ,k}(−1/pτ) = \frac{pτ^{k}}{S\overline{χ}(1)}V_{\overline{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\overline{χ}(1) = S_{χ}(1) = i^{(p−1)^{2}/4}\sqrt{p}.

We define

(2.2.7a)  U_{p,k} := U_{χ_{p},k},
(2.2.7b)  V_{p,k} := V_{χ_{p},k},
(2.2.7c)  A_{p,k} := A_{χ_{p},k}.

We now determine which η-products belong to M_{k}(p, χ). If we let √ denote the branch of the square root having nonnegative real part, then

(2.2.8)  η(−1/τ) = \sqrt{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),

−kp ≤ \frac{(p−1)}{2}m ≤ k. (2.2.10b) Then

Y_{k,m} := \frac{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}). □

Corollary (2.2.12). Let p > 3 be prime. Then

(i) X_{p} := \frac{η^{p}(pτ)}{η(τ)} ∈ M_{\frac{1}{2} (p−1)}(p, χ_{p}), (ii) c_{p}X_{p} − U_{p, \frac{1}{2} (p−1)} ∈ S_{\frac{1}{2} (p−1)}(p, χ_{p}), where c_{p} := pA_{p, \frac{1}{2} (p−1)}i^{\frac{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^{r_{p}} + …,

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, \frac{1}{2} (p−1)} is zero. Using (2.2.5)(iii), (2.2.6) and (2.2.8) we find

c_{p} := pA_{p, \frac{1}{2} (p−1)}i^{\frac{1}{4} (p−1)(p−3)},

which gives the desired result. Finally, c_{p} ≠ 0 follows from the fact that A_{p, \frac{1}{2} (p−1)} is a nonzero multiple ([Kol3, (2.26)]) of a Dirichlet L-series L(\frac{1}{2}(p − 1), χ_{p}) ([Se1, p.71]) which is nonzero. □

p  c_{p}
5  1
7  8 = 2^{3}
11  1275 = 3 · 5^{2} · 17
13  33463 = 109 · 307
17  59901794 = 2 · 19 · 1576363
19  3708443635 = 5 · 13 · 67 · 851537
23  27533989805352 = 2^{3} · 3 · 63659 · 18021797

Table I. c_{p} of Corollary (2.2.12) (ii).

We need to calculate which η-products are cusp forms for S_{\frac{1}{2} (p−1)}(p, χ_{p}), 5 ≤ p ≤ 23. For k = \frac{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.

p  η-product cusp forms
5  −
7  η^{3}(τ)η^{3}(7τ)
11  −
13  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_{\frac{1}{2} (p−1)}(p, χ_{p}) for small p.

Klyachko [Kl, p.1883] has the following explicit formula

dim S_{\frac{1}{2} (p−1)}(p, χ_{p}) = \{ \begin{matrix}[(p − 1)^{2}/24] , & if p ≢ −1 (mod 12), \\ [(p − 1)^{2}/24] − 1, & if p ≡ −1 (mod 12). \end{matrix}

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].

\begin{matrix}p & dim S_{\frac{1}{2} (p−1)}(p, χ_{p}) \\ 5 & 0 \\ 7 & 1 \\ 11 & 3 \\ 13 & 6 \\ 17 & 10 \\ 19 & 13 \\ 23 & 19 \\ 29 & 32 \end{matrix}

Table III. Dimension of certain spaces of cusp forms.

From this table we observe that

(2.2.13)  dim S_{\frac{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.

2.3 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_{n}q^{n} (q = e^{2πiτ}). For p prime, the Hecke operator T_{p} is given by T_{p}(f(τ)) = ∑_{n≥0} b_{n}q^{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⟩ := \frac{1}{[Γ(1) : Γ_{1}(N)]} ∫_{Γ_{1}(N)\H} f(τ)\overline{g(τ)}y^{k} \frac{dx dy}{y^{2}}  (τ = x + iy).

We need

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

⟨T_{n}f, g⟩ = χ(n)⟨f, T_{n}g⟩.

See [Kob, Prop.48 p.171] for a proof.

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 (\frac{n}{p}) = χ_{p}(n) = 1, (ii) λ_{n} is pure imaginary if (\frac{n}{p}) = χ_{p}(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) = \overline{f(−τ̄)}.

If f(τ) = ∑_{n≥1} a_{n}q^{n} then f^{ρ′}(τ) = ∑_{n≥1} \overline{a_{n}}q^{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 eigenvalues are λ_{n} = a_{n} ([Kob, Prop.40 p.163]). By Corollary (2.3.5) we have

(2.3.7)  a_{n}^{ρ′} = χ_{p}(n)a_{n} = (\frac{n}{p})a_{n}  if p ∤ n.

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})

ρ : S_{k}(p, χ_{p}) ⟶ S_{k}(p, χ_{p}),

so that

(2.3.8)  a_{n}^{ρ} = (\frac{n}{p})a_{n}  if p ∤ n,

where f(τ) = ∑_{n≥1} a_{n}q^{n} is any f ∈ S_{k}(p, χ_{p}). We let

(2.3.9) S_{k,p}^{+} := \{f ∈ S_{k}(p, χ_{p}) : f(τ) = ∑_{n≥1} a_{n}q^{n}, a_{n} = 0 for (\frac{n}{p}) = −1\} ,

(2.3.10) S_{k,p}^{−} := \{f ∈ S_{k}(p, χ_{p}) : f(τ) = ∑_{n≥1} a_{n}q^{n}, a_{n} = 0 for (\frac{n}{p}) = +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 = \frac{1}{2}(f + f^{ρ}) + \frac{1}{2}(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_{n}q^{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}^{−}.

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_{\frac{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_{\frac{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 ϵ = ±1 be fixed. Suppose p is prime with 7 ≤ p ≤ 23. Let d := dim S_{\frac{1}{2} (p−1)}(p, χ_{p}). Suppose f ∈ S_{\frac{1}{2} (p−1)}(p, χ_{p}) and f(τ) = ∑_{n≥1} a_{n}q^{n}. If

a_{n} = 0  for (\frac{n}{p}) = ϵ and n ≤ d,

then

a_{n} = 0  for all n with (\frac{n}{p}) = ϵ.

Remark: A table for the d’s is given in Table III in §2.2.

Proof. Let g := f + ϵf^{ρ} = ∑_{n≥1} b_{n}q^{n}, so that g ∈ S_{\frac{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_{\frac{1}{2} (p−1),p}^{−ϵ} and the result follows. □

Example. For p = 7, d = 1 and a basis for S_{3}(7, χ_{7}) is

η^{3}(τ)η^{3}(7τ) = q + ⋯ = ∑_{n≥1} a_{n}q^{n}.

Hence, a_{n} = 0 for (\frac{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_{\frac{1}{2} (p−1)}(p^{2}, χ_{p}) rather than S_{\frac{1}{2} (p−1)}(p, χ_{p}). Clearly the smaller space S_{\frac{1}{2} (p−1)}(p, χ_{p}) and ρ are preferable.

The action of Hecke operators on Eisenstein series is easily calculated.

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)  T_{r}(U_{p,k}) = (χ_{p}(r) + p^{k−1})U_{p,k},
(ii)  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 ϵ = + or − we define the operator Ψ_{ϵ} = Ψ_{ϵ,p} that acts on a series ∑_{n≥0} a_{n}q^{n} and picks out those terms in which χ_{p}(n) = ϵ; i.e.

(2.3.18)  Ψ_{ϵ} (∑_{n≥0} a_{n}q^{n}) = ∑_{\begin{matrix}n≥1 \\ χ_{p}(n)=ϵ \end{matrix}} a_{n}q^{n}.

We note that Ψ_{ϵ} does not necessarily preserve the space M_{k}(p, χ_{p}). However, the action of Ψ_{ϵ} on Eisenstein series is nice. Lemma (2.3.19). Let p be an odd prime then for ϵ = ± we have

(2.3.20)  Ψ_{ϵ} (U_{p,k}) = ϵΨ_{ϵ} (V_{p,k}) .

Proof. Let r be prime, r ≠ p. Let u(n) (v(n) resp.) be the n-th coefficient in the qexpansion of U_{p,k} (V_{p,k} resp.) at ∞. An easy calculation gives

u(r^{α}) = χ_{p}(r^{α})v(r^{α}).

So, by multiplicativity, we have

u(n) = χ_{p}(n)v(n) for p ∤ n,

and the result follows. □

Finally, for r > 1 we define Φ_{r} to be the operator that acts on a series ∑_{n≥0} a_{n}q^{n} and picks out those terms in which n ≢ 0 (mod r); i.e.

(2.3.21)  Φ_{r} (∑_{n≥0} a_{n}q^{n}) = ∑_{\begin{matrix}n≥1 \\ n≢0 (mod r) \end{matrix}} 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.

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 Φ_{r} (W^{r}) ≡ 0 (mod r), (i) Φ_{r} (T_{s} (W^{r})) ≡ 0 (mod r). (ii)

Proof. (i) is trivial. We have

W^{r} = ∑_{n≥0} a_{n}q^{n} ∈ M_{k}(p, χ_{p}),

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 (\frac{n}{p}) = −1 and n ≢ 0 (mod d) then there is a prime q, q ≠ d, such that (\frac{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)} + ⋯ − 1
≡ 0 (mod d),

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_{\frac{1}{2} (p−1)}(p, χ_{p}) for p prime, 7 ≤ p ≤ 23. Let d := dim S_{\frac{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)  B_{j} ∈ ℤ[[q]],
(2.4.1b)  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_{n}q_{p}^{n} 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) = ∑_{n≥ν_{∞}} a_{n}q^{n},  q = e^{2πiτ} , some ν_{∞} ∈ ℤ

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_{ν_{0}} ≠ 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).

When p is a prime > 3 the genus of Γ_{0}(p) is given by

(2.4.3)  g = \{ \begin{matrix}[p/12] − 1, & if p ≡ 1 (mod 12), \\ [p/12] , & if p ≡ 5, 7 (mod 12), \\ [p/12] + 1, & if p ≡ 11 (mod 12). \end{matrix}

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_{\frac{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)  f_{j} is zero at τ = 0 (i.e. ν_{0} ≥ 1).

We call such a basis good.

Let p > 3 be prime. Following Fine [F] we define

(2.4.5)  W_{k}(τ) = W_{k,p}(τ) := q^{\frac{6k^{2}}{p} −k} ∏_{m≥1} \frac{(1 − q^{pm−4k})(1 − q^{pm+4k−p})}{(1 − q^{pm−2k})(1 − q^{pm+2k−p})},

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 W_{k} = W_{−k} = W_{k+p}, (i)

W_{k} | [γ]_{0} = exp(12πik^{2}ab/p)W_{ak}, for γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(p), (ii)

W_{k}(−1/pτ) = 2 cos(2kπ/p) + O(q). (iii)

From (i) there are \frac{1}{2}(p − 1) distinct W_{k} namely W_{1}, W_{2}, …, W_{\frac{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 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 ≤ \frac{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_{p}^{a}, 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) := (\frac{η(τ)}{η(pτ)})^{ϱ} .

Then

(2.4.9)  w_{p}(τ) = q^{−(p−1)ϱ/24} + …,

and by (2.2.8) we have

(2.4.10)  w_{p}(−1/pτ) = p^{ϱ/2}q^{(p−1)ϱ/24} + ….

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): p = 5, g = 0 :

f_{1} = w_{5} = (\frac{η(τ)}{η(5τ)})^{6} = q^{−1} − 6 + 9 q + 10 q^{2} − 30 q^{3} + 6 q^{4} − 25 q^{5} + ….

p = 7, g = 0 :

f_{1} = w_{7} = (\frac{η(τ)}{η(7τ)})^{4} = q^{−1} − 4 + 2 q + 8 q^{2} − 5 q^{3} − 4 q^{4} − 10 q^{5} + ….

p = 11, g = 1 :

f_{2} = W_{1}^{3}W_{4}W_{5} + W_{2}^{3}W_{3}W_{1} + W_{3}^{3}W_{1}W_{4} + W_{4}^{3}W_{5}W_{2} + W_{5}^{3}W_{2}W_{3} − 17

= q^{−2} + 2 q^{−1} − 12 + 5 q + 8 q^{2} + q^{3} + 7 q^{4} + …,
f_{3} = 2 − (W_{1}^{6}W_{4} + W_{2}^{6}W_{3} + W_{3}^{6}W_{1} + W_{4}^{6}W_{5} + W_{5}^{6}W_{2})
= q^{−3} + q^{−1} − 12 + 2 q + 2 q^{2} + 16 q^{3} + 16 q^{4} + ….

p = 13, g = 0 :

f_{1} = w_{13} = (\frac{η(τ)}{η(13τ)})^{2} = q^{−1} − 2 − q + 2 q^{2} + q^{3} + 2 q^{4} − 2 q^{5} + ….

p = 17, g = 1 :

f_{2} = W_{1}W_{2}^{2}W_{5} + W_{2}W_{4}^{2}W_{7} + W_{3}W_{6}^{2}W_{2} + W_{4}W_{8}^{2}W_{3} + W_{5}W_{7}^{2}W_{8}

+ W_{6}W_{5}^{2}W_{4} + W_{7}W_{3}^{2}W_{1} + W_{8}W_{1}^{2}W_{6} − 9
= q^{−2} + q^{−1} − 6 + q + 2 q^{2} + 2 q^{3} + 3 q^{4} + …,
f_{3} = W_{1}^{3}W_{2}W_{5}W_{6} + W_{2}^{3}W_{4}W_{7}W_{5} + W_{3}^{3}W_{6}W_{2}W_{1} + W_{4}^{3}W_{8}W_{3}W_{7}

+ W_{5}^{3}W_{7}W_{8}W_{4} + W_{6}^{3}W_{5}W_{4}W_{2} + W_{7}^{3}W_{3}W_{1}W_{8} + W_{8}^{3}W_{1}W_{6}W_{3} − 36

= q^{−3} + 3 q^{−2} + 3 q^{−1} − 24 + 5 q + 8 q^{2} + 3 q^{3} + ….

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^{−2} − 4 + q + 2 q^{2} + 2 q^{3} − q^{4} + …,

f_{3} = w_{19} = (\frac{η(τ)}{η(19τ)})^{4} = q^{−3} − 4 q^{−2} + 2 q^{−1} + 8 − 5 q − 4 q^{2} − 10 q^{3} + ….

p = 23, g = 2 :
f_{3} = −8 − (W_{1}^{2}W_{6}W_{10} + W_{2}^{2}W_{11}W_{3} + W_{3}^{2}W_{5}W_{7} + W_{4}^{2}W_{1}W_{6} + W_{5}^{2}W_{7}W_{4}
+ W_{6}^{2}W_{10}W_{9} + W_{7}^{2}W_{4}W_{1} + W_{8}^{2}W_{2}W_{11} + W_{9}^{2}W_{8}W_{2} + W_{10}^{2}W_{9}W_{8}
+W_{11}^{2}W_{3}W_{5})
= q^{−3} + 2 q^{−2} + q^{−1} − 12 + 2 q + 2 q^{2} + 6 q^{4} + …,
f_{4} = 7 − (W_{1}^{3}W_{5}W_{8} + W_{2}^{3}W_{10}W_{7} + W_{3}^{3}W_{8}W_{1} + W_{4}^{3}W_{3}W_{9} + W_{5}^{3}W_{2}W_{6}
+ W_{6}^{3}W_{7}W_{2} + W_{7}^{3}W_{11}W_{10} + W_{8}^{3}W_{6}W_{5} + W_{9}^{3}W_{1}W_{3} + W_{10}^{3}W_{4}W_{11}
+W_{11}^{3}W_{9}W_{4})
= q^{−4} − q^{−3} − q^{−2} + 2 q^{2} + 4 q^{3} − 6 q^{4} + …,
f_{5} = −32 − (W_{1}^{4}W_{6}W_{11} + W_{2}^{4}W_{11}W_{1} + W_{3}^{4}W_{5}W_{10} + W_{4}^{4}W_{1}W_{2} + W_{5}^{4}W_{7}W_{9}
+ W_{6}^{4}W_{10}W_{3} + W_{7}^{4}W_{4}W_{8} + W_{8}^{4}W_{2}W_{4} + W_{9}^{4}W_{8}W_{7} + W_{10}^{4}W_{9}W_{5}
+W_{11}^{4}W_{3}W_{6})
= q^{−5} + q^{−4} + q^{−3} + q^{−2} + 2 q^{−1} − 24 + 3 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_{\frac{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_{\frac{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_{\frac{1}{2} (p−1)}(p, χ_{p}) has a good linear basis B = \{B_{1}, B_{2}, …, B_{d}\} where

B_{j} = X_{p}f_{r_{p}−j},  if d − g ≤ j ≤ d,
B_{j} = X_{p}f_{g+1}^{(t−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_{\frac{1}{2} (p−1)}(p, χ_{p}) with ν_{∞}(f) > r_{p} − g − 1. Then f/X_{p} ∈ F_{p} since X_{p} ∈ M_{\frac{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_{\frac{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_{\frac{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_{\frac{1}{2} (p−1)}(p, χ_{p}),

since X_{p} ∈ M_{\frac{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

m = t(g + 1) + s,

where 0 ≤ s < g + 1, so that

m = (t − 1)(g + 1) + (s + g + 1)

and

g + 1 ≤ s + g + 1 ≤ 2g + 1.

In this case we take

B_{j} := X_{p}f_{g+1}^{(t−1)}f_{s+g+1},

which has the desired properties.

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).

Corollary (2.4.14). Let p be prime, 7 ≤ p ≤ 23, and d := dim S_{\frac{1}{2} (p−1)}(p, χ_{p}). Then there exists a good basis C = \{C_{1}, C_{2}, …, C_{d}\} for S_{\frac{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} : (\frac{i}{p}) = 1\} is a good basis for S_{\frac{1}{2} (p−1),p}^{+}, (iii) C^{−} = \{C_{i} : (\frac{i}{p}) = −1\} is a good basis for S_{\frac{1}{2} (p−1),p}^{−}.

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 \frac{1}{2}(p − 1). In this section we construct modular forms f of weight \frac{(p−1)}{2r} such that f^{r} ∈ M_{\frac{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_{n}q^{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 = \frac{1}{2}(p − 1) = 5. Since 11 ≡ 3 (mod 4) we have, by Proposition (2.2.5),

(2.5.1)  W := 2V_{11,1} = 1 + 2 ∑_{m,n≥1} (\frac{n}{11})q^{nm} ∈ M_{1}(11, χ_{11}).

Hence

W^{5} ∈ M_{5}(11, χ_{11}).

If we define

(2.5.2)  V := 11V_{11,5} = 1275 + 11 ∑_{m,n≥1} (\frac{n}{11})n^{4}q^{nm}

and

(2.5.3)  U := U_{11,5} = ∑_{m,n≥1} (\frac{n}{11})m^{4}q^{nm},

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

In this case \frac{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_{13,3}^{3} = χ_{13} = (\frac{·}{13}). Let ω = exp(πi/3). More explicitly we may define χ_{13,3} 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

F (τ) = q^{ν} + …

with ν ≥ 2. But F^{3} ∈ S_{6}(13, χ_{13}) so that

F^{3}(τ) = cB_{6} = c η(τ)η^{11}(13τ)

for some constant c ≠ 0. By (2.2.8) we have

[τ^{−2}F (−1/13τ)]^{3} = c\sqrt{13}η(13τ)η^{11}(τ) = c\sqrt{13} q + ….

This is impossible since τ^{−2}F (−1/13τ) ∈ S_{2}(13, \overline{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

C(τ) := \frac{13}{6 + 18ω} \{−A_{χ,2} \frac{η^{2}(τ)}{η^{2}(13τ)}U_{χ,2} + V_{χ,2}\} ∈ S_{2}(13, χ). (2.5.7)
= q − (1 + ω) q^{2} + 2(ω − 1) q^{3} + ω q^{4} + (1 − 2ω) q^{5} + 2(2 − ω) q^{6} + ….

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)\frac{η^{2}(13τ)}{η^{2}(τ)} = U_{χ,2} − C(τ).

24 FRANK G. GARVAN ^{∗}

Hence

(2.5.8)  C(τ) = U_{χ,2} [1 + (3 + 2ω)\frac{η^{2}(13τ)}{η^{2}(τ)}]^{−1} ,

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} = (\frac{·}{13}). In the notation of (2.1.9) χ = χ_{13,2}. Lemma (2.5.9). If χ = χ_{13,2} then

(2.5.10)  D(τ) := \frac{1 + i}{2} \{(27 + 60i) \frac{η^{2}(τ)}{η^{2}(13τ)}U_{χ,3} + 13V_{χ,3}\} ∈ S_{3}(13, χ)

and the coefficient set of D is in ℤ[i]. Proof. Noting that

A_{χ,3} = − (\frac{27 + 60i}{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) := (\frac{2}{1 + i}) D(τ),

so that the coefficient set of D_{1} is in ℤ[i]. A MAPLE calculation gives

(2.5.11)  D_{1}^{2} = (2058 i − 21560) B_{2} + (66374 i − 208852) B_{3}
+ (352600 i − 761288) B_{4} + (699296 i − 1048944) B_{5}.

Hence in ℤ[i] we have

D_{1}^{2} ≡ ∑_{n≥1} d_{n}^{2}q^{2n} ≡ 0 (mod 2),

where D_{1}(τ) = ∑_{n≥1} d_{n}q^{n}. Now for each n, d_{n} = a_{n} + b_{n}i for some a_{n}, b_{n} ∈ ℤ. We know d_{n}^{2} ≡ 0 (mod 2) which implies a_{n} ≡ b_{n} (mod 2) so that \frac{1}{2}(1 + i)d_{n} ∈ ℤ[i] and the coefficient set of D is in ℤ[i]. □

p = 17

In this case \frac{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} = (\frac{·}{17}). In the notation of (2.1.9) χ = χ_{17,8}. 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 := \frac{V_{χ,1}}{A_{χ,1}} = 1 + (ω_{8}^{2} − ω_{8}^{3} + ω_{8}^{5} − ω_{8}^{7}) ∑_{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 := \frac{W^{8}}{(1 − ω_{8})}.

Since

(2.5.14)  (ω_{8}^{2} − ω_{8}^{3} + ω_{8}^{5} − ω_{8}^{7}) = (1 − ω_{8})(ω_{8}^{2} + ω_{8}^{5} + ω_{8}^{6})

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_{8} + ⋯ + a_{7}ω_{8}^{7} (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) = \frac{1}{8} ∑_{ρ∈Frob} α^{ρ}.

For ρ ∈ Frob we define W^{ρ} := ∑_{n≥0} a_{n}^{ρ}q^{n} where W = ∑_{n≥0} a_{n}q^{n}. It is clear that W^{ρ} ∈ M_{1}(17, χ^{ρ}) and (χ^{ρ})^{8} = χ_{17} = (\frac{·}{17}). Hence

(2.5.15)  Z := CT (Y) = CT(W^{8}/(1 − ω_{8})) = \frac{1}{8} ∑_{ρ∈Frob} \frac{(W^{8})^{ρ}}{(1−ω_{8})^{ρ}} ∈ M_{8}(17, χ_{17}).

Via MAPLE, we have

(2.5.16) Z = \frac{1}{2} + 8 q^{2} + 112 q^{3} − 552 q^{4} + 1344 q^{5} − 4108 q^{6} + 4104 q^{7}
− 12577 q^{8} + 12544 q^{9} − 27536 q^{10} − 39912 q^{11} − 16730 q^{12}
− 112264 q^{13} − 327000 q^{14} − 493408 q^{15} − 667184 q^{16} − 937824 q^{17}
− 1361376 q^{18} − 1649208 q^{19} − 2705028 q^{20} + ….

Some combination of U_{17,8}, V_{17,8} and Z is a cusp form. We find

(2.5.17)  C := 262387 U_{17,8}/ 2 − 17 V_{17,8}/ 2 + 59901794 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 = 131185 B_{1} + 495087737 B_{2} + 4024494968 B_{3} − 52619639118 B_{4}
+ 389220577648 B_{5} − 2898533428798 B_{6} + 6549413846776 B_{7}
− 39849989479818 B_{8} + 31228607702837 B_{9} − 169818268113767 B_{10}.
p = 19

In this case \frac{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}):

(2.5.19)  [η(19τ)η^{5}(τ)]^{3} = q^{3} + …

and

(2.5.20)  [η(τ)η^{5}(19τ)]^{3} = q^{12} + ….
p = 23

In this case r = \frac{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^{11} + ….

3. 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.

SOME CONGRUENCES FOR PARTITIONS THAT ARE p-CORES 27

p = 5, 7, 19, 23

Let r be a prime divisor of \frac{1}{2}(p − 1). In this first case ϵ_{p} = −1 and our goal is to prove

(3.1)  a_{p}(n − δ_{p}) ≡ 0 (mod r)  if (\frac{n}{p}) = −1 and n ≢ 0 (mod r),

Φ_{r} ∘ Ψ_{−}(X_{p}) ≡ 0 (mod r). i.e. We recall from Corollary (2.2.12) that

(3.2)  C := c_{p}X_{p} − U_{p, \frac{1}{2} (p−1)} ∈ S_{\frac{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 gcd (c_{p}, r) = 1, (3.3a) (r − 1) | (\frac{1}{2}(p − 1) − 1). (3.3b) Hence, from Lemma (2.3.23), we have

(3.4)  Φ_{r} ∘ Ψ_{−}(U_{p, \frac{1}{2} (p−1)}) ≡ 0 (mod r).

We need the existence of a special basis for S_{\frac{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_{\frac{1}{2} (p−1)}(p, χ_{p}), and r is a prime divisor of \frac{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 (\frac{i}{p}) = 1 then D_{i} ∈ S_{\frac{1}{2} (p−1),p}^{+}, (iii) If (\frac{i}{p}) = −1 and i ≢ 0 (mod r) then D_{i} ∈ S_{\frac{1}{2} (p−1),p}^{−}, (iv) If (\frac{i}{p}) = −1 and i ≡ 0 (mod r) then Φ_{r}(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).

For p = 19, d = 13 and r = 3. If 1 ≤ i ≤ 13, (\frac{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, \frac{1}{2} (p−1)} = ∑_{i=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))

(3.7)  α_{i} ≡ 0 (mod r)  if (\frac{i}{p}) = −1 and i ≢ 0 (mod r).

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

In this case r = 2 or 3 and ϵ_{p} = 1. Our goal is to prove

(3.8a)  a_{13}(n − 7) ≡ 0 (mod 2)  if (\frac{n}{13}) = 1 and n is odd,

(3.8b)  a_{13}(n − 7) ≡ 0 (mod 3)  if (\frac{n}{13}) = 1 and n ≢ 0 (mod 3);
i.e.  Φ_{r} ∘ Ψ_{+}(X_{13}) ≡ 0 (mod r) for r = 2 or 3.

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} = \frac{η^{6}(13τ)}{η^{6}(τ)}(B_{3} + B_{4}) ≡ [\frac{η^{2}(13τ)}{η^{2}(τ)}C(τ)]^{3}  (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} = (1029 + 10780 i) B_{2} + (33187 + 104426 i) B_{3}
+ (176300 + 380644 i) B_{4} + (349648 + 524472 i) B_{5},

so that

(3.14)  D^{2} ≡ B_{2} + B_{3}  (mod 2).

But

(3.15)  X_{13} + B_{6} = \frac{η^{8}(13τ)}{η^{8}(τ)}(B_{2} + B_{3}) ≡ [\frac{η^{4}(13τ)}{η^{4}(τ)}D(τ)]^{2}  (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

∑_{n≥0} b_{n}q^{n} = ∏_{m=1}^{∞} \frac{(1 − q^{m})}{(1 − q^{13m})} + q ∏_{m=1}^{∞} \frac{(1 − q^{13m})}{(1 − q^{m})},

and r = 2 or 3 then

b_{n} ≡ 0  (mod r)  if n ≢ 0 (mod r).
p = 11, 17

For p = 11, r = 5 and ϵ_{p} = 1, and our goal is to prove

(3.17)  a_{11}(n − 5) ≡ 0 (mod 5)  if (\frac{n}{11}) = 1 and n ≢ 0 (mod 5).

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 ϵ_{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 (\frac{n}{17}) = −1 and n ≢ 0 (mod 4).

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} = (\frac{·}{p}) for some Dirichlet character χ, so that W^{(p−1)/2} ∈ M_{\frac{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)  d_{ϵ} := dim S_{\frac{1}{2} (p−1),p}^{ϵ_{p}},
(3.20)  d := dim S_{\frac{1}{2} (p−1)}(p, χ_{p}).

We can easily calculate d_{ϵ} using Corollary (2.4.14).

\begin{matrix}p & ϵ_{p} & d_{ϵ} & d \\ 11 & 1 & 2 & 3 \\ 17 & −1 & 5 & 10 \end{matrix}

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_{\frac{1}{2} (p−1)}(p, χ_{p}) so that

(3.21)  S_{\frac{1}{2} (p−1)}(p, χ_{p}) = S′ ⊕ S_{\frac{1}{2} (p−1),p}^{−ϵ_{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_{\frac{1}{2} (p−1),p}^{−ϵ} = \{0\}. This can be done by showing the Ψ_{ϵ_{p}}(C) is linearly independent. For p = 11 this could (in principle) be done by hand. We give some details. We find

(3.22)  C = 16 (−713 q − 1950 q^{2} − 1091 q^{3} + 292 q^{4} + 3737 q^{5} + 5850 q^{6} + …),

and

(3.23) T_{2}(C) = 2^{5} · 3 · 5^{2} · 13 (− q + 6 q^{2} + 3 q^{3} + 14 q^{4} + …).

Hence

(3.24)  Ψ_{+}(C) = −16 (713 q + 1091 q^{3} + …),
(3.25)  Ψ_{+}(T_{2}(C)) = 2^{5} · 3 · 5^{2} · 13 (− q + 3 q^{3} + …),

which are clearly linearly independent. Since Ψ_{+} annihilates S_{5,11}^{−}, 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)  15808 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)  T_{11}^{α}(F) = 11^{α} (T_{3} − 2 T_{2} + 4)^{α}(F)

and

(3.30)  T_{11}^{α}(F) ≡ 0 (mod 11^{α}).

From Lemma (2.3.17) we have

(3.31)  T_{11}^{α}(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 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. The methods of this section seem inadequate.

4. The Proof of (1.11)

In this section we prove

(4.1)  a_{17}(n − 12) ≡ 0 (mod 8) if (\frac{n}{17}) = −1 and n ≢ 0 (mod 2).

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} b_{n}q^{n} := \frac{η^{41}(17τ)}{η(τ)} ∈ S_{20}(17, χ_{17})

by Proposition (2.2.9). Now

(4.3)  B(τ) = η^{24}(τ)\frac{η^{17}(17τ)}{η(τ)}

= q^{17} ∏_{m=1}^{∞}(1 − q^{17m})^{24} ∑_{n≥12} a_{17}(n − 12)q^{n}

≡ ∏_{m=1}^{∞}(1 − q^{34m})^{12} ∑_{n≥12} a_{17}(n − 12)q^{n+17} (mod 8),

since (1 − x)^{24} ≡ (1 − x^{2})^{12} (mod 8). It follows that (4.1) is equivalent to

(4.4)  b_{n} ≡ 0 (mod 8) if (\frac{n}{17}) = −1 and n ≡ 0 (mod 2),

i.e.

(4.5)  b_{2n} ≡ 0 (mod 8) if (\frac{n}{17}) = −1,

since (\frac{2}{17}) = 1. Now we consider the Hecke operator T_{2} 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} + (\frac{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)  B_{0} := \frac{η^{17}(τ)}{η(17τ)} = 1 + …,
(4.9)  B_{12} := X_{17} = \frac{η^{17}(17τ)}{η(τ)} = q^{12} + ….

We note that there is no form in M_{8}(17, χ_{17}) with a q-expansion of the form

q^{11} + …,

since dim M_{8}(17, χ_{17}) = 10 + 2 = 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 = 1 + 34 q^{13} + 102 q^{15} + 170q^{16} + ….

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)  C_{j} = q^{j} + β_{j} q^{28} + O(q^{30})  (for 1 ≤ j ≤ 27 and β_{j} ∈ ℤ),
(4.12c)  C_{29} = q^{29} + O(q^{30}).

We define

(4.13)  Δ(τ) := η^{24}(τ) = q + ….

As is well-known Δ ∈ S_{12}(Γ(1)). It follows, from [Kob, Prop. 17 p.127], that

(4.14)  Δ_{17} := Δ(17τ) = q^{17} + ⋯ ∈ S_{12}(17, χ_{0}).

In addition Δ_{17} has no zeros in H. If C ∈ S_{20}(17, χ_{17}) and ν_{∞}(C) ≥ 17 then Δ_{17}^{−1}C ∈ M_{8}(17, χ_{17}). It follows that there is no cusp form in S_{20}(17, χ_{17}) with ν_{∞}(C) = 17+11 = 28 and that for C ∈ S_{20}(17, χ_{17}), C ≠ 0, we have

(4.15)  ν_{∞}(C) ≤ 29.

We take

(4.16)  C_{29} := B = \frac{η^{41}(17τ)}{η(τ)},
(4.17)  C_{17} := Δ_{17}D,

where D is the form in M_{8}(17, χ_{17}) described before so that

(4.18)  C_{17} = q^{17} + 34 q^{30} + 102 q^{32} + 170 q^{33} + ….

We next construct C′_{i} ∈ S_{20}(17, χ_{17}), 1 ≤ i ≤ 27 i ≠ 17, with

C′_{i} = q^{i} + ⋯ ∈ ℤ[[q]],

as in Proposition (2.4.13). We simply set

C′_{i} = C_{29}f_{2}^{k}f_{3}^{ℓ},

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

ρ(C_{17}) = γ q^{17} + 34 q^{30} + …,

for some γ ∈ ℂ. Hence

ρ(C_{17}) − γ C_{17} = O(q^{30}) ∈ S_{20}(17, χ_{17}).

From (4.15) it follows that

ρ(C_{17}) = γ C_{17} = C_{17}

and C_{17} ∈ S_{20,17}^{+}.

Propostion (4.19).

(a) \{C_{j} : 1 ≤ j ≤ 26 and (\frac{j}{17}) = 1 or j = 17\} is a basis for S_{20,17}^{+}.

(b) For certain constants γ_{j},

\{C_{j} − γ_{j}C_{17} : 3 ≤ j ≤ 29 j ≠ 28 and (\frac{j}{17}) = −1\} is

a basis for S_{20,8}^{−}.

Proof. Suppose 1 ≤ j ≤ 26 and (\frac{j}{17}) = 1. We know

C_{j} = q^{j} + β_{j} q^{28} + O(q^{30})

for some β_{j} ∈ ℤ, so that

ρ(C_{j}) = q^{j} + 2γ_{j} q^{17} − β_{j} q^{28} + O(q^{30})

for some γ_{j} ∈ ℂ. We have

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},

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 (\frac{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

ρ(C_{j} − γ_{j}C_{17}) = −C_{j} + γ_{j}C_{17},

so that

C_{j} − γ_{j}C_{17} ∈ S_{20,17}^{−}.

The result follows from (2.3.13) since dim S_{20}(17, χ_{17}) = 28. □

We need to verify (4.1) for n ≤ 41 (i.e. n = 23, 27, 29, 31, 37, 39, 41).Via MAPLE we have

X_{17} := q^{12} + q^{13} + 2 q^{14} + 3 q^{15} + 5 q^{16} + 7 q^{17} + 11 q^{18} + 15 q^{19} + 22 q^{20} + 30 q^{21} (4.20)
+ 42 q^{22} + 56 q^{23} + 77 q^{24} + 101 q^{25} + 135 q^{26} + 176 q^{27} + 231 q^{28} + 280 q^{29}
+ 368 q^{30} + 456 q^{31} + 576 q^{32} + 707 q^{33} + 883 q^{34} + 1068 q^{35} + 1320 q^{36}
+ 1584 q^{37} + 1926 q^{38} + 2296 q^{39} + 2766 q^{40} + 3256 q^{41} + 3887 q^{42}
+ 4547 q^{43} + 5357 q^{44} + 6216 q^{45} + 7380 q^{46} + …
≡ q^{12} + q^{13} + 2 q^{14} + 3 q^{15} + 5 q^{16} + 7 q^{17} + 3 q^{18} + 7 q^{19} + 6 q^{20} + 6 q^{21}
+ 2 q^{22} + 5 q^{24} + 5 q^{25} + 7 q^{26} + 7 q^{28} + 3 q^{33} + 3 q^{34} + 4 q^{35}
+ 6 q^{38} + 6 q^{40} + 7 q^{42} + 3 q^{43} + 5 q^{44} + 4 q^{46} + … (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) = ∑_{\begin{matrix}j=15 \\ j≠28 \end{matrix}}^{29} α_{j}C_{j}.

By Proposition (4.19) we have

(4.22)  Ψ_{−}(T_{2}(B)) = ∑_{\begin{matrix}j=20 \\ (\frac{j}{17})=−1 \\ j≠28 \end{matrix}}^{29} α_{j}Ψ_{−}(C_{j}).

Let 20 ≤ j ≤ 29, (\frac{j}{17}) = −1 and j ≠ 28. From (4.3), (4.6), (4.12) and (4.20) it follows that α_{j} ≡ 0 (mod 8) and we have (4.7) as required. □

5. Further results

There are some congruences for b_{p}(n), defined in (1.13), due to Morris Newman. If n is a nonnegative integer, define p_{s}(n) as the coefficient of q^{n} in

(5.1)  ∏_{n=1}^{∞}(1 − q^{n})^{s};

otherwise define p_{s}(n) as 0. Newman [N1], [N2] has shown the following result:

Suppose tha s is even, 0 < s ≤ 24. Let r be a prime such that δ := s(r − 1)/24 is an integer. Then for all integral n

(5.2)  p_{s}(rn + δ) = p_{s}(n)p_{s}(δ) − r^{\frac{1}{2} s−1}p_{s}((n − δ)/r).

For p prime we have b_{p}(n) ≡ p_{p−1}(n) (mod p). Hence we have the following result:

Suppose p is prime, 3 ≤ p ≤ 23. Let r be a prime such that θ := (p − 1)(r − 1)/24 is an integer. Then for all integral n

(5.3)  b_{p}(rn + θ) ≡ b_{p}(n)b_{p}(θ) − r^{(p−3)/2}b_{p}((n − θ)/r) (mod p).

A nice instance is when p = r = 13.

(5.4)  b_{13}(13n + 6) ≡ 11 b_{13}(n) (mod 13).

We have found a weak analog of (5.3) for a_{p}(n). It seems that the following holds: Suppose p is prime with 5 ≤ p ≤ 23, and let r be any prime. Then for all integral n

(5.5) a_{p}(prn − δ_{p}) + δ_{p}a_{p}(pr − δ_{p})a_{p}(pn − δ_{p}) + r^{(p−2)}a_{p}(p(n/r) − δ_{p}) ≡ 0 (mod p).

We note that for 5 ≤ p ≤ 11 (5.5) follows trivially from (1.6)–(1.8). Although we have not carried out the details, it would seem likely that T_{p}(X_{p}) is congruent to a Hecke eigenform modulo p. A nice consequence of (5.5) is

For p prime, 5 ≤ p ≤ 23 and all integral n

(5.6)  a_{p}(p^{2}n − δ_{p}) ≡ −δ_{p}a_{p}(p^{2} − δ_{p})a_{p}(pn − δ_{p}) (mod p).

Acknowledgments

I would like to thank the following for helpful discussions: Oliver Atkin, Winfried Kohnen, Morris Newman, J.B. Olsson and S.S. Rangachari.

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