p-ADIC PROPERTIES OF COEFFICIENTS OF WEAKLY HOLOMORPHIC MODULAR FORMS

DARRIN DOUD, PAUL JENKINS

Abstract. We examine the Fourier coefficients of modular forms in a canonical basis for the spaces of weakly holomorphic modular forms of weights 4, 6, 8, 10, and 14, and show that these coefficients are often highly divisible by the primes 2, 3, and 5.

1. Introduction

Let Δ(τ) be the unique normalized cusp form of weight 12 for SL_{2}(ℤ) given by

Δ(τ) = q ∏_{n=1}^{∞}(1 − q^{n})^{24} = ∑_{n=1}^{∞} τ(n)q^{n}.

Here, as is traditional, τ is a complex number in the upper half plane, q = e^{2πiτ}, and τ(n) is Ramanujan’s tau-function. Let

j(τ) = q^{−1} + 744 + 196884q + … = q^{−1} + 744 + ∑_{n≥1} c(n)q^{n}

be the standard modular j-function, a weakly holomorphic modular form of weight 0 for SL_{2}(ℤ).

The arithmetic properties of the Fourier coefficients τ(n) of Δ and c(n) of j have been studied extensively. For instance, Ramanujan [12] proved that

τ(2n) ≡ 0 (mod 2), τ(3n) ≡ 0 (mod 3), τ(5n) ≡ 0 (mod 5),

and Lehner [9, 10] proved that

c(2^{a}3^{b}5^{c}7^{d}n) ≡ 0 (mod 2^{3a+8}3^{2b+3}5^{c+1}7^{d}).

In fact, Lehner’s results are much more general; he proved similar congruences for the coefficients of many weakly holomorphic modular functions on Γ_{0}(p). In this paper we will prove such divisibility results for the Fourier coefficients of a large class of weakly holomorphic modular forms of positive integer weight on SL_{2}(ℤ).

Recall that a weakly holomorphic modular form of weight k ∈ 2ℤ satisfies the modular equation

f (\frac{aτ + b}{cτ + d}) = (cτ + d)^{k}f(τ)

for all ( \begin{matrix}a & b \\ c & d \end{matrix}) in some finite index subgroup Γ of SL_{2}(ℤ) and is holomorphic on the upper half plane, but may have poles at any or all of the cusps of Γ. We denote the space of holomorphic modular forms by M_{k}(N) and denote by M_{k}^{!}(N) the space of weakly holomorphic modular forms on Γ_{0}(N) = \{( \begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL_{2}(ℤ) : c ≡ 0 (mod N)\}. When N = 1 we write simply M_{k} and M_{k}^{!}, and note that any form f ∈ M_{k}^{!} is uniquely determined by its Fourier expansion ∑_{n≥n_{0}} a(n)q^{n}, where n_{0} ≥ 0 if f ∈ M_{k}.

Date: October 29, 2009.

Because SL_{2}(ℤ) has only one cusp (at ∞), there is a very nice canonical basis for the space M_{k}^{!}, indexed by the order of the pole at ∞. To define this basis (as in [5]) write k = 12ℓ + k′, where ℓ ∈ ℤ and k′ ∈ {0, 4, 6, 8, 10, 14}, so that if ℓ ≥ 0, the space of cusp forms of weight k has dimension ℓ. For every integer m ≥ −ℓ, there exists a unique weakly holomorphic modular form f_{k,m} ∈ M_{k}^{!} with a q-expansion of the form

f_{k,m}(τ) = q^{−m} + O(q^{ℓ+1});

together these f_{k,m} form a basis for M_{k}^{!}. It is straightforward to see that any modular form f = ∑ a(n)q^{n} in M_{k}^{!}, if its first few Fourier coefficients are known, may easily be written in terms of these basis elements as

f = ∑_{n_{0}≤n≤ℓ} a(n)f_{k,−n}.

The basis elements f_{k,m} may be directly constructed from Δ, j, and the Eisenstein series E_{k′}, where we let E_{0} = 1. From the standard valence formula, we know that

(1)  ord_{∞}(f) ≤ ℓ

for all f ∈ M_{k}^{!}. It is also clear that the form Δ^{ℓ}E_{k′} has order ℓ at ∞, so it must be f_{k,−ℓ}; it is unique because the difference of any two such forms will have a Fourier expansion that is O(q^{ℓ+1}), which must be zero by (1). We can then construct the f_{k,m} iteratively by multiplying f_{k,m−1} by j(τ) ∈ M_{0}^{!} to get a form of weight k with Fourier expansion beginning q^{−m}, and then subtracting appropriate integer multiples of f_{k,i} (where −ℓ < i < m) to eliminate the q^{−i} terms in the Fourier expansion. This construction shows that

f_{k,m} = Δ^{ℓ}E_{k′}F_{k,D}(j),

where F_{k,D}(x) is a monic polynomial in x of degree D = ℓ + m with integer coefficients. We define the Fourier coefficients a_{k}(m, n) of these basis elements by

f_{k,m}(τ) = q^{−m} + ∑_{n} a_{k}(m, n)q^{n},

noting that a_{k}(m, n) = 0 when m < −ℓ or n ≤ ℓ or when m or n are not integers. Since each of E_{k′}, Δ, j, and F_{k,m} has integer coefficients, it follows that a_{k}(m, n) ∈ ℤ.

These basis elements are studied extensively in [5]; for instance, for basis elements which are not cusp forms, the zeros in the fundamental domain are all shown to lie on the unit circle. Additionally, the following theorem is proved.

Theorem 1.1 ([5], Theorem 2). For any even integer k, the basis elements f_{k,m} satisfy the generating function

∑_{m≥−ℓ} f_{k,m}(z)q^{m} = \frac{f_{k,−ℓ}(z)f_{2−k,1+ℓ}(τ)}{j(τ) − j(z)}.

Replacing k with 2 − k and switching τ and z, we find the following beautiful duality of Fourier coefficients.

Corollary 1.2 ([5], Corollary 1). For any even integer k and any integers m, n, the equality

a_{k}(m, n) = −a_{2−k}(n, m)

holds for the Fourier coefficients of the modular forms f_{k,m} and f_{2−k,n}.

We note that these bases for M_{k}^{!} closely parallel those of half integral weight defined by Zagier in his work on traces of singular moduli [17]. For the weights 1/2 and 3/2, Zagier constructed modular forms on Γ_{0}(4) satisfying a plus space condition and with Fourier expansions

\begin{matrix}f_{d} = q^{−d} + O(q) ∈ M_{1/2}^{!} & (0 ≤ d ≡ 0, 1 (mod 4)), \\ g_{D} = q^{−D} + O(1) ∈ M_{3/2}^{!} & (0 ≤ D ≡ 0, 3 (mod 4)). \end{matrix}

These modular forms have a generating function and satisfy a duality theorem very similar to Theorem 1.1 and Corollary 1.2, and bases with similar properties for all half integral weights are constructed in [4].

The Fourier coefficients of Zagier’s f_{d} and g_{D} can be interpreted as traces and twisted traces of singular moduli, and have been widely studied. For instance, Ahlgren and Ono [1] proved many congruences for these traces (and their associated half integral weight Fourier coefficients) modulo p^{s}, and gave an elementary argument that if p splits in ℚ(\sqrt{−d}), then Tr(p^{2}d) is congruent to 0 (mod p). Edixhoven [6] extended their observation, proving that if (\frac{−d}{p}) = 1, then Tr(p^{2n}d) ≡ 0 (mod p^{n}). An elementary proof of this result using Hecke operators was given by the second author in [8], and Boylan [3] exactly computed Tr(2^{2n}d), obtaining even stronger congruences and divisibility results for p = 2.

With these congruences for Fourier coefficients of forms of half integral weight as a model, it is natural to ask whether similar divisibility results exist for the Fourier coefficients a_{k}(m, n) of the integral weight basis elements f_{k,m}. A result similar to Edixhoven’s appears in [5], where it is proved that for the weights k = 4, 6, 8, 10, and 14, if p^{r}|n and p ∤ m, it is true that p^{(k−1)r}|a_{k}(m, n). Thus, if (m, n) = 1, we have n^{k−1}|a_{k}(m, n). This theorem seems to be sharp if p > 7. However, looking at the divisibility of these Fourier coefficients by smaller primes, it appears that the a_{k}(m, n) are divisible by higher powers of these primes. For instance, certain coefficients of

f_{4,1}(τ) = E_{4}(τ)(j(τ) − 984) = q^{−1} + ∑_{n≥1} a_{4}(1, n)q^{n}

factor in the following way.

\begin{aligned}a_{4}(1, 2) & = & 2^{10} · 5 · 13327 \\ a_{4}(1, 4) & = & 2^{13} · 3^{2} · 13 · 113 · 2543 \\ a_{4}(1, 5) & = & 2^{3} · 5^{4} · 19 · 9931 · 7639 \\ a_{4}(1, 8) & = & 2^{16} · 3 · 5^{2} · 293 · 15918317 \\ a_{4}(1, 10) & = & 2^{11} · 3^{6} · 5^{4} · 2184176461 \\ a_{4}(1, 20) & = & 2^{14} · 5^{4} · 29243 · 235531684534847 \\ a_{4}(1, 25) & = & 2^{2} · 3^{2} · 5^{7} · 11491 · 102481 · 4609259 · 4679867 \end{aligned}

While Theorem 3 in [5] predicts that if p^{r}∥n, then p^{3r}|a_{4}(1, n), we see that in fact there are extra powers of 2 and 5 dividing these coefficients. For instance, we expect to see that 2^{3}|a_{4}(1, 2) and 5^{3}|a_{4}(1, 5), while in fact the stronger divisibility results 2^{10}|a_{4}(1, 2) and 5^{4}|a_{4}(1, 5) are actually true. Further computation reveals similar divisibility by small primes for other modular forms of these weights.

In this paper, we will prove the following theorem making these divisibility results more explicit. For an integer N, let v_{p}(N) be the p-adic valuation of N, or the largest integer s such that p^{s}|N.

Theorem 1.3. Let k ∈ {4, 6, 8, 10, 14} and let p ∈ {2, 3, 5}. Then for

ϵ_{k,p} = \begin{cases}7 & if p = 2, k = 4, 6, 14, \\ 8 & if p = 2, k = 8, 10, \\ 2 & if p = 3, k = 4, 10, \\ 3 & if p = 3, k = 6, 8, 14, \\ 1 & if p = 5, \end{cases}

we have for all m, n > 0,

v_{p}(a_{k}(m, n)) ≥ \begin{cases}ϵ_{k,p} & if v_{p}(m) > v_{p}(n), \\ (v_{p}(n) − v_{p}(m))(k − 1) + ϵ_{k,p} & if v_{p}(n) > v_{p}(m). \end{cases}

We note that the theorem makes no prediction about divisibility if v_{p}(m) = v_{p}(n). If (m, 30) = 1, the theorem implies that for a, b, c > 0,

a_{k}(m, 2^{a}3^{b}5^{c}n) ≡ 0 (mod 2^{(k−1)a+7}3^{(k−1)b+2}5^{(k−1)c+1}).

Because any weakly holomorphic modular form of weight k ∈ {4, 6, 8, 10, 14} is a linear combination of the f_{k,m}(τ) and the Eisenstein series E_{k}(τ), we can use this result to derive divisibility results for many coefficients of more general weakly holomorphic modular forms. For instance, if

f(τ) = c_{−2}q^{−2} + c_{−1}q^{−1} + ∑_{n>0} c_{n}q^{n} = c_{−2}f_{4,2}(τ) + c_{−1}f_{4,1}(τ)

is any modular form of weight 4 with integer coefficients, constant coefficient 0, and a pole of order 2 at the cusp, we can deduce easily that for 3|n, v_{3}(c_{n}) ≥ 3v_{3}(n) + 2.

2. Definitions

In this section, we define various modular forms and operators that will be used throughout the paper.

2.1. Modular forms derived from Eisenstein series. We define the standard normalized Eisenstein series of even weight k ≥ 4 as

E_{k}(τ) = 1 + A_{k} ∑_{n=1}^{∞} σ_{k−1}(n)q^{n},

where A_{k} = −2k/B_{k} and B_{k} is the kth Bernoulli number. In addition, we will need the forms

S_{k,p}(τ) = \frac{E_{k}(τ) − E_{k}(pτ)}{A_{k}}, and T_{k,p}(τ) = \frac{p^{k}E_{k}(pτ) − E_{k}(τ)}{p^{k} − 1}.

For k ∈ {4, 6, 8, 10, 14} and any p, it is well known that A_{k} is an integer and it is easily seen that S_{k,p} has integral Fourier coefficients. For k ∈ {4, 6} and p = 2 and for k = 4 and p = 3, one checks easily that T_{k,p} has integer coefficients. Often, when p is clear from context, we will write S_{k} and T_{k} in place of S_{k,p} and T_{k,p}. Note that S_{k,p} and T_{k,p} are modular forms of weight k and level p, with S_{k,p} vanishing at ∞ and T_{k,p} vanishing at 0.

In weight 2, we will need the Eisenstein series E_{2}(τ) = 1 − 24 ∑_{n=1}^{∞} σ_{1}(n)q^{n}, which, although not modular itself, is used to produce the modular form

\frac{E_{2}(τ) − pE_{2}(pτ)}{1 − p}

of weight 2 and level p [14, p. 88].

2.2. Newforms. In level 2, we use the following notation for the normalized newforms of weight 8 and 10:

Ξ_{8}(τ) = (η(τ)η(2τ))^{8} and Ξ_{10}(τ) = S_{4,2}(τ)T_{6,2}(τ).

In level 3, we use the following notation for the normalized newform of weight 6:

Ω_{6}(τ) = (η(τ)η(3τ))^{6}.

In level 5, we use the following notation for the normalized newforms of weights 4 and 6:

Λ_{4}(τ) = (η(τ)η(5τ))^{4} and Λ_{6}(τ) = \frac{5E_{2}(5τ) − E_{2}(τ)}{4}Λ_{4}(τ).

These newforms can all be found in William Stein’s online tables of modular forms [15], and we check there that the sign of the Fricke involution is 1 for Ξ_{8} and Λ_{4} and −1 for Ξ_{10}, Ω_{6}, and Λ_{6}.

2.3. Weakly holomorphic forms of weight 0. Following Apostol [2, pg. 87], for p ∈ {2, 3, 5} we define λ = λ_{p} = 24/(p − 1),

Φ(τ) = Φ_{p}(τ) = (\frac{η(pτ)}{η(τ)})^{λ} ,

and ψ(τ) = 1/Φ(τ). Although λ, Φ, and ψ depend on p, we often omit this dependence from the notation, since it will be clear from context.

We recall from [2] that both ψ and Φ are in M_{0}^{!}(p), both have integer Fourier coefficients, Φ has a zero at ∞ and a pole at 0, ψ has a pole at ∞ and a zero at 0, and

ψ (\frac{−1}{pτ}) = p^{λ/2}Φ(τ).

2.4. Operators on modular forms. For p a prime, and f(τ) = ∑_{n≥n_{0}} a_{n}q^{n} a weakly holomorphic modular form, we define the U_{p} operator by

(f|U_{p})(τ) = ∑_{n≥n_{0}/p} a_{pn}q^{n}.

An alternative definition of f|U_{p} that we will find useful is [2, Thm. 4.5]

(f|U_{p})(τ) = \frac{1}{p} ∑_{j=0}^{p−1} f (\frac{τ + j}{p} ) .

If f has weight k and level N, then f|U_{p} is again a weakly holomorphic modular form of weight k and level N (if p|N) or level pN (if p ∤ N) [11, Prop. 2.22].

We also define the operators V_{p} by (f|V_{p})(τ) = f(pτ) and the Hecke operators T_{p} by (f|T_{p}) = (f|U_{p}) + p^{k−1}(f|V_{p}). Note that V_{p} takes modular forms of weight k and level N to forms of weight k and level Np, while T_{p} preserves both weight and level [11, Prop. 2.2, Thm. 4.5].

3. Reduction to the case where p|m, (n, p) = 1

In this section, we will prove that if Theorem 1.3 is true for the special case of Fourier coefficients a_{k}(m, n) with p|m and (n, p) = 1, then it is true in all cases. We begin by citing a result of Duke and Jenkins relating certain Fourier coefficients, from which the divisibility result of [5] cited in the introduction follows immediately.

Proposition 3.1 ([5], Lemma 1). Let p be a prime and k ∈ {4, 6, 8, 10, 14}. Then for m, n, s ∈ ℤ, with m, n, s > 0 and p a prime,

a_{k}(m, np^{s}) = p^{s(k−1)} (a_{k}(mp^{s}, n) − a_{k}(mp^{s−1}, n/p)) + a_{k}(m/p, np^{s−1}).

Applying induction to this proposition, we obtain the following.

Corollary 3.2. Let (m, p) = (n, p) = 1, r, s ≥ 0, k ∈ {4, 6, 8, 10, 14}. Then for 0 ≤ t ≤ min(r, s − 1),

a_{k}(mp^{r}, np^{s}) = a_{k}(mp^{r−t−1}, np^{s−t−1}) + ∑_{j=0}^{t} p^{(s−j)(k−1)}a_{k}(mp^{r+s−2j}, n).

Applying this corollary, we obtain the following reduction.

Theorem 3.3. Let p ∈ {2, 3, 5} and let k ∈ {4, 6, 8, 10, 12}. Assume that v_{p}(a_{k}(a, b)) ≥ ϵ_{k,p} for all a, b > 0 having p|a and p ∤ b. Let m, n > 0 be relatively prime to p and r, s ≥ 0. Then if r > s,

v_{p}(a_{k}(mp^{r}, np^{s})) ≥ ϵ_{k,p},

and if r < s,

v_{p}(a_{k}(mp^{r}, np^{s})) ≥ (s − r)(k − 1) + ϵ_{k,p}.

Proof. For r > s, we apply Corollary 3.2 with t = s − 1 to a_{k}(mp^{r}, np^{s}), and note that each term obtained is divisible by p^{ϵ_{k,p}}.

For r < s, we apply Corollary 3.2 with t = r. In this case, we note that the term outside the sum vanishes, and each term inside the sum is divisible by p^{(s−r)(k−1)+ϵ_{k,p}}. □

Hence, we see that in order to prove Theorem 1.3, we need only prove it for a_{k}(m, n) when p|m and p ∤ n.

4. Poles at zero of weakly holomorphic modular forms

In this section, we derive a formula which will allow us to compute the Fourier expansion at 0 of f|U_{p}, where f is a weakly holomorphic modular form for which we know the Fourier expansion at ∞. This will allow us to use the expansion at 0 of f|U_{p} to obtain information about the Fourier coefficients in the expansion of f at ∞.

Lemma 4.1. Let f be a meromorphic modular form of weight k on SL_{2}(ℤ), and let f_{p} = f|U_{p}. Then

pτ^{−k}f_{p}(−1/τ) = −f(τ/p) + pf_{p}(τ) + p^{k}f(pτ).

Note the similarity of the lemma to Theorem 4.6 of [2].

Proof. For an integer j with 1 ≤ j ≤ p − 1, we will denote by j′ the unique integer with −(p − 1) ≤ j ≤ −1 such that jj′ ≡ 1 (mod p), and we will write b_{j} = (jj′ − 1)/p. We note that

pτ^{−k}f_{p}(−1/τ) = pτ^{−k} \frac{1}{p} ∑_{j=0}^{p−1} f (\frac{−1/τ + j}{p} )

= τ^{−k} ∑_{j=0}^{p−1} f (\frac{jτ − 1}{pτ})

= τ^{−k} ∑_{j=1}^{p−1} f ((\begin{matrix}j & b \\ p & j′ \end{matrix}) (\begin{matrix}1 & −j′ \\ 0 & p \end{matrix}) τ) + τ^{−k}f(−1/(pτ))

= τ^{−k} ∑_{j=1}^{p−1} (p (\frac{τ − j′}{p} ) + j′)^{k} f (\frac{τ − j′}{p} ) + τ^{−k}(pτ)^{k}f(pτ))

= −f(τ/p) + ∑_{j=0}^{∞} f (\frac{τ + j}{p} ) + p^{k}f(pτ)

= −f(τ/p) + pf_{p}(τ) + p^{k}f(pτ).

□

Corollary 4.2. Let f be a meromorphic modular form of weight k on Γ, and let f_{p} = f|U_{p}. Then

p(pτ)^{−k}f_{p}(−1/pt) = −f(τ) + pf_{p}(pτ) + p^{k}f(p^{2}τ).

Further, p(pτ)^{−k}f_{p}(−1/pt) is modular of weight k and level p.

Proof. The equality follows immediately by replacing τ by pτ in Lemma 4.1. The statement about the weight and level follows from the fact that pf_{p}(pτ) + p^{k}f(p^{2}τ) = p((f|T_{p})|V_{p}), along with the fact that T_{p} preserves weight and level, while V_{p} raises levels by a factor of p. □

5. Integral bases for spaces of modular forms

As seen in the previous section, modular forms on Γ_{0}(p) can be used to study associated modular forms on SL_{2}(ℤ). Therefore, for p = 2, 3, 5 we now construct integral bases for the spaces M_{k}(p), or bases having the property that the modular forms in M_{k}(p) with integer coefficients are exactly the integer linear combinations of the basis elements. These bases allow us to study divisibility properties of Fourier coefficients by studying the first several coefficients of a given form.

Lemma 5.1. Let p ∈ {2, 3, 5}, let k ≥ 0 be even, and let d = dim M_{k}(p). Then there is a basis {B_{n,k,p} : 0 ≤ n < d} of M_{k}(p) such that each B_{n,k,p} = q^{n} + O(q^{d}) and each B_{n,k,p} has integer coefficients.

Proof. To construct the basis, for each weight k we will find a modular form f in M_{k}(p) with integer coefficients and leading coefficient 1 that vanishes with order d − 1 at ∞. We can then multiply f by ψ(τ) and subtract off an appropriate integer multiple of f to get a form with q-expansion beginning q^{d−2} + O(q^{d}). Repeating this process of multiplying by ψ and subtracting earlier basis elements, we generate a basis for M_{k}(p) of modular forms with integer coefficients and the desired Fourier expansions.

To construct these modular forms vanishing to order d−1, we note that standard dimension formulas [14, Prop. 6.1] yield the following values for d = dim(M_{k}(p):

dim(M_{k}(2)) = ⌊\frac{k}{4}⌋ + 1, dim(M_{k}(3)) = ⌊\frac{k}{3}⌋ + 1, and dim(M_{k}(5)) = 2 ⌊\frac{k}{4}⌋ + 1.

In level 2, we find that for weight 0, the constant function 1 is the desired modular form, and for weight 2, the form 2E_{2}(2τ) − E_{2}(τ) works. We then note that for k ≥ 0, dim(M_{k+4}(2)) = dim(M_{k}(2)) + 1, so the appropriate form of weight k + 4 can be obtained from the form of weight k by multiplying by S_{4,2}(τ), which is of weight 4 and has a Fourier expansion at ∞ beginning q + O(q^{2}).

For level 3, in weights 0, 2, 4 we have 1, \frac{1}{2}(3E_{2}(3τ)−E_{2}(τ)), S_{4,3} respectively. Additionally, dim(M_{k+6}(3)) = dim(M_{k}(3))+2, so multiplying by a form of weight 6 with Fourier expansion beginning with q^{2} suffices to construct the needed forms of higher weight; the form

\frac{η^{18}(3τ)}{η^{6}(τ)} = Φ_{3}(τ)Ω_{6}(τ) = q^{2} + 6q^{3} + ⋯

works.

For level 5, dim(M_{k+4}(5)) = dim(M_{k}(5)) + 2, and the weight 4 form η^{10}(5τ)η^{−2}(τ) = Φ_{5}(τ)Λ_{4}(τ), along with the constant function 1 and the weight 2 form \frac{1}{4}(5E_{2}(5τ) − E_{2}(τ)), suffice to construct the basis in all weights. □

Note that the set {B_{n,k,p}} does in fact form an integral basis of M_{k}(p), since any form

f(τ) = ∑_{n=0}^{∞} a_{n}q^{n} ∈ M_{k}(p)

with integer coefficients can clearly be written as the integer linear combination

f(τ) = ∑_{n=0}^{d−1} a_{n}B_{k,n,p}(τ).

The importance of this basis for our purposes is that it allows us to check the divisibility of all coefficients of a modular form by checking only finitely many coefficients.

Lemma 5.2. Let k ≥ 0 be even and let p ∈ {2, 3, 5}. Let d = dim M_{k}(p). If F (τ) ∈ M_{k}(p) has integer coefficients and its first d coefficients are divisible by p^{s}, then F (τ) ≡ 0 (mod p^{s}).

Proof. For F (τ) = ∑_{n=0}^{∞} a_{n}q^{n}, we have F (τ) = ∑_{n=0}^{d−1} a_{n}B_{k,n,p}(τ). If each a_{n} with 0 ≤ n ≤ d − 1 is divisible by p^{s}, then clearly every coefficient of F (τ) is divisible by p^{s}. □

6. Weakly holomorphic modular forms of negative weight with minimal poles

Because of the duality of coefficients of basis elements of the weights k and 2 − k (Corollary 1.2), for a fixed n we can study the coefficients a_{k}(mp^{s}, n) of the forms f_{k,mp^{s}} for all positive values of m and s simply by studying the Fourier coefficients of the single negative weight form f_{2−k,n}|U_{p}. To facilitate our study of these forms, we now examine negative weight weakly holomorphic modular forms of level p which are holomorphic except for a pole at exactly one of 0 and ∞. Since there are no holomorphic modular forms of negative weight, we see that in fact such a form must have a pole of order at least one at one of 0 and ∞. In fact, using a valence formula, we will see that in many cases, such a modular form must have a pole of order greater than one at one of the two cusps.

We begin by bounding the order of the pole.

Proposition 6.1. Let p ∈ {2, 3, 5}, and let k < 0 be even. Let f ∈ M_{k}^{!}(p) and suppose that f is holomorphic at 0. Denote by ord_{∞}(f) the order of vanishing of f at ∞ (so if f has a pole, this is negative). Then

ord_{∞}(f) ≤ k (\frac{ν_{2}(p)}{4} + \frac{ν_{3}(p)}{3}) ,

where ν_{2} and ν_{3} are given by [7, p. 535] (see also [13, p. 25]). If equality holds, then f must be nonvanishing in the upper half plane and at 0.

We note that ν_{2}(2) = 1, ν_{2}(3) = 0 and ν_{2}(5) = 2, while ν_{3}(2) = 0, ν_{3}(3) = 1 and ν_{3}(5) = 0. In addition, we note that reversing the roles of 0 and ∞ yields an analogous bound for modular functions holomorphic at ∞ with a pole at 0.

Proof. This follows immediately from the valence formula [7, (3.9)], by noticing that for a weakly holomorphic modular form whose only pole is at ∞, all the orders of vanishing at any point except ∞ must be nonnegative. For equality, it is clear that the order of vanishing at any non-infinite point must be 0. □

For k < 0 and even and p prime, we now define θ_{k,p} to be a weakly holomorphic modular form of weight k and level p that is holomorphic on the upper half plane and at 0, with a pole of minimal possible order at ∞ and leading coefficient 1. We define α_{k,p} in a similar way, but require that it be holomorphic at ∞ and with minimal possible pole at 0.

Note that θ_{k,p} and α_{k,p} are well-defined, since given two such objects, their difference must be a holomorphic modular form of negative weight, hence 0. In addition, for the k and p that we study, (pτ)^{−k}θ_{k,p}(−1/pτ) can easily be shown (using properties of the Fricke involution) to be a multiple of α_{k,p}. It will be convenient to write this relation as θ_{k,p}(−1/pτ) = μ_{k,p}τ^{k}α_{k,p}(τ). We also note that often, α_{k,p} ≡ 1 (mod p^{ξ_{k,p}}) for some value of ξ_{k,p}. In Tables 1, 2, and 3 we define θ_{k,p} and α_{k,p} for k ∈ {−2, −4, −6, −8, −12} and p ∈ {2, 3, 5}, and give the values of μ_{k,p} and ξ_{k,p}.

We remark that in all cases except p = 5, k = −2 or −6, the minimum order of the pole predicted by the valence formula was achieved. For p = 5, k = −2, the valence formula predicts that θ_{k,p} should have a pole of order at least 1 (since k(ν_{2}(5)/4 + ν_{3}(5)/3) = −1). If equality were achieved, then θ_{k,p} would have to be nonvanishing on the upper half plane and at 0, so that 1/θ_{k,p} would be a holomorphic modular form of weight 2 and level 5, vanishing

k  θ_{k,2}  α_{k,2} μ_{k,2} ξ_{k,2}
−2 \frac{Ξ_{10}(τ)}{Δ(2τ)} \frac{Ξ_{10}(τ)}{Δ(τ)} −2^{5} 3
−4 \frac{Ξ_{8}(τ)}{Δ(2τ)} \frac{Ξ_{8}(τ)}{Δ(τ)} 2^{4} 4
−6 \frac{T_{6}(τ)}{Δ(2τ)} \frac{S_{6}(τ)}{Δ(τ)} −2^{9} 3
−8 \frac{T_{4}(τ)}{Δ(2τ)} \frac{S_{4}(τ)}{Δ(τ)} 2^{8} 5
−12 \frac{Δ(τ)}{Δ^{2}(2τ)} \frac{Δ(2τ)}{Δ^{2}(τ)} 2^{12} 4

Table 1. Values of θ_{k,p}, α_{k,p}, μ_{k,p} and ξ_{k,p} for p = 2.

k  θ_{k,3}  α_{k,3}  μ_{k,3} ξ_{k,3}
−2 \frac{S_{4}(τ)Ω_{6}(τ)}{Δ(3τ)} \frac{T_{4}(τ)Ω_{6}(τ)}{Δ(τ)} −3^{2} 1
−4 \frac{S_{4}(τ)T_{4}(τ)}{Δ(3τ)} \frac{S_{4}(τ)T_{4}(τ)}{Δ(τ)} 3^{4} 1
−6  \frac{Ω_{6}(τ)}{Δ(3τ)}  \frac{Ω_{6}(τ)}{Δ(τ)}  −3^{3} 2
−8  \frac{T_{4}(τ)}{Δ(3τ)}  \frac{S_{4}(τ)}{Δ(τ)}  3^{5} 1
−12 \frac{Φ(τ)Δ(τ)}{Δ^{2}(3τ)} \frac{ψ(τ)Δ(3τ)}{Δ^{2}(τ)} 3^{6} 2
Table 2. Values of θ_{k,p},
α_{k,p}, μ_{k,p} and ξ_{k,p} for p = 3.

k  θ_{k,5}  α_{k,5}  μ_{k,5} ξ_{k,5}
−2 \frac{Λ_{4}(τ)Λ_{6}(τ)Φ(τ)}{Δ(5τ)} \frac{Λ_{4}(τ)Λ_{6}(τ)ψ(τ)}{Δ(τ)} −5^{2} 0
−4  \frac{Λ_{4}(τ)^{2}Φ(τ)}{Δ(5τ)}  \frac{Λ_{4}^{2}(τ)ψ(τ)}{Δ(τ)}  5 1
−6  \frac{Λ_{6}(τ)}{Δ(5τ)}  \frac{Λ_{6}(τ)}{Δ(τ)}  −5^{3} 0
−8  \frac{Λ_{4}(τ)}{Δ(5τ)}  \frac{Λ_{4}(τ)}{Δ(τ)}  5^{2} 1
−12  \frac{Φ(τ)Λ_{4}^{3}(τ)}{Δ^{2}(5τ)}  \frac{ψ(τ)Λ_{4}^{3}(τ)}{Δ^{2}(τ)}  5^{3} 1
Table 3. Values of θ_{k,p}, α_{k,p}, μ_{k,p} and ξ_{k,p} for p = 5.

at ∞. Since no such form exists, we see that θ_{−2,5} must have at least a double pole. Similarly, for p = 5 and k = −6 the bound arising from the valence formula indicates that θ_{−6,5} must have a pole of order at least 3. However, if θ_{−6,5} had a pole of order 3, then its reciprocal would be a nonzero holomorphic modular form of weight 6, level 5 that vanishes at ∞ with order 3. The basis computations in the previous section show that such an object cannot exist. Hence, the order of the pole of θ_{−6,5} must be at least 4.

The congruences on α_{k,p} follow from the following lemma.

Lemma 6.2. Let α(τ) = ∑_{n=0}^{∞} a_{n}q^{n} be a weight k weakly holomorphic modular form of level p, with k even and negative. Suppose that α has integer coefficients, is holomorphic at ∞ with a_{0} = 1, and has a pole of order m at 0. If p^{r}|a_{i} for all 1 ≤ i ≤ m and there is some F (τ) ∈ M_{−k}(p) having leading coefficient 1 and integer coefficients with F (τ) ≡ 1 (mod p^{r}), then α(τ) ≡ 1 (mod p^{r}).

Proof. Note that α(τ)F (τ) − 1 = ∑_{n=1}^{∞} b_{n}q^{n} is weakly holomorphic of weight 0 and is holomorphic at ∞, with a pole of order at most m at 0 and vanishing at ∞. As such, it must be an integer linear combination of Φ(τ), …, Φ^{m}(τ). Since F (τ) ≡ 1 (mod p^{r}), we see that each b_{n} ≡ a_{n}, so that each b_{n} is divisible by p^{r}. If we write

α(τ)F (τ) − 1 = ∑_{n=1}^{∞} b_{n}q^{n} = ∑_{n=1}^{m} c_{n}Φ^{n}(τ)

it is clear that c_{1} = b_{1}, and that c_{2} ≡ b_{2} (mod b_{1}), so that c_{2} ≡ b_{2} ≡ 0 (mod p^{r}). Inductively, we find that each c_{n} ≡ 0 (mod p^{r}), so that α(τ)F (τ) − 1 ≡ 0 (mod p^{r}). Since F (τ) ≡ 1 (mod p^{r}), we see that α(τ) ≡ 1 (mod p^{r}). □

We apply this lemma to each of the α_{k,p} with k ∈ {−2, −4, −6, −8, −12}. For k = −4, −6, −8, we use the Eisenstein series of weight −k for F (τ). For k = −12, we set F (τ) = E_{4}(τ)^{3}. For k = −2 and p = 2, 3, we let F (τ) be the unique monic form of weight 2, level p, which is given by [14, p. 88]

\frac{E_{2}(τ) − pE_{2}(pτ)}{1 − p}

and is congruent to 1 mod 8 in level 2, and congruent to 1 mod 3 in level 3. In each case, one checks easily that F (τ) ≡ 1 (mod p^{ξ_{k,p}}), and that the first m coefficients of α_{k,p} are divisible by p^{ξ_{k,p}}. Note that there is no congruence on the coefficients of α_{−2,5} modulo 5.

7. Reduction to finitely many coefficients

To prove Theorem 1.3, it remains to show that for any integer s ≥ 1, the coefficient a_{k}(mp^{s}, n) of the form f_{k,mp^{s}} is sufficiently divisible by p when p ∤ n. To accomplish this, we define

g(τ) = f_{k,mp^{s}}(τ) − f_{k,mp^{s−1}}(pτ) = ∑_{n=1}^{∞} γ_{n}q^{n}.

Note that for p ∤ n, γ_{n} = a_{k}(mp^{s}, n). We will prove that all of the γ_{n} are divisible by p^{ϵ_{k,p}}. In this section, we show that it is actually sufficient to prove that finitely many of the γ_{n} satisfy the desired divisibility.

Lemma 7.1. Let p ∈ {2, 3, 5, 7, 13}, k ∈ {4, 6, 8, 10, 14}, λ = 24/(p − 1), and m > 0. Let g(τ) = f_{k,mp^{s}} − f_{k,mp^{s−1}}(pτ) with s > 0. Then we have that

g(τ) ≡ F (τ) (mod p^{λ/2})

for some holomorphic modular form F ∈ M_{k}(p), where F has integer coefficients and vanishes at ∞.

Proof. We immediately notice that g(τ) ∈ M_{k}^{!}(p) and vanishes at ∞. We are then interested in its behavior at the cusp 0 of Γ_{0}(p).

We have g(t) = f_{k,mp^{s}}(τ) − f_{k,mp^{s−1}}(pτ). Hence,

τ^{−k}g(−1/τ) = τ^{−k} (f_{k,mp^{s}}(−1/τ) − f_{k,mp^{s−1}}(−p/τ))

= τ^{−k} (τ^{k}f_{k,mp^{s}}(τ) − (\frac{τ}{p})^{k} f_{k,mp^{s−1}}(τ/p))

= f_{k,mp^{s}}(τ) − \frac{1}{p^{k}} f_{k,mp^{s−1}}(τ/p).

Replacing τ by pτ, we obtain

(pτ)^{−k}g(−1/(pτ)) = f_{k,mp^{s}}(pτ) − \frac{1}{p^{k}} f_{k,mp^{s−1}}(τ).

= q^{−mp^{s+1}} − \frac{1}{p^{k}} q^{mp^{s−1}} + O(q)

= ∑_{i=1}^{mp^{s+1}} B_{i}ψ(τ)^{i}E_{k}(τ) + h(τ)

where p^{k}B_{i} ∈ ℤ for all i, and where h(τ) is modular on Γ_{0}(p) and holomorphic at both 0 and ∞.

We now replace τ by −1/pτ, to obtain

τ^{k}g(τ) = ∑_{i=1}^{mp^{s+1}} B_{i}ψ(−1/pτ)^{i}E_{k}(−1/pτ) + h(−1/pτ)

= ∑_{i=1}^{mp^{s+1}} B_{i}(p^{λ/2}Φ(τ))^{i}(pτ)^{k}E_{k}(pτ) + h(−1/pτ)

= ∑_{i=1}^{mp^{s+1}}(p^{k}B_{i})(p^{λ/2}Φ(τ))^{i}τ^{k}E_{k}(pτ) + h(−1/pτ)

Dividing by τ^{k}, writing A_{i} = p^{k}B_{i} ∈ ℤ, and setting F (τ) = τ^{−k}h(−1/pτ), we see that

g(τ) = p^{λ/2} ∑_{i=1}^{mp^{s+1}} A_{i}(p^{λ/2})^{i−1}Φ(τ)^{i}E_{k}(pτ) + F (τ).

We note that F (τ) has the following properties: it is modular of weight k on Γ_{0}(p) and has integer coefficients (since both g(τ) and the sum have these properties), and it is holomorphic at both cusps of Γ_{0}(p) (since h(τ) has this property). Further, since both g(τ) and the sum vanish at ∞, it is clear that F (τ) vanishes at ∞. □

Now, for a given p ∈ {2, 3, 5}, since λ/2 > ϵ_{k,p}, we can study the divisibility of the coefficients of g(τ) by p^{ϵ_{k,p}} by studying the coefficients of F (τ). Since F (τ) ∈ M_{k}(p), we can apply Lemma 5.2 to it to obtain the following corollary.

Corollary 7.2. Let k ∈ {4, 6, 8, 10, 14}, p ∈ {2, 3, 5}, m, s > 0. Let d be the dimension of M_{k}(p) If

g(τ) = f_{k,mp^{s}}(τ) − f_{k,mp^{s−1}}(pτ) = ∑_{n=1}^{∞} γ_{n}q^{n},

and the γ_{n} with 1 ≤ n ≤ d − 1 are all divisible by p^{ϵ_{k,p}}, then all of the γ_{n} are divisible by p^{ϵ_{k,p}}. In particular, if p ∤ n, we have a_{k}(mp^{s}, n) = γ_{n} and thus v_{p}(a_{k}(mp^{s}, n) ≥ ϵ_{k,p}.

Proof. By Lemma 7.1, we have that g(τ) ≡ F (τ) (mod p^{λ/2}), for some F (τ) ∈ M_{k}(p) that vanishes at infinity. Write

F (τ) = ∑_{n=1}^{∞} d_{n}q^{n}.

Since λ/2 > ϵ_{k,p}, the assumptions of the corollary show that p^{ϵ_{k,p}}|d_{n} for all 1 ≤ n < d. Lemma 5.2 then shows that p^{ϵ_{k,p}}|d_{n} for all n, and hence that p^{ϵ_{k,p}}|γ_{n} for all n, as desired. □

Corollary 7.2 allows us to test divisibility of all the γ_{n} (for a given m) by testing finitely many of them. We now prove a theorem which uses duality to allow us to test divisibility of all the γ_{n} (for arbitrary m > 0) by studying a small number of modular forms.

Theorem 7.3. Let p ∈ {2, 3, 5}, k ∈ {4, 6, 8, 10, 14}, and let d be the dimension of M_{k}(p). If

(f_{2−k,j}|U_{p})(τ) ≡ a_{2−k}(j, 0) (mod p^{ϵ_{k,p}})

for each j with 1 ≤ j ≤ d − 1 and p ∤ j, and

(f_{2−k,j}|U_{p})(τ) − f_{2−k,j/p}(τ) ≡ a_{2−k}(j, 0) − a_{2−k}(j/p, 0) (mod p^{ϵ_{k,p}})

for each j with 1 ≤ j ≤ d − 1 and p|j, then p^{ϵ_{k,p}}|a_{k}(mp^{s}, n) for all s > 0 and m, n > 0 prime to p.

Proof. By Corollary 7.2, we need to show that for any m, s > 0, p^{ϵ_{k,p}} divides each γ_{n} = a_{k}(mp^{s}, n) for 1 ≤ n ≤ d − 1 with p ∤ n and each γ_{n} = a_{k}(mp^{s}, n) − a_{k}(mp^{s−1}, n/p) for 1 ≤ n ≤ d − 1 with p|n. Using duality (Corollary 1.2), we see that this is equivalent to checking that for 1 ≤ n < d, p^{ϵ_{k,p}} divides each a_{2−k}(n, mp^{s}) when p ∤ n and each a_{2−k}(n, mp^{s}) − a_{2−k}(n/p, mp^{s−1}) when p|n. As m and s run through all positive integers, the a_{2−k}(n, mp^{s}) are just the nonconstant coefficients of f_{2−k,n}|U_{p}, and the a_{2−k}(n, mp^{s}) − a_{2−k}(n/p, mp^{s−1}) are the nonconstant coefficients of (f_{2−k,n}|U_{p}) − f_{2−k,n/p}. □

To conclude the proof of Theorem 1.3, we must verify that the modular forms in Theorem 7.3 satisfy the congruences described there. In the next section, we describe how to prove these congruences by computing finitely many coefficients of each form, and we summarize the results of these computations.

8. Conclusion

In examining the case p = 2 (as well as some weights for p = 3 or p = 5), we use the following lemma to reduce the congruences we desire to the computation of a single coefficient of a modular form. Since the modular forms in question are easily computable, Theorem 1.3 follows.

Lemma 8.1. Let p ∈ {2, 3, 5} and k ∈ {4, 6, 8, 10, 14}. Let f(τ) ∈ M_{2−k}^{!}(p) be holomorphic at ∞. Suppose that f(τ) and p^{k−2} (p(pτ)^{k−2}f(−1/pτ)) have integer Fourier coefficients, and that v_{p}(μ_{2−k,p}) + 1 − k + λ_{p}/2 ≥ ϵ_{k,2}. Let K be the constant Fourier coefficient of f(τ). If v_{p}(K) ≥ ϵ_{k,p} − ξ_{2−k,p}, then f(τ) ≡ K (mod p^{ϵ_{k,p}}).

Proof. The assumptions of the lemma indicate that (pτ)^{k−2}f(−1/pτ) has its only pole at ∞; hence, it may be written as a linear combination

(pτ)^{k−2}f(−1/pτ) = p^{1−k} ∑_{i=0}^{∞} A_{i}ψ^{i}(τ)θ_{2−k}(τ),

where the A_{i} ∈ ℤ. Replacing τ by −1/pτ, and dividing by τ^{2−k}, we find that

f(τ) = ∑_{i=0}^{∞} A_{i}μ_{2−k,p}p^{1−k+iλ_{p}/2}Φ^{i}(τ)α_{2−k}(τ).

Since v_{p}(μ_{2−k,p}) + 1 − k + λ_{p}/2 ≥ ϵ_{k,p} we see that for i > 0, the coefficient of Φ^{i}α_{2−k} is divisible by p^{ϵ_{k,p}}. Further, comparing constant coefficients, we see that K = A_{0}μ_{2−k,p}p^{1−k}. Hence, we have that

f(τ) ≡ Kα_{k}(τ) (mod p^{ϵ_{k,p}}).

Under the assumption that v_{p}(K) ≥ ϵ_{k,p} − ξ_{2−k,p}, and using that α_{2−k} ≡ 1 (mod p^{ξ_{2−k,p}}), we see that

f(τ) ≡ K (mod p^{ϵ_{k,p}}),

as desired. □

We note that all of the modular forms considered in Theorem 7.3 satisfy the holomorphicity condition of this theorem, and Corollary 4.2 guarantees that they satisfy the integrality conditions. The inequality v_{p}(μ_{2−k,p}) + 1 − k + λ_{p}/2 ≥ ϵ_{k,p} holds for p = 2 (all weights), for p = 3 in weights k = 4, 6, 10, and for p = 5 in weight k = 4. Hence, by calculating a single coefficient of each form, we prove the desired divisibility properties for these cases. In the cases where the theorem applies, we have calculated the necessary constant coefficients and verified Theorem 1.3.

As an example, for p = 2, k = 14, we compute the constant coefficients of f_{−12,1}|U_{2}, f_{−12,2}|U_{2} − f_{−12,1}, and f_{−12,3}|U_{2} to be, respectively, 24, 196608, and 38263776. Since each of these is divisible by 2^{3} and ξ_{−12,2} = 4, we see that each of the modular forms is congruent to a constant modulo 2^{7}, proving Theorem 1.3 for p = 2 in weight 14.

We remark that this lemma does not immediately work in general for p = 3 and p = 5 because we do not always have the inequality v_{p}(μ_{2−k,p}) + 1 − k + λ_{p}/2 ≥ ϵ_{k,p}. One could prove a similar theorem, involving checking the divisibility of additional coefficients, but we find it simpler to perform explicit computations, as described below.

8.1. Computational proof of the remaining cases. For p = 3, k = 8, 14 and for p = 5, k > 4 we now describe the calculations needed to prove the congruences in Theorem 1.3 and give a single illustrative example.

Let f be one of the modular forms of weight 2 − k described in Theorem 7.3. We wish to show that f satisfies the congruence stated in Theorem 7.3. We begin by using Lemma 4.1 to compute the Fourier expansion of p(pτ)^{k−2}f(−1/pτ) to high precision (say to O(q^{500})). This allows us to easily determine a linear combination

p(pτ)^{k−2}f(−1/pτ) = ∑_{i=0}^{N} A_{i}p^{2−k}ψ^{i}(τ)θ_{2−k}(τ),

with A_{i} ∈ ℤ. Replacing τ by −1/pτ and dividing by pτ^{2−k}, we obtain a linear combination

f(τ) = ∑_{i=0}^{N} A_{i}μ_{2−k,p}p^{λi/2+1−k}Φ^{i}(τ)α_{k}(τ) = ∑_{i=0}^{N} B_{i}Φ^{i}(τ)α_{k}(τ).

In each case, we find that for i > 0, v_{p}(B_{i}) ≥ ϵ_{k,p}, so that

f(τ) ≡ B_{0}α_{k}(τ) (mod p^{ϵ_{k,p}}).

We then check that v_{p}(B_{0}) ≥ ϵ_{k,p} − ξ_{2−k,p}, thereby proving that

f(τ) ≡ B_{0} (mod p^{ϵ_{k,p}}),

as desired.

As an example (the simplest) for p = 3, k = 8, we must show that f_{−6,1}|U_{3} is congruent to a constant modulo 3^{3}. Following the procedure above, we find that

(f_{−6,1}|U_{3})(τ) = ∑_{i=0}^{7} B_{i}Φ^{i}(τ)α_{−6}(τ)

with

B_{0} = −480, B_{1} = −10451430, B_{2} = −8628476076,

B_{3} = −1922380466418, B_{4} = −177993370102248, B_{5} = −7892493396961545, B_{6} = −166771816996665690, B_{7} = −1350851717672992089.

One checks easily that each B_{i} with i > 0 is divisible by 3^{ϵ_{8,3}} = 3^{3} (indeed, even by 3^{5}) and that 3|B_{0}. Since ξ_{−6,3} = 2, we see that f_{−6,1}|U_{3} ≡ −480 (mod 3^{3}).

Note that there is nothing difficult about the computations; they are nothing more than basic arithmetic with q-series. They are somewhat daunting to write down; for instance, if p = 5 and k = 14, proving that f_{−12,6} is congruent to a constant modulo 5 involves working with a linear combination of 144 terms of the form Φ^{i}α_{−12,5}, many of which have coefficients over 290 digits long. All of the computations were done using GP/PARI [16], and the scripts used are available from the authors upon request.

These computations have been performed for p = 3, k = 8, 10, 14 and p = 5, k = 4, 6, 8, 10, 14, and complete the proof that Theorem 1.3 is true.

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Department of Mathematics, Brigham Young University, Provo, UT 84602

E-mail address: doud@math.byu.edu, jenkins@math.byu.edu