INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A10

INTEGRALITY OF QUOTIENTS OF WRONSKIANS OF THE ANDREWS-GORDON SERIES

Zajj Daugherty^{1}

Department of Mathematics, University of Wisconsin, Madison, WI 53706

                           daughert@math.wisc.edu

Received: 8/18/05, Revised: 1/29/06, Accepted: 3/30/06, Published: 4/2/06

Abstract

For integers k ≥ 2, we consider the integrality of quotients of Wronskians involving certain normalizations of the Andrews-Gordon q-series (see [2])

∏_{1 ≤ n ≢ 0, ±i (mod 2k+1)} \frac{1}{1 − q^{n}} .

This study is motivated by the appearance of these series in conformal field theory as irreducible characters of Virasoro minimal modules. We establish an upper bound on primes p for which such quotients of Wronskians are non-p-integral. Also, we show that for primes p below this bound, the p-integrality of these quotients depends only on their first few coefficients.

1. Introduction and Statement of Results

Modular functions sometimes appear as characters of infinite-dimensional Lie algebras. They play this role specifically in two-dimensional conformal field theory and vertex operator algebra theory. A particularly well-known example of this is the character equation of the Moonshine Module, which is the classical modular function

j(z) − 744 = q^{−1} + 196884q + 21493760q^{2} + ⋯,

where q := e^{2πiz} throughout. Although these characters are not always modular in this way, it can instead be the case that the vector spaces spanned by these irreducible characters are invariant under actions of the modular group. For example, modular forms arise from Wronskians of bases for SL_{2}(ℤ)-modules (for a more in-depth discussion of these topics, see [2]). Here, we explore Wronskians from Virasoro vertex algebras.

1The author was supported for this research by the REU component of the University of Wisconsin Madison NSF VIGRE program

We begin by defining some notation. Throughout, let k ≥ 2 be an integer. Define c_{k} and h_{i,k} by

c_{k} := 1 − \frac{3(2k − 1)^{2}}{(2k + 1)} ,

h_{i,k} := \frac{(2(k − i) + 1)^{2} − (2k − 1)^{2}}{8(2k + 1)} ,

where i is an integer such that 1 ≤ i ≤ k. Then let a_{i,k} be defined by

a_{i,k} := h_{i,k} − \frac{c_{k}}{24} = \frac{(2(k − i) + 1)^{2}}{8(2k + 1)} − \frac{1}{24}.  (1)

Now, the character equation ch_{i,k}(q) for an irreducible lowest-weight module for the Virasoro algebra of central charge c_{k} and weight h_{i,k} is given by (see [2])

ch_{i,k}(q) = q^{a_{i,k}} · (  ∏_{1 ≤ n ≢ 0, ±i (mod 2k+1)} \frac{1}{1 − q^{n}} ) .  (2)

Notice that if p_{b}(a, n) is the number of partitions of n into parts not congruent to 0 or ±a mod b, then ch_{i,k}(q)/q^{a_{i,k}} is the generating function for p_{2k+1}(i, n). Such series have been studied extensively by Andrews and Gordon.

We then define the Wronskian and its derivative on the complete set of characters as follows. Let the k × k matrices W_{k}(q) and W′_{k}(q) be defined by

W_{k}(q) := ( \begin{matrix}ch_{1,k}(q) & ch_{2,k}(q) & ⋯ & ch_{k,k}(q) \\ ch′_{1,k}(q) & ch′_{2,k}(q) & ⋯ & ch′_{k,k}(q) \\ ⋮ & ⋮ & & ⋮ \\ ch_{1,k}^{(k−1)}(q) & ch_{2,k}^{(k−1)}(q) & ⋯ & ch_{k,k}^{(k−1)}(q) \end{matrix} ) ,  (3)

and

W′_{k}(q) := ( \begin{matrix}ch′_{1,k}(q) & ch′_{2,k}(q) & ⋯ & ch′_{k,k}(q) \\ ch_{1,k}^{(2)}(q) & ch_{2,k}^{(2)}(q) & ⋯ & ch_{k,k}^{(2)}(q) \\ ⋮ & ⋮ & & ⋮ \\ ch_{1,k}^{(k)}(q) & ch_{2,k}^{(k)}(q) & ⋯ & ch_{k,k}^{(k)}(q) \end{matrix} ) ,  (4)

where differentiation is given by

(∑ a(n)q^{n})′ := ∑ na(n)q^{n}.

Notice that this is equivalent to evaluating \frac{1}{2πi} · \frac{d}{dz} when q := e^{2πiz}. Given these matrices, we define the Wronskians W_{k}(q) and W′_{k}(q) by

\begin{matrix}W_{k}(q) := α(k) · det W_{k}(q), \\ W′_{k}(q) := β(k) · det W′_{k}(q), \end{matrix}  (5)

respectively, where α(k) and β(k) are rational numbers defined such that the resulting corresponding power series has leading coefficient of 1 whenever it is non-vanishing.

As referenced in [2], W_{k}(q) can be easily expressed in terms of Dedekind’s eta-function, which is defined for z in the upper half-plane of ℂ by

η(z) := q^{\frac{1}{24}} ∏_{n≥1}(1 − q^{n}).

Namely, if k ≥ 2, then W_{k}(q) = η(z)^{2k(k−1)}. In contrast, there is no known closed formula for W′_{k}(q). So instead, we investigate the quotient

F_{k}(q) := \frac{W′_{k}(q)}{W_{k}(q)}.  (6)

This function is itself a power series with rational coefficients. As W_{k}(q) (resp. W′_{k}(q)) is defined so to have leading coefficient of 1 (resp. 1 when it is non-vanishing), F_{k}(q) also has leading coefficient 1 when it is non-vanishing. Natural questions pertaining to this power series are to determine when F_{k}(q) vanishes, and what form its coefficients take when they are non-zero. Our first task is to characterize primes p for which F_{k}(q) is p-integral (meaning that p does not divide the denominators of its coefficients).

Theorem 1. If k ≥ 2 and

B(k) = 6k^{2} − 31k + 37,

then F_{k}(q) is p-integral for every prime p > B(k).

This bound is achieved when k = 6, where the largest prime appearing in the denominators is B(6) = 67. This is not to say, however, that all primes up to this bound can be found in the denominators of the coefficients of F_{k}(q). For example, for small k, F_{k}(q) has integer coefficients. In particular, for 2 ≤ k ≤ 5, we have

\begin{matrix}F_{2}(z) = E_{4}(z), & F_{3}(z) = E_{6}(z), \\ F_{4}(z) = E_{4}(z)^{2}, & F_{5}(z) = E_{4}(z)E_{6}(z), \end{matrix}

where E_{2k} is the standard Eisenstein series (see p. 12 of [1]). Specifically, E_{4}(z) and E_{6}(z) are given by

E_{4}(z) = 1 + 240 ∑_{n≥1} ∑_{d | n} d^{3}q^{n} and E_{6}(z) = 1 − 504 ∑_{n≥1} ∑_{d | n} d^{5}q^{n}.  (7)

Therefore, F_{k}(q) has integer coefficients for 2 ≤ k ≤ 5.

In general, we can construct F_{k}(q) from these two Eisenstein series. From [2] we know that F_{k}(q) is a weight 2k holomorphic modular form on SL_{2}(ℤ). These modular forms form a vector space whose dimension is given by (see p. 119 in [1]):

dim(M_{2k}) = \{ \begin{matrix}⌊\frac{2k}{12}⌋ & 2k ≡ 2 (mod 12), \\ ⌊\frac{2k}{12}⌋ + 1 & 2k ≢ 2 (mod 12). \end{matrix}  (8)

In Section 3, we make use of a diagonalized basis with integer coefficients for these forms. This allows us to easily describe the coefficients of F_{k}(q) in terms of the first few of its coefficients. In particular, we have the following theorem.

Theorem 2. If the first dim(M_{2k}) coefficients of F_{k}(q) are p-integral, then F_{k}(q) is pintegral.

Remark. By explicitly expanding the quotient using techniques similar to those outlined in [2] (see the proof of Lemma 5.1), one can show that when k ≢ 1 mod 4 (resp. when k ≢ 1 mod 3), F_{k}(q) is 2-integral (resp. 3-integral) and F_{k}(q) ≡ 1 mod 2 (resp. F_{k}(q) ≡ 1 mod 3). In view of Theorem 2, such congruences should not come as a surprise, as the Eisenstein series are well-behaved with respect to certain moduli. In particular, it is easy to see from (7) that both E_{4}(z) and E_{6}(z) are congruent to 1 modulo 2 and 3.

The following sections present proofs of the theorems above. In Section 2, we characterize sources of primes for which F_{k}(q) is not p-integral and prove Theorem 1. In Section 3, we provide a useful basis for M_{2k} and prove Theorem 2.

2. Proof of Theorem 1

Here we investigate the quotient

F_{k}(q) := \frac{W′_{k}(q)}{W_{k}(q)},

characterize the source of primes in the denominators of its coefficients, and prove Theorem 1. We begin with a calculation of α(k) and β(k) as given in (5).

Lemma 3. W_{k}(q) is non-vanishing for all values of k, and the scaling factor α(k) is

α(k) = ( ∏_{1≤i<j≤k}(a_{j,k} − a_{i,k}))^{−1}  (9)

= \frac{(2(2k + 1))^{\frac{k(k−1)}{2}}}{∏_{1≤i<j≤k}(j − i)(j + i − (2k + 1))}.

Additionally,

β(k) = \{ \begin{matrix}0 & if a_{i,k} = 0 for some 1 ≤ i ≤ k, \\ α(k) (∏_{1≤i≤k} a_{i,k})^{−1} & otherwise. \end{matrix}  (10)

Remark. Note that when β(k) is non-zero, W′_{k}(q) (and thus F_{k}(q)) is non-zero. Alternatively, W′_{k}(q) vanishes precisely when β(k) vanishes. Thus F_{k}(q) vanishes precisely when a_{i,k} = 0 for some 1 ≤ i ≤ k.

Proof. By applying the differentiation operator and expanding the terms of W_{k}(q) we have

W_{k}(q) = α(k) · det | \begin{matrix}q^{a_{1,k}} + ⋯ & ⋯ & q^{a_{k,k}} + ⋯ \\ a_{1,k}q^{a_{1,k}} + ⋯ & ⋯ & a_{k,k}q^{a_{k,k}} + ⋯ \\ ⋮ & & ⋮ \\ a_{1,k}^{k−1}q^{a_{1,k}} + ⋯ & ⋯ & a_{k,k}^{k−1}q^{a_{k,k}} + ⋯ \end{matrix} | .

Notice that the smallest power of q in the resulting power series can be calculated exclusively in terms of these leading values. Therefore the leading term of det(W_{k}(q)) can be calculated exclusively in terms of the leading coefficients of each of these entries. In other words, the leading coefficient c of det(W_{k}(q)) (if det(W_{k}(q)) ≠ 0, then c = 1/α(k) by definition) can be evaluated by

c = det | \begin{matrix}1 & ⋯ & 1 \\ a_{1,k} & ⋯ & a_{k,k} \\ ⋮ & & ⋮ \\ a_{1,k}^{k−1} & ⋯ & a_{k,k}^{k−1} \end{matrix} | .

This is a Vandermonde determinant, whose evaluation is well known. Precisely, we have

c = ∏_{1≤i<j≤k}(a_{j,k} − a_{i,k}).

We can clearly see from this relationship that since each a_{i,k} is distinct as i varies, c never evaluates to zero. Thus, W_{k}(q) itself never evaluates to zero, and

α(k) = ( ∏_{1≤i<j≤k}(a_{j,k} − a_{i,k}))^{−1}

= \frac{(2(2k + 1))^{\frac{k(k−1)}{2}}}{∏_{1≤i<j≤k}(j − i)(j + i − (2k + 1))}.

Observe that the leading term of each entry of W′_{k}(q) is the coefficient of the corresponding entry in W_{k}(q) scaled by a factor of a_{i,k} in column i. Thus, the leading term c′ of det(W′_{k}(q)) is

c′ = \frac{1}{α(k)} ∏_{1≤i≤k} a_{i,k}.

Since α(k) is finite, c′ clearly evaluates to zero precisely when a_{i,k} = 0 for some 1 ≤ i ≤ k. Thus,

β(k) = \{ \begin{matrix}0 & if a_{i,k} = 0 for some 1 ≤ i ≤ k, \\ α(k) (∏_{1≤i≤k} a_{i,k})^{−1} & otherwise. \end{matrix}

Thus concludes our proof.

Lemma 4. The primes p exceeding 2k + 1 for which the coefficients of F_{k}(q) are non-p-integral are bounded by the prime factors of the numerators of {a_{i,k}}_{1≤i≤k}.

Proof. First note that the coefficients of the character functions ch_{i,k}(q), as given in (2), are integral. Thus, the only source of non-integral coefficients in the entries of W_{k}(q) and W′_{k}(q) are powers of a_{i,k} introduced by differentiation.

Let R_{k} be the power series ring P_{k}[[q]], where P_{k} := ℤ[\{\frac{1}{p_{i}} | prime p_{i} ≤ (2k + 1)\}]. In other words, R_{k} is the ring of power series whose coefficients are rational numbers with denominators whose prime factors are bounded above by 2k + 1. Our claim is that both α(k) · det(W′_{k}(q)) and W_{k}(q) are elements of R_{k}:

The prime factors of the denominators of each a_{i,k} are bounded above by 2k + 1, so ch_{i,k}^{(j)}(q) ∈ R_{k} for all i and j. The determinants of W_{k}(q) and W′_{k}(q) are both linear combinations (with integral coefficients) of these character equations, so det(W_{k}(q)) and det(W′_{k}(q)) are elements of R_{k}. By Lemma 3, α(k) is an element of P_{k}, so α(k) · det(W′_{k}(q)), W_{k}(q) ∈ R_{k}. W_{k}(q) has leading coefficient 1 by definition, so its inverse is also in R_{k}.

So finally, if F_{k}(q) is defined by

F_{k}(q) := α(k) · det(W′_{k}(q)) · W_{k}(q)^{−1},

(so F_{k}(q) = (∏_{1≤i≤k} a_{i,k})^{−1} · F_{k}(q) by Lemma 3), then F_{k}(q) is in the power series ring R_{k}.

However, we are not guaranteed that (∏_{1≤i≤k} a_{i,k})^{−1} is an element of P_{k}. Indeed, for even k = 2,

(∏_{i≤2} a_{i,2})^{−1} = −3600/11 ∉ P_{2}.

Therefore, there may be primes p dividing the denominators of the coefficients of F_{k}(q) which exceed 2k + 1. If there are, it follows then that p must be a prime divisor of the numerator of some a_{i,k}.

Proof of Theorem 1. Following from Lemmas 3 and 4, we need only find bounds for primes resulting from numerators of a_{i,k}. From (1) we can see that a_{i,k} is strictly decreasing over 1 ≤ i ≤ k, so is maximal at i = 1. Simplifying a_{i,k} we have

a_{i,k} = \frac{6i^{2} − 6(2k + 1)i + (2k + 1)(3k + 1)}{12(2k + 1)}.

Thus, the largest primes appearing in the denominators of the terms in F_{k}(q) are bounded above by the largest value of

b_{k}(i) := 6i^{2} − 6(2k + 1)i + (2k + 1)(3k + 1).

However

b_{k}(1) = (6k − 1)(k − 1)

and

b_{k}(2) = (6k − 13)(k − 1),

which do not have large prime factors when evaluated at k since the polynomials themselves factor. The largest b_{k}(i) which does not factor as a polynomial in k is

b_{k}(3) = 6k^{2} − 31k + 37.

Note that for k > 5, b_{k}(3) > (6k−1), and as demonstrated in the example following Theorem 1, F_{k}(q) is p-integral for all primes when k ≤ 5. Thus we have that all primes p for which F_{k}(q) is not p-integral are bounded above by B(k) := b_{k}(3) = 6k^{2}−31k+37 for all k ≥ 2.

3. Proof of Theorem 2

Here we use facts about modular forms over SL_{2}(ℤ) to understand the coefficients of F_{k}(q) (see [1] for background). As mentioned in Section 1, we can describe M_{2k} (the set of holomorphic modular forms of weight 2k over SL_{2}(ℤ)) as a vector space over ℂ. We know that the dimension of M_{2k} is given by (8). Recall that E_{4} and E_{6}, the standard Eisenstein series given by (7), are elements of M_{4} and M_{6} respectively, and both have integer coefficients. Note also that these each have leading term 1. We will also make use of the standard delta function, given by

Δ(z) = \frac{E_{4}(z)^{3} − E_{6}(z)^{2}}{1728} = q − 24q^{2} + 252q^{3} ….  (11)

This function is also modular, and is an element of M_{12} with integer coefficients. The integrality of its coefficients follow from the formulas for E_{4}(z) and E_{6}(z), or from its infinite product expansion, given by

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

From these forms, we will define a basis B_{k} for M_{2k} as follows.

For 2 ≤ k ≤ 7, let B_{k} be defined by

\begin{matrix}B_{2} := \{E_{4}(z)\}, & B_{3} := \{E_{6}(z)\}, & B_{4} := \{E_{4}(z)^{2}\}, \\ B_{5} := \{E_{4}(z)E_{6}(z)\}, & B_{6} := \{E_{4}(z)^{3}, Δ(z)\}, & B_{7} := \{E_{4}(z)^{2}E_{6}(z)\}. \end{matrix}

Notice that each of these bases are diagonalized. That is, for each 0 ≤ i ≤ dim(M_{2k}) − 1, there is one basis element which has minimal degree i. From these, we define B_{k} recursively as follows. If B_{k} is given by

B_{k} = {b_{0}(z), …, b_{dim(M_{2k})−1}(z)},

where the leading degree of b_{i}(z) is i, then let B_{k+6} be defined by

B_{k+6} := {E_{4}(z)^{3}b_{0}(z), Δ(z)b_{0}(z), Δ(z)b_{1}(z), …, Δ(z)b_{dim(M_{2k})−1}(z)}. (12)

As a quick check, we can see first that by (8) that B_{k+6} should have exactly one more element than B_{k}. Also, since both E_{4}(z)^{3} and Δ(z) have weight 12, multiplying by these will increase the weights of basis elements by the desired amount. Finally, if B_{k} is diagonalized, then B_{k+6} is diagonalized, and thus contains linearly independent elements. Using this basis, we can easily proceed into a proof of Theorem 2.

Proof of Theorem 2. We know that F_{k}(q) is an element of M_{2k}, and thus can be expressed as a linear combination of elements of B_{k}. Let

F_{k}(q) = f_{0}b_{0}(z) + ⋯ + f_{dim(M_{2k})−1}b_{dim(M_{2k})−1}(z),

where f_{i} ∈ ℚ. Since every element of B_{k} has integer coefficients, if F_{k}(q) is not p-integral for some prime p, then there must be some f_{i} which is not p-integral. Since B_{k} is diagonalized, then it follows that the i^{th} coefficient of F_{k}(q) must not be p-integral. Therefore, if F_{k}(q) is p-integral for the first dim(M_{2k}) coefficients, then it must be p-integral overall.

Acknowledgements. The author would like to extend thanks to Ken Ono, Karl Mahlburg, and the referee for their guidance and thoughtful editorial comments.

References

[1] T. M. Apostol, Modular functions and Dirichlet series in number theory, Graduate Texts in Mathematics, No. 41, Springer-Verlag, New York, 1976.

[2] A. Milas, E. Mortenson, K. Ono, Number Theoretic Properties of Wronskians of Andrews-Gordon Series, (preprint).