Lacunarity of Han-Nekrasov-Okounkov $q$-series

A power series is called lacunary if `almost all' of its coefficients are zero. Integer partitions have motivated the classification of lacunary specializations of Han's extension of the Nekrasov-Okounkov formula. More precisely, we consider the modular forms \[F_{a,b,c}(z) := \frac{\eta(24az)^a \eta(24acz)^{b-a}}{\eta(24z)},\] defined in terms of the Dedekind $\eta$-function, for integers $a,c \geq 1$ where $b \geq 1$ is odd throughout. Serre determined the lacunarity of the series when $a = c = 1$. Later, Clader, Kemper, and Wage extended this result by allowing $a$ to be general, and completely classified the $F_{a,b,1}(z)$ which are lacunary. Here, we consider all $c$ and show that for ${a \in \{1,2,3\}}$, there are infinite families of lacunary series. However, for $a \geq 4$, we show that there are finitely many triples $(a,b,c)$ such that $F_{a,b,c}(z)$ is lacunary. In particular, if $a \geq 4$, $b \geq 7$, and $c \geq 2$, then $F_{a,b,c}(z)$ is not lacunary. Underlying this result is the proof the $t$-core partition conjecture proved by Granville and Ono.


A series
Two examples of generating functions which are lacunary come from identities due to Euler and Jacobi: These combinatorial identities all have partition-theoretic interpretations. For example, Euler's identity (1.1) gives a recurrence for calculating p(n), the number of partitions of n. Further, they are all related to modular forms by the Dedekind η-function, where q = e 2πiz and Im(z) > 0. It is well-known that η(24z) is a weight 1 2 modular cusp form on Γ 0 (576) with Nebentypus character 12 n . For background on modular forms, see [8].
In light of (1.1) and (1.2), it is natural to investigate the lacunarity of The series f r (z) is closely related to modular forms via the Dedekind η-funtion. Indeed, η(24z) r = q r f r (24z) is a modular cusp form of weight r 2 on Γ 0 (576). Via the connection of f r (z) to modular forms, Serre [10] investigated the lacunarity of η(24z) r and thus f r (z): he proved that, if r is even, then f r (z) is lacunary if and only if r ∈ {2, 4, 6, 8, 10, 14, 26}.
Remark. If r is odd, the lacunarity of f r (z) is not as well-understood since, in this case, η(24z) r has half-integral weight, and so the methods developed by Serre are not applicable. Of course, it is evident from (1.1) and (1.2) that both f 1 (z) and f 3 (z) are lacunary.
At first glance, it is not obvious that the series f r (z) is always a relative of the partition generating function However, work by Nekrasov and Okounkov [7] provides the key connection via an extension of this generating function. Recall that for a partition λ of an integer n ≥ 1, we can label each box u of its Ferrers Diagram with a positive integer called the hooklength, the number of boxes v such that v = u, v lies in the same column and below u, or v lies in the same row and to the right of u. For example, when n = 11, we have the following Ferrers diagram for the partition λ ′ = 6 + 4 + 1: We define the multiset of all hook lengths of λ to be H(λ). For λ ′ , we have H(λ ′ ) = {1, 1, 1, 2, 2, 3, 4, 5, 5, 6, 8}. Nekrasov and Okounkov [7] generalized a result of Macdonald [6] to create a partition-theoretic generating function. They showed that, for any b ∈ C, one has Given an integer a ≥ 1, we consider the subset H a (λ) ⊆ H(λ) defined by A partition λ is said to be an a-core if H a (λ) = ∅. For λ ′ , we have that H 2 (λ ′ ) = {2, 2, 4, 6, 8} and that λ ′ is an a-core for a = 7 and all a ≥ 9. Han [5] generalized Nekrasov and Okounkov's formula to incorporate this set H a (λ): where b, y ∈ C. When y = a = 1, one recovers the identity of Nekrasov and Okounkov. We are interested in the case where y = q a(c−1) for c ≥ 1. Hence, for a, b, c ≥ 1 we define where r := abc + a 2 − a 2 c − 1.
Remark. In Lemma 2.1, we show that F a,b,c (z) is a weakly holomorphic modular form of weight b−1 2 whenever b is odd. From now on, we assume a, b, c ≥ 1 are all integers and b is odd.
Note that this is a generalization of Clader, Kemper and Wage's [2] work for which they took c = 1, corresponding to y = 1. In view of Serre's work and Han's partition-theoretic interpretation, they investigated the lacunarity of F a,b,1 (z) and showed that there are finitely many pairs (a, b) for which F a,b,1 (z) is lacunary. The complete list they gave is replicated in the {9, 15} We now focus our attention to the case where c ≥ 1. It turns out that when we allow c to vary, there are infinitely many lacunary F a,b,c (z). In fact, our first theorem shows that for a ∈ {1, 2, 3}, there exist infinite families of triples (a, b, c) such that F a,b,c (z) is lacunary. Remark. Note that this is not a complete classification of lacunary F a,b,c (z) for a ∈ {1, 2, 3}.
However, if a ≥ 4, then we obtain a different phenomenon. Namely, there are only finitely many triples (a, b, c) such that F a,b,c (z) is lacunary. To obtain this result, we make use of the theory of modular forms with complex multiplication, Granville and Ono's [4] proof of the t-core partition conjecture, and the Pólya-Vinogradov Inequality.

Preliminaries
In this section we recall the modularity properties of η-quotients as well as various conditions on modular forms for lacunarity.
) denote the complex vector space of holomorphic (resp. cuspidal) modular forms of weight k with Nebentypus character χ on Γ 0 (N). An important example of a cuspidal modular form is the Dedekind η-function, defined in (1.3). Products of powers of the Dedekind η-function, η-quotients, are central to this paper. Formally, an η-quotient is any function f that can be written as for some N and each r δ ∈ Z. Theorem 1.64 of [8] gives conditions for when certain ηquotients are weakly holomorphic, which means that the poles are supported at cusps. In particular, if there is N such that We can also compute the order of vanishing of f (z) at a cusp x y , for y | N and gcd(x, y) = 1 using the formula from Theorem 1.65 of [8]: By considering the order of vanishing for all possible y | N, one can determine whether f (z) is holomorphic or cuspidal. Applying this theory to our series F a,b,c (z), we have the following:

) is holomorphic if and only if b ≥ a and cuspidal if and only if b > a.
Remark. We choose to include a factor of 24 in the definition of F a,b,c (z) to guarantee that the conditions mentioned above are satisfied for all triples (a, b, c). This implies that F a,b,c (z) is a weakly holomorphic modular form on Γ 0 (N) where N = 576ac. In general, this N is not necessarily optimal. The optimal N may require altering the factor of 24 in the definition of F a,b,c (z) to some smaller m satisfying mr ≡ 0 (mod 24) for r defined in (1.6). The optimal level of this new modular form is then N opt := lcm(a, c)nm where the integer n ≥ 1 is chosen minimally to satisfy lcm(a,c) ac · n(b − a) ≡ 0 (mod 24).

Conditions on Modular Forms for Lacunarity.
The lacunarity of certain modular forms is already known. Suppose first that f (z) is a weight-one holomorphic modular form on a congruence subgroup. Deligne and Serre (Proposition 9.7 in [3]) showed that f (z) must be lacunary by expressing the coefficients in terms of the Artin L-function. Next, suppose that f (z) is a weakly holomorphic modular form of integral weight k ≥ 2. If f (z) has a pole at one of its cusps, then it cannot be lacunary. This can be shown using the Hardy-Littlewood circle method, which gives estimates for the magnitude of the Fourier coefficients of f (z) as in Chapter 5 of [1]. Additionally, if f (z) is holomorphic but not cuspidal, then it cannot be lacunary since f (z) can be expressed as a linear combination of Eisenstein series together with a cusp form. Known bounds on the coefficients of such forms preclude them from being lacunary.
In light of these remarks, we may restrict our study of the lacunarity of F a,b,c (z) to the space of cusp forms of level N and integral weight k ≥ 2. Central to this study are modular forms with complex multiplication. Serre [10] proved that an element of S k (Γ 0 (N), χ) is lacunary if and only if it is expressible as a linear combination of forms with complex multiplication. We denote the space of cusp forms of weight k with complex multiplication and Nebentypus character χ on Γ 0 (N) by S CM k (Γ 0 (N), χ) and will sometimes call forms with complex multiplication CM forms.
Recall that given any modular form f (z) ∈ M k (Γ 0 (N), χ), the action of the Hecke operator T s on f (z) is defined by where χ(m) = 0 if gcd(N, m) = 1. When s = p is prime, this definition reduces to where we define a(m/p) := 0 whenever p ∤ m. We now state an extension of a well-known lemma by Serre [10] which can be seen in [2]. It says that given a cusp form with complex multiplication, the action of certain Hecke operators is zero.

Proofs of Theorems
Proof of Lemma 2.1. For F a,b,c (z), take N = 576ac and δ 1 = 24a, δ 2 = 24ac, δ 3 = 24 as seen in §2.1. So r δ 1 = a, r δ 2 = b − a, and r δ 3 = −1. Then F a,b,c (z) satisfies the appropriate conditions to be a weakly holomorphic modular form with Nebentypus character Properties of the Kronecker symbol give the desired χ F for a even and odd.
As the numerator is nonnegative when b ≥ a, F a,b,c (z) is holomorphic under this condition.
To show the conditions of Lemma 2.1 precisely, we take y = 24, for which the numerator is nonnegative for b ≥ a and strictly positive for b > a. Therefore, F a,b,c (z) is only holomorphic for b ≥ a and cuspidal for b > a.

Proof of Theorem 1.2.
To prove this theorem, we first show that for a fixed pair (a, c), there are only finitely many b such that F a,b,c (z) is lacunary. Next, we show that there are finitely many pairs (a, c) such that F a,b,c (z) is lacunary by bounding the product ac. This implies that there are finitely many triples (a, b, c) such that F a,b,c (z) is lacunary. Finally, we review further steps for eliminating triples from being lacunary. In §4, we discuss a method that one could use to complete this classification. Proof. Choose s satisfying the conditions of Lemma 2.2. As seen in §2, a result of Serre [10] states that F a,b,c (z) lacunary if and only if it is expressible as a linear combination of CM forms. By Lemma 2.2, F a,b,c (z) is not lacunary unless F a,b,c (z)|T s = 0. We show only finitely many odd b satisfy this condition. Suppose for simplicity that s = p is prime. This is possible since s is square-free and T α T β = T αβ whenever gcd(α, β) = 1. Define the coefficients A a,b,c (m) by the expansion for r as defined in (1.6). Therefore, acting on F a,b,c (z) with T p , we obtain Choose m 0 ≥ 0 to be the minimal m so that p | 24m + r. We must have 0 ≤ m 0 ≤ p − 1.
We show that this term cannot appear in the second sum, so A a,b,c (m 0 ) is the coefficient of q 24m 0 +r p in the Fourier expansion of F a,b,c (z)|T p 1 . First note that for m ≥ 1, by the inequality m 0 ≤ p − 1. Next, looking at the first term of the second sum, we show that 24m 0 +r p = pr. If this were true, using the inequality 0 ≤ m 0 ≤ p − 1 again, one obtains: which gives 24 ≥ (p + 1)r = (p + 1)(abc + a 2 − a 2 c − 1). This inequality can only hold if p ≤ 23. Each prime p ≤ 23 yields a finite set of triples (a, b, c) for which the inequality holds. We reach a contradiction for each triple by calculating m 0 and checking equality of the first exponents of each of the two sums.
Therefore, it suffices to show A a,b,c (m 0 ) = 0 for finitely many b. Notice from (3.1) that Since a and c are fixed, the coefficient A a,b,c (m) is a polynomial in b with degree at most m ac . Let's observe this phenomenon for a = 4 and c = 1: By considering the possible ways that terms in this product could multiply to q 96+r , we deduce that Hence A 4,b,1 (4) is a polynomial in b of degree at most 4 4·1 = 1. We now show that for a ≥ 4, A a,b,c (m) is a nonzero polynomial in b for all m. Therefore A a,b,c (m 0 ) = 0 for finitely many b. Plugging b = a into (3.1), we have: We show A a,a,c (m) = 0 for m ≥ 0. Note that from Corollary 1.9 in [5] we have the following: #{λ : |λ| = m and λ is an a-core }q 24m .
It then follows from Granville and Ono's [4] proof of the t-core partition conjecture that #{λ : |λ| = m and λ is an a-core } ≥ 1, since we are assuming a ≥ 4. Thus, A a,b,c (m) is a nonzero polynomial in b for all m ≥ 0.
The following lemma is a direct generalization of Theorem 1.1 in [2]. In order to prove this lemma, we recall the Pólya-Vinogradov Inequality which states that given a nontrivial Dirichlet character χ with modulus m, the following holds: ≥ 4 there exists finitely many triples (a, b, c) so that F a,b,c (z) is lacunary.
Proof. Fix a ≥ 4 and c ≥ 1. Let a ′ be the square-free part of 576ac, the level of F a,b,c (z). We will show that there exists an s ∈ Z satisfying the conditions of Lemma 2.2. In order to do this, it suffices to show that there exists an s ∈ Z such that −1  For each pair (a, c), there exists an s satisfying the conditions of Lemma 2.2. We show that for all but finitely many pairs, there is an s < ac. By (3.1), the set of odd b for which F a,b,c (z) is lacunary is a subset of the roots of the nonzero polynomials A a,b,c (0), . . . , A a,b,c (s − 1). All of these polynomials have degree less than or equal to s ac , so no b can exist for all but finitely many pairs (a, c).
We first assume that a ′ = 6ac. We show that there is a κ such that whenever a ′ = 6ac ≥ κ there exists an s < ac satisfying the desired conditions. It suffices to find an s ≡ 23 (mod 24) such that −p Thus, it is sufficient to find an s satisfying g ac (s) = 1 and s ≡ 23 (mod 24). We will do this by using the Pólya-Vinogradov Inequality to show that, if ac is sufficiently large, then Note that ψ d • χ is a nontrivial Dirichlet character modulo 24d. So, by applying the Pólya-Vinogradov Inequality to the innermost sum: If m ≥ 12, then ac 24 − 1 − 2 m+1 √ 24ac log(24ac) > 0. Hence, there exists an integer s < ac satisfying the conditions of Lemma 2.2. So in this case, the set of b for which F a,b,c (z) is lacunary is empty. For each m < 12, the inequality (3.5) can only fail if ac < κ, where κ = 5 · 7 · . . . · 43 ≈ 2.18 × 10 15 . So if 6ac is square-free, there are only finitely many pairs (a, c) for which there could exist odd b where F a,b,c (z) is lacunary. Now we turn to the case where 6ac is not square-free. If a ′ 6 has a corresponding s ≤ a ′ 6 < ac, then F a,b,c (z) is not lacunary for any choice of b. If a ′ 6 does not have an s ≤ a ′ 6 , then F a,b,c (z) can only be lacunary for pairs (a, c) such that s ≤ ac where s is chosen minimally. By the above, there are only finitely many possible a ′ , each of which yields finitely many pairs (a, c) for which there could exist odd b such that F a,b,c (z) is lacunary. It follows from Lemma 3.1 that there can be only finitely many pairs (a, b, c) for which F a,b,c (z) is lacunary.
We now prove Theorem 1.2.
Proof. We demonstrate the process of showing that the only possible triples (a, b, c) for which F a,b,c (z) could be lacunary are (4, 5, 3), (4,5,5), and (4, 5, 11). We have the following algorithm: (1) From Lemma 3.2 we know that there are finitely many triples (a, b, c) such that F a,b,c (z) is lacunary. Further, all such triples have the property that a ′ ≤ 6κ where a ′ and κ are defined above. For each a ′ ≤ 6κ, we check whether there exists an s ≡ 23 (mod 24) with s < a ′ 6 and g a ′ /6 (s) = 1. For all a ′ with such an s, the possible pairs (a, c) corresponding to a ′ yield no odd b for which F a,b,c (z) is lacunary. For all a ′ without such an s, there are finitely many pairs (a, c) such that F a,b,c (z) could be lacunary. Let S ac be the set of all such pairs.
(2) For all (a, c) ∈ S ac , choose s minimally and use Lagrange interpolation to construct the polynomials A a,b,c (ac), . . . , A a,b,c (s − 1) 2 . Now find all of the odd positive integer roots of these polynomials. By repeating this processes for all (a, c) with corresponding s, we have a set of all possible (a, b, c). Denote this set by S abc .
(3) For all (a, b, c) ∈ S abc , choose prime p minimally satisfying p ≡ 23 (mod 24) and g a ′ /6 (p) = 1. Recall the definition of m 0 in Lemma 3.1. Find m 0 ∈ Z/pZ such that 24m 0 ≡ −r (mod p). If A a,b,c (m 0 ) = 0 then F a,b,c (z) cannot be lacunary. Denote the set of remaining triples S ′ abc .

Discussion of Theorem 1.2
Due to our results in §3, we conjecture that the series F 4,5,3 (z), F 4,5,5 (z), and F 4,5,11 (z) are lacunary. In theory, it is not difficult to prove this conjecture. Namely, one has to systematically compute all of the weight 2 modular forms with complex multiplication on a suitable level. Our calculations reveal that if this conjecture is true, then these F a,b,c (z) will be linear combinations of CM forms corresponding to fields in the table given below 3 . We have been unable to resolve this issue due to limits on our computational power.
Optimal level of F a,b,c (z) CM fields of F a,b,c (z)