Proof of conjecture regarding the level of Rose's generalized sum-of-divisor functions

In a recent paper, Rose proves that certain generalized sum-of-divisor functions are quasi-modular forms for some congruence subgroup and conjectures that these forms are quasi-modular for $\Gamma_1(n)$. Here, we prove this conjecture.


Introduction
In [3], Rose studies generalizations of MacMahon's sum-of-divisor functions and shows that they are quasi-modular forms. For the context and role of these functions, we refer the reader to this paper and references cited therein. Rose's generalized functions are defined with respect to symmetric sets S ⊂ {1, . . . , n} of residues modulo n satisfying the property that for all ℓ ∈ S, we also have −ℓ ∈ S (mod n). For such a set S and any positive integer k, define A S,n,k (q) := 0<m 1 <···<m k m i ∈S (mod n) q m 1 +...+m k (1 − q m 1 ) 2 · · · (1 − q m k ) 2 .
Rose proves that A S,n,k (q) is a quasi-modular form of mixed weight for some congruence subgroup Γ ⊆ SL 2 (Z) and conjectures that A S,n,k (q) is actually quasi-modular for Γ 1 (n). We prove that this is indeed the case. More precisely, our result is the following.
Theorem 1.1. The functions A S,n,k (q) are quasi-modular forms of mixed weight at most 2k for Γ 1 (n).
We present an example of the forms that arise when n = 5 and k = 1.
Remark. The constant summands above are permitted as the weight-0 components of A S,n,k (q).
Rose proves that the A S,n,k (q) are quasi-modular by relating them to the Taylor coefficients of a particular Jacobi form. Since the Taylor coefficients of a Jacobi form are quasi-modular for the same group, we study the modularity properties of this Jacobi form to deduce the desired properties of A S,n,k (q). In Section 2, we review the theory of Jacobi forms as in [1] and recall Rose's results in terms of these functions. Then in Section 3, we study the modularity properties of the Jacobi forms and nearly-holomorphic modular forms that arise in Rose's work to prove the conjecture.
Acknowledgements. This research was carried out during the 2015 REU at Emory University. The author would like to thank Ken Ono for pointing out Rose's recent work, Michael Mertens, Sarah Trebat-Leader, Eric Larson, and Michael Griffin for useful conversations, and the NSF for its support. The author also thanks an anonymous referee for helpful comments on an earlier version of this paper.

Jacobi forms and Rose's work
Given a matrix γ = ( a b c d ) ∈ SL 2 (Z), the weight-k slash operator acts on functions on the upper half-plane H by A nearly-holomorphic modular form of weight k on a subgroup Γ ⊆ SL 2 (Z) is a function f : H → C which satisfies (f | k γ)(τ ) = f (τ ) for all γ ∈ Γ and is expressible as a polynomial in 1 y , where y := Im(τ ), with holomorphic coefficients. A quasi-modular form is the constant term, with respect to 1 y , of a nearly-holomorphic modular form. The Jacobi group is the set of triples where e(x) := e 2πix . Similar to the slash operator, given integers k and m, the Jacobi group acts on the space of holomorphic functions φ : Remark. One also requires certain growth conditions at the cusps. We refer the reader to [1] for more details on Jacobi forms.
The Jacobi theta function, defined by In particular, the above implies that the function defined by is a Jacobi form of weight 0 and index 1 2 on the subgroup ( 1 2 As in [3], let Similarly, define where the last term is written to match Rose's notation. In addition, as in [3], let With this notation, we can restate the relevant parts of Rose's results as follows (see proofs of Theorems 1.11 and 1.12 in [3]).
Theorem 2.1 (Rose). Let S ⊂ {1, . . . , n} be a symmetric set of residues modulo n. The function A S,n,k (q) is a quasi-modular form of mixed weight at most 2k. In particular, when n / ∈ S, the weight 2w part of A S,n,k (q) is a multiple of , and when n ∈ S, the weight 2w part is a multiple of The key relationship between Jacobi forms and quasi-modular forms is that the Taylor coefficients for the expansion around z = 0 of a Jacobi form are quasi-modular forms for the same group (see equation (6) on p. 31 of [1]). More precisely, we have the following. Lemma 2.2 (Corollary 2.3 of [3]). Given any Jacobi form φ(τ, z) of weight k for some group Γ ⊆ SL 2 (Z) and any positive integer m, the function is a quasi-modular form of weight k + m for the same group Γ. That is, there is a nearly-holomorphic modular form Φ (m) (τ ) of weight k + m on Γ whose constant term with respect to 1 y is the function above.
For a symmetric set S and positive integer w, we define Φ (2w) S (τ ) to be the nearlyholomorphic modular form whose constant term with respect to 1 y is ∂ ∂z To prove Rose's conjecture, we need to show that Φ (2w) S (nτ ) is modular for Γ 1 (n).

Proof of conjecture
By Lemma 2.2, the modularity properties of Φ (2w) S (τ ) are determined by those of the Jacobi form φ S (τ, z). We first determine congruence conditions on matrices which are sufficient for them to fix φ S (τ, z), and hence Φ . In particular, for any positive integer w, we have Proof. First note that Γ(2) ⊂ ( 1 2 0 1 ) , ( 0 −1 1 0 ) , so (ψ|[( a b c d )]) (τ, z) = ψ(τ, z). Note also that the congruence conditions imposed on a, b, c, d ensure that when n is even, a nb c/n d satisfies these conditions for any ( a b c d ) ∈ Γ 0 (2n) ∩ Γ 1 (n). In addition, Lemma 3.2 implies that Φ (2w) S (nτ ) is fixed by ( 1 0 n 1 ) and ( 1 1 0 1 ). We now consider the cases when n is odd and n is even separately.
Finally, when n ∈ S, we also need to know that the term involving the Dedekind eta-function is quasi-modular for Γ 1 (n). It is well-known that derivatives of modular forms are quasi-modular for the same group (see Section 5 of [4]). We can view the Dedekind eta-function as a modular form of weight 1 2 on SL 2 (Z) with a multiplier system. Once we take the quotient, the additional multiplier cancels and we find that with the change of variable τ → nτ , this term is quasi-modular for Γ 1 (n).
Thus, A S,n,k (q) is a mixed-weight quasi-modular form for Γ 1 (n).