For an Abelian surface $A$ with a symplectic action by a finite group $G$, one can define the partition function for $G$-invariant Hilbert schemes \[Z_{ A, G}(q) = \sum_{ d=0}^{ \infty} e(\text{ Hilb}^{ d}(A)^{ G})q^{ d}.\] We prove the reciprocal $Z_{ A,G}^{ -1}$ is a modular form of weight $\frac{ 1}{ 2}e(A/G)$ for the congruence subgroup $\Gamma_{ 0}(|G|)$, and give explicit expressions in terms of eta products. Refined formulas for the $\chi_{ y}$-genera of $\text{ Hilb}(A)^{ G}$ are also given. For the group generated by the standard involution $\tau : A \to A$, our formulas arise from the enumerative geometry of the orbifold Kummer surface $[A/\tau]$. We prove that a virtual count of curves in the stack is governed by $\chi_{ y}(\text{ Hilb}(A)^{ \tau})$. Moreover, the coefficients of $Z_{ A, \tau}$ are true (weighted) counts of rational curves, consistent with hyperelliptic counts of Bryan, Oberdieck, Pandharipande, and Yin.
Categories: math.AG