G-invariant Hilbert schemes on Abelian surfaces and enumerative geometry of the orbifold Kummer surface

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