Gromov-Witten theory of $\mathsf{Hilb}^n(\mathbb{C}^2)$ and Noether-Lefschetz theory of $\mathcal{A}_g$

We calculate the genus 1 Gromov-Witten theory of the Hilbert scheme $\mathsf{Hilb}^n(\mathbb{C}^2)$ of points in the plane. The fundamental 1-point invariant (with a divisor insertion) is calculated using a correspondence with the families local curve Gromov-Witten theory over the moduli space $\overline{\mathcal{M}}_{1,1}$. The answer exactly matches a parallel calculation related to the Noether-Lefschetz geometry of the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties. As a consequence, we prove that the associated cycle classes satisfy a homomorphism property for the projection operator on $\mathsf{CH}^*(\mathcal{A}_g)$. The fundamental 1-point invariant determines the full genus 1 Gromov-Witten theory of $\mathsf{Hilb}^n(\mathbb{C}^2)$ modulo a nondegeneracy conjecture about the quantum cohomology. A table of calculations is given.