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

We calculate the genus 1 Gromov-Witten theory of the Hilbert scheme ${\mathsf{H}\mathsf{i}\mathsf{l}\mathsf{b}}^n({\mathbb{C}}^2)$ 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 ${\bar{\mathcal{M}}}_{1,1}$. 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 ${\mathsf{C}\mathsf{H}}^{\ast }({\mathcal{A}}_g)$. The fundamental 1-point invariant determines the full genus 1 Gromov-Witten theory of ${\mathsf{H}\mathsf{i}\mathsf{l}\mathsf{b}}^n({\mathbb{C}}^2)$ modulo a nondegeneracy conjecture about the quantum cohomology. A table of calculations is given.