Gromov-Witten invariants of the Hilbert schemes of points of a K3 surface

We study the enumerative geometry of rational curves on the Hilbert schemes of points of a K3 surface. Let S be a K3 surface and let Hilb(d) (S) be the Hilbert scheme of d points of S. In the case of elliptically fibered K3 surfaces S -> P-1, we calculate genus-0 Gromov-Witten invariants of Hilb(d)(S), which count rational curves incident to two generic fibers of the induced Lagrangian fibration Hilb(d) (S) -> P-d. The generating series of these invariants is the Fourier expansion of a power of the Jacobi theta function times a modular form, hence of a Jacobi form. We also prove results for genus-0 Gromov-Witten invariants of Hilb(d)(S) for several other natural incidence conditions. In each case, the generating series is again a Jacobi form. For the proof we evaluate Gromov-Witten invariants of the Hilbert scheme of two points of P-1 x E, where E is an elliptic curve. Inspired by our results, we conjecture a formula for the quantum multiplication with divisor classes on Hilb(d) (S) with respect to primitive curve classes. The conjecture is presented in terms of natural operators acting on the Fock space of S. We prove the conjecture in the first nontrivial case Hilb(d) (S). As a corollary, we find that the full genus-0 Gromov-Witten theory of Hilb(d) (S) in primitive classes is governed by Jacobi forms. We present two applications. A conjecture relating genus-1 invariants of Hilb(d) (S) to the Igusa cusp form was proposed in joint work with R Pandharipande. Our results prove the conjecture when d = 2. Finally, we present a conjectural formula for the number of hyperelliptic curves on a K3 surface passing through two general points.