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--- abstract: | 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 $\operatorname{\mathsf{Hilb}}^d(S)$ be the Hilbert scheme of $d$ points of $S$. In case of elliptically fibered K3 surfaces $S \to {\mathbb{P}}^1$, we calculate genus $0$ Gromov-Witten invariants of $\operatorname{\mathsf{Hilb}}^d(S)$, which count rational curves incident to two generic fibers of the induced Lagrangian fibration $\operatorname{\mathsf{Hilb}}^d(S) \to {\mathbb{P}}^d$. 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 $\operatorname{\mathsf{Hilb}}^d(S)$ 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 $2$ points of ${\mathbb{P}}^1 \times E$, where $E$ is an elliptic curve. Inspired by our results, we conjecture a formula for the quantum multiplication with divisor classes on $\operatorname{\mathsf{Hilb}}^d(S)$ 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 non-trivial case $\operatorname{\mathsf{Hilb}}^2(S)$. As a corollary, we find that the full genus $0$ Gromov-Witten theory of $\operatorname{\mathsf{Hilb}}^2(S)$ in primitive classes is governed by Jacobi forms. We present two applications. A conjecture relating genus $1$ invariants of $\operatorname{\mathsf{Hilb}}^d(S)$ to the Igusa cusp form was proposed in joint work with R. Pandharipande in [@K3xE]. Our results prove the conjecture in case $d=2$. Finally, we present a conjectural formula for the number of hyperelliptic curves on a K3 surface passing through $2$ general points. author: - Georg Oberdieck date: October 2015 title: | Gromov-Witten invariants of\ the Hilbert schemes of points of a K3 surface --- Introduction {#Intro} ============ The Yau-Zaslow formula ---------------------- Let $S$ be a smooth projective K3 surface, and let $\beta_h \in H_2(S,{{\mathbb{Z}}})$ be a primitive effective curve class of self-intersection $\beta_h^2 = 2 h - 2$. The Yau-Zaslow formula [@YZ] predicts the number $\mathsf{N}_h$ of rational curves in class $\beta_h$ in the form of the generating series $$\sum_{h \geq 0} \mathsf{N}_h\, q^{h-1} = \frac{1}{q} \prod_{m \geq 1} \frac{1}{(1-q^m)^{24}} \,. \label{YZ}$$ The right hand side is the reciprocal of the Fourier expansion of a classical modular form of weight 12, the modular discriminant $$\Delta(\tau) = q \prod_{m \geq 1} (1-q^m)^{24} \label{Delta}$$ where $q = \exp(2 \pi i \tau)$ and $\tau \in {{\mathbb{H}}}$. The prediction was proven by Beauville [@Beauville2] and Chen [@ChenK3] using the compactified Jacobian, and by Bryan and Leung [@BL] using Gromov-Witten theory. It is the starting point of further research into the enumerative geometry of algebraic curves on K3 surfaces, see for example [@MPT; @KKV; @KKP] and the references therein. The Hilbert scheme of $d$ points on $S$, denoted $$\operatorname{\mathsf{Hilb}}^d(S),$$ is the moduli space of zero-dimensional subschemes of $S$ of length $d$, see [@Lehn; @Nakajima] for an introduction. It is a non-singular projective variety of dimension $2d$, which