Let S be a nonsingular projective K3 surface. Motivated by the study of the Gromov-Witten theory of the Hilbert scheme of points of S, we conjecture a formula for the Gromov-Witten theory (in all curve classes) of the Calabi-Yau 3-fold S × E where E is an elliptic curve. In the primitive case, our conjecture is expressed in terms of the Igusa cusp form χ10 and matches a prediction via heterotic duality by Katz, Klemm, and Vafa. In imprimitive cases, our conjecture suggests a new structure for the complete theory of descendent integration for K3 surfaces. Via the Gromov-Witten/Pairs correspondence, a conjecture for the reduced stable pairs theory of S × E is also presented. Speculations about the motivic stable pairs theory of S × E are made.

The reduced Gromov-Witten theory of the Hilbert scheme of points of S is much richer than S × E. The 2-point function of Hilb^d(S) determines a matrix with trace equal to the partition function of S × E. A conjectural form for the full matrix is given.