  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 x E where E is an elliptic curve. In the primitive case, our
conjecture is expressed in terms of the Igusa cusp form chi_{10} 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 x E is
also presented. Speculations about the motivic stable pairs theory of S x E are
made.
  The reduced Gromov-Witten theory of the Hilbert scheme of points of S is much
richer than S x E. The 2-point function of Hilb(S,d) determines a matrix with
trace equal to the partition function of S x E. A conjectural form for the full
matrix is given.
