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Curve counting on K3 x E, the Igusa cusp form chi_{10}, and descendent integration

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.