  Oberdieck and Pandharipande conjectured that the curve counting invariants of
$S\times E$, the product of a $K3$ surface and an elliptic curve, is given by
minus the reciprocal of the Igusa cusp form of weight 10. For a fixed primitive
curve class in $S$ of square $2h-2$, their conjecture predicts that the
corresponding partition functions are given by meromorphic Jacobi forms of
weight $-10$ and index $h-1$. We calculate the partition functions for
primitive classes of square -2 and of square 0.
  Our computation uses reduced Donaldson-Thomas invariants which are defined as
the Behrend function weighted Euler characteristics of the quotient of the
Hilbert scheme of curves in $S\times E$ by the action of $E$. Our technique is
a mixture of motivic and toric methods (developed with Martijn Kool) which
allows us to express the partition functions in terms of the topological vertex
and subsequently in terms of Jacobi forms. We compute the partition functions
for both Behrend function weighted Euler characteristics and for unweighted
Euler characteristics. The results for the Behrend function weighted case
depends on Conjecture 18 from https://arxiv.org/abs/1608.07369
