  For a Calabi-Yau threefold admitting both a $K3$ fibration and an elliptic
fibration (with some extra conditions) we discuss candidate asymptotic
expressions of the genus 0 and 1 Gromov-Witten potentials in the limit
(possibly corresponding to the perturbative regime of a heterotic string) where
the area of the base of the $K3$ fibration is very large. The expressions are
constructed by lifting procedures using nearly holomorphic Weyl-invariant
Jacobi forms. The method we use is similar to the one introduced by Borcherds
for the constructions of automorphic forms on type IV domains as infinite
products and employs in an essential way the elliptic polylogarithms of
Beilinson and Levin. In particular, if we take a further limit where the base
of the elliptic fibration decompactifies, the Gromov-Witten potentials are
expressed simply by these elliptic polylogarithms. The theta correspondence
considered by Harvey and Moore which they used to extract the expression for
the perturbative prepotential is closely related to the Eisenstein-Kronecker
double series and hence the real versions of elliptic polylogarithms introduced
by Zagier.
