  We associate a Jacobi form over a rank s lattice to N=2, D=4 heterotic string
compactifications which have s Wilson lines at a generic point in the vector
multiplet moduli space. Jacobi forms of index m=1 and m=2 have appeared earlier
in the context of threshold corrections to heterotic string couplings. We
emphasize that higher index Jacobi forms as well as Jacobi forms of several
variables over more generic even lattices also appear and construct models in
which they arise. In particular, we construct an orbifold model which can be
connected to models that give index m=3, 4 or 5 Jacobi forms through the
Higgsing process. Constraints from being a Jacobi form are then employed to get
threshold corrections using only partial information on the spectrum. We apply
this procedure for index m=3, 4 or 5 Jacobi form examples and also for Jacobi
forms over A_2 and A_3 root lattices. Examples with a single Wilson line are
examined in detail and we display the relation of Siegel forms over a
paramodular group \Gamma_m to these models, where \Gamma_m is associated with
the T-duality group of the models we study. Finally, results on the heterotic
string side are used to clarify the linear mapping of vector multiplet moduli
to Type IIA duals without using the one-loop cubic part of the prepotential on
the Type II side, and also to give predictions for the geometry of the dual
Calabi-Yau manifolds.
