  An explicit formula is derived for the generating function of vertical
D4-D2-D0 bound states on smooth K3 fibered Calabi-Yau threefolds, generalizing
previous results of Gholampour and Sheshmani. It is also shown that this
formula satisfies strong modularity properties, as predicted by string theory.
This leads to a new construction of vector valued modular forms which exhibits
some of the features of a generalized Hecke transform.

High Energy Physics - Theory;Algebraic Geometry