  The Gromov-Witten theory of threefolds admitting a smooth K3 fibration can be
solved in terms of the Noether-Lefschetz intersection numbers of the fibration
and the reduced invariants of a K3 surface. Toward a generalization of this
result to families with singular fibers, we introduce completed
Noether-Lefschetz numbers using toroidal compactifications of the period space
of elliptic K3 surfaces. As an application, we prove quasi-modularity for some
genus 0 partition functions of Weierstrass fibrations over ruled surfaces, and
show that they satisfy a holomorphic anomaly equation.
