  We apply the modular approach to computing the topological string partition
function on non-compact elliptically fibered Calabi-Yau 3-folds with higher
Kodaira singularities in the fiber. The approach consists in making an ansatz
for the partition function at given base degree, exact in all fiber classes to
arbitrary order and to all genus, in terms of a rational function of weak
Jacobi forms. Our results yield, at given base degree, the elliptic genus of
the corresponding non-critical 6d string, and thus the associated BPS
invariants of the 6d theory. The required elliptic indices are determined from
the chiral anomaly 4-form of the 2d worldsheet theories, or the 8-form of the
corresponding 6d theories, and completely fix the holomorphic anomaly equation
constraining the partition function. We introduce subrings of the known rings
of Weyl invariant Jacobi forms which are adapted to the additional symmetries
of the partition function, making its computation feasible to low base wrapping
number. In contradistinction to the case of simpler singularities, generic
vanishing conditions on BPS numbers are no longer sufficient to fix the modular
ansatz at arbitrary base wrapping degree. We show that to low degree, imposing
exact vanishing conditions does suffice, and conjecture this to be the case
generally.
