  We study topological string theory on elliptically fibered Calabi-Yau
threefolds using mirror symmetry. We compute higher genus topological string
amplitudes and express these in terms of polynomials of functions constructed
from the special geometry of the moduli space. The polynomials are fixed by the
holomorphic anomaly equations supplemented by the expected behavior at the
boundary in moduli space. We further expand the amplitudes in the base moduli
of the elliptic fibration and find that the fiber moduli dependence is captured
by a finer polynomial structure in terms of the modular forms of the modular
group of the elliptic curve. We further find a recursive equation which governs
this finer structure and which can be related to the anomaly equations for
correlation functions.
