  We show that the stringy K\"ahler moduli space of a generic genus one curve
of degree $N$, for $N\le 5$, is the $\Gamma_1(N)$ modular curve $X_1(N)$. This
implies a correspondence between the cusps of the modular curves and certain
large volume limits in the stringy K\"ahler moduli spaces of genus one fibered
Calabi-Yau manifolds with $N$-sections. Using Higgs transitions in M-theory and
F-theory as well as modular properties of the topological string partition
function, we identify these large volume limits with elements of the
Tate-Shafarevich group of the genus one fibration. Singular elements appear in
the form of non-commutative resolutions with a torsional B-field at the
singularity. The topological string amplitudes that arise at the various large
volume limits are related by modular transformations. In particular, we find
that the topological string partition function of a smooth genus one fibered
Calabi-Yau threefold is transformed into that of a non-commutative resolution
of the Jacobian by a Fricke involution. In the case of Calabi-Yau threefolds,
we propose an expansion of the partition functions of a singular fibration and
its non-commutative resolutions in terms of Gopakumar-Vafa invariants that are
associated to BPS states with discrete charges. For genus one fibrations with
5-sections, this provides an enumerative interpretation for the partition
functions that arise at certain irrational points of maximally unipotent
monodromy.
