Abstract: We calculate the generating functions of BPS indices using their modular propertiesin Type II and M-theory compactifications on compact genus one fibered CY 3-foldswith singular fibers and additional rational sections or just N-sections, in order tostudy string dualities in four and five dimensions as well as rigid limits in which gravitydecouples. The generating functions are Jacobi-forms of Γ_1(N) with the complexifiedfiber volume as modular parameter. The string coupling λ, or the \epsilon_{±} parameters inthe rigid limit, as well as the masses of charged hypermultiplets and non-Abelian gaugebosons are elliptic parameters. To understand this structure, we show that specificauto-equivalences act on the category of topological B-branes on these geometries andgenerate an action of Γ1(N) on the stringy K¨ahler moduli space. We argue that theseactions can always be expressed in terms of the generic Seidel-Thomas twist with respectto the 6-brane together with shifts of the B-field and are thus monodromies. This impliesthe elliptic transformation law that is satisfied by the generating functions. We useHiggs transitions in F-theory to extend the ansatz for the modular bootstrap to genusone fibrations with N-sections and boundary conditions fix the all genus generatingfunctions for small base degrees completely. This allows us to study in depth a widerange of new, non-perturbative theories, which are Type II theory duals to the CHLZN orbifolds of the heterotic string on K3 × T2. In particular, we compare the BPSdegeneracies in the large base limit to the perturbative heterotic one-loop amplitudewith $R^2_+F_+^{2g−2}  insertions for many new Type II geometries. In the rigid limit we canrefine the ansatz and obtain the elliptic genus of superconformal theories in 5d.