Topological string on elliptic Calabi--Yau $3$-folds with $N$-sections and modular forms

The all genus topological string amplitudes on Calabi--Yau $3$-folds are generating functions of the symplectic invariants of holomorphic curves. We argue that the Fourier--Mukai transform on the A-model category as well as the holomorphic anomaly of the B-model restrict these amplitudes to be meromorphic Jacobi forms of $\Gamma_0(N)$ subgroups of ${\rm SL}(2,\mathbb{Z})$. Vanishing conditions and Nakajima's blow up equations allow then to fix these amplitudes explicitly.