The modularity conjecture for Gromov–Witten potentials of elliptic fibrations

Let $X$ be a smooth projective elliptic variety with flat proper morphism $\pi:X\to B$ to a smooth variety $B$, generic fiber a smooth genus-one curve, and regular section $z:B\to X$. For a primitive class $\beta\in H_2(B,\mathbb Z)$, define the relative potential function

$$F^X_{g,\beta}(q)=\sum_{\pi_*\alpha=\beta}N^X_{g,\alpha}q^{\alpha\cdot z_*[B]+\frac{k}{2}}.$$

Here $k=\beta\cdot c_1(\pi_*\omega_{X/B})$, and

$$\Delta(q)=q\prod_{m\geq 1}(1-q^m)^{24}.$$

The modularity conjecture. The relative potential has the form

$$F^X_{g,\beta}(q)=\varphi(q)\cdot\Delta(q)^{-k},$$

where $\varphi(q)$ is a quasi-modular form for $SL_2(\mathbb Z)$. This conjecture predicts that the Gromov–Witten theory of elliptic fibrations is controlled by quasi-modular forms and the modular discriminant; the supplied text gives no evidence resolving it.