  We study the quasimap invariants of elliptic and K3 fibrations. Oberdieck and
Pixton conjectured that the Gromov-Witten potentials of elliptic fibrations are
quasi-modular forms. Analogously, we propose similar conjecture for the
quasimap potentials of elliptic fibrations. We also conjecture some finite
generation properties of quasimap potentials of K3 fibrations. Via
wall-crossing conjecture, this will imply some quasi-modularity of the
Gromov-Witten potentials of K3 fibrations. We provide some evidences for our
conjectures through several examples. The method here can be further
generalized to arbitrary n-dimensional Calabi-Yau fibrations.
