K3 mirror symmetry, Legendre family and Deligne's conjecture for the Fermat quartic

In this paper, we will study the connections between mirror symmetry of K3 surfaces and geometry Legendre family elliptic curves. We prove that map Dwork is equal to period family. This result provides an interesting explanation modularities counting functions for from point view. also discuss relations arithmetic smooth fibers Fermat pencil (Dwork family) family, e.g. Shioda-Inose structures, zeta functions, etc. particular, quartic, which modular with a weight-3 form ?(4z)6, curve over ?=2 whose weight-2 newform labeled as 32.2.a.a in LMFDB. compute Deligne's periods are given by special values theta function ?3. Then numerically verify they satisfy predictions conjecture on L-functions critical motives.