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

In this paper, we will study the connections between the mirror symmetry of K3 surfaces and the geometry of the Legendre family of elliptic curves. We will prove that the mirror map of the Dwork family is equal to the period map of the Legendre family. This result provides an interesting explanation to the modularities of counting functions for K3 surfaces from the mirror symmetry point of view. We will also discuss the relations between the arithmetic geometry of smooth fibers of the Fermat pencil (Dwork family) and that of the smooth fibers of the Legendre family, e.g. Shioda-Inose structures, zeta functions, etc. In particular, we will study the relations between the Fermat quartic, which is modular with a weight-3 modular form $\eta(4z)^6$, and the elliptic curve over $\lambda=2$ of the Legendre family, whose weight-2 newform is labeled as \textbf{32.2.a.a} in LMFDB. We will also compute the Deligne's periods of the Fermat quartic, which are given by special values of the theta function $\theta_3$. Then we will numerically verify that they satisfy the predictions of Deligne's conjecture on the special values of $L$-functions of critical motives.