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In this paper, we will study the connections between the mirror symmetry ofK3 surfaces and the geometry of the Legendre family of elliptic curves. We willprove that the mirror map of the Dwork family is equal to the period map of theLegendre family. This result provides an interesting explanation to themodularities of counting functions for K3 surfaces from the mirror symmetrypoint of view. We will also discuss the relations between the arithmeticgeometry of smooth fibers of the Fermat pencil (Dwork family) and that of thesmooth fibers of the Legendre family, e.g. Shioda-Inose structures, zetafunctions, etc. In particular, we will study the relations between the Fermatquartic, which is modular with a weight-3 modular form $\eta(4z)^6$, and theelliptic curve over $\lambda=2$ of the Legendre family, whose weight-2 newformis labeled as \textbf{32.2.a.a} in LMFDB. We will also compute the Deligne'speriods of the Fermat quartic, which are given by special values of the thetafunction $\theta_3$. Then we will numerically verify that they satisfy thepredictions of Deligne's conjecture on the special values of $L$-functions ofcritical motives.

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