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 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.

在本文中，我们将研究K3曲面的镜像对称性与Legendre椭圆曲线族的几何之间的联系。我们将证明Dwork家族的镜像图等于Legendre家族的周期图。从镜像对称的角度来看，此结果为K3表面的计数函数的模块化提供了有趣的解释。我们还将讨论费马铅笔（Dwork系列）的光滑纤维与勒让德家族的光滑纤维的算术几何之间的关系，例如Shioda-Inose结构，zeta函数等。特别是，我们将研究Fermat四次方之间的关系，这是权重为3的模块化形式$\eta {（4ž）}^6$，并且椭圆曲线超过 $\lambda =2$是Legendre家族的一员，在LMFDB中权重为2的新形式被标记为32.2.aa。我们还将计算费马四次方程的Deligne周期，这些周期由theta函数的特殊值给出${\theta }_3$。然后，我们将数值验证它们满足关键动机L函数的特殊值对Deligne猜想的预测。