论文标题
K3镜子对称性，Legendre家族和Deligne的猜想
K3 mirror symmetry, Legendre family and Deligne's conjecture for Fermat quartic
论文作者
论文摘要
在本文中，我们将研究K3表面的镜像对称性与椭圆曲线的Legendre家族的几何形状之间的连接。我们将证明DWork家族的镜像图等于Legendre家族的时期图。从镜像对称的角度来看，该结果对K3表面的计数函数的模块化提供了一个有趣的解释。我们还将讨论Fermat铅笔光滑纤维的算术几何形状与Legendre家族的光滑纤维的算术几何形状，例如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 $η(4z)^6$, and the elliptic curve over $λ=2$ of the Legendre family, whose weight-2 newform is labeled as \textbf{32.2.a.a} in LMFDB.我们还将计算Fermat四分之一的deligne期间，这些时期由Theta函数的特殊值$θ_3$给出。然后，我们将在数字上验证它们是否满足了DeLigne对关键动机的特殊值的猜想的预测。
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 $η(4z)^6$, and the elliptic curve over $λ=2$ 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 $θ_3$. Then we will numerically verify that they satisfy the predictions of Deligne's conjecture on the special values of $L$-functions of critical motives.
