AI 中文总结
本文构造了附属于柏拉图立体的特殊有理椭圆曲面，研究其代数几何性质等，发现需改进Dwork展开以确定其纤维化纤维的欧拉因子。
AI 中文摘要
我们构造了某些特殊的有理椭圆曲面，它们具有四个约化奇异纤维，自然附属于柏拉图立体，属于补充著名的半稳定Beauville曲面的一类曲面。我们描述了这些曲面的基本代数几何性质、它们相关的Picard-Fuchs算子、整数序列、Laurent多项式表示以及Apéry常数。在算术方面，我们发现需要对用于寻找这些纤维化纤维欧拉因子的Dwork展开进行改进，这通过比较来自椭圆曲线的q坐标与Picard-Fuchs方程的q坐标来解释，也直接反映在Frobenius基中的单值矩阵的形状上。
英文摘要
We construct certain special rational elliptic surfaces with four reduced singular fibres that are naturally attached to the platonic solids and belong to a group of surfaces complementing the well-known semi-stable Beauville surfaces. We describe the basic algebraic geometric properties of these surfaces, their associated Picard-Fuchs operators, integer sequences, Laurent polynomial representations and Apéry constants. On the arithmetic side, we discover the need for a refinement of the Dwork-expansion used to find the Euler factors of the fibres of these fibrations. This is explained by comparing the $q$-coordinates coming from the elliptic curve and the $q$-coordinate of the Picard-Fuchs equation and is also directly reflected in the shape of the monodromy matrices in the Frobenius basis.
Comments74 pages, 13 figures and two appendices
