Arithmetic of Calabi-Yau threefolds with mirror symmetry

本研究的目的是研究镜像对称的Calabi-Yau三重数的算术性质，并探讨数论与物理的关系。本文的主要研究对象是有限域和数域上的Calabi-Yau三重数列。特别是，对加权射影空间中的Calabi-Yau三重数和具有K3纤丝的数进行了详细的研究。在整个项目中，我与加拿大安大略省皇后大学的Yi Noriko教授合作，在这个项目的各个方面，除了L系列的特殊价值之外，我都能取得预期的结果。结果在四次研讨会/讲习班演讲中公布，我写了一篇被接受的论文和三份预印本。下面描述了我的结果的细节：1.我考虑了由加权Delsarte三折叠和具有K3原纤的Calabi-Yau三折叠构造的Calabi-Yau三折叠，并计算了它们的上同调群和精确形式的…计算了Calabi-Yau三重形式群的高度，改进了已知的形式群高度公式。3.考虑了镜像对称性对Calabi-Yau三重函数的Zeta函数和形式群的影响。结果表明，镜面对称性对形式群没有影响，但对Zeta函数有很强的影响。这一结果被用于形式群高度的计算和Zeta-函数的刻画。这实际上给出了数论和物理之间的重要关系。4.我计算了一些四维变种的Zeta函数和L级数，并将它们与三重数的结果进行了比较。最后，我注意到我与Yui教授的合作是通过电子邮件和五次密集的面对面会议进行的。较少

The purpose of this research was to study the arithmetic properties of Calabi-Yau threefolds with mirror symmetry and investigate the relationships between number theory and physics. The main object of this research was Calabi-Yau threefolds over finite fields and number fields. In particular, detailed studies were conducted for Calabi-Yau threefolds in weighted projective spaces and for those having K3 fibrations. Throughout the project, I had collaboration work with Professor Noriko Yui at Queen's University in Ontario, Canada.In every aspect of this project, except for the part concerning the special values of L-series, I was able to obtain results as expected. The results were presented in four seminar/workshop talks and I wrote one accepted paper and three preprints. The following describe details of my results :1.I considered Calabi-Yau threefolds constructed from weighted Delsarte threefolds and those having K3 fibrations, and computed their cohomology groups and the exact form … More of their zeta-functions and L-series.2.The height of the formal groups of Calabi-Yau threefolds was calculated and I refined the known formula for the height of the formal groups. Also, I found many Calabi-Yau threefolds with large height which had not been discovered earlier.3.I considered the effects of mirror symmetry on the zeta-functions and formal groups of Calabi-Yau threefolds. It was found that the mirror symmetry does not have any influence on the formal groups, while it has strong effects on the zeta-functions. This result was used for the calculations of the height of formal groups and for the characterization of zeta-functions. This gives, in fact, an important relationship between number theory and physics.4.I computed the zeta-functions and L-series of some 4-dimensional varieties and compared them with those of threefolds. Consequently, the difference and similarities between these varieties became clear.Finally, I note that my collaboration with Professor Yui was carried out by email and in five intensive meetings in person. Less