Arithmetic of Calabi-Yau threefolds with mirror symmetry

点击翻译按钮获取中文摘要

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