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Arithmetic of Calabi-Yau threefolds with mirror symmetry

镜像对称的 Calabi-Yau 三重算术

基本信息

批准号：
15540001
负责人：
GOTO Yasuhiro
金额：
$ 2.18万
依托单位：
Hokkaido University of Education
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

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

本研究的目的是研究具有镜像对称性的卡-丘三重数的算术性质，并探讨数论与物理学之间的关系。本文的主要研究对象是有限域和数域上的Calabi-Yau三重性。特别是，对加权射影空间中的Calabi-Yau三重和具有K3纤维化的空间进行了详细的研究。在整个项目中，我与加拿大安大略皇后大学的结衣纪子教授进行了合作，除了关于L系列的特殊值的部分外，其他方面都得到了预期的结果。结果在四个研讨会/研讨会上发表，我写了一篇被接受的论文和三篇预印本。本文的主要工作如下：1.考虑了由加权Delsarte三重和具有K3纤维化的三重构成的Calabi-Yau三重，计算了它们的上同调群和精确形式 ...更多信息 2.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三折叠的zeta函数和形式群的影响。发现镜像对称性对形式群没有任何影响，但对zeta函数有很强的影响。这一结果被用于计算正式基团的高度和表征zeta函数。事实上，这给出了数论和物理之间的一个重要关系。4.我计算了一些四维簇的ζ函数和L级数，并将它们与三维簇的ζ函数和L级数进行了比较。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。少