Period domains and related studies in arithmetic

K.Kato与C.Nakayama和S.Usui共同构造了退化混合Hodge结构的模空间。这推广了经典的厄米对称域的环形紧致理论。他还得到了它的p-进类似物。这些关于时期域的工作涉及到许多有趣的领域。调节器映射和局部高度对的渐近行为，动机的退化，物理中出现的分歧的渐近行为，Hodge猜想，以及岩泽理论。与岩泽理论的关系来自于调节映射和Zeta值是相关的，也来自于模簇的紧化的边界在推广Sharifi猜想中起着重要作用，而Sharifi猜想给出了岩泽主要猜想的精化。退化的研究还涉及到方案的分支理论。提出者希望研究这些领域。在对Sharifi猜想的推广中，他与共同首席研究员T.Fukay.这项研究提供了Hodge理论、算术和物理之间的新的相互作用。通过这项研究，不同地区的人们可以进入其他领域。提出者预计，许多研究生将选择在与本建议书中的材料相关的领域撰写论文。

The proposer K. Kato constructed moduli spaces of degenerating mixed Hodge structures in joint works with C. Nakayama and S. Usui. This extends the classical theory of toroidal compactifications of Hermitian symmetric domains. He also obtained the p-adic analogue of it. These works on period domains are related to many interesting areas. Asymptotic behaviors of regulator maps and local height pairings, degeneration of motives, asymptotic behaviors of divergences which occur in physics, Hodge conjecture, and also to Iwasawa theory. The relation to Iwasawa theory comes from the fact that the regulator maps and zeta values are related, and also comes from the fact that the boundary of the compactifications of modular varieties play important roles in the generalization of Sharifi conjectures which give refinements of Iwasawa main conjecture. The study of degeneration is also related to ramification theory of schemes. The proposer wishes to study these areas. In the generalization of Sharifi conjectures, he collaborates with the co-principal investigator T. Fukaya.This study provides new interactions between Hodge theory, arithmeticand physics. People in different areas can enter the other fields via this study. The proposer anticipates that many graduate students will choose to write theses in areas related to the materials in this proposal.