Period domains and related studies in arithmetic

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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.