Integral structures of arithmetic differential equations and geometries behind them

2014 Fiscal Year Final Research Report

• Project/Area Number: 24654002
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Tohoku University
• Principal Investigator: TSUZUKI Nobuo 東北大学, 理学(系)研究科(研究院), 教授 (10253048)
• Co-Investigator(Kenkyū-buntansha): YAMAUCHI Takuya 鹿児島大学, 教育学部, 准教授 (90432707)
• Co-Investigator(Renkei-kenkyūsha): TAKAHASHI Nobuyoshi 広島大学, 大学院理学研究科, 准教授 (60301298)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 超幾何関数 / 剛性カラビ・ヤウ多様体 / 半安定的退化 / 整係数コホモロジー / ガロア表現 / 保型性 / 2進還元

Outline of Final Research Achievements

We studied properties of the arithemtic family of Calabi-Yau varieties, constructed by the representative of this research, for which the period integral is a generalized hypergeometric functions. In particular, if the dimension is odd, we found a semistable family around a degenerated fiber such that the number of irreducible components of the special fiber is two among which the one is rational and the other has an interesting natures with respect to arithmetic geometry. In particuler, we proved the modularity of the special fiber in dimension 3.
In the case of dimension 2, we constructed a semistable family over an extension
of Z which is ramified only at 2 and got a K3 surface over an algebraic number field such that it has a good reduction everywhere.

Free Research Field

数論幾何学