Arithmetic geometry related to the rigidity of hyperbolic algebraic curves

2020 Fiscal Year Final Research Report

• Project/Area Number: 18K03239
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Kyoto University
• Principal Investigator: Hoshi Yuichiro 京都大学, 数理解析研究所, 准教授 (50456761)
• Project Period (FY): 2018-04-01 – 2021-03-31
• Keywords: 双曲的代数曲線 / 遠アーベル幾何学 / p進タイヒミュラー理論

Outline of Final Research Achievements

One important result obtained in this research project is an affirmative solution to the absolute version of the anabelian conjecture for quasi-tripods over fields that belong to a certain wide class of generalized sub-p-adic fields. As an application of this result, one may conclude that an arbitrary smooth algebraic variety over a field that belongs to the wide class of generalized sub-p-adic fields admits an open basis that consists of absolutely anabelian algebraic varieties.
In this research project, as a joint work with Mochizuki, Fesenko, Minamide, and Porowski, a certain explicit Diophantine inequality has been established by refining some portions of inter-universal Teichmuller theory.
Moreover, as a joint work with Mochizuki and Tsujimura, we have established a certain group-theoretic construction of the absolute Galois group of the field of rational numbers from the point of view of combinatorial anabelian geometry.

Free Research Field

双曲的代数曲線の数論幾何学の研究

Academic Significance and Societal Importance of the Research Achievements

遠アーベル幾何学，通常曲線の理論やp進タイヒミュラー理論のいずれも，先行研究は数少なく，本研究で得られた様々な成果には，それら研究領域の今後の研究の指針になり得るものも含まれていると評価している．また，より具体的には，本研究によって，例えば，広いクラスの基礎体の上で定義された任意の非特異代数多様体が遠アーベル多様体による開基を持つこと，その上，その絶対版の成立が明らかになった．基礎体が有理数体の有限生成拡大体である場合のこの問題は，遠アーベル幾何学における古典的な問題の1つである．そのような古典的な問題のより一般的な場合の解決，そして，その絶対版の解決には，充分な学術的意義があると考えられる．