Relationship between the geometric properties of hyperbolic algebraic curves and the group-theoretic properties of the arithmetic fundamental groups of curves

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540016
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyoto University
• Principal Investigator: HOSHI Yuichiro 京都大学, 数理解析研究所, 講師 (50456761)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 遠アーベル幾何学 / 組み合わせ論的遠アーベル幾何学 / 多重双曲的曲線 / Hodge-Tate性 / 穏やかな点 / 合同部分群問題 / p進Teichmuller理論 / 単遠アーベル幾何学

Outline of Final Research Achievements

(1) I obtained Grothendieck conjecture-type results for hyperbolic polycurves over sub-p-adic fields, p-adic local fields, and hyperbolic curves over Kummer-faithful fields. (2) By a joint work with Shinichi Mochizuki, we developed combinatorial anabelian geometry. (3) I studied the kernels and images of the outer Galois actions associated to hyperbolic curves and proved the finiteness of the moderate rational points of a hyperbolic curve over a number field. (4) As a joint work with Yu Iijima, we studied a pro-p version of the congruence subgroup problem. (5) I studied nilpotent admissible indigenous bundles, as well as nilpotent ordinary indigenous bundles, which is a central object in p-adic Teichmuller theory, in characteristic three. (6) I developed mono-anabelian geometry for number fields.

Free Research Field

数物系科学