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

• 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
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000); Fiscal Year 2014: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• 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 / Hodge-Tate保存性 / 組み合わせ論的セクション予想 / サイクルの標準持ち上げ / 双曲的代数曲線 / 写像類群 / 三点基 / 組み合わせ論的カスプ化 / テンパード基本群 / グロタンディーク予想 / ホッジ・テイト保存性

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.

Research Products

• [Journal Article] On the field-theoreticity of homomorphisms between the multiplicative groups of number fields (2014)
  • Author(s): Yuichiro Hoshi
  • Journal Title: Publications of the Research Institute for Mathematical Sciences Volume: 50 Issue: 2 Pages: 269-285
  • DOI: 10.4171/prims/133
  • NAID: 110009815002
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] The Grothendieck conjecture for hyperbolic polycurves of lower dimension (2014)
  • Author(s): Yuichiro Hoshi
  • Journal Title: Journal of Mathematical Sciences, The University of Tokyo Volume: 21 Pages: 153-219
  • NAID: 120006906118
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] On a problem of Matsumoto and Tamagawa concerning monodromic fullness of hyperbolic curves: Genus zero case (2014)
  • Author(s): Yuichido Hoshi
  • Journal Title: Tohoku Mathematical Journal Volume: 65 Pages: 231-242
  • NAID: 130005562142
  • Peer Reviewed
• [Journal Article] A note on the geometricity of open homomorphisms between the absolute Galois groups of p-adic local fields (2014)
  • Author(s): Yuichido Hoshi
  • Journal Title: Kodai Mathematical Journal Volume: 36 Pages: 284-298
  • NAID: 120005327026
  • Peer Reviewed
• [Journal Article] Corrigendum to “Tame-blind extension of morphisms of truncated Barsotti-Tate group schemes” (2012)
  • Author(s): Yuichiro Hoshi
  • Journal Title: Journal of Mathematical Sciences, The University of Tokyo Volume: 19 Pages: 309-312
  • NAID: 120006906080
  • Peer Reviewed
• [Journal Article] Topics surrounding the combinatorial anabelian geometry of hyperbolic curves I: Inertia groups and profinite Dehn twists (2012)
  • Author(s): Yuichiro Hoshi and Shinichi Mochizuki
  • Journal Title: Galois-Teichmuller Theory and Arithmetic Geometry (Adv. Stud. Pure Math.) Volume: 63 Pages: 659-811
  • Peer Reviewed
• [Presentation] 数体の単遠アーベル的復元 (2015)
  • Author(s): 星裕一郎
  • Organizer: 宇宙際タイヒミューラー理論の検証と更なる発展
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2015-03-09
  • Invited
• [Presentation] Reconstruction of a Number Field from the Absolute Galois Group (2014)
  • Author(s): 星裕一郎
  • Organizer: 第 18 回早稲田大学整数論研究集会
  • Place of Presentation: 早稲田大学
  • Invited
• [Presentation] Grothendieck-Teichmuller Groups in the Combinatorial Anabelian Geometry (2013)
  • Author(s): 星裕一郎
  • Organizer: Grothendieck-Teichmuller theory and Multiple Zeta Values
  • Place of Presentation: Isaac Newton Institute for Mathematical Sciences in Cambridge University （イギリス）
  • Invited
• [Presentation] 低次元多重双曲曲線に対する Grothendieck 予想 (2013)
  • Author(s): 星裕一郎
  • Organizer: Okayama Anabelful Days
  • Place of Presentation: 岡山大学
  • Invited
• [Presentation] Moderate Points of Hyperbolic Curves (2013)
  • Author(s): 星裕一郎
  • Organizer: Fundamental groups in arithmetic and algebraic geometry
  • Place of Presentation: Ennio De Giorgi Mathematical Research Center （イタリア）
  • Invited
• [Presentation] 遠アーベル幾何学概論 (2013)
  • Author(s): 星裕一郎
  • Organizer: 第10回城崎新人セミナー
  • Place of Presentation: 兵庫県豊岡市城崎市民センター
  • Invited
• [Presentation] On the Grothendieck conjecture for hyperbolic polycurves (2012)
  • Author(s): 星裕一郎
  • Organizer: Mini-workshop on arithmetic geometry and related topics
  • Place of Presentation: 京都大学
  • Invited
• [Presentation] Conditional results on the birational section conjecture over small number fields (2012)
  • Author(s): 星裕一郎
  • Organizer: Galois covers and deformations
  • Place of Presentation: Universite de Bordeaux 1 （フランス）
  • Invited
• [Presentation] 双曲的な代数的リーマン面の配置空間の基本群の間の同型射について (2012)
  • Author(s): 星裕一郎
  • Organizer: 代数幾何とその周辺
  • Place of Presentation: 北海道大学
  • Invited
• [Remarks] 星裕一郎のホームページ
  • URL: http://www.kurims.kyoto-u.ac.jp/~yuichiro/index.html
• [Remarks] 星裕一郎のホームページ
  • URL: http://www.kurims.kyoto-u.ac.jp/~yuichiro/index.html
• [Remarks] 星 裕一郎 の ホームページ
  • URL: http://www.kurims.kyoto-u.ac.jp/~yuichiro/index.html