Computational Algebraic approach to anabelian geometry

• Project/Area Number: 24654006
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Osaka University (2013-2014); Okayama University (2012)
• Principal Investigator: NAKAMURA Hiroaki 大阪大学, 理学(系)研究科(研究院), 教授 (60217883)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 遠アーベル幾何 / 曲面写像類群 / グロタンディークデッサン / 楕円曲線 / 分岐被覆 / 国際研究者交流 / ベリー関数 / 連分数 / 代数学 / トポロジー / ジョンソン準同型 / 表現論 / アルゴリズム

Outline of Final Research Achievements

We studied computational aspects of irreducible decomposition of graded modules arising from weight filtrations of mapping class groups of surfaces. In particular, we obtained results that fix certain parts of the stable image of the Johnson homomorphism in the case of genus zero. We also advanced a computational algebraic approach to X-shape, Y-shaped plane trees defined over the rational number field, and found several remarkable series of Pakovich-Zapponi type examples of Grothendieck dessins of genus one associated with them.

Research Products

• [Journal Article] On arithmetic monodromy representations of Eisenstein type in fundamental groups of once punctured elliptic curves (2013)
  • Author(s): H.Nakamura
  • Journal Title: Publ. RIMS, Kyoto University Volume: 49 Issue: 3 Pages: 413-496
  • DOI: 10.4171/prims/110
  • NAID: 110009623779
  • Peer Reviewed
• [Presentation] Monodromy of elliptic curves and Mordell transformations in Grothendieck-Teichmueller theory (2014)
  • Author(s): Hiroaki Nakamura
  • Organizer: Workshop on Multiple Zeta Values, Modular Forms and Elliptic Motives II
  • Place of Presentation: Instituto de Ciencias Mateaticas, Madrid
  • Year and Date: 2014-12-01 – 2014-12-05
  • Invited
• [Presentation] On Grothendieck dessins and elliptic curves (2014)
  • Author(s): Hiroaki Nakamura
  • Organizer: Workshop "Braids and Arithmetic"
  • Place of Presentation: CIRM, Luminy, Marseille
  • Year and Date: 2014-10-13 – 2014-10-17
  • Invited
• [Presentation] Some observation in Johnson homomorphisms (2013)
  • Author(s): H. Nakamura
  • Organizer: Workshop "Johnson homomorphisms
  • Place of Presentation: 東京大学数理科学研究科
• [Presentation] Arithmetic and topological problems in universal monodromy representation of Galois-Teichmueller groups (2013)
  • Author(s): H. Nakamura
  • Organizer: Introductory Workshop "Grothendieck-Teichmueller Groups, Deformation and Operads"
  • Place of Presentation: Cambridge, UK