Further developments of inter-universal Teichmuller theory

• Project/Area Number: 15K04781
• 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: Yamashita Go 京都大学, 数理解析研究所, 講師 (70444453)
• Co-Investigator(Kenkyū-buntansha): 望月 新一 京都大学, 数理解析研究所, 教授 (10243106)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 宇宙際Teichmuller理論 / 遠アーベル幾何 / Hodge-Arakelov理論 / ゼータ関数 / L関数 / Siegel零点 / 宇宙際幾何 / Riemannゼータ

Outline of Final Research Achievements

The co-investigator Mochizuki showed that we can generalise inter-universal Teichmuller theory for
elliptic curves with complex multiplications. Hence, he obtained a relationship between inter-universal Teichmuller theory and the zeros of Dirichlet L-functions. This can be regarded as the first step of studies of the zeros of zeta functions via further developments of inter-universal geometry.

Academic Significance and Societal Importance of the Research Achievements

ゼータ関数の零点の研究は極めて困難であるが、宇宙際Teichmuller理論によるabc予想の証明においてはいわゆる「一元体上の微分」に相当する現象が起こっているため、宇宙際幾何学の手法によるアプローチは有力であると思われる。今回、宇宙際Teichmuller理論とDirichlet L関数の零点の間に関係が生まれたことはゼータ関数の零点の研究にとって大きな第一歩である。ゼータ関数の零点に関するRiemann予想はクレイによって挙げられている21世紀に解決すべき7つの問題の1つであり、まだRiemann予想までの道のりは遠いが最初の第一歩を踏み出せたことは社会的意義も大きいと感じる。

Research Products

• [Journal Article] A small remark on finite multiple zeta values and $p$-adic multiple zeta values. (2017)
  • Author(s): Go Yamashita
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B68 Pages: 171-174
  • Peer Reviewed
• [Journal Article] A small remark on the filtered $\phi$-module of Fermat varieties and Stickelberger's theorem (2016)
  • Author(s): Go Yamashita
  • Journal Title: Tsukuba J. Math. Volume: 40 Issue: 1 Pages: 119-124
  • DOI: 10.21099/tkbjm/1474747490
  • Peer Reviewed
• [Journal Article] Bogomolov’s proof of the geometric version of the Szpiro Conjecture from the point of view of inter-universal Teichm¨uller theory (2016)
  • Author(s): Shinichi Mochizuki
  • Journal Title: Res. Math. Sci. Volume: 3 Issue: 1
  • DOI: 10.1186/s40687-016-0057-x
  • Peer Reviewed
• [Journal Article] Bogomolov’s Proof of the Geometric Version of the Szpiro Conjecture from the Point of View of Inter-universal Teichm¨uller Theory (2016)
  • Author(s): Shinichi Mochizuki
  • Journal Title: Res. Math. Sci. Volume: 3
  • Peer Reviewed
• [Journal Article] Topics in Absolute Anabelian Geometry III: Global Reconstruction Algorithms (2015)
  • Author(s): Shinichi Mochizuki
  • Journal Title: J. Math. Sci. Univ. Tokyo Volume: 22 Pages: 939-1156
  • Peer Reviewed
• [Journal Article] Monomorphisms in Categories of Log Schemes (2015)
  • Author(s): Shinichi Mochizuki
  • Journal Title: Kodai Math. J. Volume: 38 Pages: 365-429
  • NAID: 130005475916
  • Peer Reviewed
• [Presentation] A Proof of the ABC Conjecture after Mochizuki (2018)
  • Author(s): Go Yamashita
  • Organizer: Pan Asia Number Theory Conference 2018
  • Int'l Joint Research / Invited
• [Presentation] Reductions of crystalline representations and Hypergeometric polynomials (2018)
  • Author(s): Go Yamashita
  • Organizer: Pan Asia Number Theory Conference 2018
  • Int'l Joint Research / Invited
• [Presentation] A Proof of the ABC Conjecture after Mochizuk (2018)
  • Author(s): Go Yamashita
  • Organizer: seminaire de theorie des nombres at Institut de Mathematiques de Jussieu
  • Invited
• [Presentation] ABC Conjecture and Inter-universal Teichmuller Theory (2018)
  • Author(s): Go Yamashita
  • Organizer: 集中講義於慶應義塾大学
  • Invited
• [Presentation] The mathematics of mutually alien copies: from Gaussian integrals to inter-universal Teichm¨uller theory (2016)
  • Author(s): Shinichi Mochizuki
  • Organizer: Inter-universal Teichmuller Theory Summit 2016
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2016-07-19
  • Int'l Joint Research
• [Presentation] IUTchIII-IV with remarks on the function-theoretic roots of the theory (2016)
  • Author(s): Go Yamashita
  • Organizer: Inter-universal Teichmuller Theory Summit 2016
  • Place of Presentation: 京都大学数理解析研究所
  • Int'l Joint Research / Invited
• [Presentation] IUT-III (2015)
  • Author(s): Go Yamashita
  • Organizer: Clay Math. Institute Workshop: IUT Theory of Shinichi Mochizuki
  • Place of Presentation: Oxford大学
  • Year and Date: 2015-12-11
  • Int'l Joint Research / Invited
• [Presentation] IUT-III and IUT-IV section 1 with some remarks on the language of species (2015)
  • Author(s): Go Yamashita
  • Organizer: Clay Math. Institute Workshop: IUT Theory of Shinichi Mochizuki
  • Place of Presentation: Oxford大学
  • Year and Date: 2015-12-11
  • Int'l Joint Research / Invited
• [Remarks] Go Yamashita Official Homepage
  • URL: http://www.kurims.kyoto-u.ac.jp/~gokun/
• [Remarks] 望月新一＠数理研
  • URL: http://www.kurims.kyoto-u.ac.jp/~motizuki/top-japanese.html
• [Funded Workshop] Inter-universal Teichmuller Theory Summit 2016 (2016)
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2016-07-18