Geometric approach to the local Langlands correspondence

• Project/Area Number: 26707003
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Partial Multi-year Fund
• Research Field: Algebra
• Research Institution: The University of Tokyo
• Principal Investigator: Imai Naoki 東京大学, 大学院数理科学研究科, 准教授 (90597775)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥9,750,000 (Direct Cost: ¥7,500,000、Indirect Cost: ¥2,250,000); Fiscal Year 2017: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000); Fiscal Year 2016: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000); Fiscal Year 2015: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000); Fiscal Year 2014: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
• Keywords: Langlands 対応 / 代数学

Outline of Final Research Achievements

We studied a geometric realization of the local Langlands correspondence.
More concretely, we constructed affinoids in the Lubin-Tate perfectoid space, which is a limit space of Lubin-Tate spaces, using CM points and showed that the etale cohomology of the reductions of formal models of the affinoids realizes the local Langlands correspondence and the local Jacquet-Langlands correspondence for simple supercuspidal representations.

Research Products

• [Journal Article] Stable models of Lubin-Tate curves with level three (2017)
  • Author(s): Naoki Imai and Takahiro Tsushima
  • Journal Title: Nagoya Math. J. Volume: 225 Pages: 100-151
  • Peer Reviewed
• [Journal Article] Affinoids in Lubin-Tate surfaces with exponential full level two (2017)
  • Author(s): Naoki Imai and Takahiro Tsushima
  • Journal Title: Contemp. Math. Volume: 691 Pages: 157-180
  • Peer Reviewed
• [Journal Article] Local Jacquet-Langlands correspondences for simple supercuspidal representations (2017)
  • Author(s): Naoki Imai and Takahiro Tsushima
  • Journal Title: Kyoto J. Math. Volume: N/A
  • Peer Reviewed
• [Journal Article] Stable models of Lubin-Tate curves with level three (2016)
  • Author(s): Naoki Imai and Takahiro Tsushima
  • Journal Title: Nagoya Math. J. Volume: N/A
  • Peer Reviewed
• [Journal Article] Cohomology of rigid curves with semi-stable coverings (2015)
  • Author(s): Naoki Imai and Takahiro Tsushima
  • Journal Title: Asian J. Math. Volume: 19 Pages: 637-650
  • Peer Reviewed
• [Journal Article] Local root numbers of elliptic curves over dyadic fields (2015)
  • Author(s): Naoki Imai
  • Journal Title: J. Math. Sci. Univ. Tokyo Volume: 22 Pages: 247-260
  • NAID: 120006906132
  • Peer Reviewed
• [Journal Article] Explicit construction of semi-stable models of Lubin-Tate curves with low level (2014)
  • Author(s): Naoki Imai and Takahiro Tsushima
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: 51 Pages: 15-31
  • Peer Reviewed
• [Presentation] Non-semi-stable loci in Hecke stacks and Fargues' conjecture (2018)
  • Author(s): Naoki Imai
  • Organizer: UK-Japan Winter School 2018 on Number Theory
  • Int'l Joint Research / Invited
• [Presentation] Non-semi-stable loci in Hecke stacks and Fargues' conjecture (2017)
  • Author(s): Naoki Imai
  • Organizer: Master Lectures - The Legacy of Carl Friedrich Gauss
  • Int'l Joint Research / Invited
• [Presentation] Simple epipelagic local Galois representations (2016)
  • Author(s): Naoki Imai
  • Organizer: New Developments in Representation Theory
  • Place of Presentation: Institute for Mathematical Sciences, National University of Singapore
  • Year and Date: 2016-03-14
  • Int'l Joint Research / Invited
• [Presentation] Deligne-Lusztig construction for a pair of forms (2016)
  • Author(s): Naoki Imai
  • Organizer: MIT Lie Group Seminar
  • Place of Presentation: MIT
  • Int'l Joint Research / Invited
• [Presentation] Heckeスタックの非半安定部分とFarguesの予想 (2016)
  • Author(s): Naoki Imai
  • Organizer: 整数論&保型形式セミナー
  • Place of Presentation: 大阪大学
  • Invited
• [Presentation] Affinoids in the Lubin-Tate perfectoid space and simple epipelagic representations (2015)
  • Author(s): Naoki Imai
  • Organizer: Reductions of Shimura Varieties
  • Place of Presentation: Mathematisches Forschungsinstitut Oberwolfach
  • Year and Date: 2015-08-26
  • Int'l Joint Research / Invited
• [Presentation] Affinoids in the Lubin-Tate perfectoid space and simple epipelagic representations (2015)
  • Author(s): Naoki Imai
  • Organizer: Arithmetic geometry, representation theory and applications
  • Place of Presentation: Centre International de Rencontres Mathematiques
  • Year and Date: 2015-06-26
  • Int'l Joint Research / Invited
• [Presentation] Good reduction of affinoids for epipelagic representations in the Lubin-Tate perfectoid space (2015)
  • Author(s): Naoki Imai
  • Organizer: MIT Number Theory Seminar
  • Place of Presentation: MIT
  • Year and Date: 2015-02-03
  • Invited
• [Presentation] The p-adic and mod p local Langlands correspondence for GL(2,Q_p) (2015)
  • Author(s): Naoki Imai
  • Organizer: Winter school on p-adic Hodge theory
  • Place of Presentation: Korea Institute for Advanced Study
  • Year and Date: 2015-01-12 – 2015-01-14
  • Invited
• [Presentation] Good reduction of affinoids for epipelagic representations in the Lubin-Tate perfectoid space (2014)
  • Author(s): Naoki Imai
  • Organizer: Berkeley Number Theory Seminar
  • Place of Presentation: University of California, Berkeley
  • Year and Date: 2014-11-19
  • Invited
• [Funded Workshop] Workshop on Shimura varieties, representation theory and related topics (2016)
  • Place of Presentation: 京都大学
  • Year and Date: 2016-11-21