p-adic Langlands correspondence and p-adic geometry of Shimura varieties

• Project/Area Number: 26610003
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: 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 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 志村多様体 / 代数学

Outline of Final Research Achievements

We studied p-adic geometry of Shimura varieties. In particular, we construct potentially good reduction locus, where motives don't degenerate, as an open subspace of the adic space associated to a Shimura variety over p-adic field. Further, we studied it's cohomology.
We studied also on the Kramer-Tunnel conjecture, which describe the epsilon factor of an elliptic curve over an local field by rational points of the elliptic curve. The conjecture was open in the characteristic two case. We showed the conjecture by reducing it to the characteristic zero case.

Research Products

• [Journal Article] The remaining cases of the Kramer-Tunnell conjecture (2016)
  • Author(s): Kestutis Cesnavicius and Naoki Imai
  • Journal Title: Compos. Math. Volume: 152 Pages: 2255-2268
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] 完備コホモロジーと p 進局所 Langlands 対応 (2016)
  • Author(s): 今井直毅
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: N/A
  • Peer Reviewed
• [Journal Article] The remaining cases of the Kramer-Tunnell conjecture (2016)
  • Author(s): Kestutis Cesnavicius and Naoki Imai
  • Journal Title: Compositio Math. Volume: N/A
  • 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] Cohomology of rigid curves with semi-stable coverings (2015)
  • Author(s): Naoki Imai and Takahiro Tsushima
  • Journal Title: Asian J. Math. Volume: 未定
  • Peer Reviewed
• [Journal Article] 完備コホモロジーと p 進局所 Langlands 対応 (2015)
  • Author(s): 今井 直毅
  • Journal Title: 数理解析研究所講究録別冊 Volume: 未定
  • Peer Reviewed
• [Presentation] Potentially good reduction loci of Shimura varieties (2016)
  • Author(s): Naoki Imai
  • Organizer: Boston University Number Theory Seminar
  • Place of Presentation: Boston University
  • Year and Date: 2016-02-22
  • 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