Geometric Iwasawa Theory

• Project/Area Number: 15K04793
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Sophia University
• Principal Investigator: TRIHAN FABIEN 上智大学, 理工学部, 准教授 (60738300)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: Iwasawa / Elliptic curve / Function fields / abelian variety / function field / function fields / Geometric / Hida / function / Iwasawa Theory / abelian varieties

Outline of Final Research Achievements

We have established a foundation work generalizing the previous work of Kato -Trihan (this is now a submitted paper of more than 200 pages), proved the Iwasawa Main conjecture in two important cases : semistable abelian varieties over the arithmetic extension and constant ordinary abelian varieties over any Zp^d extensions ramifying at a finite set of places.

Academic Significance and Societal Importance of the Research Achievements

The Iwasawa theory for abelian variety over the number fields was initiated in the 70s by Mazur. We give after Kato-Urban-Skinner the second case where the conjecture holds.

Research Products

• [Journal Article] Pontryagin duality for Iwasawa modules and abelian varieties. (2018)
  • Author(s): Lai, King Fai; Longhi, Ignazio; Tan, Ki-Seng; Trihan, Fabien
  • Journal Title: Trans. Amer. Math. Soc. Volume: 370 Issue: 3 Pages: 1925-1958
  • DOI: 10.1090/tran/7016
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] The Iwasawa Main conjecture of constant ordinary abelian varieties over function fields (2016)
  • Author(s): Lai, King Fai; Longhi, Ignazio; Tan, Ki-Seng; Trihan, Fabien
  • Journal Title: Proc. Lond. Math. Soc. Volume: 112 Pages: 1040-1058
• [Journal Article] , The Iwasawa main conjecture for semistable abelian varieties over function fields (2016)
  • Author(s): Lai, King Fai; Longhi, Ignazio; Tan, Ki-Seng; Trihan, Fabien
  • Journal Title: Math. Z. Volume: 282 Pages: 485-510
• [Journal Article] The Iwasawa main conjecture for semistable abelian varieties over function fields (2016)
  • Author(s): K.F. Lai, I. Longhi, K.-S. Tan , F. Trihan
  • Journal Title: Mathematische Zeitschrift Volume: 282 Issue: 1-2 Pages: 485-510
  • DOI: 10.1007/s00209-015-1550-4
  • Peer Reviewed / Int'l Joint Research
• [Presentation] On the equivariant Tamagawa number conjecture for abelian variety (2019)
  • Author(s): Fabien Trihan
  • Organizer: Seminar of the National Taiwan University
  • Invited
• [Presentation] On the equivariant Tamagawa number conjecture for abelian variety (2018)
  • Author(s): Fabien Trihan
  • Organizer: International conference on arithmetic geometry over global fields, Hanoi
  • Int'l Joint Research
• [Presentation] Equivariant Tamagawa Number conjecture for abelian varieties over function fields (2018)
  • Author(s): Trihan Fabien
  • Organizer: Colloquium of Number theory, Stellenbosch University, South Africa
  • Invited
• [Presentation] Equivariant Tamagawa Number conjecture for abelian varieties over function fields (2017)
  • Author(s): Trihan Fabien
  • Organizer: Colloquium of Number theory, Rennes University
  • Invited
• [Presentation] Equivariant Tamagawa Number conjecture for abelian varieties over function fields (2017)
  • Author(s): Trihan Fabien
  • Organizer: Colloquium University of Chile
  • Invited
• [Presentation] Geometric Iwasawa Theory (2016)
  • Author(s): Fabien Trihan
  • Organizer: Colloquium of Strasbourg University
  • Place of Presentation: Strasbourg University, Alsace, France
  • Year and Date: 2016-04-04
  • Invited
• [Presentation] Geometric Iwasawa Theory (2016)
  • Author(s): Fabien Trihan
  • Organizer: Colloquium of Harish Chandra Institute
  • Place of Presentation: Harish Chandra Institute, Allahabad, India
  • Year and Date: 2016-02-26
  • Invited