Study on regulator maps using the theory of Arakelov geometry

• Project/Area Number: 25400017
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyushu University
• Principal Investigator: Takeda Yuichiro 九州大学, 数理学研究院, 准教授 (30264584)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥3,250,000 (Direct Cost: ¥2,500,000、Indirect Cost: ¥750,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); 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)
• Keywords: アラケロフ幾何学 / チェック理論 / レギュレーター写像 / チャーン指標 / リーマン・ロッホの定理

Outline of Final Research Achievements

We have made researches in order to formulate and prove higher arithmetic Riemann-Roch theorem. To be more precise, we have tried to construct the theory of higher analytic torsion forms associated with Hermitian vector bundles on an iterated double. We have found a sufficient condition under which the proof of the higher arithmetic Riemann-Roch theorem goes well.
We have made researches on a special class of partitions of natural numbers called t-core. We have found quadratic forms which are closely related to t-core partitions. We have given a purely combinatorial method to prove congruence conditions on numbers of t-core partitions. It relies on the theory of quadratic forms and geometry on finite fields.

Research Products

• [Presentation] Higher arithmetic Chern character
  • Author(s): ＹｕｉｃｈｉｒｏTakeda
  • Organizer: Low dimensional topology and number theory
  • Place of Presentation: Fukuoka
  • Invited