Milnor type K-group attached to algebraic groups and arithmetic geometry

• Project/Area Number: 25800019
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Hiroshima University
• Principal Investigator: Hiranouchi Toshiro 広島大学, 理学研究科, 助教 (30532551)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 類体論 / 類数 / 数論的基本群 / 局所体 / 楕円曲線 / Chow 群

Outline of Final Research Achievements

The results obtained during the research period are as follows: 1. For some elliptic curves over a p-adic field, we show a conjecture due to Somekawa on the Galois symbol map. Results have been published in Funct. Approx. Comment. Math. 2. We investigate the class field theory for an open curve over a local field. 3. We show the finiteness of etale coverings of such a variety with given degree whose ramification bounded along an effective Cartier divisor. This can be thought of a higher dimensional analogue of the classical Hermite-Minkowski theorem. 4. We give a lower bound of the class number of the number field associated to the p-power division points of elliptic curves over Q

Research Products

• [Journal Article] A Hermite-Minkowski type theorem of varieties over finite fields (2017)
  • Author(s): Toshiro Hiranouchi
  • Journal Title: Journal of Number Theory Volume: 176 Pages: 473-499
  • DOI: 10.1016/j.jnt.2016.12.023
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] MILNOR K-GROUPS ATTACHED TO ELLIPTIC CURVES OVER A p-ADIC FIELD (2016)
  • Author(s): Toshiro Hiranouchi
  • Journal Title: Functiones et Approximatio Volume: 54 Issue: 1 Pages: 39-55
  • DOI: 10.7169/facm/2016.54.1.4
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] Finiteness of certain products of algebraic groups over a finite field (2014)
  • Author(s): T. Hiranouchi
  • Journal Title: RIMS kokyuroku Volume: 未定
  • Peer Reviewed
• [Journal Article] An additive variant of Somekawa's K-groups and Kaehler differentials (2014)
  • Author(s): T. Hiranouchi
  • Journal Title: J. K-theory Volume: 未定 Issue: 3 Pages: 481-516
  • DOI: 10.1017/is014003007jkt257
  • Peer Reviewed
• [Presentation] A Hermite-Minkowski type theorem of function fields over finite fields (2016)
  • Author(s): 平之内俊郎
  • Organizer: 仙台広島整数論集会
  • Place of Presentation: 広島大学
• [Presentation] Class field theory for curves over local fields (2015)
  • Author(s): 平之内俊郎
  • Organizer: East Asia Number Theory Conference
  • Place of Presentation: Chuncheon, Korea
  • Year and Date: 2015-08-24 – 2015-08-28
  • Invited
• [Presentation] Class field theory of curves over local fields (2015)
  • Author(s): 平之内俊郎
  • Organizer: East Asia Number Theory Conference
  • Place of Presentation: 春川 (韓国)
  • Year and Date: 2015-08-24
  • Int'l Joint Research / Invited
• [Presentation] Class field theory for curves over local fields (2015)
  • Author(s): 平之内俊郎
  • Organizer: 岡山広島代数学シンポジウム
  • Place of Presentation: 広島大学
  • Year and Date: 2015-03-27
  • Invited
• [Presentation] Barwick’s K-theory of higher categories (2014)
  • Author(s): 平之内俊郎
  • Organizer: Workshop on noncommutative motives and derived algebraic geometry
  • Place of Presentation: 中央大学
  • Invited
• [Presentation] Computations of K-groups of reciprocity functors (2013)
  • Author(s): 平之内俊郎
  • Organizer: Workshop on reciprocity sheaves
  • Place of Presentation: Yatsugatake Natural and Culture Park
  • Invited