Applications of class field theory for curves over local fields

2019 Fiscal Year Final Research Report

• Project/Area Number: 17K05174
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyushu Institute of Technology
• Principal Investigator: Hiranouchi Toshiro 九州工業大学, 大学院工学研究院, 准教授 (30532551)
• Project Period (FY): 2017-04-01 – 2020-03-31
• Keywords: 類体論 / 楕円曲線 / 類数

Outline of Final Research Achievements

The results in this project are mainly the following four: 1. First, I completed the class theory on open curves over local fields (which may has positive characteristic). 2. The lower bound of the class number associated with an elliptic curve over a number field is given by the rank of the Mordell-Weil group of the elliptic curve. 3. For a curve over a p-adic field, when the associated Jacobian variety has a good ordinary reduction, we obtain an explicit computation of the "class group" of the curve. 4. We discussed conditions under which the Somekawa K-group associated with two elliptic curves on a p-adic field becomes p-divisible.

Free Research Field

整数論

Academic Significance and Societal Importance of the Research Achievements

局所体上の曲線に対する類体論そのものは1980年代に完成していたが、付随する「類群」の計算についての結果はこれまでそれほど多くはなかった。今回の研究成果により、こうした「類群」を幾つかの場合は具体的に計算することが分かった。将来的な発展の余地も大きいと思われる。