Study of the class group in the class field theory for curves over local fields

2022 Fiscal Year Final Research Report

• Project/Area Number: 20K03536
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• 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): 2020-04-01 – 2023-03-31
• Keywords: 類体論 / 楕円曲線 / イデアル類群

Outline of Final Research Achievements

The purpose of this study was to explicitly compute the finite part of the "class group" which is a finitely generated abelian group (analogous to the classical ideal class group) in the context of the class field theory of curves over p-adc fields. In this study, by studying the structure of the Milnor type K-group, the so-called "Somekawa K-group," associated with Abelian varieties with good redcution and the multiplicative groups, we were able to provide the Abelian group structure of the "class group" associated with a curve and the upper and lower bounds of the order of the group.

Free Research Field

数論幾何学

Academic Significance and Societal Importance of the Research Achievements

今回の研究対象であるp進体上の曲線の類体論は、もともと高次元の多様体に対する類体論（いわゆる高次元類体論）を証明する過程で生まれた理論である。これまではどちらかと言えばその理論的側面の研究に主眼が置かれていたように思われる。しかし、今回の研究で具体的な曲線に対してより具体的に計算を行うことができた。その結果、古典的な類体論と同じように岩澤理論のような他の分野への応用・発展にも貢献するものと考えられる。