Unramified cohomology groups and rationality problem for fields of invariants

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03418
• 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: Niigata University
• Principal Investigator: Hoshi Akinari 新潟大学, 自然科学系, 教授 (50434262)
• Co-Investigator(Kenkyū-buntansha): 山崎 愛一 京都大学, 理学研究科, 准教授 (10283590)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 有理性問題 / ネーター問題 / 不分岐コホモロジー / 半単項式作用 / 安定有理性 / レトラクト有理性 / 双有理分類 / ハッセ原理

Outline of Final Research Achievements

About rationality problem for fields of invariants, we studied (1) Hasse norm principle for fields of degree <= 15 except for 12; (2) Davenport and Hasse's theorems and lifts of multiplication matrices of Gaussian periods; (3) Hasse norm principle for fields of degree 12; (4) multiplicative invariant fields of dimension <= 6; (5) a two-dimensional rationality problem and intersections of two quadrics; (6) three-dimensional purely quasi-monomial actions; (7) splitting fields of Lecacheux's family of quintic polynomials and rational points of associated elliptic curves; (8) norm one tori; (9) relation modules for dihedral groups; (10) degree three unramified cohomology; (11) algebraic tori of small dimensions, and we got several new results.

Free Research Field

数論、代数学、代数幾何学

Academic Significance and Societal Importance of the Research Achievements

不変体の有理性問題に関する研究を行った。得られた研究成果は代数幾何、数論幾何、数論、群論、環論、表現論、計算代数の各分野に関連しており、各分野において重要な結果をいくつも含んでいる。ここで得られた具体例を元にして、理論が具体例の計算を生み、具体例の計算が理論を生む、好循環が生まれることが期待される。