Adelic new methods on arithemetic geometry and their applications to p-adic Hodge theory and multiple L-functions

2019 Fiscal Year Final Research Report

• Project/Area Number: 15H03610
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Osaka University
• Principal Investigator: Yasuda Seidai 大阪大学, 理学研究科, 准教授 (90346065)
• Co-Investigator(Kenkyū-buntansha): 古庄 英和 名古屋大学, 多元数理科学研究科, 教授 (60377976) ; 山下 剛 京都大学, 数理解析研究所, 講師 (70444453) ; 岩成 勇 東北大学, 理学研究科, 准教授 (70532547)
• Project Period (FY): 2015-04-01 – 2020-03-31
• Keywords: ガロア表現 / p 進 Hodge 理論 / 関数体の数論 / 標数2の代数曲線 / 多重ゼータ値 / トポスの理論

Outline of Final Research Achievements

The research representative and Go Yamashita have constructed families of Wach modules of rank two and applied them to the study of crystalline deformation rings of dimension two. He and Satoshi Kondo have constructed lifts of the zeta elements in motivic cohomologies of Drinfeld modular varieties to their integral models satisfying norm relations, and have constructed a theory of topoi related to monoids. He and Yusuke Sugiyama have introduced a new notion of pseudo-tameness and, by using them, have proved that any algebraic curve over an algebraically closed field has a tame morphism to the projective line. He has introduced the derived double shuffle spaces and has applied them to show a double shuffle analogue of Broadhurst-Kreimer conjecture in depth four. He has found that a suitable quotient of a Hilber modular surface related to the L-function of a certain curve of genus is a Kummer surface.

Free Research Field

整数論

Academic Significance and Societal Importance of the Research Achievements

上に述べた通り研究成果は多岐にわたっているが、いずれも整数論の当該分野の研究において、超幾何多項式、トポスの理論、pseudo-tame関数の概念、高次複シャッフル空間の考察など、新しい手法を用いて従来であまり進展がなかった方向への結果を得ているという点が画期的である。また、特別な種数2の代数曲線のL関数についての成果は限定的なものであるが、regulator 写像と L 関数の特殊値の関係の幾何的理解に役立ち、わかっている場合の少ない Beilinson 予想の進展に役立つと期待される。