New developments of automorphy of Galois representations and Serre conjecture.

2023 Fiscal Year Final Research Report

• Project/Area Number: 19H01778
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Tohoku University
• Principal Investigator: Yamauchi Takuya 東北大学, 理学研究科, 准教授 (90432707)
• Co-Investigator(Kenkyū-buntansha): 都築 暢夫 東北大学, 理学研究科, 教授 (10253048) ; 山名 俊介 大阪公立大学, 大学院理学研究科, 教授 (50633301) ; 宮内 通孝 岡山大学, 教育学域, 准教授 (70533644)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: ガロア表現 / 保型表現 / ジーゲル保型形式

Outline of Final Research Achievements

In order to first formulate the Sale Conjecture for the problem of the coercivity of Galois representations, which is the theme of this research, the weight part was formulated successfully in the case of a high dimensional algebraic group called GSp4. This is to identify the weight of the corresponding preserving form of a preserving Galois representation. We formulated the potential cohomology problem and deformation theory by developing a technique to reduce it to Barsochte deformations. We also gave a generalization of the so-called Sale weight to the case of GSp4, where the weight is explicitly determined from the bifurcation data by a detailed computation of Galois cohomology. I also proved the preservation of modulo Galois representations in the case of Dwork families. He also gave other results on the theory of preserving representations, such as composition problems and equidistribution problems.

Free Research Field

整数論

Academic Significance and Societal Importance of the Research Achievements

ガロア表現の保型性問題を解決するためには問題そのものよりも、その周辺の数学発展が高い水準で発展させることが重要な問題となる。具体的にはガロア表現のp進ホッジ論的性質、保型表現の分類や構成、等の発展が望まれる。また,保型的であるガロア表現を既存理論に見合うだけ豊富に構成することも重要である。成果は前半の重さの対応(局所的なラングランズ対応の一部)と上記後半の内容に対して寄与・意義があると考えている。