Relative unipotent completion of fundamental groups of modular curves and non-commutative motives

2019 Fiscal Year Final Research Report

• Project/Area Number: 18K13391
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Kyoto University
• Principal Investigator: Sakugawa Kenji 京都大学, 数理解析研究所, 特別研究員(PD) (80784214)
• Project Period (FY): 2018-04-01 – 2020-03-31
• Keywords: モジュラー曲線 / 相対的冪単基本群

Outline of Final Research Achievements

The objectives of this study are to (1) give a motivic consturuction of the relative completion of the topological fundamental groups of modular curves (2) Determine the concrete structure of the maximal mixed Tate quotient of the motivic relative completion constructed in (1) (3) Using the results of (2), we are going to study the Galois representations associated with elliptic newforms.
For (1), we have a somewhat satisfactory construction in the category of Arapura's motivic local systems. In the case (2), the problem was more difficult than expected and no result was obtained. On the other hand, some results about computations of the associated Galois representations with newforms were obtained without conditions.

Free Research Field

数論幾何学

Academic Significance and Societal Importance of the Research Achievements

DeligneとGoncharovのモチーフ的冪単基本群の構成は, その後の混合テイトモチーフの研究に於いて欠かせない基本的な道具となっている. 本研究の結果はその結果の楕円的な一般化であり, 楕円曲線は現在様々な場所で現れる基本的数学的対象であることから, 今後様々な研究のための基本的道具となると思われる. また, 保型形式に対応するガロワ表現を係数に持つガロワコホモロジーは保型形式の基本的な不変量である. 保型形式も楕円曲線同様に様々な場面で現れる数学的対象であり, 従ってその基本的な対象の不変量が部分的にでも計算できたことは今後の研究の大切なステップとなるかと思われる.