Infinite product presentation of the Mumford form and special values of geometric zeta functions

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400018
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Saga University
• Principal Investigator: ICHIKAWA Takashi 佐賀大学, 工学(系)研究科(研究院), 教授 (20201923)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 代数曲線 / モジュライ空間 / Chern-Simons不変量 / Arakelov理論 / Deligne-Riemann-Roch同型写像 / Schottky群 / Ruelleゼータ関数

Outline of Final Research Achievements

By the arithmetic Schottky-Mumford uniformization theory, we proved the arithmeticity of Chern-Simons invariants. Using this result together with the Arakelov theory in arithmetic geometry and the theory of Zograf, Mcintyre-Takhatajan on the classical Liouville field theory, we gave an infinite product presentation of the Deligne-Riemann-Roch isomorphism which expresses the Chern-Simons line bundle. As its application, we express the special values of the Ruelle zeta functions of Schottky groups as the products of period integrals and discriminants. This result gives an analog of the Deligne conjecture on
such geometric zeta values.

Free Research Field

数論幾何