Arithmetic refinement of mirror symmetry for K3 surfaces

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540004
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Hokkaido University of Education
• Principal Investigator: GOTO YASUHIRO 北海道教育大学, 教育学部, 教授 (40312425)
• Research Collaborator: YUI Noriko クイーンズ大学, 数理科学研究科, 教授
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 数論幾何 / Ｋ３曲面 / ミラー対称性 / デルサルト型曲面 / ゼータ関数 / 形式群 / モジュラリティ

Outline of Final Research Achievements

We studied mirror symmetry for K3 surfaces over finite fields and number fields, and aimed to refine their mirror-symmetric properties from arithmetic viewpoints. The K3 surfaces we considered are algebraic surfaces in weighted projective 3-spaces with non-symplectic involution and defined by the equations of Delsarte type or by one-dimensional deformations of them. For such K3 surfaces, we computed the height of their formal Brauer groups and refined mirror symmetry for K3 surfaces using the height of formal groups. We then observed mirror symmetry among the zeta-functions of such K3 surfaces and also in their special values described in terms of Jacobi sums etc. Furthermore, we showed the modularity of L-functions of the K3 surfaces we considered.

Free Research Field

数論的代数幾何