Studies of Mordell-Weil Lattices

2016 Fiscal Year Final Research Report

• Project/Area Number: 25400052
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Rikkyo University
• Principal Investigator: SHIODA Tetsuji 立教大学, 名誉教授, 名誉教授 (00011627)
• Co-Investigator(Renkei-kenkyūsha): AOKI Noboru 立教大学, 理学部, 教授 (30183130)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: 代数幾何学 / モーデル・ヴェイユ格子 / ガロア表現 / 有理楕円曲面 / フェルマー曲面 / ネロン・セヴェリ格子 / 高種数ファイブレーション

Outline of Final Research Achievements

The study of Mordell-Weil lattices has been focused on the following subjects.
1. The multiplicative excellent family of rational elliptic surfaces has the defining Weierstrass equation such that the coefficients form a set of fundamental invariants of the Weyl group in a Laurent polynomial ring. So the Mordell-Weil lattices, originally of Diophantine nature, can have a direct connection with the fundamental representations of Lie groups of the corresponding type, which admits various applications.
2. We have examined the structure of Mordell-Weil lattices of higher genus fibration on a Fermat surface. If the degree is relatively prime to 6, the structure (rank, height pairing etc.) is determined. More generally, the method applies to smooth surfaces containing a line.

Free Research Field

代数幾何学