Research on rational points on moduli of abelian varieties and related topics

2015 Fiscal Year Final Research Report

• Project/Area Number: 25800025
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Tokyo Denki University
• Principal Investigator: Arai Keisuke 東京電機大学, 未来科学部, 准教授 (80422393)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: 有理点 / モジュライ / アーベル多様体 / ガロア表現

Outline of Final Research Achievements

The problem on rational points is a basic problem in number theory; it is related to solving equations defined by polynomials. On the other hand, the problem on rational points on moduli spaces is an important subject in arithmetic geometry; it is also related to classifying geometric objects corresponding to rational points.
In this research, we have determined possible characters appearing in the Galois representations associated to abelian varieties. Also, by using this result, we have obtained criteria of non-existence of rational points on a Shimura curve, which is a certain moduli space of abelian varieties. Furthermore, we have several numerical examples, which make the results visible.

Free Research Field

数論幾何