Arithmetic and the Abel Jacobi map on elliptic surfaces and their applications

2016 Fiscal Year Final Research Report

• Project/Area Number: 25610007
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Tokyo Metropolitan University
• Principal Investigator: TOKUNAGA HIROO 首都大学東京, 理工学研究科, 教授 (30211395)
• Research Collaborator: BANNAI SHINZO 茨城工業高等専門学校, 講師 (20732556) ; SHIRANE TAKETO 宇部工業高等専門学校, 准教授 (70615161) ; Guerville-Balle Benoit 東京学芸大学 ; Tumenbayar Kuhlan National University of Mongolia
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: 楕円曲面 / Mordell Weil群 / 整切断 / 2重切断 / Zariski ペア / contact conic

Outline of Final Research Achievements

We study arithmetic of rational points and Abel-Jacobi map for an elliptic curve　appeared as the generic fiber of an elliptic surface S over a projective line. More precisely, we study curves on S given by the sum of two rational points or the duplication of a rational point. We also study bisections on S and curves determined by the bisections.
As applications, we study quasi torus decompositions of plane curves and Zariski N plet. Precisely, we give examples of Zariski pairs for conic-line arrangements, Zariski N-plets for conic arrangements, and weak contact conics for plane quartic curves.

Free Research Field

代数学(代数幾何学)