A study of algorithms on the arithmetic of algebraic curves and Abelian varieties

2016 Fiscal Year Final Research Report

• Project/Area Number: 25800023
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Tokyo Metropolitan University
• Principal Investigator: UCHIDA Yukihiro 首都大学東京, 理工学研究科, 准教授 (90533258)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: アーベル多様体 / ヤコビ多様体 / 高さ関数 / 楕円曲線 / 超楕円曲線 / 暗号

Outline of Final Research Achievements

The arithmetic of algebraic curves and Abelian varieties is a very important subject in both theory and applications. We studied algorithms and their applications on the arithmetic of algebraic curves and Abelian varieties defined over a number field or a finite field. We obtained results on computationally hard problems for cryptography, attacks against a certain cryptosystem, computation of heights on Jacobian varieties, secret sharing using elliptic curves, Diophantine equations related to Waring’s problem, and counting points on curves of genus 2.

Free Research Field

数論幾何学