Elliptic Curves

• Grant number: 23/08279-6
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: September 01, 2023
• End date: August 31, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Roberto Carlos Alvarenga da Silva Junior
• Grantee: João Gabriel Halabi
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

Elliptic curves are central objects in mathematics. They are actually omnipresent in all the mathematics areas. In this project, we shall probe the theory of elliptic curves in the point of view of its structure as algebraic curves. It means, to study since its first geometric properties as a projective curve until the abelian group structure of its set of rational points. Hence, explore the celebrated Mordel-Weil Theorem and the problems behind the rank of an elliptic curve. Established the theory, the project aims some applications in cryptography and integers' factorization.