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MATH529 ELLIPTIC CURVES

Course Objectives

The objective of this course is to provide a deep and unified introduction to the theory of elliptic curves, covering their algebraic structure, arithmetic properties, and applications. Beginning with the group law and Weierstrass equations, the course explores topics such as isogenies, torsion points, and division polynomials. Students will study elliptic curves over finite fields, including point counting and Schoof's algorithm, as well as applications to primality proving. The course further develops the theory over the rational and complex numbers, introducing the Mordell–Weil Theorem, complex multiplication, and the integrality of j-invariants, culminating in an overview of the role of elliptic curves in the proof of Fermat’s Last Theorem. The course aims to equip students with both theoretical foundations and computational tools, bridging modern algebra, number theory, and arithmetic geometry. This course provides a rigorous mathematical treatment of the subject while at the same time addressing issues of algorithmic implementations. This includes a thorough treatment of elliptic curve-related algorithms over finite fields and the theory of complex multiplication. These concepts are crucial to many theoretical and practical applications of elliptic curves.

Course Content

The group law and Weierstrass equation. Isogenies. Torsion points and division polynomials. Elliptic curve over finite fields. Determining the group order. Schoof's algorithm. Elliptic curve primality proving. Elliptic curves over rational numbers. Mordell-Weil Theorem. Elliptic Curves over complex numbers. Complex multiplication. Integrality of j-invariants. Fermat's last theorem.

Course Learning Outcomes

By the end of this course, students will be able to:

• Apply both theoretical and algorithmic tools to solve concrete problems involving elliptic curves.
• Define an elliptic curve using the Weierstrass equation and describe the group law on its points.
• Analyze and compute isogenies between elliptic curves and understand their properties.
• Determine and study torsion points on elliptic curves and compute division polynomials.
• Describe the structure of the group of rational points on an elliptic curve over a finite field.
• Use point-counting techniques to determine the order of the group and understand the Hasse bound.
• Implement or analyze Schoof's algorithm for efficient point counting over finite fields.
• Understand the principles of elliptic curve primality proving (ECPP) and its applications in computational number theory.
• Describe the Mordell–Weil theorem and understand the structure of the group of rational points on an elliptic curve over the rational numbers.
• Perform computations involving rank, torsion subgroups, and rational points on elliptic curves.
• Understand the analytic theory of elliptic curves over ? via complex tori and the theory of complex multiplication.
• Explain the j-invariant, its integrality properties, and its role in classifying elliptic curves.

Program Outcomes Matrix

0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution