18.783 | Spring 2021 | Undergraduate

Elliptic Curves

Lecture Notes and Worksheets

Lecture 1: Introduction to Elliptic Curves

Lecture 2: The Group Law and Weierstrass and Edwards Equations

Lecture 3: Finite Field Arithmetic

Lecture 4: Isogenies

Lecture 5: Isogeny Kernels and Division Polynomials

Lecture 6: Endomorphism Rings

Lecture 7: Hasse’s Theorem and Point Counting

Lecture 8: Schoof’s Algorithm

Lecture 9: Generic Algorithms for the Discrete Logarithm Problem

Lecture 10: Index Calculus, Smooth Numbers, and Factoring Integers

Lecture 11: Elliptic Curve Primality Proving (ECPP)

Lecture 12: Endomorphism Algebras

Lecture 13: Ordinary and Supersingular Curves

Lecture 14: Elliptic Curves over C (Part I)

Lecture 15: Elliptic Curves over C (Part II)

Lecture 16: Complex Multiplication (CM)

Lecture 17: The CM Torsor

Lecture 18: Riemann Surfaces and Modular Curves

Lecture 19: The Modular Equation

Lecture 20: The Hilbert Class Polynomial

Lecture 21: Ring Class Fields and the CM Method

Lecture 22: Isogeny Volcanoes

Lecture 23: The Weil Pairing

Lecture 24: Modular Forms and L-Functions

Lecture 25: Fermat’s Last Theorem

Course Info

As Taught In
Spring 2021
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Problem Sets
Lecture Notes
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