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18.783 | Fall 2025 | Undergraduate

# Elliptic Curves

All lecture notes in one file: Elliptic Curves (PDF)

### Lecture 1: Introduction to Elliptic Curves

- Introduction to Elliptic Curves (notes) (PDF)
- Introduction to Elliptic Curves (slides) (PDF)

### Lecture 2: The Group Law and Weierstrass and Edwards Equations

- Elliptic Curves as Abelian Groups (notes) (PDF)
- [slides not available]
- Proof of Associativity (worksheet) (IPYNB)
- Group Law on Edwards Curves (worksheet) (IPYNB)

### Lecture 3: Finite Field Arithmetic

- Finite Field Arithmetic (notes) (PDF)
- Finite Field Arithmetic (slides) (PDF)
- Root-Finding in Finite Fields (worksheet) (IPYNB)
- Root-Finding and Polynomial Factorization over Finite Fields (YouTube)

### Lecture 4: Isogenies

- Isogenies (notes) (PDF)
- Isogenies (slides) (PDF)

### Lecture 5: Isogeny Kernels and Division Polynomials

- Isogeny Kernels and Division Polynomials (notes) (PDF)
- Isogeny Kernels and Division Polynomials (slides) (PDF)
- Division Polynomials (worksheet) (IPYNB)

### Lecture 6: Torsion Subgroups and Endomorphism Rings

- Torsion Subgroups and Endomorphism Rings (notes) (PDF)
- Torsion Subgroups and Endomorphism Rings (slides) (PDF)

### Lecture 7: Point Counting

- Point Counting (notes) (PDF)
- Point Counting (slides) (PDF)

### Lecture 8: Schoof’s Algorithm

- Schoof’s Algorithm (notes) (PDF)
- Schoof’s Algorithm (slides) (PDF)
- Schoof’s Algorithm (worksheet) (IPYNB)

### Lecture 9: The Discrete Logarithm Problem

- The Discrete Logarithm Problem (notes) (PDF)
- The Discrete Logarithm Problem (slides) (PDF)

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

- Index Calculus, Smooth Numbers, and Factoring Integers (notes) (PDF)
- Index Calculus, Smooth Numbers, and Factoring Integers (slides) (PDF)
- Index Calculus (worksheet) (IPYNB)
- Pollard p-1 Algorithm (worksheet) (IPYNB)
- Montgomery Elliptic Curve Factorization Method (worksheet) (IPYNB)

### Lecture 11: Primality Proving

- Primality Proving (notes) (PDF)
- Primality Proving (slides) (PDF)

### Lecture 12: Endomorphism Algebras

- Endomorphism Algebras (notes) (PDF)
- Endomorphism Algebras (slides) (PDF)

### Lecture 13: Ordinary and Supersingular Elliptic Curves

- Ordinary and Supersingular Elliptic Curves (notes) (PDF)
- Ordinary and Supersingular Elliptic Curves (slides) (PDF)

### Lecture 14: Elliptic Curves over C (Part I)

- Elliptic Curves over C (Part I) (notes) (PDF)
- Elliptic Curves over C (Part I) (slides) (PDF)

### Lecture 15: Elliptic Curves over C (Part II)

- Elliptic Curves over C (Part II) (notes) (PDF)
- Elliptic Curves over C (Part II) (slides) (PDF)

### Lecture 16: Complex Multiplication (CM)

- Complex Multiplication (CM) (notes) (PDF)
- Complex Multiplication (CM) (slides) (PDF)

### Lecture 17: The CM Torsor

- The CM Torsor (notes) (PDF)
- The CM Torsor (slides) (PDF)

### Lecture 18: Riemann Surfaces and Modular Curves

- Riemann Surfaces and Modular Curves (notes) (PDF)
- Riemann Surfaces and Modular Curves (slides) (PDF)

### Lecture 19: The Modular Equation

- The Modular Equation (notes) (PDF)
- The Modular Equation (slides) (PDF)

### Lecture 20: The Hilbert Class Polynomial

- The Hilbert Class Polynomial (notes) (PDF)
- The Hilbert Class Polynomial (slides) (PDF)

### Lecture 21: Ring Class Fields and the CM Method

- Ring Class Fields and the CM Method (notes) (PDF)
- Ring Class Fields and the CM Method (slides) (PDF)

### Lecture 22: Isogeny Volcanoes

- Isogeny Volcanoes (notes) (PDF)
- Isogeny Volcanoes (slides) (PDF)

### Lecture 23: Divisors and the Weil Pairing

- Divisors and the Weil Pairing (notes) (PDF)
- Divisors and the Weil Pairing (slides) (PDF)

### Lecture 24: Modular Forms and L-Functions

- Modular Forms and L-Functions (notes) (PDF)
- Modular Forms and L-Functions (slides) (PDF)

### Lecture 25: Fermat’s Last Theorem

- Fermat’s Last Theorem (notes) (PDF)
- Fermat’s Last Theorem (slides) (PDF)

## Course Info

##### Instructor

- Andrew V. Sutherland

##### Departments

- Mathematics

##### As Taught In

Fall 2025

##### Level

Undergraduate

##### Topics

- Engineering
  - Computer Science
- Mathematics
  - Algebra and Number Theory
  - Computation

##### Learning Resource Types

notes Lecture Notes
menu_book Open Textbooks
assignment Problem Sets
auto_stories Readings

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