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18.783 | Fall 2025 | Undergraduate

# Elliptic Curves

Lecture 1: Introduction to Elliptic Curves

Problem Set 1 out

Lecture 2: The Group Law and Weierstrass and Edwards Equations

Lecture 3: Finite Field Arithmetic

Problem Set 1 due

Problem Set 2 out

Lecture 4: Isogenies

Lecture 5: Isogeny Kernels and Division Polynomials

Problem Set 2 due

Problem Set 3 out

Lecture 6: Endomorphism Rings

Lecture 7: Hasse’s Theorem and Point Counting

Problem Set 3 due

Problem Set 4 out

Lecture 8: Schoof’s Algorithm

Lecture 9: Generic Algorithms for the Discrete Logarithm Problem

Problem Set 4 due

Problem Set 5 out

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

Problem Set 5 due

Lecture 11: Elliptic Curve Primality Proving (ECPP)

Problem Set 6 out

Lecture 12: Endomorphism Algebras

Problem Set 6 due

Lecture 13: Ordinary and Supersingular Curves

Problem Set 7 out

Lecture 14: Elliptic Curves over C (Part I)

Problem Set 7 due

Lecture 15: Elliptic Curves over C (Part II)

Problem Set 8 out

Lecture 16: Complex Multiplication (CM)

Lecture 17: The CM Torsor

Problem Set 8 due

Problem Set 9 out

Lecture 18: Riemann Surfaces and Modular Curves

Lecture 19: The Modular Equation

Problem Set 9 due

Problem Set 10 out

Lecture 20: The Hilbert Class Polynomial

Lecture 21: Ring Class Fields and the CM Method

Problem Set 10 due

Problem Set 11 out

Lecture 22: Isogeny Volcanoes

Lecture 23: The Weil Pairing

Problem Set 11 due

Problem Set 12 out

Lecture 24: Modular Forms and L-Functions

Lecture 25: Fermat’s Last Theorem

Problem Set 12 due

## 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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