MIT OpenCourseWare

Browse materialsMIT 18.704 · Fall 2004

Seminar in Algebra and Number Theory: Rational Points on Elliptic Curves

Subjects: Algebra · Linear Algebra · Mathematics

Art: Johannes Kepler, Konjukce Saturnu a Jupiteru Originál, 1596 · Public domain

Instructor: Dr. Daniel Rogalski
Term: Fall 2004
Level: Undergraduate
What's inside: 1 lecture note · 1 reading · 92 other files · 4 web pages
Materials open: MIT OCW

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About this course

Source: MIT OpenCourseWare 18.704 Seminar in Algebra and Number Theory: Rational Points on Elliptic Curves, Fall 2004, Dr. Daniel Rogalski — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

Materials are grouped the way MIT organized them. A verified, course-wide order has not been established here.

Before you start

Prerequisites are not listed here. Check MIT's own syllabus for this course.

This course’s cited books, in our reading lists: Rational Points on Elliptic Curves

How this course was organized

What's here, and what isn't

A lecture sequence of 36 items is confirmed. Assignment and exam materials exist too, but MIT's own order for them could not be verified, so they're shown as materials below rather than a sequence.

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1 item from the source could not be safely included:

Referenced in a table, but without MIT's own verified order, so it isn't used to claim a sequence.

Materials by section

Grouped by section, as MIT organized them. Not sequenced or ordered.

Lecture Notes

39 files

Assignments

20 files

Projects

8 files

Other materials from this course

25 files

MIT OpenCourseWare, Dr. Daniel Rogalski, 18.704 Seminar in Algebra and Number Theory: Rational Points on Elliptic Curves, Fall 2004.

License: CC BY-NC-SA 4.0 · Edition: Fall 2004 · Open the original course on OCW ↗

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Seminar in Algebra and Number Theory: Rational Points on Elliptic Curves

Before you start

What's here, and what isn't

Not included here

Materials by section

Lecture Notes

Assignments

Projects

Other materials from this course

Lecture 1: The Projective Plane

Lecture 2: Curves in the Projective Plane

Lecture 3: Rational Points on Conics

Lecture 4: Geometry of Cubic Curves

Lecture 5: Weierstrass Normal Form

Lecture 6: Explicit Formulas for the Group Law

Lecture 7: Points of Order Two and Three

Lecture 8: The Discriminant Points of Finite Order have Integer Coordinates - Part 1

Lecture 9: Points of Finite Order have Integer Coordinates - Part 2

Lecture 10: Points of Finite Order have Integer Coordinates - Part 3 The Nagell-Lutz Theorem

Lecture 11: Real and Complex Points on Cubics

Lecture 12: Heights and Descent

Lecture 13: Height of P + P_0

Lecture 14: Height of 2P

Lecture 15: A Useful Homomorphism - Part 1

Lecture 16: A Useful Homomorphism - Part 2

Lecture 17: Mordell’s Theorem - Part 1

Lecture 18: Mordell’s Theorem - Part 2 Examples - Part 1

Lecture 19: Examples - Part 2

Lecture 20: Examples - Part 3

Lecture 21: Singular Cubics

Lecture 22: Rational Points over Finite Fields

Lecture 23: Gauss’s Theorem - Part 1

Lecture 24: Gauss’s Theorem - Part 2

Lecture 25: Points of Finite Order Revisited

Lecture 26: Factorization using Elliptic Curves - Part 1

Lecture 27: Factorization using Elliptic Curves - Part 2

Lecture 28: Integer Points on Cubics Taxicabs - Part 1

Lecture 29: Taxicabs - Part 2 Thue’s Theorem - Part 1

Lecture 30: Thue’s Theorem - Part 2

Lecture 31: Construction of an Auxiliary Polynomial

Lecture 32: The Auxiliary Polynomial is Small

Lecture 33: The Auxiliary Polynomial Does Not Vanish

Lecture 34: Proof of the DAT Further Developments

Lecture 35: Congruent Numbers and Elliptic Curves I: Koblitz - Part 1

Lecture 36: Congruent Numbers and Elliptic Curves II: Koblitz - Part 2