Syllabus

Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Prerequisite

A course in algebra covering groups, rings, and fields, such as 18.703 Modern Algebra.

Corequisite

18.702 Algebra II (should be taken at the same time)

Description

This course is a computationally focused introduction to elliptic curves, with applications to number theory and cryptography. While this is an introductory course, we will (gently) work our way up to some fairly advanced material, including an overview of the proof of Fermat's Last Theorem.

Each of the topics listed below corresponds to roughly one week of lectures (a total of three hours).

  1. Introduction
  2. Efficient computation
  3. Isogenies and endomorphisms
  4. Elliptic curves over finite fields
  5. The discrete logarithm problem
  6. Integer factorization and primality proving
  7. Endomorphism rings
  8. Elliptic curves over the complex numbers
  9. Modular curves
  10. The theory of complex multiplication
  11. Isogeny graphs
  12. Divisors and pairings
  13. Modular forms and Fermat’s Last Theorem

Textbook and Notes

There is no required text; lecture notes will be provided.

We will make reference to material in the five books listed below. We will follow the Washington text most closely in the early stages of the course and rely more heavily on Milne and Silverman as we move into more advanced topics. The text by Cox gives a wonderful exposition of the theory of complex multiplication that really cannot be found anywhere else; we will use portions of it.

Washington, Lawrence C. Elliptic Curves: Number Theory and Cryptography. Chapman & Hall / CRC, 2008. ISBN: 9781420071467. (This resource may not render correctly in a screen reader.Errata (PDF)) [Preview with Google Books]. Online version.

Milne, James S. Elliptic Curves. BookSurge Publishing, 2006. ISBN: 9781419652578. (This resource may not render correctly in a screen reader.Addendum / erratum (PDF)). This resource may not render correctly in a screen reader.Online version (PDF - 1.5MB).

Silverman, Joseph H. The Arithmetic of Elliptic Curves. Springer-Verlag, 2009. ISBN: 9780387094939. (This resource may not render correctly in a screen reader.Errata (PDF)) [Preview with Google Books]. Online version.

———. Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, 1994. ISBN: 9780387943251. (This resource may not render correctly in a screen reader.Errata (PDF)). Online version.

Cox, David A. Primes of the Form x2 + ny2: Fermat, Class Field Theory, and Complex Multiplication. Wiley-Interscience, 1989. ISBN: 9780471506546. (This resource may not render correctly in a screen reader.Errata (PDF)). Online version.

The following two books give quite accessible introductions to elliptic curves from very different perspectives. You may find them useful as supplemental reading, but we will not use them in the course.

Blake, Ian F., Gadiel Seroussi, and Nigel P. Smart. Elliptic Curves in Cryptography. Cambridge University Press, 1999. ISBN: 9780521653749. [Preview with Google Books]

Silverman, Joseph H., and John T. Tate. Rational Points on Elliptic Curves. Springer-Verlag, 1994. ISBN: 9780387978253. [Preview with Google Books]

The following references provide introductions to algebraic number theory and complex analysis; neither of these topics is an official prerequistie for this course, but we will occasionally need to make use of their results.

Algebraic Number Theory Course Notes by James S. Milne.

Lang, Serge. Complex Analysis. Springer-Verlag, 2003. ISBN: 9780387985923.

Software

Some of the theorems and algorithms presented in lecture will be demonstrated using the Sage computer algebra system, which is based on Python. Most of the problem sets will contain at least one computationally focused problem, which you will likely want to use Sage to solve, but you are free to use other packages, or to simply write your own code, if you wish. In any case, you will be graded on your results, not your code.

Problem Sets

There will be weekly problem sets, each of which typically contains three to five multi-part problems. Typically you will not be required to solve all of the problems; you will be given the option to choose a subset that sums to 100 points. Some problems are purely theoretical in nature, while others are more computationally focused; those who prefer proofs to programming (or vice versa) can choose problems that appeal to their interests.

Problem sets are to be prepared in typeset form (typically via LaTeX) and submitted electronically as PDF files. Collaboration is permitted, but you must write up your own solutions and identify any collaborators, as well as any resources you used that are not listed above; there will be computational problems for which the correct answer will be different for every student, based on a unique identifier derived from your MIT ID.

Grading

Your grade will be determined by your average problem set score. Provided you fill in the weekly surveys, your lowest score will be dropped when computing the average. There are no exams and no final.