Lectures: 2 sessions / week, 1.5 hours / session

A first course in algebra covering groups, rings, and fields such as *18.701 Algebra I*.

May be taken concurrently with *18.702 Algebra II* or *18.703 Modern Algebra*.

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).

- Introduction
- Efficient computation
- Isogenies and endomorphisms
- Elliptic curves over finite fields
- The discrete logarithm problem
- Integer factorization and primality proving
- Endomorphism rings
- Elliptic curves over the complex numbers
- Modular curves
- The theory of complex multiplication
- Divisors and pairings
- Modular forms and Fermat's Last Theorem

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. (errata (PDF)) [Preview with Google Books]

Milne, J. S. *Elliptic Curves*. BookSurge Publishing, 2006. ISBN: 9781419652578. (This book is also available online at the author's website, along with addendum / erratum (PDF).)

Silverman, Joseph H. *The Arithmetic of Elliptic Curves*. Springer-Verlag, 2009. ISBN: 9780387094939. (errata (PDF)) [Preview with Google Books]

———. *Advanced Topics in the Arithmetic of Elliptic Curves*. Springer-Verlag, 1994. ISBN: 9780387943251. (errata (PDF))

Cox, David A. *Primes of the Form X ^{2} + Ny^{2}: Fermat, Class Field Theory, and Complex Multiplication*. Wiley-Interscience, 1989. ISBN: 9780471506546.

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

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

The following references provide introductions to algebraic number theory, which is not part of this course per se, but may be useful background for those who are interested.

Algebraic Number Theory Course Notes by J. S. Milne.

Stewart, Ian, and David Orme Tall. *Algebraic Number Theory and Fermat's Last Theorem*. A. K. Peters / CRC Press, 2001. ISBN: 9781568811192.

The book by Stewart and Tall also includes some introductory material on elliptic curves and the proof of Fermat's last theorem, which are topics we will cover, but in greater depth.

Some of the theorems and algorithms presented in lecture are demonstrated using Sage, an open-source computer algebra system with extensive support for computing with elliptic curves. You can download a copy of Sage to run on your own machine if you wish, or create an account for free on the SageMathCloud™.

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 the code that produced them.

There will be weekly problem sets, each of which typically contain three to five multi-part problems. Typically you will not be required to solve all of the problems, you 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 there 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.

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.