18.783 - Elliptic Curves

# SYLLABUS

The course outline is now available. There is an OpenCourseWare version of the course as it was taught in 2013, but we will deviate from this a bit.

## Text Book

There is no required text; lecture notes will be provided. We will make reference to material in the following books, all of which can be accessed on-line from MIT (see MIT Libraries web page for offisite access). 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 any where else; we will use portions of it.

Elliptic Curves: Number Theory and Cryptography, Second Edtion, Lawrence C. Washington. (errata)

Elliptic Curves, J.S. Milne. (errata)

The Arithmetic of Elliptic Curves, Joseph H. Silverman. (errata)

Advanced Topics in the Arithmetic of Elliptic Curves, Joseph H. Silverman. (errata)

Primes of the form x2+ny2: Fermat, Class Field Theory, and Complex Multiplication, David A. Cox. (errata)

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 of them in the course.

Elliptic Curves in Cryptography, Blake, Seroussi, and Smart.

Rational Points on Elliptic Curves, Joseph H. Silverman and John Tate.

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, J.S. Milne.

Algebraic number theory and Fermat's last theorem, Ian Stewart and David Tall.

(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).

## 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 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 via e-mail as pdf-files by 4pm 10pm on the day they are due (the first problem set is due Friday, February 13). 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.

## Late Policy

Late problem sets turned in within 24 hours of the deadline will be accepted but incur a penalty of 1 point (out of 100) for each hour they are late. No problem sets will be accepted more than 24 hours after the deadline without prior arrangement or documentation from Student Support Services. If you have an outside conflict that you can anticipate (for example, visiting a prospective graduate school or traveling for a job interview) you can request an extension. Reasonable extension requests will typically be granted, provided they are made at least three days in advance.