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02/10/05 Course notes posted Some of the notes from the course are posted below, in the notes section below.
01/18/05 Updated updated class time Please ignore all previous announcements! The class will now meet from 11-12:15 on Tuesdays and Thursdays. The meeting room will be the third floor conference room on Tuesdays, and Thackeray 524 on Thursdays.
01/10/05 Updated class time Starting from Monday January 17, the class will meet at 8am on Monday, Wednesday and Friday in room 524, Thackeray building.
01/10/05 Class meeting times We finally have a room and a time to meet! We'll meet on Tuesday and Thursdays from 11-12:15 in the 3rd floor conference room, Thackeray building.
01/04/05 Preferred meeting times needed I've sent out an email asking for preferred meeting times to everybody registered for the course. If you didn't get this email and you're intending to attend, please let me know as soon as possible.
01/03/05 Course website now online A first skeletal version of the course website now exists, at http://www.pitt.edu/~dickinsm/3065-052. The site will expand as the semester progresses; any feedback would be very welcome.
Course description Elliptic curves have applications in a surprisingly wide range of areas. One of those areas is algebraic number theory, where elliptic curves over the field rational numbers played a central role in Andrew Wiles' proof of Fermat's Last Theorem. A second apparently unrelated area where elliptic curves have had a major impact is cryptography, where cryptosystems based on elliptic curves over finite fields have some significant advantages over older public key cryptosystems. In this course we will study elliptic curves, with emphasis on their applications to the two areas described above. We'll spend some time introducing elliptic curves over an arbitrary field, and then develop their applications to number theory and cryptography.
Prerequisites The prerequisites for this class are fairly minimal: students should have a good grasp of basic abstract algebra (groups, rings, fields, vector spaces etc.), some memories of elementary number theory, and some degree of mathematical maturity.
Assessment The course grade will be based on class attendance and on a final presentation. There won't be any exams for this course. I will try to put some problem sets together every week or so. If you want to follow the course properly then it's highly recommended that you spend some time looking at these problem sets. However, they will be optional, and won't form part of the course grade.
Books See also the references below. The course will not be strongly tied to any particular book, and there's no required text for the course. But the book ‘Elliptic Curves: Number Theory and Cryptography’ by Lawrence C. Washington will serve as a good guide to what we plan to cover, and I'd recommend buying it if you can. The bible for elliptic curves in number theory is Silverman's two-volume Graduate Texts in Mathematics series: The Arithmetic of Elliptic Curves (GTM 106) and Advanced Topics in Elliptic Curves (GTM 151); the second volume is beyond the scope of the course. If you haven't seen any algebraic geometry before then the first couple of chapters of volume 1 are pretty tough; after that it gets easier. For elliptic curves in cryptography, one of the main sources I'll use will be ‘Elliptic Curves and Cryptography’ by Ian Blake, Gadiel Seroussi and Nigel Smart. As with Silverman's books, this text is a little above the level of the course, but it's one of the classic references. See the references below for more books.
Here's a list of books related to the course.
| [B] | Joe P. Buhler. Elliptic Curves, Modular Forms, and Applications. In ‘Arithmetic Algebraic Geometry&rsquo, edited by Brian Conrad and Karl Rubin. American Mathematical Society, 2001. ISBN 0821821733. Introduces elliptic curves, modular forms and their roles in number theory. This is the write-up of a series of lectures delivered at a 2000 conference in Park City, Utah (in fact, the conference at which the completed proof of the Shimura-Taniyama conjecture was announced). |
| [BSS] | Elliptic Curves in Cryptography. Ian Blake, Gadiel Seroussi and Nigel Smart. Cambridge University Press, 1999. ISBN 0521653746. This will be the main reference for the cryptography parts of the course. |
| [C] | Henri Cohen. A Course in Computational Algebraic Number Theory. Springer, 1993. ISBN 0387556400. Algorithms for algebraic number theory, from Euclid's algorithm all the way up to the Number Field Sieve (the fastest known general factoring algorithm). Contains algorithms for elliptic curves over the rationals as well as for elliptic curves modulo a prime. |
| [CSS] | Gary Cornell, Joseph H. Silverman and Glenn Stevens (editors). Modular Forms and Fermat's Last Theorem. Springer, 1997. ISBN 0387946098. Papers from a conference devoted to the proof of Fermat's Last Theorem. |
| [CP] | Richard Crandall and Carl Pomerance. Prime Numbers: a Computational Perspective. Springer, 2001. ISBN 0387947779. Contains sections on elliptic curve primality proving, the elliptic curve method for factorization, and point-counting on elliptic curves. |
| [H] | Dale Husemöller. Elliptic Curves, second edition. Springer, 2004. ISBN 0387954902. Elliptic curves, from definition through to some current research areas. Contains an appendix by Otto Forster on elliptic curves in algorithmic number theory and cryptography. |
| [Ki] | Frances Kirwan. Complex Algebraic Curves. Cambridge University Press, 1992. ISBN 052142353. A good low-level introduction to plane algebraic curves, though it treats only curves over the complex numbers. |
| [Ko] | Neal Koblitz. A Course in Number Theory and Cryptography, second edition. Springer, 1994. ISBN 0387942939. A readable introduction to elementary number theory and cryptography. Elliptic curve cryptography is introduced in the last chapter, along with information about elliptic-curve based primality tests and factorization methods. |
| [S0] | Joseph H. Silverman. A Friendly Introduction to Number Theory, second edition. Prentice Hall, 2001. ISBN 0130309540. A course in elementary number theory, ending with a discussion of points on elliptic curves modulo a prime. |
| [S1] | Joseph H. Silverman. The Arithmetic of Elliptic Curves. Springer, 1986. ISBN 0387962034. A thorough introduction to elliptic curves, covering elliptic curves over finite fields, the complex numbers, local fields and number fields. |
| [S2] | Joseph H. Silverman. Advanced Topics in the Arithmetic of Elliptic Curves. Springer, 1994. ISBN 0387943285. Successor to [S1]. Too advanced for this course, but an essential reference if you want to learn more about elliptic curves. |
| [TW] | Wade Trappe and Lawrence C. Washington. Introduction to Cryptography with Coding Theory. Prentice Hall, 2002. ISBN 0130618144. Includes a chapter on elliptic curves modulo a prime, with applications to cryptography and factoring. |
| [W] | Lawrence C. Washington. Elliptic Curves: Number Theory and Cryptography. Chapman & Hall/CRC, 2003. ISBN 1584883650. This book will be the main guide for the topics in the course. |
Here are some external links of varying relevance to the course. Let me know of anything that you think should be added here!
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