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Math 686R Section 1

Winter 2014

323 TMCB
3:00-3:50 PM MWF

Instructor: Darrin Doud
Office: 282 TMCB
Office Hours: MWF 4:00 or by appointment

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Course Description

This semester we will study the 1995 proof by Andrew Wiles of Fermat's Last Theorem. Although we will probably not be able to get into a great deal of technical detail, we will attempt to study the major tools that Wiles used to achieve his proof. In particular, we will study elliptic curves, modular forms and Galois representations. Although our efforts will be focused on understanding the proof of FLT, we will occasionally discuss other aspects of these three topics which are of interest, such as the Birch and Swinnerton-Dyer conjecture, and connections arithmetic cohomology and modular forms/Galois representations. The course will be primarily a discussion course--students will prepare material to present, but much of the time in class will be taken up with questions and discussion of the answers. A list of topics will be posted as the semester progresses.

Course Documents

Syllabus

Course Resources

Note that many of these may only be accessible from on-campus computers.

Modular elliptic curves and Fermat's Last Theorem by Andrew Wiles

Ring-theoretic properties of certain Hecke algebras by Richard Taylor and Andrew Wiles

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The Arithmetic of Elliptic Curves by Joseph Silverman

A First Course in Modular Forms by Fred Diamond and Jerry Shurman

Algorithms for Modular Elliptic Curves by J. E. Cremona

Lectures on Serre's conjectures by Kenneth A. Ribet and William A. Stein

The proof of Fermat's Last Theorem by Nigel Boston

Fermat's Last Theorem by Henri Darmon, Fred Diamond, and Richard Taylor

The Shimura-Taniyama conjecture (d'apres Wiles) by Henri Darmon

Wiles' theorem and the arithmetic of elliptic curves by Henri Darmon

Deforming Galois representations by Barry Mazur

Explicit Construction of Universal Deformation Rings by Bart de Smit and H. W. Lenstra, Jr. (See also this video

Explicit deformations of Galois representations by Nigel Boston

Hecke algebras and Galois representations by Burcu Baran

On A⁴+B⁴+C⁴=D⁴ by Noam Elkies