The Super-Fermat Equation

Abstract

This chapter gives a detailed survey of the work done on the super-Fermat equation by many authors, assuming without proof the most difficult results. The parametrizations given for the elliptic case were initially obtained (with a few errors and omissions) by F. Beukers and D. Zagier, and completed by J. Edwards for the most difficult and interesting icosahedral case. Although I have included this chapter in the part dealing with “modern methods,” most of its contents is the treatment of the elliptic cases not including the icosahedral case. This is not at all modern, but is exactly the type of reasoning done using simple algebraic number theory that we have employed many times in Chapter 6. Sections 14.2, 14.3, and 14.4 should therefore not be studied directly (it would probably be rather boring to do so), but considered instead as exercises that the reader is invited to solve by himself without looking at the completely detailed solutions given in these sections. On the other hand, the solution to the icosahedral case, due to Beukers and Edwards, uses classical invariant theory, but in a very original manner linked to the modern theory of Grothendieck dessins d’enfants, and the results on the hyperbolic case use modern methods for finding rational points on curves of higher genus (Chapter 13), and the modular method of Ribet-Wiles (Chapter 15).