Deformations and Hecke Curves

Following Andrew Wiles' announcement of his work, in June 1993, and in view of the immense interest it met with, I felt it reasonable to give a two semester graduate course 1 covering topics that might be helpful to people who want to understand, and possibly pursue, the ~. / subject. In the Fall semester I covered the basic deformation theory of Galois representations (in detail) and Fontaine's theory (very briefly), all the material covered having been already available in published papers. The Spring semester was entirely devoted to a study the methods of Flach's article2 in some depth, these methods following on the work of Kolyvagin. The pages below comprise a (still very rough) draft of a compilation of course notes for the Spring semester. I would like to extend these notes in future drafts, writing the unwritten sections, completing the unfinished sections3, and adding a good deal more material (including a certain amount of numerical data). Nevertheless, incomplete as these notes are at present, I'm distributing them in the hope that they may be useful. Please let me know if you spot any errors, regret any omissions, or want any clarifications to be included in later drafts.