Recent Developments in Automorphic Forms and Applications

In the past decade there has been tremendous progress in the area of automorphic forms. The crown jewel was Wiles’ spectacular proof of Fermat’s Last Theorem in 1994; this was a by-product of his proof of the Taniyama- Shimura conjecture for semi-stable elliptic curves defined over Q. The remaining case of the Taniyama-Shimura conjecture was completely settled through the joint effort of Breuil, Conrad, Diamond, and Taylor. In another direction, the proof of the Local Langlands conjecture for GL(n) in all characteristics is now complete, with the finite characteristic case proved by Laumon, Rapoport and Stuhler and the characteristic zero case proved by Harris and Taylor and by Henniart. Finally, the proof of the Global Langlands conjecture for GL(n) over function fields is well underway. In addition to these developments, substantial progress has been made on many related subjects.