ABSTRACT
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 TaniyamaShimura conjecture for semi-stable elliptic curves defined over https://www.w3.org/1998/Math/MathML"> Q https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429258978/3c59d1b3-c997-442a-81c4-7652a0f9e6a5/content/eq2897.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . 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 https://www.w3.org/1998/Math/MathML"> G L ( n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429258978/3c59d1b3-c997-442a-81c4-7652a0f9e6a5/content/eq2898.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 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 https://www.w3.org/1998/Math/MathML"> G L ( n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429258978/3c59d1b3-c997-442a-81c4-7652a0f9e6a5/content/eq2899.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> over function fields is well underway. In addition to these developments, substantial progress has been made on many related subjects.
