A big theme in number theory in the last 50 years has been the relationship between automorphic forms, Galois representations and objects from algebraic geometry. There is an extensive web of extraordinary conjectures (for instance the Artin conjecture, the Shimura-Taniyama conjecture, Langlands' conjectures,Serre's conjecture and the Fontaine-Mazur conjecture) linking these three seemingly very different subjects (which relate to analysis, algebra and geometry respectively). Progress on these conjectures is currently very exciting. Under previous NSF grants the PI, with various collaborators, completed the proof of the Shimura-Taniyama conjecture; proved the local Langlands conjecture for GL(n) over a p-adic field; proved the Sato-Tate conjecture for elliptic curves over totally real fields; proved the first general automorphy lifting theorems and potential automorphy theorems for Galois representations of arbitrary dimension; and proved that the L-function of any polarized, regular, irreducible motive over a CM field has meromorphic continuation to the whole complex plane and satisfies the expected functional equation. The PI proposes to continue to improve the currently available automorphy lifting and potential automorphy theorems; to relate the cohomology of Rapoport- Zink spaces to the local Langlands conjecture for groups other than GL(n); with Kevin Buzzard and Joe Rabinoff to prove the Artin conjecture for odd degree two representations of the Galois group of a totally real field in which 5 splits completely; and to think about more speculative problems relating Galois representations and automorphic forms, for instance how to understand the case of very degenerate Hodge-Tate numbers/infinitesimal character. In addition the PI will continue his work with post-docs and, particularly, with graduate students.
This circle of ideas is the one that led to Andrew Wiles' celebrated proof of Fermat's last theorem after over 300 years. They fall into the general area of arithmetic geometry - a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.
