Galois Representations and Modular Forms

Harvard University, Cambridge MA

Investigators

Abstract

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 his previous NSF grant the PI proved (with Michael Harris) the local Langlands conjecture for GL(n) of a p-adic field; completed (with Christophe Breuil, Brian Conrad and Fred Diamond) the proof of the Shimura-Taniyama conjecture; found (with Kevin Buzzard, Mark Dickinson and Nick Shepherd-Barron) the first infinite families of non-soluble irreducible Artin representations for which one could prove the Artin conjecture; and established the meromorphic continuation and functional equation of the L-functions of all abelian varieties ``of GL(2)-type''. The PI will continue to work on related problems. In particular he will work with Michael Harris to generalise the work of Wiles and of Wiles and the PI from GL(2) to GL(n). There seems to be just one significant problem remaining in the way of a really useful result. If this problem can be overcome the PI will work with Michael Harris and Nick Shepherd-Barron on applications to the arithmetic of elliptic curves. The PI will also look for generalisations of his work with Michael Harris on the local Langlands conjecture for GL(n) to other reductive groups. 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.