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 (particularly recent work of Kisin and Khare-Wintenberger). The PI proposes (with K. Buzzard and T. Gee) to prove the Artin conjecture for odd, two dimensional representations of the absolute Galois group of the rational numbers. He also proposes to investigate generalisations of Ihara's lemma to higher rank unitary groups. As the PI (with L. Clozel, M. Harris and N. Shepherd-Barron) has recently shown, this would imply the Sato-Tate conjecture for rational elliptic curves with multiplicative reduction somewhere. 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.