Galois Representations and Their Families
The study of systems of polynomial equations with rational coefficients has its origins in the mathematics of ancient Greece, and lies at the very heart of algebraic number theory and arithmetic geometry. Rather than studying one equation at a time, it is far more fruitful to study the set of all solutions to all equations at once: this set has an enormous wealth of structure, which is encoded by its group of symmetries, the absolute Galois group. This group is extremely complicated and full of mysteries, so to glimpse its inner workings we are led to study its continuous representations. Over the real or complex numbers, any such representation necessarily has finite image, and unfortunately much of the structure of the Galois group is lost. There is however a remarkable substitute for the real and complex numbers as coefficients: the field of p-adic numbers (for a prime p). In dramatic contrast, representations of the Galois group over the p-adic numbers rarely have finite image and their study is full of riches: indeed, p-adic Galois representations have played a pivotal role in some of the most spectacular advances in modern number theory, including the recent proofs of Fermat's Last Theorem, the Shimura-Taniyama Conjecture, Serre's Conjecture, and the Sato-Tate conjecture. The key feature of p-adic Galois representations (in no way shared by their real or complex counterparts) underlying these achievements is the fact that they can be deformed in continuous families.
In this talk, I will survey the exciting world of p-adic Galois representations and the groundbreaking work of H. Hida from the 1980's which first proved the existence of certain interesting families of them. I will then outline a new and purely geometric approach to Hida's work via recent progress in integral p-adic Hodge theory, and will give some indications as to how such an approach might be used to attack some unsolved problems in number theory.
This talk is aimed at a broad mathamatical audience. In particular, no prior acquaintance with Galois representations, Hodge theory (p-adic or otherwise), or the p-adic numbers will be assumed.
Colloquium Tea will be held at 3:30 in the first floor Commons Room