Extensions of Global Fields with Restricted Ramification and the Fontaine-Mazur Conjecture

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

The investigator and his collaborators will investigate certain "arithmetic fundamental groups," that is to say the Galois group of a global field (finite extension of the rational numbers or the function field of a curve over a finite field) with a specified (finite) set of branch points. They will study, in particular, extensions in which the filtration of the Galois group by higher ramification groups has fixed bounded length, especially in connection with a conjecture of Fontaine and Mazur. The proposed activity includes a computational component which seeks to construct infinite fundamental groups with extremal properties. Such explicit constructions give a good qualitative understanding of the asymptotic behavior of discriminant with respect to the degree (number fields) and genus with respect to the number of points (curves over finite fields). The latter has applications to the efficiency of certain codes made from number fields and function fields. This is a proposal in number theory, the oldest branch of mathematics where the fundamental question is that of solving polynomial equations. Knowledge regarding solutions of polynomial equations is of fundamental importance in cryptology and coding theory, the science of secure and efficient transmission of data. Solutions of polynomial equations in one unkown are equipped with a kind of algebraic symmetry and the study of these symmetries is called "Galois theory." These symmetries are encoded in a mathematical object called the Absolute Galois Group, which contains such a wealth of information about equations in not just one but any number of variables that it is sometimes known as the "Holy Grail of Number Theory." This proposal is a theoretical and computational investigation of certain aspects of the Absolute Galois Group.