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

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The investigator and his collaborators will investigate certain"arithmetic fundamental groups," that is to say the Galois group ofa global field (finite extension of the rational numbers or the functionfield of a curve over a finite field) with a specified (finite) set ofbranch points. They will study, in particular, extensions in which thefiltration of the Galois group by higher ramification groups has fixedbounded length, especially in connection with a conjecture of Fontaine andMazur. The proposed activity includes a computational component whichseeks to construct infinite fundamental groups with extremal properties. Such explicit constructions give a good qualitative understanding of theasymptotic behavior of discriminant with respect to the degree (numberfields) and genus with respect to the number of points (curves over finitefields). The latter has applications to the efficiency of certain codesmade from number fields and function fields.This is a proposal in number theory, the oldest branch of mathematicswhere the fundamental question is that of solving polynomial equations.Knowledge regarding solutions of polynomial equations is of fundamentalimportance in cryptology and coding theory, the science of secure andefficient transmission of data. Solutions of polynomial equations in oneunkown are equipped with a kind of algebraic symmetry and the study ofthese symmetries is called "Galois theory." These symmetries are encodedin a mathematical object called the Absolute Galois Group, which containssuch a wealth of information about equations in not just one but anynumber of variables that it is sometimes known as the "Holy Grail ofNumber Theory." This proposal is a theoretical and computationalinvestigation of certain aspects of the Absolute Galois Group.