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

调查员和他的合作者将调查某些"算术基本群"，也就是说，伽罗瓦集团的全球领域（有限的扩展的有理数或功能领域的曲线在有限领域）与一个指定的（有限）一套分支点。 他们将研究，特别是，扩展其中的过滤伽罗瓦群的高分歧群体具有fixedbounded长度，特别是在连接的猜想方丹和马祖尔。 拟议的活动包括一个计算组件，旨在构建具有极值性质的无限基本群。 这种明确的结构给出了一个很好的定性理解的渐近行为的判别式的程度（数域）和亏格的数量点（曲线在有限域）。 后者应用于由数域和函数域构成的某些代码的有效性。这是数论中的一个建议，数论是数学最古老的分支，其基本问题是解多项式方程。关于多项式方程解的知识在密码学和编码理论中具有根本的重要性，密码学和编码理论是安全和有效传输数据的科学。 未知多项式方程的解具有一种代数对称性，对这种对称性的研究称为"伽罗瓦理论"。这些对称性被编码在一个称为绝对伽罗瓦群的数学对象中，它包含了关于方程的丰富信息，不仅仅是一个变量，而是任何数量的变量，有时被称为“数论的圣杯”。“这个提议是对绝对伽罗瓦群某些方面的理论和计算研究。

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.