Tree Representations and Probabilistic Zeta Functions

DMS-0300321波士顿，尼日利亚摘要标题：树表示研究人员和他的合作者正在发展一种关于根树上的伽罗华群作用的理论，类似于在p-进向量空间上的这种作用的成熟理论。Fontaine-Mazur猜想及其推广表明，对于在p处分解的数域扩张的Galois群，后者具有有限象，而应存在具有无限象的树作用。研究人员的程序将识别这些行为，从而识别这些(仍是神秘的)Galois群，从而允许在这些情况下直接验证Fontaine-Mazur。可能的副产品包括改进的根判别界限和沿着Cohen-Lenstrauristic的Fontaine-Mazur的量化版本，以及对Pro-p群理论家的应用，例如新的分支Pro-p群族。近年来，“伽罗瓦表示法”的使用使数论发生了革命性的变化，最著名的是威尔斯在他的费马大定理的证明中。特别是，他的合著者泰勒继续将这些技术应用于许多其他长期存在的问题。唯一的缺点是，这些方法只适用于一半的情况，即“p-分支”情况。这一建议发展了一种新的伽罗瓦表示理论，适用于处理另一半情况。Taylor和Wiles的工作证明了基本的Fontaine-Mazur猜想的情况，在Fontaine-Mazur猜想的另一半中，费马最后定理和类似方程的解简单地跟随着Fontaine-Mazur猜想的另一半，有许多显著的结果，新的理论提出了一个验证该猜想的程序，从而证明了它的推论。

DMS-0300321Boston, NigelAbstractTitle: Tree Representations The investigator and his collaborators are developing a theory ofGalois group actions on rooted trees, in analogy to the well-establishedtheory of such actions on p-adic vector spaces. The Fontaine-Mazur conjectureand generalizations of it predict that for Galois groups of number fieldextensions unramified at p the latter actions have finite image whereas there should exist tree actions with infinite image. The investigator'sprogram will identify these actions and hence these (as yet mysterious) Galoisgroups, allowing direct verification of Fontaine-Mazur in these cases.Possible spin-offs of this include improved root-discriminant bounds anda quantitative version of Fontaine-Mazur along the lines of Cohen-Lenstraheuristics, together with applications for the pro-p group theorists suchas new families of branch pro-p groups. Number theory has been revolutionized in recent years by the use of "Galois representations", most notably by Wiles in his proof of Fermat'sLast Theorem. In particular his co-author, Taylor, has gone on to applythese techniques to many other longstanding problems. The only drawbackis that these methods only work in one half of cases, the "p-ramified" ones.This proposal develops a new theory of Galois representations suited tohandling the other half. The work of Taylor and Wiles proves cases of thefundamental Fontaine-Mazur conjecture, from which solutions to Fermat'sLast Theorem and similar equations simply follow - in the other halfthe Fontaine-Mazur conjecture has many striking consequences and the newtheory presents a program for verifying the conjecture and hence itscorollaries.