Tree Representations and Probabilistic Zeta Functions

DMS-0300321 Boston，Nigel摘要标题：树表示法 研究者和他的合作者正在发展一种伽罗瓦群作用于有根树的理论，类似于这种作用于p-adic向量空间的成熟理论。Fontaine-Mazur定理及其推广预言，对于数域扩张在p处非分歧的Galois群，后者的作用有有限象，而应该存在有无限象的树作用.研究者的程序将识别这些行为，从而识别这些（迄今为止神秘的）伽罗瓦群，从而允许在这些情况下直接验证Fontaine-Mazur。其可能的衍生产品包括改进的根判别界和Fontaine-Mazur的量化版本，沿着科恩-伦斯特拉启发式的路线，以及支持p群理论家的应用，例如分支支持p群的新家族。 近年来，由于使用了 “伽罗瓦表示”，最著名的是怀尔斯在他的证明费马大定理。特别是他的合著者，泰勒，已经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-分歧”的情况。泰勒和怀尔斯的工作证明了方丹-马祖尔猜想的基本情况下，从该解决方案的费马的最后定理和类似的方程简单地遵循-在另一半方丹-马祖尔猜想有许多惊人的后果和新的理论提出了一个程序，以验证猜想，因此它的推论。

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.