Tree Representations and Probabilistic Zeta Functions

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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.