The investigator and his collaborators are developing a theory of Galois group actions on rooted trees, in analogy to the well-established theory of such actions on p-adic vector spaces. The Fontaine-Mazur conjecture and generalizations of it predict that for Galois groups of number field extensions unramified at p the latter actions have finite image whereas there should exist tree actions with infinite image. The investigator's program will identify these actions and hence these (as yet mysterious) Galois groups, allowing direct verification of Fontaine-Mazur in these cases. Possible spin-offs of this include improved root-discriminant bounds and a quantitative version of Fontaine-Mazur along the lines of Cohen-Lenstra heuristics, together with applications for the pro-p group theorists such as 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's Last Theorem. In particular his co-author, Taylor, has gone on to apply these techniques to many other longstanding problems. The only drawback is that these methods only work in one half of cases, the "p-ramified" ones. This proposal develops a new theory of Galois representations suited to handling the other half. The work of Taylor and Wiles proves cases of the fundamental Fontaine-Mazur conjecture, from which solutions to Fermat's Last Theorem and similar equations simply follow - in the other half the Fontaine-Mazur conjecture has many striking consequences and the new theory presents a program for verifying the conjecture and hence its corollaries.
