Tree representations of Galois groups, by Nigel Boston
Much work has gone into matrix representations of Galois groups, but
there is a whole new class of naturally occurring representations that
have as yet gone almost unnoticed. In fact, it is well-known in various
areas of mathematics that the main sources of totally disconnected groups are
matrix groups over local fields AND automorphism groups of locally finite
trees. It is perhaps surprising then that representations of Galois
groups into the latter have been almost ignored, while at the same time
Galois representations into the former have been enormously effective
in resolving long-standing problems in number theory.
These ``tree'' representations are
important as regards topics such as the unramified Fontaine-Mazur conjecture.
This conjecture states that any p-adic representation
of the Galois group of an extension unramified at p (and ramified at
only finitely many primes)
should have finite image. In other words, p-adic representations say
little about such Galois groups. This paper proposes
the conjecture that these Galois groups should, on the other hand, have
large image (measured by Hausdorff dimension) in the automorphism group of
a rooted tree.
Thus, they might allow us
to investigate the structure of the Galois group of infinite pro-p extensions (such
as Hilbert p-class towers), something unapproachable by standard
p-adic representation methods.
Nigel Boston <email@example.com>