### Explicit computation of Galois p-groups unramified at p, by Nigel Boston and Charles Leedham-Green

In this paper we introduce a new method for finding Galois groups by computer.
This
is particularly effective in the case of Galois groups of p-extensions
ramified at finitely many primes but unramified at the primes above p.
Such Galois groups have been regarded as amongst the most mysterious objects in
number
theory. Very little has hitherto been discovered regarding them despite
their importance in studying p-adic Galois representations unramified
at p. The conjectures of Fontaine-Mazur say that they should have no
p-adic analytic quotients (equivalently, the images of the Galois representations
should always be finite), and there are generalizations due to the
first author, suggesting that they should instead have `large' actions on
certain trees.
The idea is
to modify the p-group generation algorithm so that as it goes along,
it uses number-theoretical information to eliminate groups that cannot
arise as suitable quotients of the Galois group under investigation.
In the best cases we obtain a short list of candidates for the Galois
group. This leads to various conjectures.

Nigel Boston and Charles Leedham-Green <boston@math.uiuc.edu,C.R.Leedham-Green@qmw.ac.uk>