Explicit computation of Galois p-groups unramified at p

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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.