On $p$-class groups and the Fontaine-Mazur conjecture

Answering a question by Stark, we show that for an infinite unramified pro-$p$-extension of a number field $k$, the $p$-class numbers of its finite subextensions tend to infinity. This is proven by means of a group-theoretical result on compact $p$-adic analytic groups. Furthermore, we provide an equivalent formulation of the Fontaine-Mazur conjecture for $p$-extensions unramified outside a finite set of primes not containing any prime above $p$.

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Published 13 October 2014