Mathematical Research Letters

Volume 21 (2014)

Number 3

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

Pages: 469 – 477

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n3.a5

Author

Jochen Gärtner (Mathematisches Institut, Universität Heidelberg, Germany)

Abstract

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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