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- Let p be a prime number and K an algebraic number field. What is thearithmetic structure of Galois extensions L/K having p-adic analytic Galoisgroup $\Gamma$ = Gal(L/K)? The celebrated Tame Fontaine-Mazur conjecturepredicts that such extensions are either deeply ramified (at some primedividing p) or ramified at an infinite number of primes. In this work, we takeup a study (initiated by Boston) of this type of question under the assumptionthat L is Galois over some subfield k of K such that [K : k] is a prime = p.Letting $\sigma$ be a generator of Gal(K/k), we study the constraints posed onthe arithmetic of L/K by the cyclic action of $\sigma$ on $\Gamma$, focusing onthe critical role played by the fixed points of this action, and their relationto the ramification in L/K. The method of Boston works only when there are nonon-trivial fixed points for this action. We show that even in the presence ofarbitrarily many fixed points, the action of $\sigma$ places severe arithmeticconditions on the existence of finitely and tamely ramified uniform p-adicanalytic extensions over K, which in some instances leads us to be able todeduce the non-existence of such extensions over K from their non-existenceover k.

Analytic lie extensions of number fields with cyclic fixed points and tame ramification