# Journal of the Ramanujan Mathematical Society

Volume 37, Issue 1, March 2022 pp. 63–85.

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

**Authors**:
Farshid Hajir and Christian Maire

**Author institution:**Department of Mathematics & Statistics, University of Massachusetts, Amherst MA 01003, USA

**Summary: **
Let p be a prime number and K an algebraic number field. What is the arithmetic
structure of infinite Galois extensions L/K having p-adic analytic Galois group
Γ = Gal(L/K)? The celebrated Tame Fontaine-Mazur conjecture predicts that such
extensions are either deeply ramified (at some prime dividing p) or ramified at
an infinite number of primes. In this work, we take up a study (initiated by
Boston) of this type of question under the assumption that L is Galois over some
subfield k of K such that [K : k] is a prime l ≠ p. Letting σ be a generator of
Gal(K/k), we study the constraints posed on the arithmetic of L/K by the cyclic
action of σ on Γ, focusing on the critical role played by the fixed points of
this action, and their relation to the ramification in L/K. The method of Boston
works only when there are no non-trivial fixed points for this action. We show
that even in the presence of arbitrarily many fixed points, the action of σ
places severe arithmetic conditions on the existence of finitely and tamely
ramified uniform p-adic analytic extensions over K, which in some instances
leads us to be able to deduce the non-existence of such extensions over K from
their non-existence over k.

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