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

Abstract : — Let p be a prime number and K an algebraic number field. What is the arithmetic structure of 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 = 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.
Type de document :
Pré-publication, Document de travail
2017
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https://hal.archives-ouvertes.fr/hal-01622014
Contributeur : Christian Maire <>
Soumis le : mercredi 25 octobre 2017 - 14:53:15
Dernière modification le : vendredi 6 juillet 2018 - 15:18:04
Document(s) archivé(s) le : vendredi 26 janvier 2018 - 12:34:38

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  • HAL Id : hal-01622014, version 1
  • ARXIV : 1710.09214

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Farshid Hajir, Christian Maire. Analytic lie extensions of number fields with cyclic fixed points and tame ramification. 2017. 〈hal-01622014〉

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