2017/10/25 by Farshid Hajir, Hajir, Farshid, Christian Maire +1
Computer Science · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1710.09214
arxiv created 2017/10/25 · openalex publication_date 2017/10/25 · arxiv updated 2017/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
- 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.