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- 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 $Gamma$ = 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 $sigma$ be a generator of Gal(K/k), we study the constraints posed on the arithmetic of L/K by the cyclic action of $sigma$ on $Gamma$, 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 $sigma$ 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.
Let $K$ be a number field and $S$ a finite set of places of $K$. We study the kernels $Sha_S$ of maps $H^2(G_S,fq_p) rightarrow oplus_{vin S} H^2(G_v,fq_p)$. There is a natural injection $Sha_S hookrightarrow CyB_S$, into the dual $CyB_S$ of a certai
Given a natural number n and a number field K, we show the existence of an integer ell_0 such that for any prime number ellgeq ell_0, there exists a finite extension F/K, unramified in all places above ell, together with a principally polarized abeli
Let E/F be a CM field split above a finite place v of F, let l be a rational prime number which is prime to v, and let S be the set of places of E dividing lv. If E_S denotes a maximal algebraic extension of E unramified outside S, and if u is a plac
We give the complete proof of a conjecture of Georges Gras which claims that, for any extension $K/k$ of number fields in which at least one infinite place is totally split, every ideal $I$ of $K$ principalizes in the compositum $Kk^{ab}$ of $K$ with the maximal abelian extension $k^{ab}$ of $k$
In this article we explicitly describe irreducible trinomials X^3-aX+b which gives all the cyclic cubic extensions of Q. In doing so, we construct all integral points (x,y,z) with GCD(y,z)=1, of the curves X^2+3Y^2 = 4DZ^3 and X^2+27Y^2=4DZ^3 as D va