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Register a new userPrincipalization of ideals in abelian extensions of number fields
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Authors
Sebastien Bosca
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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 $k$
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Let $k$ be a field of characteristic $q$, $cac$ a smooth geometrically connected curve defined over $k$ with function field $K:=k(cac)$. Let $A/K$ be a non constant abelian variety defined over $K$ of dimension $d$. We assume that $q=0$ or $>2d+1$. Let $p e q$ be a prime number and $cactocac$ a finite geometrically textsc{Galois} and etale cover defined over $k$ with function field $K:=k(cac)$. Let $(tau,B)$ be the $K/k$-trace of $A/K$. We give an upper bound for the $bbz_p$-corank of the textsc{Selmer} group $text{Sel}_p(Atimes_KK)$, defined in terms of the $p$-descent map. As a consequence, we get an upper bound for the $bbz$-rank of the textsc{Lang-Neron} group $A(K)/tauB(k)$. In the case of a geometric tower of curves whose textsc{Galois} group is isomorphic to $bbz_p$, we give sufficient conditions for the textsc{Lang-Neron} group of $A$ to be uniformly bounded along the tower.
Let $pequiv 1,(mathrm{mod},9)$ be a prime number and $zeta_3$ be a primitive cube root of unity. Then $mathrm{k}=mathbb{Q}(sqrt[3]{p},zeta_3)$ is a pure metacyclic field with group $mathrm{Gal}(mathrm{k}/mathbb{Q})simeq S_3$. In the case that $mathrm{k}$ possesses a $3$-class group $C_{mathrm{k},3}$ of type $(9,3)$, the capitulation of $3$-ideal classes of $mathrm{k}$ in its unramified cyclic cubic extensions is determined, and conclusions concerning the maximal unramified pro-$3$-extension $mathrm{k}_3^{(infty)}$ of $mathrm{k}$ are drawn.
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 certain readily computable Kummer group $V_S$, which is always an isomorphism in the wild case. The tame case is much more mysterious. Our main result is that given a finite $X$ coprime to $p$, there exists a finite set of places $S$ coprime to $p$ such that $Sha_{Scup X} stackrel{simeq}{hookrightarrow} CyB_{Scup X} stackrel{simeq}{twoheadleftarrow} CyB_X hookleftarrow Sha_X$. In particular, we show that in the tame case $Sha_Y$ can {it increase} with increasing $Y$. This is in contrast with the wild case where $Sha_Y$ is nonincreasing in size with increasing $Y$.
- 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 X be an algebraic curve over Q and t a non-constant Q-rational function on X such that Q(t) is a proper subfield of Q(X). For every integer n pick a point P_n on X such that t(P_n)=n. We conjecture that, for large N, among the number fields Q(P_1), ..., Q(P_N) there are at least cN distinct. We prove this conjecture in the special case when t defines a geometrically abelian covering of the projective line, and the critical values of t are all rational. This implies, in particular, that our conjecture follows from a famous conjecture of Schinzel.
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