Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the Shafarevich Group of Restricted Ramification Extensions of Number Fields in the Tame Case
187
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 $Y$.
rate research
Read More
- 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.
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 abelian variety A of dimension n over F such that the resulting ell-torsion representation rho_{A,ell} from G_F to GSp(A[ell](bar{F})) is surjective and everywhere tamely ramified. In particular, we realize GSp_{2n}(mathbb{F}_ell) as the Galois group of a finite tame extension of number fields F/F such that F is unramified above ell.
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 place of E dividing v, we show that any field embedding E_S to bar{E_u} has a dense image. The unramified outside S number fields we use are cut out from the l-adic cohomology of the simple Shimura varieties studied by Kottwitz and Harris-Taylor. The main ingredients of the proof are then the local Langlands correspondence for GL_n, the main global theorem of Harris-Taylor, and the construction of automorphic representations with prescribed local behaviours. We explain how stronger results would follow from the knowledge of some expected properties of Siegel modular forms, and we discuss the case of the Galois group of a maximal algebraic extension of Q unramified outside a single prime p and infinity.
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 $k$
For a number field $F$ and a prime number $p$, the $mathbb{Z}_p$-torsion module of the Galois group of the maximal abelian pro-$p$ extension of $F$ unramified outside $p$ over $F$, denoted as $mathcal{T}_p(F)$, is an important subject in abelian $p$-ramification theory. In this paper we study the group $mathcal{T}_2(F)=mathcal{T}_2(m)$ of the quadratic field $F=mathbb{Q}(sqrt{ m})$. Firstly, assuming $m>0$, we prove an explicit $4$-rank formula for $mathcal{T}_2(-m)$. Furthermore, applying this formula and exploring the connection of $mathcal{T}_2(-m)$ to the ideal class group of $mathbb{Q}(sqrt{-m})$ and the tame kernel of $mathbb{Q}(sqrt{m})$, we obtain the $4$-rank density of $mathcal{T}_2$-groups of imaginary quadratic fields. Secondly, for $l$ an odd prime, we obtain results about the $2$-divisibility of orders of $mathcal{T}_2(pm l)$ and $mathcal{T}_2(pm 2l)$. In particular we find that $#mathcal{T}_2(l)equiv 2# mathcal{T}_2(2l)equiv h_2(-2l)bmod{16}$ if $lequiv 7bmod{8}$ where $h_2(-2l)$ is the $2$-class number of $mathbb{Q}(sqrt{-2l})$. We then obtain density results for $mathcal{T}_2(pm l)$ and $mathcal{T}_2(pm 2l)$. Finally, based on our density results and numerical data, we propose distribution conjectures about $mathcal{T}_p(F)$ when $F$ varies over real or imaginary quadratic fields for any prime $p$, and about $mathcal{T}_2(pm l)$ and $mathcal{T}_2(pm 2 l)$ when $l$ varies, in the spirit of Cohen-Lenstra heuristics. Our conjecture in the $mathcal{T}_2(l)$ case is closely connected to Shanks-Sime-Washingtons speculation on the distributions of the zeros of $2$-adic $L$-functions and to the distributions of the fundamental units.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
