Related papers: The m-step Solvable Mono-anabelian Geometry of Num…

In this paper, we develop a new method to reconstruct an abelian number field $K$ from the maximal $6$-step solvable quotient of $G_K$ group- theoretically. The new aspect of this paper is that the results in this paper are independent from…

Given a number field $K$ and an integer $m\geq 0$, let $K_m$ denote the maximal $m$-step solvable Galois extension of $K$ and write $G_K^m$ for the maximal $m$-step solvable Galois group Gal$(K_m/K)$ of $K$. In this paper, we prove that the…

Let $K$ be a mixed-characteristic local field. For an integer $m \geq 0$, we denote by $K^m / K$ the maximal $m$-step solvable extension of $K$, and by $G_K^m$ the maximal $m$-step solvable quotient of the absolute Galois group $G_K$ of…

In this paper, we develop the main step in the global theory for the mod-$\ell$ analogue of Bogomolov's program in birational anabelian geometry for higher-dimensional function fields over algebraically closed fields. More precisely, we…

A main problem in Galois theory is to characterize the fields with a given absolute Galois group. We apply a K-theoretic method for constructing valuations to study this problem in various situations. As a first application we obtain an…

In \cite{Ho3}, Hoshi proved that open homomorphisms between solvably closed Galois groups of number fields which are compatible with the cyclotomic characters arise from field embeddings. In this paper, we will prove an $m$-step solvable…

In the present paper, we show a new result on the geometrically $2$-step solvable Grothendieck conjecture for genus $0$ curves over finitely generated fields. More precisely, we show that two genus $0$ hyperbolic curves over a finitely…

In this paper, we present some partial results for the geometrically m-step solvable Grothendieck conjecture in anabelian geometry. Among other things, we prove the geometrically 3-step solvable Grothendieck conjecture for genus 0 curves…

In this paper, we apply Hoshi's mono-anabelian reconstruction of number fields to establish a group-theoretic reconstruction of a number field K together with its maximal unramified outside S extension K_S for a density 1 subset of primes…

Anabelian geometry suggests that, for suitably geometric objects, their \'etale fundamental groups determine the geometric objects up to isomorphism. From a group-theoretic viewpoint, this philosophy requires rigidity properties, which…

The main purpose of this paper is to extend results on isomorphism types of the abelianized absolute Galois group $\mathcal G_K^{ab}$, where $K$ denotes imaginary quadratic field. In particular, we will show that if the class number $h_K$…

We investigate unramified extensions of number fields with prescribed solvable Galois group and certain extra conditions. In particular, we are interested in the minimal degree of a number field $K$, Galois over $\mathbb{Q}$, such that $K$…

We prove the existence of two non-isomorphic number fields $K$ and $L$ such that the maximal two-step nilpotent quotients of their absolute Galois groups are isomorphic. In particular, one may take $K$ and $L$ to be any of the imaginary…

In this paper, we present some new results on the geometrically m-step solvable Grothendieck conjecture in anabelian geometry. Specifically, we show the (weak bi-anabelian and strong bi-anabelian) geometrically m-step solvable Grothendieck…

Let (k1,k2,k3,k4) be a quartet of cyclic cubic number fields sharing a common conductor c=pqr divisible by exactly three prime(power)s p,q,r. For those components k of the quartet whose 3-class group Cl(3,k) = Z/3Z x Z/3Z is elementary…

We compute the Galois group of the splitting field $F$ of any irreducible and separable polynomial $f(x)=x^6+ax^3+b$ with $a,b\in K$, a field with characteristic different from two. The proofs require to distinguish between two cases:…

The absolute Galois group Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$ of the field $\mathbb{Q}$ of rational numbers can be presented as a highly computable object, under the notion of type-2 Turing computation. We formalize such a presentation…

Let $K$ be a number field and $\mathcal{C}$ a full class of finite groups. We write $K^{\mathcal{C}}/K$ for the maximal pro-$\mathcal{C}$ Galois extension of $K$, and $G_K^{\mathcal{C}}$ for its Galois group. In this paper, we deal with the…

The aim of Bogomolov's programme is to prove birational anabelian conjectures for function fields $K|k$ of varieties of dimension $\geq 2$ over algebraically closed fields. The present article is concerned with the 1-dimensional case. While…

Let $K$ be a number field and $K_{ur}$ be the maximal extension of $K$ that is unramified at all places. In a previous article, the first author found three real quadratic fields $K$ such that $Gal(K_{ur}/K)$ is finite and nonabelian simple…