Related papers: On Abelianized Absolute Galois Group of Global Fun…

Let F be a finitely generated field of characteristic zero and \Gamma<GL_n(F) a finitely generated subgroup. For an element g in \Gamma, let Gal(F(g)/ F) be the Galois group of the splitting field of the characteristic polynomial of g over…

Let $K$ be a number field and $A/K$ be an abelian variety of dimension $g$. Assuming that the image $G_{\ell^\infty}$ of the natural Galois representation attached to the Tate module $T_\ell(A)$ is $\operatorname{GSp}_{2g}(\mathbb{Z}_\ell)$…

We study the inverse Galois problem with local conditions. In particular, we ask whether every finite group occurs as the Galois group of a Galois extension of $\mathbb{Q}$ all of whose decomposition groups are cyclic (resp., abelian). This…

For a list $\cal{L}$ of finite groups and for a profinite group $G$, we consider the intersection $T(G)$ of all open normal subgroups $N$ of $G$ with $G/N$ in $\cal{L}$. We give a cohomological characterization of the epimorphisms…

Let $K$ be the function field of a smooth, irreducible curve defined over $\overline{\mathbb{Q}}$. Let $f\in K[x]$ be of the form $f(x)=x^q+c$ where $q = p^{r}, r \ge 1,$ is a power of the prime number $p$, and let $\beta\in \overline{K}$.…

A number field $K$ is Hilbert-Speiser if all of its tame abelian extensions $L/K$ admit NIB (normal integral basis). It is known that $\mathbb{Q}$ is the only such field, but when we restrict $\text{Gal}(L/K)$ to be a given group $G$, the…

We study the Galois symbol map associated to the multiplicative group and an abelian variety which has good ordinary reduction over a $p$-adic field. As a byproduct, one can calculate the "class group" in the view of the class field theory…

Let $p$ be an odd prime and $L/K$ a $p$-adic Lie extension whose Galois group is of the form $\mathbb{Z}_p^{d-1}\rtimes \mathbb{Z}_p$. Under certain assumptions on the ramification of $p$ and the structure of an Iwasawa module associated to…

Given a global field K and a positive integer n, there exists an abelian extension L/K (of exponent n) such that the local degree of L/K is equal to n at every finite prime of K, and is equal to two at the real primes if n=2. As a…

For a number field $K$, we consider $K^{\rm ta}$ the maximal tamely ramified algebraic extension of~$K$, and its Galois group $G^{\rm ta}_K= Gal(K^{ta}/K)$. Choose a prime $p$ such that $\mu_p \not \subset K$. Our guiding aim is to…

Associated to an abelian variety $A$ of dimension $g$ over a number field $K$ is a Galois representation $\rho_A\colon Gal(\bar{K}/K)\to GL_{2g}(\hat{\mathbb{Z}})$. The representation $\rho_A$ encodes the Galois action on the torsion points…

This paper studies the structure of the Mordell--Weil groups of semiabelian varieties over algebraic extensions of number fields whose absolute Galois group is finitely generated, with particular emphasis on that generated by a single…

We prove that if $f:X \rightarrow A$ is a morphism from a smooth projective variety $X$ to an abelian variety $A$ over a number field $K$, and $G$ is a subgroup of automorphisms of $X$ satisfying certain properties, and if a prime $p$…

Given a field $k$ and a finite group $G$, the Beckmann--Black problem asks whether every Galois field extension $F/k$ with group $G$ is the specialization at some $t_0 \in k$ of some Galois field extension $E/k(T)$ with group $G$ and $E…

We study the groups of rational points of abelian varieties defined over a finite field $ \mathbb{F}_q$ whose endomorphism rings are commutative, or, equivalently, whose isogeny classes are determined by squarefree characteristic…

Let $F$ be a finite field of characteristic $p>0$ with $q = p^{n}$ elements. In this paper, a complete characterization of the unit groups $U(FG)$ of group algebras $FG$ for the abelian groups of order $32$, over finite field of…

We classify all cubic function fields over any finite field, particularly developing a complete Galois theory which includes those cases when the constant field is missing certain roots of unity. In doing so, we find criteria which allow…

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…

We obtain a classification of the finite two-generated cyclic-by-abelian groups of prime-power order. For that we associate to each such group $G$ a list $\inv(G)$ of numerical group invariants which determines the isomorphism type of $G$.…

For every finite field F and every positive integer r, there exists a finite extension F' of F such that either SO(2r+1,F') or its simple derived group can be realized as a Galois group over Q. If the characteristic of F is 3 or 5 (mod 8),…