Related papers: On a variant of the Beckmann--Black problem

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…

We consider an infinite extension $K$ of a local field of zero characteristic which is a union of an increasing sequence of finite extensions. $K$ is equipped with an inductive limit topology; its conjugate $\bar{K}$ is a completion of $K$…

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 field of positive characteristic p and let G be a finite group. In this paper we study the category TsG of finitely generated commutative k-algebras A on which G acts by algebra automorphisms with surjective trace. If A = k[X],…

Let K/F be a finite Galois extension of global fields with Galois group G and let M be a 1-motive over F. We discuss the kernel and cokernel of the restriction map Sha^{i}(F,M) --> Sha^{i}(K,M)^{G} for i=1 and 2, independently of any…

We show that every linear algebraic group over an algebraically closed field of characteristic zero is the differential Galois group of a regular singular linear differential equation with rational function coefficients.

Let $k$ be a field of characteristic zero and ${\bar k}$ an algebraic closure of $k$. For a geometrically integral variety $X$ over $k$, we write ${\bar k}(X)$ for the function field of ${\bar X}=X\times_k{\bar k}$. If $X$ has a smooth…

Let $d \geq 1$, $k \geq 2$ and $n\geq d+1$ be integers. A $d$-dimensional smooth complex algebraic variety $M$ is called a generalized Fermat variety of type $(d;k,n)$ if there is a Galois holomorphic branched covering $\pi:M \to {\mathbb…

We discuss the following conjecture of Kitaoka: if a finite subgroup $G$ of $GL_{n}(O_{K})$ is invariant under the action of $Gal(K/\Bbb Q)$ then it is contained in $GL_{n}(K^{ab})$. Here $O_{K}$ is the ring of integers in a finite, Galois…

Let $K$ be a complete discrete valuation field of characteristic zero with residue field $k_K$ of characteristic $p>0$. Let $L/K$ be a finite Galois extension with Galois group $G=\Gal(L/K)$ and suppose that the induced extension of residue…

Consider a number field $K$ and a rational function $f$ of degree greater than 1 over $K$. By taking preimages of $\alpha\in K$ under successive iterates of $f$, an infinite $d$-ary tree $T_\infty$ rooted at $\alpha$ can be constructed. An…

For any algebraically closed field $K$ and any endomorphism $f$ of $\mathbb{P}^1(K)$ of degree at least 2, the automorphisms of $f$ are the M\"obius transformations that commute with $f$, and these form a finite subgroup of…

We introduce a condition for Hopf-Galois extensions that generalizes the notion of Kummer Galois extension. Namely, an $H$-Galois extension $L/K$ is $H$-Kummer if $L$ can be generated by adjoining to $K$ a finite set $S$ of eigenvectors for…

Let k be a global field, p an odd prime number different from char(k) and S, T disjoint, finite sets of primes of k. Let G_S^T(k)(p)=Gal(k_S^T(p)|k) be the Galois group of the maximal p-extension of k which is unramified outside S and…

Let $k$ be a global field, $K/k$ be a finite separable field extension and $L/k$ be the Galois closure of $K/k$ with Galois groups $G={\rm Gal}(L/k)$ and $H={\rm Gal}(L/K)\lneq G$. In 1931, Hasse proved that if $G$ is cyclic, then the Hasse…

Let k be a global field, p an odd prime number different from char(k) and S, T disjoint, finite sets of primes of k. Let G_S^T(k)(p)=Gal(k_S^T(p)|k) be the Galois group of the maximal p-extension of k which is unramified outside S and…

We extend to the context of algebraic groups a classic result on extensions of abstract groups relating the set of isomorphism classes of extensions of $G$ by $H$ with that of extensions of $G$ by the center $Z$ of $H$. The proof should be…

We describe the Segal $K$-theory of the symmetric monoidal category of finite-dimensional vector spaces over a perfect field $\mathbb{F}$ together with an automorphism, or, equivalently, the group-completion of the $E_\infty$-algebra of…

We prove that every place of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension F' of F. We show that F'|F can be chosen to be normal. If K is perfect and P is of rank 1, then…

We show that each connected group scheme of finite type over an arbitrary ground field is isomorphic to the component of the identity inside the automorphism group scheme of some projective, geometrically integral scheme. The main…