相关论文: The $m$-step solvable anabelian geometry of number…

We give some positive answers to the following problem: Given a field $K$ and a continuous Galois representation $\rho:G_K \to GL_n(\mathbf{Q})$, construct an abelian variety $J/K$ of small dimension such that $\rho$ is a sub-representation…

For any odd prime $p$ and any imaginary quadratic field $K$, the $p$-tower group $G_K$ associated to $K$ is the Galois group over $K$ of the maximal unramified pro-$p$-extension of $K$. This group comes with an action of a finite group…

Given a generic field extension F/k of degree n>3 (i.e. the Galois group of the normal closure of F is isomorphic to the symmetric group $S_n$), we prove that the norm torus, defined as the kernel of the norm map $N:R_{F/k}(G_m)\to\G_m$, is…

Given a number field $K$ one associates to it the set $\Lambda_K$ of Dedekind zeta-functions of finite abelian extensions of $K$. In this short note we present a proof of the following Theorem: for any number field $K$ the set $\Lambda_K$…

Let $G$ be an algebraic group, $X$ a generically free $G$-variety, and $K=k(X)^G$. A field extension $L$ of $K$ is called a splitting field of $X$ if the image of the class of $X$ under the natural map $H^1(K, G) \mapsto H^1(L, G)$ is…

We study toric varieties over a field k that split in a Galois extension K/k using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation…

Let $K$ be a number field and $G$ a finite abelian group. We study the asymptotic behaviour of the number of tamely ramified $G$-extensions of $K$ with ring of integers of fixed realisable class as a Galois module.

Let $k\subseteq K$ be a finite Galois extension of fields with Galois group $G$. Let $\mathscr{G}$ be the automorphism $k$-group scheme of $K$. We construct a canonical $k$-subgroup scheme $\underline{G}\subset\mathscr{G}$ with the property…

Given an elliptic curve $E/k$ and a Galois extension $k'/k$, we construct an exact functor from torsion-free modules over the endomorphism ring ${\rm End}(E_{k'})$ with a semilinear ${\rm Gal}(k'/k)$ action to abelian varieties over $k$…

By exploiting the arithmetic homotopy of the moduli spaces of curves, Galois-Teichm\"uller theory stands at the interface of braid-mapping class groups and of anabelian geometry. Starting from the classical braid-theoretic construction of…

We study the trace form $q_L$ of $G$-Galois algebras $L/K$ when $G$ is a finite group and $K$ is a field of characteristic different from $2$. We introduce in this paper the category of $2$-reduced groups and, when $G$ is such a group, we…

We employ methods from homotopy theory to define new obstructions to solutions of embedding problems. By using these novel obstructions we study embedding problems with non-solvable kernel. We apply these obstructions to study the…

Let $K$ be a 2-dimensional global field of characteristic $\neq 2$, and let $V$ be a divisorial set of places of $K$. We show that for a given $n \geqslant 5$, the set of $K$-isomorphism classes of spinor groups $G = \mathrm{Spin}_n(q)$ of…

We show that for any given field $k$ and natural number $r\geq2$, every continuous extension of the absolute Galois group $\mathrm{Gal}_k$ by a finite group is the arithmetic fundamental group of a geometrically connected smooth projective…

Let $U/K$ be a smooth affine curve over a number field and let $L$ be an irreducible rank 3 $\overline{\mathbb Q}_{\ell}$-local system on $U$ with trivial determinant and infinite geometric monodromy around a cusp. Suppose further that $L$…

The isomorphism type of the Galois group G of finite 3-class field towers of quadratic number fields with 3-class group of type (9,9) is determined by means of Artin patterns which contain information on the transfer of 3-classes to…

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…

The K-theory of a functor may be viewed as a relative version of the K-theory of a ring. In the case of a Galois extension of a number field F/L with rings of integers A/B respectively, this K-theory of the "norm functor" is an extension of…

By a classical result of Neukirch and Uchida, a number field K is determined by the structure of its absolute Galois group Gal(K). We show that K is not determined by the structure of the Sylow subgroups of Gal(K), answering a question…

Let K be a cyclic Galois extension of degree f over k and T a generator of the Galois group. For any v=(v_1,... , v_m)\in K^m such that v is linearly independent over k, and any 0< d < m the Gabidulin-like code C(v, T , d) is a maximum rank…