Let $F$ be a number field with ring of integers $o_F$. For a fixed prime number $p$ and $i\ge 2$ the étale wild kernels $W{K}_{2i-2}^{\acute{\mathrm{e}}\mathrm{t}}\left(F\right)$ are defined as kernels of certain localization maps on the $i$-fold twist of the $p$-adic étale cohomology groups of $\mathrm{spec}{o}_F[\frac{1}{p}]$. These groups are finite and coincide for $i=2$ with the $p$-part of the classical wild kernel $W{K}_2\left(F\right)$. They play a role similar to the $p$-part of the $p$-class group of $F$. For class groups, Galois co-descent in a cyclic extension $L/F$ is described by the ambiguous class formula given by genus theory....

Our main result combines three topics: it contains a Grunwald-Wang type conclusion, a version of Hilbert’s irreducibility theorem and a $p$-adic form à la Harbater, but with good reduction, of the Regular Inverse Galois Problem. As a consequence we obtain a statement that questions the RIGP over $\mathbb{Q}$. The general strategy is to study and exploit the good reduction of certain twisted models of the covers and of the associated moduli spaces.

We give a classification of finite group actions on a $K3$ surface giving rise to $K3$ quotients, from the point of view of their fixed points. It is shown that except two cases, each such group gives rise to a unique type of fixed point set.

This paper considers some refined versions of the Inverse Galois Problem. We study the local or global behavior of rational specializations of some finite Galois covers of ${\mathbb{P}}_{\mathbb{Q}}^1$.

Very little is known regarding the Galois group of the maximal $p$-extension unramified outside a finite set of primes $S$ of a number field in the case that the primes above $p$ are not in $S$. We describe methods to compute this group when it is finite and conjectural properties of it when it is infinite.

For a prime number l and for a finite Galois l-extension of function fields L / K over an algebraically closed field of characteristic p <> l, it is obtained the Galois module structure of the generalized Jacobian associated to L, l and the ramified prime divisors. In the cyclic case an implicit integral representation of the Jacobian is obtained and this representation is compared with the explicit representation.

Let $E/F$ be a Galois extension of number fields with $\Gamma =$ Gal$(E/F)$ and with property that the divisors of $(E:F)$ are non-ramified in $E/\mathbf{Q}$. We denote the ring of integers of $E$ by ${𝒪}_E$ and we study ${𝒪}_E$ as a $\mathbf{Z}\Gamma$-module. In particular we show that the fourth power of the (locally free) class of ${𝒪}_E$ is the trivial class. To obtain this result we use Fröhlich’s description of class groups of modules and his representative for the class of ${ℰ}_E$, together with new determinantal congruences for cyclic group rings and corresponding congruences...

The main results of this paper may be loosely stated as follows.Theorem.— Let $N$ and $N^{\text{'}}$ be sums of Galois algebras with group $\Gamma$ over algebraic number fields. Suppose that $N$ and $N^{\text{'}}$ have the same dimension $\mathbb{Q}$ and that they are identical at their wildly ramified primes. Then (writing ${𝒪}_N$ for the maximal order in $N$)$${𝒪}_N\oplus {𝒪}_N\oplus \mathbb{Z}\Gamma {\cong }_{\mathbb{Z}\Gamma }\oplus {𝒪}_N^{\text{'}}\oplus {𝒪}_{N^{\text{'}}}\oplus \mathbb{Z}\Gamma .$$In many cases ${𝒪}_N{\cong }_{\mathbb{Z}\Gamma }{𝒪}_{N^{\text{'}}}.$The role played by the root numbers of $N$ and $N^{\text{'}}$ at the symplectic characters of $\Gamma$ in determining the relationship between the $\mathbb{Z}\Gamma$-modules ${𝒪}_N$ and ${𝒪}_{N^{\text{'}}}$ is described. The theorem includes...