Related papers: The Local Lifting Problem for $D_4$

Linear differential equations with polynomial coefficients over a field $K$ of positive characteristic $p$ with local exponents in the prime field have a basis of solutions in the differential extension $\mathcal{R}_p=K(z_1, z_2,…

Let $L/K$ be a finite Galois extension of local fields. The Hasse-Arf theorem says that if Gal$(L/K)$ is abelian then the upper ramification breaks of $L/K$ must be integers. We prove the following converse to the Hasse-Arf theorem: Let $G$…

For a given positive integer $n$ and $K/\mathbb{Q}_p$ a finite extension of ramification degree $e$, we determine the number of finite Galois extensions $L/K$ with inertia degree $f$ and a single nonnegative ramification jump at $n$ as long…

We consider the fourth-order Schr\"odinger equation $$ i\partial_tu+\Delta^2 u+\mu\Delta u+\lambda|u|^\alpha u=0, $$ where $\alpha>0,\mu=\pm1$ or $0$ and $\lambda\in\mathbb{C}$. Firstly, we prove local well-posedness in…

We consider the class of complete discretely valued fields such that the residue field is of prime characteristic p and the cardinality of a $p$-base is 1. This class includes two-dimensional local and local-global fields. A new definition…

Suppose $K$ is a finite field extension of $\mathbb{Q} _p$ containing a primitive $p$-th root of unity. Let $\Gamma _{<p}$ be the Galois group of a maximal $p$-extension of $K$ with the Galois group of period $p$ and nilpotent class $<p$.…

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…

Let $K$ be a finite extension of $\mathbf{Q}_p$. The field of norms of a strictly APF extension $K_\infty/K$ is a local field of characteristic $p$ equipped with an action of $\mathrm{Gal}(K_\infty/K)$. When can we lift this action to…

We prove a lifting theorem for odd Frattini covers of finite groups. Using this, we characterize solvable groups and more generally p-solvable groups in terms of containing a triple of elements of distinct prime power orders with product 1.…

Given a $G$-Galois branched cover of the projective line over a number field $K$, we study whether there exists a closed point of $\mathbb{P}^1_K$ with a connected fiber such that the $G$-Galois field extension induced by specialization…

Let $R$ be a 2-dimensional normal excellent henselian local domain in which 2 is invertible and let $L$ and $k$ be respectively its fraction field and residue field. Let $\Omega_R$ be the set of rank 1 discrete valuations of $L$…

Let K be an algebraically closed field. We prove that a polynomial K-derivation $D$ in two variables is locally nilpotent if and only if the subgroup of polynomial K-automorphisms which commute with D admits elements whose degree is…

Special covers are metacyclic covers of the projective line, with Galois group of order pm, which have a specific type of bad reduction to characteristic p. Such covers arise in the study of the arithmetic of Galois covers of the projective…

Two extension problems are solved. First, the class of locally matricial algebras over an arbitrary field is closed under extensions. Second, the class of locally finite dimensional semisimple algebras over a fixed field is closed under…

Let $k$ be a number field and let ${\mathcal{A}}$ be a ${\rm GL}_2$-type variety defined over $k$ of dimension $d$. We show that for every prime number $p$ satisfying certain conditions (see Theorem 2), if the local-global divisibility…

Let $K$ be a local field of characteristic $p$ and let $L/K$ be a totally ramified Galois extension such that Gal$(L/K)\cong C_{p^n}$. In this paper we find sufficient conditions for $L/K$ to admit a Galois scaffold. This leads to…

In this article, we discuss the local rigidity of Clifford-Klein forms of homogeneous spaces of 1-connected completely solvable Lie groups. In fact, we introduce a splitting of the local rigidity: vertical rigidity and horizontal rigidity.…

Let $K$ be a local field of characteristic 0 with residue characteristic $p$. Let $G$ be an extraspecial $p$-group and let $L/K$ be a totally ramified $G$-extension. In this paper we find sufficient conditions for $L/K$ to admit a Galois…

We investigate the local descents for special orthogonal groups over p-adic local fields of characteristic zero, and obtain an explicit spectral decomposition of the local descents at the first occurrence index in terms of the local…

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…