Related papers: A local field approach to the Riemann Hypothesis

We define generalizations of classical invariants of wild ramification for coverings on a variety of arbitrary dimension over a local field. For an l-adic sheaf, we define its Swan class as a 0-cycle class supported on the wild ramification…

M. Levine proved an enrichment of the classical Riemann-Hurwitz formula to an equality in the Grothendieck-Witt group of quadratic forms. In its strongest form, Levine's theorem includes a technical hypothesis on ramification relevant in…

We give a characterization of finitely ramified $\omega$-pseudo complete valued fields of mixed characteristic $(0, p)$, with fixed residue field $k$ and value group $G$ of cardinality $\aleph_{1}$, in terms of a Hahn-like construction over…

We study the following question: given a global field $F$ and finite group $G$, what is the minimal $r$ such that there exists a finite extension $K/F$ with $\mathrm{Aut}(K/F)\cong G$ that is ramified over exactly $r$ places of $F$? We…

We construct explicitly APF extensions of finite extensions of $\qp$ for which the Galois group is not a p-adic Lie group and which do not have any open subgroup with $\zp$-quotient.

We give a characterization of ramification groups of local fields with imperfect residue fields, using those for local fields with perfect residue fields. As an application, we reprove an equality of ramification groups for abelian…

We show that the recent result of Casta\~neda and Wu about the ramification filtration in certain $p$-extensions of function fields of prime characteristic $p$ is equally valid over local fields of mixed characteristic $(0,p)$. Apart from…

Let k be a field of characteristic zero, let X be a geometrically integral k-variety of dimension n and let K be its field of fractions. Under the assumption that K contains all r-th roots of unity for an integer r, we prove that, given an…

We give a refinement of the local class field theory of Serre and Hazewinkel. This refinement allows the theory to treat extensions that are not necessarily totally ramified. Such a refinement was obtained and used in the authors' paper on…

We study on finite unramified extensions of global function fields (function fields of one valuable over a finite field). We show two results. One is an extension of Perret's result about the ideal class group problem. Another is a…

K. Kato's conjecture about the cohomological Hasse principle for regular connected schemes $\mathfrak X$ which are flat and proper over the complete discrete valuation rings $\mathcal O_N$ of higher local fields $F_N$ is proven. This…

For a smooth surface X over an algebraically closed field of positive characteristic, we consider the ramification of an Artin-Schreier extension of X. A ramification at a point of codimension 1 of X is understood by the Swan conductor. A…

The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and…

The focus of this note is on the Chow group problem over ramified regular local rings $(R, m)$. Our goal is threefold: i) to introduce a characterization of a ramified regular local ring essentially of finite type over a dvr, ii) to address…

Let X be a noetherian scheme defined over an algebraically closed field of positive characteristic p, and G be a finite group, of order divisible by p, acting on X. We introduce a refinement of the equivariant K-theory of X to take into…

We give a criterion whether given Eisenstein polynomials over a local field K define the same extension over K in terms of a certain non-Archimedean metric on the set of polynomials. The criterion and its proof depend on ramification…

For any prime power $q$, a polynomial $f(X)\in\F_q[X]$ is ``exceptional'' if it induces bijections of $\F_{q^k}$ for infinitely many $k$; this condition is known to be equivalent to $f(X)$ inducing a bijection of $\F_{q^k}$ for at least one…

Given a field $k$ of characteristic zero and an indeterminate $T$, the main topic of the paper is the construction of specializations of any given finite extension of $k(T)$ of degree $n$ that are degree $n$ field extensions of $k$ with…

Given a hilbertian field $k$ of characteristic zero and a finite Galois extension $E/k(T)$ with group $G$ such that $E/k$ is regular, we produce some specializations of $E/k(T)$ at points $t_0 \in \mathbb{P}^1(k)$ which have the same Galois…

This article discusses ramification and the structure of relative K\"ahler differentials of extensions of valued fields. We begin by surveying the theory developed in recent work with Franz-Viktor Kuhlmann and Anna Rzepka constructing the…