Related papers: Diophantine problems over tamely ramified fields

We generalize the results in [16] to higher dimensional ramified spaces. For this purpose we introduce ramified manifolds and, as special cases, locally elementary polygonal ramified spaces (LEP spaces). On LEP spaces we develop a theory of…

Formal orbifolds are defined in higher dimension. Their \'etale fundamental groups are also defined. It is shown that the fundamental groups of formal orbifolds have certain finiteness property and it is also shown that they can be used to…

The classical Artin--Whaples approximation theorem allows to simultaneously approximate finitely many different elements of a field with respect to finitely many pairwise inequivalent absolute values. Several variants and generalizations…

Let $F/K$ be a finite Galois totally & wildly ramified extension of complete discrete valuation fields. We say that the extension has the Hasse-Arf property if the ramification jumps in upper numbering are integers. We give necessary…

Let $\phi(z)$ be a non-isotrivial rational function in one-variable with coefficients in $\overline{\mathbb{F}}_p(t)$ and assume that $\gamma\in\mathbb{P}^1(\overline{\mathbb{F}}_p(t))$ is not a post-critical point for $\phi$. Then we prove…

The Eynard-Orantin topological recursion relies on the geometry of a Riemann surface S and two meromorphic functions x and y on S. To formulate the recursion, one must assume that x has only simple ramification points. In this paper we…

We give an intrinsic parametrisation of the set of tamely ramified extensions of a local field with finite residue field and bring to the fore the role played by group cohomology. We show that two natural definitions of the cohomology class…

This article is devoted to studying the ramification of Galois torsors and of $\ell$-adic sheaves in characteristic $p>0$ (with $\ell\not=p$). Let $k$ be a perfect field of characteristic $p>0$, $X$ be a smooth, separated and quasi-compact…

We show that bounding ramification at infinity bounds fierce ramification. This answers positively a question of Deligne posed to the first named author.

We determine the distribution of discriminants of wildly ramified elementary-abelian extensions of local and global function fields in characteristic $p$. For local and rational function fields, we also give precise formulae for the number…

A finite \'etale map between irreducible, normal varieties is called tame, if it is tamely ramified with respect to all partial compactifications whose boundary is the support of a strict normal crossings divisor. We prove that if the…

Let $k$ be an infinite finitely generated field of characteristic $p>0$. Fix a separated scheme $X$ smooth, geometrically connected, and of finite type over $k$ and a smooth proper morphism $f:Y\rightarrow X$. The main result of this paper…

We prove a relative decidability result for perfectoid fields. This applies to show that the fields $\mathbb{Q}_p(p^{1/p^{\infty}})$ and $\mathbb{Q}_p(\zeta_{p^{\infty}})$ are (existentially) decidable relative to the perfect hull of $…

We consider positive solutions of a fractional Lane-Emden type problem in a bounded domain with Dirichlet conditions. We show that uniqueness and nondegeneracy hold for the asymptotically linear problem in general domains. Furthermore, we…

Fix a number field k. We prove that k* - k*^2 is diophantine over k. This is deduced from a theorem that for a nonconstant separable polynomial P(x) in k[x], there are at most finitely many a in k* modulo squares such that there is a…

We study the general problem of extremality for metric Diophantine approximation on submanifolds of matrices. We formulate a criterion for extremality in terms of a certain family of algebraic obstructions and show that it is sharp. In…

We prove, assuming resolution of singularities in positive characteristic, an analogue of Siegel's theorem on sum of squares in positive characteristic. The method of proof combines techniques from central simple algebras with model theory…

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…

We provide a characterisation of differentially large fields in arbitrary characteristic and a single derivation in the spirit of Blum axioms for differentially closed fields. In the case of characteristic zero, we use these axioms to…

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…