Related papers: Finite ramification for preimage fields of postcri…

Let $K$ be a field of characteristic zero, $X$ and $Y$ be smooth $K$-varieties, and let $G$ be a algebraic $K$-group. Given two algebraic morphisms $\varphi:X\rightarrow G$ and $\psi:Y\rightarrow G$, we define their convolution…

Let G be a connected, real, semisimple Lie group contained in its complexification G_C, and let K be a maximal compact subgroup of G. We construct a K_C-G double coset domain in G_C, and we show that the action of G on the K-finite vectors…

We prove the Zariski dense orbit conjecture in positive characteristic for endomorphisms of $\mathbb{G}_a^N$ defined over $\overline{\mathbb{F}_p}$.

The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field $(K,\nu)$ and an extension $\omega$ of $\nu$ to a finite extension $L$ of $K$.…

We show that for k a perfect field of characteristic p, there exist endomorphisms of the completed algebraic closure of k((t)) which are not bijective. As a corollary, we resolve a question of Fargues and Fontaine by showing that for p a…

Let X be a normal variety such that $K_X$ is Q-Cartier, and let $f: X \rightarrow X$ be a finite surjective morphism of degree at least two. We establish a close relation between the irreducible components of the locus of singularities that…

In a previous paper, we stated a general almost purity theorem in the style of Faltings: if R is a ring for which the Frobenius maps on finite p-typical Witt vectors over R are surjective, then the integral closure of R in a finite \'etale…

Let $G$ be a connected reductive group defined over a finite field $\mathbb{F}_q$ of characteristic $p$, with Deligne--Lusztig dual $G^\ast$. We show that, over $\overline{\mathbb{Z}}[1/pM]$ where $M$ is the product of all bad primes for…

We show that a projective variety with an int-amplified endomorphism of degree invertible in the base field satisfies Bott vanishing. This is a new way to analyze which varieties have nontrivial endomorphisms. In particular, we extend some…

We study the model theory of deeply ramified fields of positive characteristic. Generalizing the perfect case treated in work by Jahnke and Kartas on the model theory of perfectoid fields, we obtain Ax-Kochen/Ershov principles for certain…

Let $K=k(C)$ be the function field of a curve over a field $k$ and let $X$ be a smooth, projective, separably rationally connected $K$-variety with $X(K)\neq\emptyset$. Under the assumption that $X$ admits a smooth projective model $\pi:…

Let ${\mathfrak{X}}$ be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Denote by ${\mathrm{k}}_{\mathfrak{X}}(G)$ the number of conjugacy classes ${\mathfrak{X}}$-maximal subgroups of a finite…

We prove that infinite p-adically discrete sets have Diophantine definitions in large subrings of some number fields. First, if K is a totally real number field or a totally complex degree-2 extension of a totally real number field, then…

Let $K$ be a number field and $S$ a set of primes of $K$. We write $K_S/K$ for the maximal extension of $K$ unramified outside $S$ and $G_{K,S}$ for its Galois group. In this paper, we prove the following generalization of the…

We define a certain finite set in set theory $\{x\mid\varphi(x)\}$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any…

Given a $p$-adic field $K$ and a prime number $\ell$, we count the total number of the isomorphism classes of $p^\ell$-extensions of $K$ having no intermediate fields. Moreover for each group that can appear as Galois group of the normal…

Let $k$ be an algebraically closed field of characteristic two, and let $G$ be isomorphic to $\mathbb{Z}/2\times\mathbb{Z}/2$. Suppose $X$ is a smooth projective irreducible curve over $k$ with a faithful $G$-action, and assume that the…

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…

Let $p$ be a prime number, $K$ a finite unramified extension of $\mathbb{Q}_p$ and $\mathbb{F}$ a finite extension of $\mathbb{F}_p$. Using perfectoid spaces we associate to any finite-dimensional continuous representation $\overline{\rho}$…

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…