Related papers: Tame class field theory over local fields

In anabelian geometry, we consider to what extent the \'{e}tale or tame fundamental groups of schemes reflect geometric properties of the schemes. Although there are many known results (mainly for smooth curves) in this area, general…

Let $A$ be an abelian variety defined over a field $K.$ We study finite generation properties of the profinite group $\mathrm{Gal}(\Omega/K)$ and of certain closed normal subgroups thereof, where $\Omega$ is the torsion field of $A$ over…

We study the group Tame($\mathbf A^3$) of tame automorphisms of the 3-dimensional affine space, over a field of characteristic zero. We recover, in a unified and (hopefully) simplified way, previous results of Kuroda, Shestakov, Umirbaev…

Let $G$ be the fundamental group of a finite graph of groups with Noetherian edges and locally tame vertices. We prove that $G$ is locally tame. It follows that if a finitely presented group $H$ has a non-trivial $JSJ$-decomposition over…

Let $F$ be a non-archimedean local field of residue characteristic $p\neq 2$. Let $G$ be a connected reductive group over $F$ that splits over a tamely ramified extension of $F$. Yu constructed types which are called tame supercuspidal…

Let $K$ be a number field and $S$ a finite set of places of $K$. We study the kernels $\Sha_S$ of maps $H^2(G_S,\fq_p) \rightarrow \oplus_{v\in S} H^2(\G_v,\fq_p)$. There is a natural injection $\Sha_S \hookrightarrow \CyB_S$, into the dual…

Anabelian geometry with etale homotopy types generalizes in a natural way classical anabelian geometry with etale fundamental groups. We show that, both in the classical and the generalized sense, any point of a smooth variety over a field…

We first give simplified and corrected accounts of some results in \cite{PiRCP} on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing \cite{Turing} to give a simplified proof that any definable…

We define a class of pre-ordered abelian groups that we call finite-by-Presburger groups, and prove that their theory is model-complete. We show that certain quotients of the multiplicative group of a local field of characteristic zero are…

Let k be a non-archimedean local field with residual characteristic p. Let G be a connected reductive group over k that splits over a tamely ramified field extension of k. Suppose p does not divide the order of the Weyl group of G. Then we…

We introduce a notion of tame ramification for general finite covers. When specialized to the separable case, it extends to higher dimensions the classical notion of tame ramification for Dedekind domains and curves and sits nicely in…

These are notes of my lectures at the summer school "Higher-dimensional geometry over finite fields" in Goettingen, June--July 2007. We present a proof of Tate's theorem on homomorphisms of abelian varieties over finite fields (including…

Tame abstract elementary classes are a broad nonelementary framework for model theory that encompasses several examples of interest. In recent years, progress toward developing a classification theory for them have been made. Abstract…

We study a class of tame $\mathcal{L}$-theories $T$ of topological fields and their $\mathcal{L}_\delta$-extension $T_{\delta}^*$ by a generic derivation $\delta$. The topological fields under consideration include henselian valued fields…

We prove that the dual fine Selmer group of an abelian variety over the unramified $\mathbb{Z}_{p}$-extension of a function field is finitely generated over $\mathbb{Z}_{p}$. This is a function field version of a conjecture of…

Let X be a smooth variety over a field k, and l be a prime number invertible in k. We study the (\'etale) unramified H^3 of X with coefficients Q_l/Z_l(2) in the style of Colliot-Th\'el\`ene and Voisin. If k is separably closed, finite or…

In a recent paper it has been established that over an Artinian ring R all two-dimensional polynomial automorphisms having Jacobian determinant one are tame if R is a Q-algebra. This is a generalization of the famous Jung-Van der Kulk…

We prove an A'Campo type formula for the tame monodromy zeta function of a smooth and proper variety over a discretely valued field $K$. As a first application, we relate the orders of the tame monodromy eigenvalues on the $\ell$-adic…

In this note, we treat two dimensional complete local rings which are called "semi-stable local rings" and discuss the tame class filed theory. In 1987, Professor S. Saito completed the unramified class field theory of the general two…

Let $X$ be a smooth projective integral variety over a finitely generated field $k$ of characteristic $p>0$. We show that the finiteness of the exponent of the $p$-primary part of $\mathrm{Br}(X_{k^s})^{G_k}$ is equivalent to the Tate…