Tame Extensions Research Articles

Tame key polynomials

We introduce a new method of constructing complete sequences of key polynomials for simple extensions of tame fields. In our approach the key polynomials are taken to be the minimal polynomials over the base field of suitably constructed elements in its algebraic closure, with the extensions generated by them forming an increasing chain. In the case of algebraic extensions, we generalize the results to countably generated infinite tame extensions over henselian but not necessarily tame fields. In the case of transcendental extensions, we demonstrate the central role that is played by the implicit constant fields, which reveals the tight connection with the algebraic case.

The epsilon constant conjecture for higher dimensional unramified twists of (1)

AbstractLet $N/K$ be a finite Galois extension of p-adic number fields, and let $\rho ^{\mathrm {nr}} \colon G_K \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ be an r-dimensional unramified representation of the absolute Galois group $G_K$ , which is the restriction of an unramified representation $\rho ^{\mathrm {nr}}_{{{\mathbb Q}}_{p}} \colon G_{{\mathbb Q}_{p}} \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ . In this paper, we consider the $\mathrm {Gal}(N/K)$ -equivariant local $\varepsilon $ -conjecture for the p-adic representation $T = \mathbb Z_p^r(1)(\rho ^{\mathrm {nr}})$ . For example, if A is an abelian variety of dimension r defined over ${{\mathbb Q}_{p}}$ with good ordinary reduction, then the Tate module $T = T_p\hat A$ associated to the formal group $\hat A$ of A is a p-adic representation of this form. We prove the conjecture for all tame extensions $N/K$ and a certain family of weakly and wildly ramified extensions $N/K$ . This generalizes previous work of Izychev and Venjakob in the tame case and of the authors in the weakly and wildly ramified case.

Tame torsion, the tame inverse Galois problem, and endomorphisms

Fix a positive integer g and rational prime p. We prove the existence of a genus g curve C/mathbb {Q} such that the mod p representation of its Jacobian is tame by imposing conditions on the endomorphism ring. As an application, we consider the tame inverse Galois problem and are able to realise general symplectic groups as Galois groups of tame extensions of mathbb {Q}.

Convergence of p-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces

Let K be a non-archimedean local field, X a smooth and proper K-scheme, and fix a pluricanonical form on X. For every finite extension K ′ of K, the pluricanonical form induces a measure on the K ′ -analytic manifold X(K ′ ). We prove that, when K ′ runs through all finite tame extensions of K, suitable normalizations of the pushforwards of these measures to the Berkovich analytification of X converge to a Lebesgue-type measure on the temperate part of the Kontsevich–Soibelman skeleton, assuming the existence of a strict normal crossings model for X. We also prove a similar result for all finite extensions K ′ under the assumption that X has a log smooth model. This is a non-archimedean counterpart of analogous results for volume forms on degenerating complex Calabi–Yau manifolds by Boucksom and the first-named author. Along the way, we develop a general theory of Lebesgue measures on Berkovich skeleta over discretely valued fields.

Good reduction of K3 surfaces

Let$K$be the field of fractions of a local Henselian discrete valuation ring${\mathcal{O}}_{K}$of characteristic zero with perfect residue field$k$. Assuming potential semi-stable reduction, we show that an unramified Galois action on the second$\ell$-adic cohomology group of a K3 surface over$K$implies that the surface has good reduction after a finite andunramifiedextension. We give examples where this unramified extension is really needed. Moreover, we give applications to good reduction after tame extensions and Kuga–Satake Abelian varieties. On our way, we settle existence and termination of certain flops in mixed characteristic, and study group actions and their quotients on models of varieties.

Galois module structure of the square root of the inverse different in even degree tame extensions of number fields

Let G be a finite group and let N/E be a tamely ramified G-Galois extension of number fields whose inverse different CN/E is a square. Let AN/E denote the square root of CN/E. Then AN/E is a locally free Z[G]-module, which is in fact free provided N/E has odd order, as shown by Erez. Using M. Taylor's theorem, we can rephrase this result by saying that, when N/E has odd degree, the classes of AN/E and ON (the ring of integers of N) in Cl(Z[G]) are equal (and in fact both trivial). We show that the above equality of classes still holds when N/E has even order, assuming that N/E is locally abelian. This result is obtained through the study of the Fröhlich representatives of the classes of some torsion modules, which are independently introduced in the setting of cyclotomic number fields. Jacobi sums, together with the Hasse–Davenport formula, are involved in this study. Finally, when G is the binary tetrahedral group, we use our result in conjunction with Taylor's theorem to exhibit a tame G-Galois extension whose square root of the inverse different has nontrivial class in Cl(Z[G]).

The model theory of separably tame valued fields

A henselian valued field K is called separably tame if its separable-algebraic closure Ksep is a tame extension, that is, the ramification field of the normal extension Ksep|K is separable-algebraically closed. Every separable-algebraically maximal Kaplansky field is a separably tame field, but not conversely. In this paper, we prove Ax–Kochen–Ershov Principles for separably tame fields. This leads to model completeness and completeness results relative to the value group and residue field. As the maximal immediate extensions of separably tame fields are in general not unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax–Kochen–Ershov Principles. Our approach also yields alternate proofs of known results for separably closed valued fields.

The algebra and model theory of tame valued fields

Abstract A henselian valued field K is called a tame field if its algebraic closure K ~ ${\tilde{K}}$ is a tame extension, that is, the ramification field of the normal extension K ~ | K ${\tilde{K}|K}$ is algebraically closed. Every algebraically maximal Kaplansky field is a tame field, but not conversely. We develop the algebraic theory of tame fields and then prove Ax–Kochen–Ershov Principles for tame fields. This leads to model completeness and completeness results relative to value group and residue field. As the maximal immediate extensions of tame fields will in general not be unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax–Kochen–Ershov Principles. The results of this paper have been applied to gain insight in the Zariski space of places of an algebraic function field, and in the model theory of large fields.

A remark on Tate's algorithm and Kodaira types

We remark that Tate's algorithm to determine the minimal model of an elliptic curve can be stated in a way that characterises Kodaira types from the minimum of $v(a_i)/i$. As an application, we deduce the behaviour of Kodaira types in tame extensions of l

Units in real cyclic fields

Let $N/\mathbb{Q}$ be a real cyclic and tame extension of prime degree $l$ with $\Gamma=\mathscr{G}al(N/\mathbb{Q})$. We give the Hom description of the class of the torsion-free part of the group of units in $N$ in the class group of the order $\mathbb{Z}\Gamma/ (\sum_{\gamma \in \Gamma}\gamma )$. This representation depends only on the structure of the ideal class group of $N$ and determines the Galois module structure of the torsion-free part of the group of units in $N$ as an ideal of the $l$th cyclotomic field. Using this approach we derive necessary and sufficient conditions for all real and tame cyclic fields of prime degree to have Minkowski units. We extend also the class of known cyclic real fields with Minkowski units.

Springer’s theorem for tame quadratic forms over Henselian fields

A quadratic form over a Henselian-valued field of arbitrary residue characteristic is tame if it becomes hyperbolic over a tame extension. The Witt group of tame quadratic forms is shown to be canonically isomorphic to the Witt group of graded quadratic forms over the graded ring associated to the filtration defined by the valuation, hence also isomorphic to a direct sum of copies of the Witt group of the residue field indexed by the value group modulo 2.

Steinitz classes of tamely ramified Galois extensions of algebraic number fields

The Steinitz class of a number field extension K / k is an ideal class in the ring of integers O k of k, which, together with the degree [ K : k ] of the extension determines the O k -module structure of O K . We call R t ( k , G ) the set of classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group. We define A ′ -groups inductively, starting with abelian groups and then considering semidirect products of A ′ -groups with abelian groups of relatively prime order and direct products of two A ′ -groups. Our main result is that the conjecture about realizable Steinitz classes for tame extensions is true for A ′ -groups of odd order; this covers many cases not previously known. Further we use the same techniques to determine R t ( k , D n ) for any odd integer n. In contrast with many other papers on the subject, we systematically use class field theory (instead of Kummer theory and cyclotomic descent).

FINITE CYCLIC TAME EXTENSIONS OF kp((t))

Let p be a prime number and let ℚ/ℤ′ be the elements in ℚ/ℤ of order prime to p. Let [Formula: see text], where c is a prime power of p. We use characters and valuation theory to prove that Δ is a parameter space for the cyclic tame extensions of the formal Laurent series field kp((t)) of degree prime to p. Furthermore, we construct the cyclic tame extension corresponding to a given triple in Δ. The structure of finite cyclic tame extensions of the p-adic number fields was thoroughly investigated by A. A. Albert in 1935. Here we get the same result as consequence of our main theorem.

ON FINITE TAME EXTENSIONS OF VALUED FIELDS#

ABSTRACT Let v be a henselian valuation of arbitrary rank of a field K and v¯ be its unique prolongation to a fixed algebraic closure K¯ of K. For any α belonging to K¯\\K, let Δ K (α) (resp. ω K (α)) denote the invariants defined to be the minimum (resp. maximum) of the set {v¯(α − α’) | α’ ≠ α runs over K-conjugates of α}. In 1998, while correcting a result of James Ax, Khanduja proved that every finite extension of (K, v) contained in K¯ is tame, if and only if to each α ∈ K¯\\K, there corresponds a ∈ K such that v¯(α − a) ≥ Δ K (α). It was also shown that the analogue of the above result does not hold in general when K¯ is replaced by a finite extension of K (see J. Alg. 201, 1998, pp. 647–655). In this paper, similar characterizations of finite tame extensions are given, and some of the invariants associated with such an extension are explicitly determined.

A class of p-adic Galois representations arising from abelian varieties over

Let V be a p-adic representation of the absolute Galois group G of Open image in new window that becomes crystalline over a finite tame extension, and assume p≠2. We provide necessary and sufficient conditions for V to be isomorphic to the p-adic Tate module Vp(Open image in new window) of an abelian variety Open image in new window defined over Open image in new window. These conditions are stated on the filtered (φ,G)-module attached to V.

Central embedding problems, the arithmetic lifting property, and tame extensions of ℚ

Central embedding problems, the arithmetic lifting property, and tame extensions of ℚ

EXTENSIONS OF NUMBER FIELDS WITH WILD RAMIFICATION OF BOUNDED DEPTH

We consider p-extensions of number fields such that the filtration of the Galois group by higher ramification groups is of prescribed finite length. We extend well-known properties of tame extensions to this more general setting; for instance, we show that these towers, when infinite, are asymptotically good (an explicit bound for the root discriminant is given). We study the difficult problem of bounding the relation-rank of the Galois groups in question. Results of Gordeev and Wingberg imply that the relation-rank can tend to infinity when the set of ramified primes is fixed but the length of the ramification filtration becomes large. We show that all p-adic representations of these Galois groups are potentially semistable; thus, a conjecture of Fontaine and Mazur on the structure of tamely ramified Galois p-extensions extends to our case. Further evidence in support of this conjecture is presented.

A Characterization of Finite Tame Extensions

Let v be a henselian valuation of a field K. In this paper it is proved that any finite extension (K′, v′) of (K, v) is tame if and only if there exists α ≠ 0 in K′ such that v′(α) = v(Tr K′/K (α)) using elementary results of valuation theory. A special case of this result, when the characteristic of the residue field of v is ρ > 0 and (K′, v′)/(K, v) is an extension of degree ρ, was proved in 1990 by J. P. Tignol (J. Reine Angew. Math. 404 (1990) 1–38). 1991 Mathematics Subject Classification 12J10, 12J25, 13A18.

Metrizable extensions of dynamical systems from function algebras

Peters and Pennings introduced and studied in [9] and [11] a kind of extensions of dynamical systems induced by certain C*-algebras, which they calltame extensions. In this paper we study the behaviour of metrizable tame extensions when some of the most important dynamical concepts related to the metric (such as sources, sinks, saddles, the shadowing property and distallity) are considered. We will confirm that these extensions can be troublesome in this context. We show, however, that sources and sinks are preserved under certain conditions.

On Extensions of Dynamical Systems by Function Algebras

A kind of extensions of dynamical systems by function algebras, which are called tame extensions, was introduced in [5] and [6]. In this paper, we search for sufficient conditions under which tame extensions preserve recurrence, periodicity, almost periodicity, semisimplicity, etc. To this end we introduce and study a new (stronger) kind of recurrence, which may be of independent interest. We also provide some examples which show that our hypotheses cannot, in general, be relaxed.