Henselian Valuation Research Articles

Having the same wild ramification is preserved by the direct image

Let S be the spectrum of an excellent henselian discrete valuation ring of residue characteristic p and X a separated scheme over S of finite type. Let $$\Lambda $$ and $$\Lambda '$$ be finite fields of characteristics $$\ell \ne p$$ and $$\ell '\ne p$$ respectively. For elements $$\mathcal {F}\in K_{c}(X,\Lambda )$$ and $$\mathcal {F}'\in K_{c}(X,\Lambda ')$$ of the Grothendieck groups of the categories of constructible sheaves of $$\Lambda $$ -modules and $$\Lambda '$$ -modules on X respectively, we introduce the notion that $$\mathcal {F}$$ and $$\mathcal {F}'$$ have the same wild ramification and prove that this condition is preserved by four of Grothendieck’s six operations except the derived tensor product and $$R \mathcal {H}om$$ .

Characterizing diophantine henselian valuation rings and valuation ideals

We give a characterization, in terms of the residue field, of those henselian valuation rings and those henselian valuation ideals that are diophantine. This characterization gives a common generalization of all the positive and negative results on diophantine henselian valuation rings and diophantine valuation ideals in the literature. We also treat questions of uniformity and we apply the results to show that a given field can carry at most one diophantine nontrivial equicharacteristic henselian valuation ring or valuation ideal.

Defining Coarsenings of Valuations

AbstractWe study the question of which Henselian fields admit definable Henselian valuations (with or without parameters). We show that every field that admits a Henselian valuation with non-divisible value group admits a parameter-definable (non-trivial) Henselian valuation. In equicharacteristic 0, we give a complete characterization of Henselian fields admitting a parameter-definable (non-trivial) Henselian valuation. We also obtain partial characterization results of fields admitting -definable (non-trivial) Henselian valuations. We then draw some Galois-theoretic conclusions from our results.

A Cohomological Hasse Principle Over Two-dimensional Local Rings

Let $K$ be the fraction field of a two-dimensional henselian, excellent, equi-characteristic local domain. We prove a local-global principle for Galois cohomology with finite coefficients over $K$. We use classical machinery from \'etale cohomology theory, drawing upon an idea in Saito's work on two-dimensional local class field theory. This approach works equally well over the function field of a curve over an equi-characteristic henselian discrete valuation field, thereby giving a different proof of (a slightly generalized version of) a recent result of Harbater, Hartmann and Krashen. We also present two applications. One is the Hasse principle for torsors under quasi-split semisimple simply connected groups without $E_8$ factor. The other gives an explicit upper bound for the Pythagoras number of a Laurent series field in three variables. This bound is sharper than earlier estimates.

A definable Henselian valuation with high quantifier complexity

We give an example of a parameter‐free definable Henselian valuation ring which is neither definable by a parameter‐free ‐formula nor by a parameter‐free ‐formula in the language of rings. This answers a question of Prestel.

On the quantifier complexity of definable canonical Henselian valuations

We discuss definability in the language of rings without parameters of the unique canonical Henselian valuation of a field. We show that in most cases where the canonical Henselian valuation is definable, it is already definable by a universal‐existential or an existential‐universal formula.

Uniform definability of henselian valuation rings in the Macintyre language:

We discuss definability of henselian valuation rings in the Macintyre language L Mac , the language of rings expanded by nth power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian residue field are uniformly ∃- ∅-definable in L Mac , and henselian valuation rings with value group Z are uniformly ∃ ∀ - ∅-definable in the ring language, but not uniformly ∃- ∅-definable in L Mac . We apply these results to local fields Q p and F p ( ( t ) ) , as well as to higher dimensional local fields.

Characterizing Distinguished Pairs by Using Liftings of Irreducible Polynomials

AbstractLet v be a henselian valuation of any rank of a field K and let be the unique extension of v to a fixed algebraic closure of K. In 2005, we studied properties of those pairs (θ,α) of elements of with where α is an element of smallest degree over K such thatSuch pairs are referred to as distinguished pairs. We use the concept of liftings of irreducible polynomials to give a different characterization of distinguished pairs.

Uniformly defining p-henselian valuations

Admitting a non-trivial p-henselian valuation is a weaker assumption on a field than admitting a non-trivial henselian valuation. Unlike henselianity, p-henselianity is an elementary property in the language of rings. We are interested in the question when a field admits a non-trivial 0-definable p-henselian valuation (in the language of rings). We give a classification of elementary classes of fields in which the canonical p-henselian valuation is uniformly 0-definable. We then apply this to show that there is a definable valuation inducing the (t-)henselian topology on any (t-)henselian field which is neither separably closed nor real closed.

DEFINABLE HENSELIAN VALUATIONS

AbstractIn this note we investigate the question when a henselian valued field carries a nontrivial ∅-definable henselian valuation (in the language of rings). This is clearly not possible when the field is either separably or real closed, and, by the work of Prestel and Ziegler, there are further examples of henselian valued fields which do not admit a ∅-definable nontrivial henselian valuation. We give conditions on the residue field which ensure the existence of a parameter-free definition. In particular, we show that a henselian valued field admits a nontrivial henselian ∅-definable valuation when the residue field is separably closed or sufficiently nonhenselian, or when the absolute Galois group of the (residue) field is nonuniversal.

Fibrés principaux sur les corps valués henséliens

Let K be the field of fractions of a henselian valuation ring A. Assume that the completion b K is a separable extension of K. Let Y be a K-variety, let G be an algebraic group over K, and let f : X! Y be a G-torsor over Y . We consider the induced map X(K)! Y (K), which is continuous for the topologies deduced from the valuation. If I denotes the image of this map, we prove that I is locally closed in Y (K); moreover, the induced surjection X(K)! I is a principal bundle with group G(K) (also topologized by the valuation).

Constructing complete distinguished chains with given invariants

Let v be a henselian valuation of arbitrary rank of a field K with value group G(K) and residue field R(K) and [Formula: see text] be the unique extension of v to a fixed algebraic closure [Formula: see text] of K with value group [Formula: see text]. It is known that a complete distinguished chain for an element θ belonging to [Formula: see text] with respect to (K, v) gives rise to several invariants associated to θ, including a chain of subgroups of [Formula: see text], a tower of fields, together with a sequence of elements belonging to [Formula: see text] which are the same for all K-conjugates of θ. These invariants satisfy some fundamental relations. In this paper, we deal with the converse: Given a chain of subgroups of [Formula: see text] containing G(K), a tower of extension fields of R(K), and a finite sequence of elements of [Formula: see text] satisfying certain properties, it is shown that there exists a complete distinguished chain for an element [Formula: see text] associated to these invariants. We use the notion of lifting of polynomials to construct it.

Definable non-divisible Henselian valuations

On a Henselian valued field (K, V), where V is the valuation ring, if the value group contains a convex p-regular subgroup that is not p-divisible, then V is definable in the language of rings. A Henselian valuation ring with a regular non-divisible value group is always 0-definable. In particular, some results of Ax and of Konenigmann are generalized.

ON LIFTINGS OF POWERS OF IRREDUCIBLE POLYNOMIALS

Let v be a henselian valuation of arbitrary rank of a field K with valuation ring Rv having maximal ideal Mv. Using the canonical homomorphism from Rv onto Rv/Mv, one can lift any monic irreducible polynomial with coefficients in Rv/Mv to yield monic irreducible polynomials over Rv. Popescu and Zaharescu extended this approach and introduced the notion of lifting with respect to a residually transcendental prolongation w of v to a simple transcendental extension K(x) of K. As it is well known, the residue field of such a prolongation w is [Formula: see text], where [Formula: see text] is the residue field of the unique prolongation of v to a finite simple extension L of K and Y is transcendental over [Formula: see text] (see [V. Alexandru, N. Popescu and A. Zaharescu, A theorem of characterization of residual transcendental extension of a valuation, J. Math. Kyoto Univ.28 (1988) 579–592]). It is known that a lifting of an irreducible polynomial belonging to [Formula: see text] with respect to w, is irreducible over K. In this paper, we give some sufficient conditions to ensure that a given polynomial in K[x] satisfying these conditions which is a lifting of a power of some irreducible polynomial belonging to [Formula: see text] with respect to w, is irreducible over K. Our results extend Eisenstein–Dumas and generalized Schönemann irreducibility criteria.

Levi Factors of the Special Fiber of a Parahoric Group Scheme and Tame Ramification

Let \(\cal{A}\) be a Henselian discrete valuation ring with fractions K and with perfect residue field k of characteristic p > 0. Let G be a connected and reductive algebraic group over K, and let \(\cal{P}\) be a parahoric group scheme over \(\cal{A}\) with generic fiber \({\cal{P}}_{/K} = G\). The special fiber \({\cal{P}}_{/k}\) is a linear algebraic group over k. If G splits over an unramified extension of K, we proved in some previous work that the special fiber \({\cal{P}}_{/k}\) has a Levi factor, and that any two Levi factors of \({\cal{P}}_{/k}\) are geometrically conjugate. In the present paper, we extend a portion of this result. Following a suggestion of Gopal Prasad, we prove that if G splits over a tamely ramified extension of K, then the geometric special fiber \({\cal{P}}_{/k_{\rm{alg}}}\) has a Levi factor, where k alg is an algebraic closure of k.

On Invariants and Strict Systems of Irreducible Polynomials Over Henselian Valued Fields

Let g(x) be a monic irreducible defectless polynomial over a henselian valued field (K, v), i.e., K(θ) is a defectless extension of (K, v) for any root θ of g(x). It is known that a complete distinguished chain for θ with respect to (K, v) gives rise to several invariants associated with g(x). Recently Ron Brown studied certain invariants of defectless polynomials by introducing strict systems of polynomial extensions. In this article, the authors establish a one-to-one correspondence between strict systems of polynomial extensions and conjugacy classes of complete distinguished chains. This correspondence leads to a simple interpretation of various results proved for strict systems. The authors give new characterizations of an invariant γ g introduced by Brown.

Counting models of genus one curves

AbstractLetCbe a soluble smooth genus one curve over a Henselian discrete valuation field. There is a unique minimal Weierstrass equation definingCup to isomorphism. In this paper we consider genus one equations of degreendefiningC, namely a (generalised) binary quartic whenn= 2, a ternary cubic whenn= 3 and a pair of quaternary quadrics whenn= 4. In general, minimal genus one equations of degreenare not unique up to isomorphism. We explain how the number of these equations varies according to the Kodaira symbol of the Jacobian ofC. Then we count these equations up to isomorphism over a number field of class number 1.

On Brown’s constant associated with irreducible polynomials over henselian valued fields

Let v be a henselian valuation of arbitrary rank of a field K and v ̃ be the prolongation of v to the algebraic closure K ˜ of K with value group G ˜ . In 2008, Ron Brown gave a class P of monic irreducible polynomials over K such that to each g ( x ) belonging to P , there corresponds a smallest constant λ g belonging to G ˜ (referred to as Brown’s constant) with the property that whenever v ̃ ( g ( β ) ) is more than λ g with K ( β ) a tamely ramified extension of ( K , v ) , then K ( β ) contains a root of g ( x ) . In this paper, we determine explicitly this constant besides giving an important property of λ g without assuming that K ( β ) / K is tamely ramified.

Note on Conic Bundles over Henselian Discrete Valued Fields with Real Closed Residue Field

Let K be the fraction field of a Henselian discrete valuation ring. In Bazyleu et al. (2007) an explicit description of the central simple algebras of exponent 2 in the kernel of the natural map 2 Br(K(x)) → ∏ω∈Ω 2 Br(K(x)ω), (the product taken over all orderings ω and K(x)ω the real closure of K(x) at ω), is given. This allows to describe a class of conic bundle surfaces over K in terms of their local data (i.e., in terms of their degenerate fibres). Such a description is given in Theorem 3.1, the main result of this note.

Every projective Schur algebra is Brauer equivalent to a radical abelian algebra

We prove that any projective Schur algebra over a field K is equivalent in Br(K) to a radical abelian algebra. This was conjectured in 1995 by Sonn and the first author of this paper. As a consequence we obtain a characterization of the projective Schur group by means of Galois cohomology. The conjecture was known for algebras over fields of positive characteristic. In characteristic zero the conjecture was known for algebras over fields with an Henselian valuation over a local or global field of characteristic zero.