Witt Ring Research Articles

KO-rings of full flag varieties

We present type-independent computations of the K O \mathrm {KO} -groups of full flag varieties, i.e. of quotient spaces G / T G/T of compact Lie groups by their maximal tori. Our main tool is the identification of the Witt ring, a quotient of the K O \mathrm {KO} -ring, of these varieties with the Tate cohomology of their complex K \mathrm {K} -ring. The computations show that the Witt ring is an exterior algebra whose generators are determined by representations of G G .

Thermodynamic semirings

The Witt construction describes a functor from the category of Rings to the category of characteristic 0 rings. It is uniquely determined by a few associativity constraints which do not depend on the types of the variables considered, in other words, by integer polynomials. This universality allowed Alain Connes and Caterina Consani to devise an analogue of the Witt ring for characteristic one, an attractive endeavour since we know very little about the arithmetic in this exotic characteristic and its corresponding field with one element. Interestingly, they found that in characteristic one, the Witt construction depends critically on the Shannon entropy. In the current work, we examine this surprising occurrence, defining a Witt operad for an arbitrary information measure and a corresponding algebra we call a thermodynamic semiring. This object exhibits algebraically many of the familiar properties of information measures, and we examine in particular the Tsallis and Renyi entropy functions and applications to non-extensive thermodynamics and multifractals. We find that the arithmetic of the thermodynamic semiring is exactly that of a certain guessing game played using the given information measure.

On lower ramification subgroups and canonical subgroups

Let p be a rational prime, k be a perfect field of characteristic p and K be a finite totally ramified extension of the fraction field of the Witt ring of k. Let G be a finite flat commutative group scheme over O_K killed by some p-power. In this paper, we prove a description of ramification subgroups of G via the Breuil-Kisin classification, generalizing the author's previous result on the case where G is killed by p>2. As an application, we also prove that the higher canonical subgroup of a level n truncated Barsotti-Tate group G over O_K coincides with lower ramification subgroups of G if the Hodge height of G is less than (p-1)/p^n.

Milnor K-theory and the graded representation ring

AbstractLet F be a field, let G = Gal(/F) be its absolute Galois group, and let R(G,k) be the representation ring of G over a suitable field k. In this preprint we construct a ring homomorphism from the mod 2 Milnor K-theory k*(F) to the graded ring grR(G,k) associated to Grothendieck's γ-filtration. We study this map in particular cases, as well as a related map involving the W-group of F, rather than G. The latter is an isomorphism in all cases considered.Naturally this echoes the Milnor conjecture (now a theorem), which states that k*(F) is isomorphic to the mod 2 cohomology of the absolute Galois group G, and to the graded Witt ring grW(F).The machinery developed to obtain the above results seems to have independent interest in algebraic topology. We are led to construct an analog of the classical Chern character, which does not involve complex vector bundles and Chern classes but rather real vector bundles and Stiefel-Whitney classes. Thus we show the existence of a ring homomorphism whose source is the graded ring associated to the corresponding K-theory ring KO(X) of the topological space X, again with respect to the γ-filtration, and whose target is a certain subquotient of H*(X, F2).In order to define this subquotient, we introduce a collection of distinguished Steenrod operations. They are related to Stiefel-Whitney classes by combinatorial identities.

The Witt Ring of a Smooth Projective Curve Over a Finite Field

In this paper, we calculate the Witt ring W(C) of a smooth geometrically connected projective curve C over a finite field with characteristic other than 2. We view W(C) as a subring of W(k(C)) where k(C) is the function field of C. The calculation is then completed using classical results for bilinear spaces over fields.

Automorphisms of Witt rings of global fields

Automorphisms of Witt rings of global fields

On strong automorphisms of direct products of Witt rings (II)

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

More examples of non-rational adjoint groups

In this paper, we give a recursive construction to produce examples of quadratic forms qn in the n-th power of the fundamental ideal in the Witt ring whose corresponding adjoint groups PSO(qn) are not stably rational. Computations of the R-equivalence classes of adjoint classical groups by Merkurjev are used to show that these groups are not R-trivial. This extends earlier results of Merkurjev and Gille where the forms considered have non-trivial and trivial discriminants respectively.

On strong automorphisms of direct products of Witt rings (I)

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

On the 3-Pfister Number of Quadratic Forms

For a field F of characteristic different from 2, containing a square root of -1, endowed with an F×2-compatible valuation v such that the residue field has at most two square classes, we use a combinatorial analogue of the Witt ring of F to prove that an anisotropic quadratic form over F with even dimension d, trivial discriminant, and Hasse–Witt invariant can be written in the Witt ring as the sum of at most (d2)/8 3-fold Pfister forms.

Isomorphism criteria for Witt rings of real fields

We prove isomorphism criteria for Witt rings and reduced Witt rings of certain types of real fields. Refined criteria are obtained under the additional assumption that the field be SAP. This leads to a generalization of a result by Koprowski on Witt equivalence of function fields of transcendence degree 1 over a real closed field. Isomorphism criteria are also obtained for Witt rings of hermitian forms over a quadratic extension of a real base field and for Witt groups of hermitian forms over a quaternion algebra with a real field as center. All these criteria are expressed in terms of properties involving topological subspaces of the space of orderings of the base field.

The image of the natural homomorphism of Witt rings of orders in a global field

Let $R$ be a Dedekind domain whose field of fractions is a global field. Moreover, let $\mathcal O < R$ be an order. We examine the image of the natural homomorphism $\varphi \colon W\mathcal {O}\to WR$ of the corresponding Witt rings. We formulate necess

P-jets of finite algebras. II: $p$-typical Witt rings

We study the structure of the p -jet spaces of the p -typical Witt rings of the p -adics. We also study the p -jets of the comonad map. These data can be viewed as an arithmetic analogue, for the ring \mathbb Z , of the Lie groupoid of the line and hence as an infinitesimal version of the Galois group of \mathbb Q over “ \mathbb F_1 ”.

Witt rings and matroids

The study of Witt rings of formally real fields in the algebraic theory of quadratic forms has led to a particularly good understanding of the finitely generated torsion free Witt rings. In this paper, we work primarily with a somewhat more general class of rings which can be completely characterized by (binary) matroids. The different types of standard constructions and invariants coming from algebra and from combinatorics lead to previously unstudied problems for both areas; in particular, there are new invariants for Witt rings and new constructions for matroids with many open questions.

Ramification correspondence of finite flat group schemes over equal and mixed characteristic local fields

Let p>2 be a rational prime, k be a perfect field of characteristic p and K be a finite totally ramified extension of the fractional field of the Witt ring of k. Let G and H be finite flat commutative group schemes killed by p over O_K and k[[u]], respectively. In this paper, we show the upper and the lower ramification subgroups of G and H in the sense of Abbes-Saito are naturally isomorphic to each other when they are associated to the same Kisin module.

Trace forms of abelian extensions of number fields of type (1, 0)

This article is concerned with describing certain bilinear forms associated with finite abelian extensions N|K of an algebraic number field K. These abelian trace forms are described up to Witt equivalence, that is, they are described as elements in the Witt ring W(K). When the base field K has exactly one dyadic prime and no real embeddings, it is shown that the Witt class of every abelian trace form over K is a product of Witt classes of five specified types.

Uncountability of the group of strong automorphisms of Witt ring of rational numbers

Uncountability of the group of strong automorphisms of Witt ring of rational numbers

Symbol length and stability index

We show that a Pythagorean field (more generally, a reduced abstract Witt ring) has finite stability index if and only if it has finite 2-symbol length. We give explicit bounds for the two invariants in terms of one another. To approach the question whether those bounds are optimal we consider examples of Pythagorean fields.

Witt groups of Grassmann varieties

We compute the Witt groups of split Grassmann varieties, over any regular base X X . The answer is that the total Witt group of the Grassmannian is a free module over the total Witt ring of X X . We provide an explicit basis for this free module, which is indexed by a special class of Young diagrams, that we call even Young diagrams.

Autour des formes quadratiques indécomposables de l’image de l’homomorphisme W(F) → W(F(φ))

Let F be a field of characteristic not 2. In this article, we treat the quadratic forms of Im(W(F) → W(F(φ))) which are indecomposable, i.e., those which are not isometric to a sum of two nonzero forms of this image, where W(F) is the Witt ring of F-quadratic forms, and F(φ) is the function field of the affine quadric given by φ. This is related to the descent problems studied in [12, 14]. More precisely, we will focus on indecomposable quadratic forms of minimal dimension, which we detail for φ of dimension less than or equal to 8. We also include other related results.