Witt Ring Research Articles

A class of finite commutative rings constructed from Witt rings

Motivated by constructions of Witt rings in the algebraic theory of quadratic forms, the authors construct new classes of finite commutative rings and explore some of their properties. These rings are constructed as quotient rings of a special class of integral group rings for which the group is an elementary 2-group. The new constructions are compared to other rings in the literature.

Annihilators of quadratic and bilinear forms over fields of characteristic two

Let F be a field with 2 = 0 , W ( F ) the Witt ring of symmetric bilinear forms over F and W q ( F ) the W ( F ) -module of quadratic forms over F. Let I F ⊂ W ( F ) be the maximal ideal. We compute explicitly in I F m and I m W q ( F ) the annihilators of n-fold bilinear and quadratic Pfister forms, thereby answering positively, in the case 2 = 0 , certain conjectures stated by Krüskemper in [M. Krüskemper, On annihilators in graded Witt rings and in Milnor's K-theory, in: B. Jacob et al. (Eds.), Recent Advances in Real Algebraic Geometry and Quadratic Forms, in: Contemp. Math., vol. 155, 1994, pp. 307–320].

Inequalities between Pythagoras numbers and algebraic ranks in Witt rings of fields

Let K be a field of characteristic different from 2 and let σ denote a non-zero element in K. Assuming that the one-dimensional form σx 2 is algebraic (definitions given in the Introduction) we prove the following inequality between the Pythagoras number of σ and n = algebraic rank of σx 2: where δ (n + 1) is the number of 1's appearing in the binary expansion of n + 1. This result is a special case of a more general inequality relating py K (σ) to the algebraic rank of any positive integral multiple m⟨σ⟩ of σx 2. As an application, an example is given of a rank 5 form q over a Hilbertian field K with algebraic rank of at least 7. By results of Epkenhans, such examples are not possible over algebraic number fields.

Abstract theory of semiorderings

Marshall's abstract theory of spaces of orderings is a powerful tool in the algebraic theory of quadratic forms. We develop an abstract theory for semiorderings, developing a notion of a space of semiorderings which is a prespace of orderings. It is shown how to construct all finitely generated spaces of semiorderings. The morphisms between such spaces are studied, generalising the extension of valuations for fields into this context. An important invariant for studying Witt rings is the covering number of a preordering. Covering numbers are defined for abstract preorderings and related to other invariants of the Witt ring.

Witt kernels of bilinear forms for algebraic extensions in characteristic 2

Let F F be a field of characteristic 2 2 and let K / F K/F be a purely inseparable extension of exponent 1 1 . We determine the kernel W ( K / F ) W(K/F) of the natural restriction map W F → W K WF\to WK between the Witt rings of bilinear forms of F F and K K , respectively. This complements a result by Laghribi who computed the kernel for the Witt groups of quadratic forms for such an extension K / F K/F . Based on this result, we will determine W ( K / F ) W(K/F) for a wide class of finite extensions which are not necessarily purely inseparable.

Koszul complexes and symmetric forms over the punctured affine space

Let $X$ be a regular separated scheme of finite Krull dimension and let $U^{n}_{X} \subset A^{n}_{X}$ be the punctured affine $n$-space over $X$. We show that the total graded Witt ring of $U^{n}_{X}$ is a free graded module over the total graded Witt ring of $X$ with two generators $1$ and $\epsilon$. The second generator satisfies the equation $\epsilon^{2} = 1$ when $n = 1$ and $\epsilon^{2} = 0$ when $n \geq 2$.

On natural homomorphisms of Witt rings

We prove that the kernel of the ring homomorphism between the Witt ring of any order of a global field and the Witt ring of the field is a nilideal.

Witt Rings and Permutation Polynomials

Let p be a prime number. In this paper, the author sets up a canonical correspondence between polynomial functions over ℤ/p2ℤ and 3-tuples of polynomial functions over ℤ/pℤ. Based on this correspondence, he proves and reproves some fundamental results on permutation polynomials mod pl. The main new result is the characterization of strong orthogonal systems over ℤ/plℤ.

On the Size of the Wild Set

AbstractTo every pair of algebraic number fields with isomorphic Witt rings one can associate a number, called the minimum number of wild primes. Earlier investigations have established lower bounds for this number. In this paper an analysis is presented that expresses the minimum number of wild primes in terms of the number of wild dyadic primes. This formula not only gives immediate upper bounds, but can be considered to be an exact formula for the minimum number of wild primes.

Holes in [formula omitted

Let F be an arbitrary field of characteristic ≠2. We write W ( F ) for the Witt ring of F, consisting of the isomorphism classes of all anisotropic quadratic forms over F. For any element x ∈ W ( F ) , its dimension dim x is defined as the dimension of a quadratic form representing x. The elements of all even dimensions form an ideal denoted by I ( F ) . The filtration of the ring W ( F ) by the powers I ( F ) n of this ideal plays a fundamental role in the algebraic theory of quadratic forms. The Milnor conjectures, recently proved by Voevodsky and Orlov–Vishik–Voevodsky, describe the successive quotients I ( F ) n / I ( F ) n + 1 of this filtration, identifying them with Galois cohomology groups and with the Milnor K-groups modulo 2 of the field F. In the present article we give a complete answer to a different longstanding question concerning I ( F ) n , asking about the possible values of dim x for x ∈ I ( F ) n . More precisely, for any n ⩾ 1 , we prove that (*) dim I n = { 2 n + 1 − 2 i | i ∈ [ 1 , n + 1 ] } ∪ ( 2 Z ∩ [ 2 n + 1 , + ∞ ) ) , where dim I n is the set of all dim x for all x ∈ I ( F ) n and all F. Previously available partial informations on dim I n include the classical Arason–Pfister theorem (saying that ( 0 , 2 n ) ∩ dim I n = ∅ ) as well as a recent Vishik's theorem on ( 2 n , 2 n + 2 n − 1 ) ∩ dim I n = ∅ (the case n = 3 is due to Pfister, n = 4 to Hoffmann). The proof of ( * ) is based on computations in Chow groups of powers of projective quadrics (involving the Steenrod operations); the method developed can be also applied to other types of algebraic varieties.

Surjectivity Criteria for p-adic Representations, Part I

We prove general surjectivity criteria for p-adic representations. In particular, we classify all adjoint and simply connected group schemes G over the Witt ring W(k) of a finite field k such that the reduction epimorphism $G(W_2(k))\twoheadrightarrow G(k)$ has a section.

Bounds for the representation of quadratic forms

We prove first that, for fixed integers n, m⩾1, there is a uniform bound on the number of Pfister forms of degree n over any Pythagorean field F necessary to represent (in the Witt ring of F) any form of dimension m as a linear combination of such forms with non-zero coefficients in F. “Uniform” means that the bound does not depend either on the form or on the field F; it is given by a recursive function f of n and m. Similar results hold for the reduced special groups arising from preordered fields and from fields whose Pythagoras number is bounded by a fixed integer. We single out a large class of Pythagorean fields and, more generally, of reduced special groups (cf. [4]) for which f has a simply exponential bound of the form cm n−1 ( c a constant). Such a class is closed under certain—possibly infinitary—operations which preserve Marshall's signature conjecture. In the case of groups of finite stability index s, we obtain an upper bound for f which is quadratic on [ m/2 n ], where c depends on s.

Semiorderings and Witt rings

For a pythagorean fieldFwith semiorderingQand associated preorderingT, it is shown that the Witt ringWT(F) is isomorphic to the Witt ringW(K) whreKis a closure ofFwith respect toQ. For an arbitrary preorderingT, it is shown how the covering number ofTrelates to the construction ofWT(F).

Witt Rings of Global Fields

In 1989 DeMeyer et al. (DeMeyer,F.,Harrison,D.,Miranda,R. (1989). Quadratic forms over ℚ and Galois extensions of commutative rings. Mem. Amer. Math. Soc. 77(394)) described the Witt ring of the rational number field in terms of Hilbert symbols. The same has been done by the first author for global function fields. The aim of this paper is to obtain a similar result for any global field.

Linkage of Fields in Characteristic 2

Let F be a field with 2 = 0 and ϕ = ⟨⟨a 1,…,a n ⟩⟩ an n-fold anisotropic bilinear Pfister form over F with function field F(ϕ). In this paper we compute ker where I F ⊂ W(F) is the maximal ideal in the Witt ring W(F) of F. We use this computation to prove an n-linkage property of the subfields F 2(a 1,…,a n ).

Vanishing and nilpotence of locally trivial symmetric spaces over regular schemes

We prove two results about Witt rings W(−) of regular schemes. First, given a semi-local regular ring R of Krull dimension d, if U is the punctured spectrum, obtained from Spec(R) by removing the maximal ideals of height d, then the natural map \(\operatorname W(R)\to \operatorname W(U)\) is injective. Secondly, given a regular integral scheme X of finite Krull dimension, consider Q its function field and the natural map \(\operatorname W(X)\to \operatorname W(Q)\). We prove that there is an integer N, depending only on the Krull dimension of X, such that the product of any choice of N elements in \(\Ker\big(\W(X)\to \W(Q)\big)\) is zero. That is, this kernel is nilpotent. We give upper and lower bounds for the exponent N.

Milnor K-Groups and Finite Field Extensions

Let E/F be a finite separable field extension and let m denote the integral part of log2(E : F ). David Leep recently showed that if char(F ) � 2, then for n m the nth power of the fundamental ideal in the Witt ring of E satisfies the equality I n E = I n−m F · I m E. The aim of this note is to prove the analogous equality for the Milnor K-groups, that is KnE = Kn−mF · KmE for n m. In either of these equalities one may not replace m by m − 1, as examples of certain m-quadratic extensions indicate. Mathematics Subject Classifications (2000): 11E81, 12F05, 19D45.

Some model-theoretic results in the algebraic theory of quadratic forms

This paper studies some model-theoretic properties of special groups of finite type. Special groups are a first-order axiomatization of the algebraic theory of quadratic forms, introduced by Dickmann and Miraglia, which is essentially equivalent to abstract Witt rings. More precisely, we consider elementary equivalence, saturation, elementary embeddings, quantifier elimination, stability and Morley rank.

Witt Rings of Hasse Domains of Global Fields

We describe the structure of the Witt group WOS(K) of any Hasse domain OS(K) of a global field K by explicitly determining its cyclic direct summands and generators of each such summand. Using this description we find a sufficient condition for the existence of Witt ring isomorphism WOS(K)≅WOS′(L). We give also the necessary and sufficient condition for the existence of Witt ring isomorphism WK→WL of global fields K and L mapping the Witt ring WOS(K) onto the Witt ring WOS′(L).

Elliptic Crystals and Modular Motives

The goal of this paper is to investigate the crystalline properties of families of elliptic curves or, more precisely, to study the families of crystals (elliptic crystals) attached to families of elliptic curves. Elliptic crystals over a point can be defined and classified very simply in terms of semilinear algebra, and this is how we begin (1.1). Our definition is cast in the logarithmic setting, so that it applies also to semistable (degenerate) elliptic curves. Roughly speaking, an elliptic crystal over a (logarithmic) perfect field k of characteristic p is a free W-module E of rank two equipped with a Frobenius-linear endomorphism 8, a nilpotent linear endomorphism N, a two-step Hodge filtration A on k E, and an isomorphism tr: 4E W. These data are required to satisfy various compatibilities: for example, A(k E) is the kernel of idk 8. When N is not zero, such a crystal is quite rigid, and in particular its canonical coordinates are very precisely determined (1.3). Liftings of elliptic crystals from k to its Witt ring are determined by specifying a lifting B of the Hodge filtration A of k E, and the classification is made explicit in (1.5). Our first main result (2.2) concerns the relationship between deformation theory and elliptic crystals on a curve. Let X k be the reduction modulo p of a smooth log curve Y W over W, let (E, B) be an elliptic crystal on Y W, and let (E, A) be the restriction of (E, B) to X k. Following Mochizuki [8] and Gunning [6], we shall say that E is indigenous if its Kodaira-Spencer mapping is an isomorphism, a condition which can be checked either on Y W or on X k. If f: X$ X is a morphism of smooth log curves over k then a lifting g: Y$ Y of f determines a lifting g*(BEY) ( f *E)Y$ of A( f *E)X$ k . Theorem (2.2) shows that if p is odd and E is indigenous, the resulting map from the set of liftings of f to the set of liftings of the mod p Hodge filtration of f *EX is a bijection. As a consequence we show (reprising some of the results of Mochizuki) in Theorem (2.4) that (when the degree of AEX is positive and p is odd) an indigenous bundle on X k determines a unique lifting Y W, characterized by the fact doi:10.1006 aima.2001.1976, available online at http: www.idealibrary.com on