Some new examples in the theory of quadratic forms

We construct a 6-dimensional anisotropic quadratic form and a 4-dimensional quadratic form over some fieldF such that becomes isotropic over the function field but every proper subform of is still anisotropic over . It is an example of non-standard isotropy with respect to some standard conditions of isotropy for 6-dimensional forms over function fields of quadrics, known previously. Besides of that, we produce an 8-dimensional quadratic form with trivial determinant such that the index of the Clifford invariant of is 4 but can not be represented as a sum of two 4-dimensional forms with trivial determinants. Using this, we find a 14-dimensional quadratic form with trivial discriminant and Clifford invariant, which is not similar to a difference of two 3-fold Pfister forms. The proofs are based on computations of the topological filtration on the Grothendieck group of certain projective homogeneous varieties. To do these computations, we develop several methods, covering a wide class of varieties and being, to our mind, of independent interest. Received November 11, 1997; in final form June 24, 1999 / Published online May 8, 2000     

Linear Spaces on the Intersection of Two Quadratic Hypersurfaces,and Systems of <Emphasis Type="Italic">p</Emphasis>-adic Quadratic Forms

We obtain new results on linear spaces on the intersection of two quadratic forms defined over a non-dyadic p-adic field . One of our main tools is a recent result of Parimala and Suresh on isotropy of quadratic forms over functions fields over . As a corollary we also get new bounds for the number of variables necessary to always find a non-trivial p-adic zero of a system of quadratic forms.     

Théorie d'Iwasawa des corps p-rationnels et p-birationnels

We study Iwasawa theory for p-rational and p-birational fields. A classical invariant characterises them and, in the case of CM-fields, this gives an explicit characterisation. We show how to compute those fields and and give numerical examples for small degrees.

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Received: 20 May 1997 / Revised version: 9 April 1998     

Orthogonal Pfister involutions in characteristic two

We show that over a field of characteristic 2 a central simple algebra with orthogonal involution that decomposes into a product of quaternion algebras with involution is either anisotropic or metabolic. We use this to define an invariant of such orthogonal involutions that completely determines the isotropy behaviour of the involution. We also give an example of a non-totally decomposable algebra with orthogonal involution that becomes totally decomposable over every splitting field of the algebra.     

An Existential Divisibility Lemma for Global Fields

We generalize the Existential Divisibility Lemma by Th. Pheidas [7] to all global fields K of characteristic not 2, and for all sets of primes that are inert in a quadratic extension L of K. We also remove the conditions in real and ramifying primes, which were present in Pheidas’ version. As a Corollary, we recover the known fact that the set of integral elements at a prime in a global field is existentially definable. The first author is a Research Assistant of the Research Foundation – Flanders (FWO – Vlaanderen). Work partially supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00287.     

Going up of the <Emphasis Type="Italic">u</Emphasis>-invariant over formally real fields

Let F be a field of characteristic not 2, and assume that F has finite reduced stability. Let K/F be any finite extension. We prove that if the general u-invariant u(F) is finite, then u(K) is finite. This article is based on part of the author’s Ph.D. thesis, written under the supervision of Richard Elman.     

On abelian quantum invariants of links in 3-manifolds

From a finite abelian group G, a quadratic form onG and an element in , we define a topological invariant of a pair(M,L) where is a closed oriented 3-manifold and L an oriented, framedn-component link inM. The main result consists in an explicit formula for this invariant, based on a reciprocity formula for Gauss sums, which features a special linking pairing. This pairing depends on both the quadratic form q and the linking pairing of M. A necessary and sufficient condition for the invariant to vanish is described in terms of a characteristic class for this pairing. We also discuss torsion spin-structures and related structures which appear in this context. Received May 13, 1998 / Accepted November 11, 1999 / Published online February 5, 2001     

On fields of u-invariant 4

This note is motivated by the problem of determining the u-invariant of a field F of characteristic different from two when it is known that A criterion is given to decide whether u(F) ≤ 4 in this situation. Received: 25 February 2005     

Normal integral bases in quadratic and cyclic cubic extensions of quadratic fields

Let K be a number field and let G be a finite abelian group. We call K a Hilbert-Speiser field of type G if, and only if, every tamely ramified normal extension L/K with Galois group isomorphic to G has a normal integral basis. Now let C₂ and C₃ denote the cyclic groups of order 2 and 3, respectively. Firstly, we show that among all imaginary quadratic fields, there are exactly three Hilbert-Speiser fields of type $C_{2}: \mathbb{Q}(\sqrt {m})$, where $m \in \{-1, -3, -7\}$. Secondly, we give some necessary and sufficient conditions for a real quadratic field $K = \mathbb{Q}(\sqrt {m})$ to be a Hilbert-Speiser field of type C₂. These conditions are in terms of the congruence class of m modulo 4 or 8, the fundamental unit of K, and the class number of K. Finally, we show that among all quadratic number fields, there are exactly eight Hilbert-Speiser fields of type $C_{3}: \mathbb{Q}(\sqrt {m})$, where $m \in \{-11,-3, -2, 2, 5, 17, 41, 89\}$.Received: 2 April 2002     

Integral spinor norm groups over dyadic local fields

The spinor norms of integral rotations of an arbitrary quadratic lattice over an arbitrary dyadic local field are determined. The results are given in terms of BONGs, short for “bases of norm generators”. This approach provides a new way to describe lattices over dyadic local fields.     

HIGHER DEGREE HYPERBOLIC FORMS

Abstract

We define higher degree hyperbolic forms, analogous to the quadratic hyperbolic forms. We prove the following descent result. Let f be a form of degree d ≥ 3 over a field F of characteristic 0, and let K|f be a field extension. Then if f is equivalent over K to a hyperbolic form, f must already be equivalent to it over F. We also prove that in the monoid of equivalence classes of forms defined over F of a fixed degree d ≥ 3, under the tensor product, the submonoid generated by the equivalence classes of the hyperbolic forms is free. The proofs of these results involve the calculation of the centres and the Lie algebras of the higher degree hyperbolic forms. For the convenience of the reader we expound some of Harrison's seminal paper [5].     

Universal quadratic forms and indecomposables over biquadratic fields

The aim of this article is to study (additively) indecomposable algebraic integers of biquadratic number fields K and universal totally positive quadratic forms with coefficients in . There are given sufficient conditions for an indecomposable element of a quadratic subfield to remain indecomposable in the biquadratic number field K. Furthermore, estimates are proven which enable algorithmization of the method of escalation over K. These are used to prove, over two particular biquadratic number fields and , a lower bound on the number of variables of a universal quadratic forms.     

Cohomological invariants of quaternionic skew-hermitian forms

We define a complete system of invariants e _n,Q ,n ≥ 0 for quaternionic skew-hermitian forms, which are twisted versions of the invariants e _n for quadratic forms. We also show that quaternionic skew-hermitian forms defined over a field of 2-cohomological dimension at most 3 are classified by rank, discriminant, Clifford invariant and Rost invariant. Received: 30 April 2006     

Arithmetic actions on cyclotomic function fields

We derive the group structure for cyclotomic function fields obtained by applying the Carlitz action for extensions of an initial constant field. The tame and wild structures are isolated to describe the Galois action on differentials. We show that the associated invariant rings are not polynomial.     

A Hermitian analogue of the Bröcker-Prestel theorem

The Bröcker-Prestel local-global principle characterizes weak isotropy of quadratic forms over a formally real field in terms of weak isotropy over the Henselizations and isotropy over the real closures of that field. A Hermitian analogue of this principle is presented for algebras of index at most two. An improved result is also presented for algebras with a decomposable involution, algebras of Pythagorean index at most two, and algebras over SAP and ED fields.     

Generic points of quadrics and Chow groups

In this article we study the sufficient conditions for the k̅-defined element of the Chow group of a smooth variety to be k-rational (defined over k). For 0-cycles this question was addressed earlier. Our methods work for cycles of arbitrary dimension. We show that it is sufficient to check this property over the generic point of a quadric of sufficiently large dimension. Among the applications one should mention the uniform construction of fields with all known u-invariants.     

Motivic Serre invariants of curves

In this article, we generalize the theory of motivic integration on formal schemes topologically of finite type and the notion of motivic Serre invariant, to a relative point of view. We compute the relative motivic Serre invariant for curves defined over the field of fractions of a complete discrete valuation ring R of equicharacteristic zero. One aim of this study is to understand the behavior of motivic Serre invariants under ramified extension of the ring R. Thanks to our constructions, we obtain, in particular, an expression for the generating power series, whose coefficients are the motivic Serre invariant associated to a curve, computed on a tower of ramified extensions of R. We give an interpretation of this series in terms of the motivic zeta function of Denef and Loeser.     

Small generators of number fields

If K is a number field of degree n over Q with discriminant D _K and if α∈K generates K, i.e. K=Q(α), then the height of α satisfies with . The paper deals with the existence of small generators of number fields in this sense. We show: (1) For each $n$ there are infinitely many number fields K of degree $n$ with a generator α such that . (2) There is a constant d ² such that every imaginary quadratic number field has a generator α which satisfies .?(3) If K is a totally real number field of prime degree n then one can find an integral generator α with . Received: 10 January 1997 / Revised version: 13 January 1998     

Equimultiplicity,algebraic elimination,and blowing-up

Given a variety X over a perfect field, we study the partition defined on X by the multiplicity (into equimultiple points), and the effect of blowing up at smooth equimultiple centers. Over fields of characteristic zero we prove resolution of singularities by using the multiplicity as an invariant, instead of the Hilbert Samuel function.     

Indecomposable forms of higher degree

All nondegenerate indecomposable forms of higher degree over a perfect field k can be realized as traces of nondegenerate absolutely indecomposable forms of higher degree over a suitable algebraic field extension of k. With the help of trace forms of certain nonassociative algebras we construct classes of indecomposable forms of degree d≥3.