Isotropy over function fields of Pfister forms

This dissertation presents computational tools for quadratic forms over global function fields of characteristic different from 2. The majority of the algorithms we develop in this study rely on our ability to find all places dividing any coefficient of the given quadratic form. This problem is equivalent to the factorization of fractional ideals in the ring of polynomial functions of a global function field. As a result, we begin by presenting alternative approaches for factoring fractional ideals that do not rely on determining the maximum order of the global function field in question. We then propose techniques for tackling the following quadratic form theory computational problems: how to detect whether a quadratic form is isotropic or not, how to detect whether a quadratic form is hyperbolic or not, how to compute the anisotropic dimension (or equivalently the Witt index) of a quadratic form, how to construct an anisotropic part of a quadratic form, how to determine if two forms are Witt-similar or are Ono-similar or not. We further explore algorithms for computing some important field invariants that are linked to quadratic forms. Those are: the length of a sum of squares, the level of a field, the Pythagoras number of a field, as well as a Pythagoras element of a field.

Introduction Classical theory of symmetric bilinear forms and quadratic forms: Bilinear forms Quadratic forms Forms over rational function fields Function fields of quadrics Bilinear and quadratic forms and algebraic extensions $u$-invariants Applications of the Milnor conjecture On the norm residue homomorphism of degree two Algebraic cycles: Homology and cohomology Chow groups Steenrod operations Category of Chow motives Quadratic forms and algebraic cycles: Cycles on powers of quadrics The Izhboldin dimension Application of Steenrod operations The variety of maximal totally isotropic subspaces Motives of quadrics Appendices Bibliography Notation Terminology.

It seems that any deeper investigation of the algebraic theory of quadratic forms requires a thorough knowledge of the theory of Pfister forms. They are the subject of this chapter. The investigation of Pfister forms is based essentially on the use of transcendental extensions of the ground field: One considers quadratic forms Σ a ij X i X j . with indeterminates X i . This approach leads naturally to the important concept of the function field of a quadratic form which is the function field of the quadratic Σ a ij X i X j =0.

The classical quadratic Gauss sum can be thought of as an exponential sum attached to a quadratic form on a cyclic group. We introduce an equivariant version of Gauss sum for arbitrary finite quadratic forms, which is an exponential sum twisted by the action of the orthogonal group. We prove that simple arithmetic formulas hold for some basic classes of quadratic forms. In applications, such invariant appears in the dimension formula for certain vector-valued modular forms.

Small representations of integers by integral quadratic forms

We obtain new results on linear spaces on the intersection of two quadratic forms defined over a non-dyadic p-adic field \({{\Bbb Q}_p}\). One of our main tools is a recent result of Parimala and Suresh on isotropy of quadratic forms over functions fields over \({{\Bbb Q}_p}\). 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.

It is somnewhat surprising that the systematic evaluation of the heights of rational points on a curve and on its jacobian variety and particularly of their relation to each other should yield any new information. Nonetheless this appears to be the case and the result is described in this article. Although the main theorem is not even a special case of the very fascinating conjecture of Mordell, still it is an estimate that already reveals that rational points on curves of genus at least 2 are much harder to come by than on curves of genus 0 or 1. It is a quantitative limitation on the heights of such points which is well-known to be false in the case of genus 0 or 1. Incidentally, there is a good explanation why an estimate of this type can be obtained so cheaply, whereas Mordell's conjecture itself could not: namely, results obtained by our methods will more or less automatically apply to the analogous funetion case [where the ground field is a function field in one variable over a finite field, rather than an algebraic number field]. And in this case, unless further restrictions are imposed, there are curves of any genus with an infinite number of rational poilnts whose heights increase exactly at the rate which we will find. Let k be an algebraic number field of finite degree over Q. Let C be a non-singular projective curve over k of genus g at least 2. Mordell's conjecture asserts that the set of k-rational points on C is finite. Now suppose

Let K be the field of algebraic functions on an elliptic curve that can be described by an equation of the form y 2 = f ( x ) {y^2} = f(x) where f ( x ) f(x) is a quartic polynomial over a field k. Moreover, assume that the Riemann surface for K contains no points rational over k. When k is the field of real numbers it is well known that K may also be expressed as a quadratic extension of a function field L = k ( u , v ) L = k(u,v) of algebraic functions on a conic whose Riemann surface also contains no points rational over k. We extend this result to p-adic ground fields k. Moreover, we describe the various subfields of index two and genus zero (conic subfields) in terms of the k-rational points on the Jacobian of K. This is done for arbitrary ground fields. In particular, the embedding of the projective class group of K (over k) is seen to describe exactly those conic subfields that possess k-rational points.

In this chapter, we introduce the class of quadratic forms which will be required in (16.7) to describe the quadrangles of type F 4. These quadrangles (and the corresponding quadratic forms) are thus named because of their connection to certain mixed groups of type F4; see (41.20). The quadratic forms of type F 4 are defective (in fact, the defect can be of arbitrarily large dimension) and they exist only over certain imperfect fields of characteristic two.

For a field F of characteristic not 2, let \({\widehat{F}}\) denote the maximal dimension of anisotropic, weakly isotropic, non-degenerate quadratic forms over F. In this paper, we investigate the behavior of this invariant under field extensions.

We show that over any field$F$of characteristic 2 and 2-rank$n$, there exist$2^{n}$bilinear$n$-fold Pfister forms that have no slot in common. This answers a question of Becher [‘Triple linkage’,Ann.$K$-Theory, to appear] in the negative. We provide an analogous result also for quadratic Pfister forms.

A 3-fold Pfister form is associated to every involution of the second kind on a central simple algebra of degree 3. This quadratic form is associated to the restriction of the reduced trace quadratic form to the space of symmetric elements; it is shown to classify involutions up to conjugation. Subfields with dihedral Galois group in central simple algebras of arbitrary odd degree with involution of the second kind are investigated. A complete set of cohomological invariants for algebras of degree 3 with involution of the second kind is given.

The generic quadratic form of even dimension n with trivial discriminant over an arbitrary field of characteristic different from 2 containing a square root of -1 can be written in the Witt ring as a sum of 2-fold Pfister forms using n -2 terms and not less.The number of 2-fold Pfister forms needed to express a quadratic form of dimension 6 with trivial discriminant is determined in various cases.

Powers of the Fundamental Ideal in the Witt Ring

Quadratic forms isotropic over the function field of a conic