Richard Elman

The Algebraic and Geometric Theory of Quadratic Forms

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

Rigid Elements, Valuations, and Realization of Witt Rings

u

Pfister Ideals in Witt Rings

-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.

Hereditarily quadratically closed fields

to the theory over the residue class field of the valuation. This theorem generalizes to any 2-henselian valuation (cf. [Kl, Sect. 12.21). Let v be the valuation. Let a E k := F\{O 1 be such that v(a) is not divisible by two. The key idea behind the proof is that the quadratic form (1, a} only represents elements in % u a%. Such an element a is called

On quadratic forms and Galois cohomology

To a certain extent, the notion of Pfister ideals (especially 1-Pfister ideals) has already been utilized in the literature of quadratic form theory. For instance, Pfisters celebrated Local-Global Principle for formally real fields can be stated in the form that the ideal of forms of total signature zero is a 1-Pfister ideal. As another example, the Annihilator Theorem of Witt states that, for any anisotropic Pfister (or round) form Q, ann(Q)= WF is 1-Pfister. Some of the other occurrences of the notion of Pfister ideals in the literature will be cited in Sect. 2 below.

The Witt ring of an elliptic curve over a local field

Let K/F be a field extension of transcendence degree n. The theorem of Tsen-Lang (cf. [G, p. 221) says that if F is an algebraically closed field then K is a C,-field, i.e., any homogeneous polynomial of degree d over K with more than d” variables has a non-trivial solution in K. This can be restated by saying that whenever F is either a real closed field or an algebraically closed field then K(n) is a C,-field. For quadratic forms this says precisely that u( K( fi)) := @l= i (1, a,). If Z”F contains no anisotropic n-fold Pfister forms having zero signature in each real closure of F then we say that F satisfies property A,. This property is important

Quadratic forms over formally real fields and pythagorean fields

Witt rings. Specifically, suppose there is a morphism of abstract Witt rings W F —> R. Does there exist a field extension K of F such that WK ~ R and the given morphism corresponds to %KjF W F —> WK? A classical example is the case R ~ Z . Then K can be chosen to be a real closure of F relative to the ordering induced by the morphism. Unfortunately, the answer is in general negative. But in the important case when W F = R x S in the category of abstract Witt rings and the morphism is the projection map, we have a positive result. In [7] we extend the work of [18] and [30] on valuations and use this extension to show THEOREM 8. Let (j> : W F^RxS be an isomorphism of abstract Witt rings and let 7r : R x S —> R be the projection. Assume that R is not basic (i.e., R is a group Witt ring over a subring). Then there exists a 2-extension K of F and an isomorphism ip : WK^R of abstract Witt rings such that the diagram WF — ^ RxS [ j * WK ——> R

On some Hasse Principles over formally real fields

J6n Kr. Arason l, Richard Elman z,*, Bill Jacob 3,* 1 Raunvisindastofnun Hfisk61ans, University of Iceland, Reykjavik, Iceland (e-mail: jka@rhi.hi.is) 2 Department of Mathematics, University of California at Los Angeles, Los Angeles, CA 90095, USA (e-mail: rse@math.ucla.edu) -~ Department of Mathematics, University of Califomiat, Santa Barbara, Santa Barbara, CA 93106, USA (e-mail: jacob@rnath.ncsb.edu)