Algebraic and Arithmetic Theory of Quadratic Forms.

Intro -- Contents -- Preface -- Conference participants -- The Milnor sequence for WqF(x) in characteristic 2 when F is perfect -- Spinor norm for local skew-Hermitian forms -- An introduction to triangular Witt groups and a survey of applications -- Regularity properties of positive definite integral quadratic forms -- Effective results on representations of quadratic forms -- Quaternary quadratic forms and Hilbert modular surfaces -- Real semigroups and abstract real spectra. I -- The role of characteristic vectors in the neighbour lattice process -- On representation by quadratic Fq[x]-lattices -- Diagonal forms of degree p in characteristic p -- 1. Introduction -- 2. Basic results -- 3. A representation theorem à la Cassels-Pfister -- 4. The norm field, the norm degree, and quasi-Pfister forms -- 5. Behaviour of p-forms under algebraic field extensions -- 6. Similarity factors of p-forms and a "norm theorem" à la Knebusch -- 7. The behaviour of p-forms under function field extensions of p-forms -- 7.1 The definition of the function field of a p-form -- 7.2 The algebraic closure of the base field inside the function field of a p-form -- 7.3 p-splitting of a p-form over the function field of a p-form -- 7.4 Isotropy of p-forms over function fields of p-forms -- 7.5 Some remarks on height and splitting patterns of p-forms -- 7.6 Some applications to quasi-Pfister neighbors -- 8. The behaviour of differential forms under function field extensions of p-forms -- Local densities and the representations of an integer by a definite quadratic form -- Periodicity of Hermitian K-theory and Milnors K-groups -- Third proof of second gap in dimensions of quadratic forms from In -- Recent developments on universal forms -- Special Siegel modular forms and singular series polynomials of quadratic forms.

Quasi-hyperbolicity of totally singular quadratic forms -- Witt kernels of triquadratic extensions -- Anti-automorphisms of the second kind -- Reduction modulo 2 and 3 of Euclidean lattices, II -- The elementary type conjecture in quadratic form theory -- 1. Quadratic form schemes -- 2. Quaternionic mappings and Witt rings -- 3. Elementary constructions -- 4. Uniqueness results and counting -- 5. The reduced case -- 6. The case of small value sets -- 7. The case of few quaternion algebras -- 8. Quotients -- 9. Morphisms and field extensions -- 10. Connections to Galois theory -- On Hilbert's theorem about ternary quartics -- Representation by integral quadratic forms-a survey -- Representations by spinor genera of ternary quadratic forms -- A survey and a complement of fundamental Hermite constants.

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