Isotropy of unitary involutions

We prove the so-called Unitary Isotropy Theorem, a result on isotropy of a unitary involution. The analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal, symplectic, and unitary involutions are formal consequences of this theorem. A component of the proof is a detailed study of the quasi-split unitary grassmannians.Comment: final version, to appear in Acta Mat

Quadratic forms of dimension 8 with trivial discrimiand and Clifford algebra of index 4

Izhboldin and Karpenko proved in 2000 that any quadratic form of dimension 8 with trivial discriminant and Clifford algebra of index 4 is isometric to the transfer, with respect to some quadratic \'etale extension, of a quadratic form similar to a 2-fold Pfister form. We give a new proof of this result, based on a theorem of decomposability for degree 8 and index 4 algebras with orthogonal involution

Formes antihermitiennes devenant hyperboliques sur un corps de déploiement

Let H = (a, b)(F) be a division quaternion algebra over a field F of characteristic not 2. Denote by tau the canonical involution on H and by K a splitting field of H. If h is a skew-hermitian form over (H,tau) then, by extension of scalars to K and by Morita equivalence, we obtain a quadratic form h(K) over K. This gives a map of Witt groups rho :W-1(H,r) --> W(K) induced by rho (h) = h(K). When K is a generic splitting field of H we prove in this note that the map rho is injective. (C) 2001 Academie des sciences/Editions scientifiques et medicales Elsevier SAS