2012
DOI: 10.1007/s11425-012-4364-4
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Hyperbolicity of unitary involutions

Abstract: Abstract. 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.

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Cited by 14 publications
(23 citation statements)
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“…By [12, Theorem A.2] (see also [9]), we may replace F by the function field of the F -variety R K/F SB(D). Since now the F -algebra D ′ splits over the quadratic extension K/F , it is equivalent to a quaternion F -algebra Q.…”
Section: ])mentioning
confidence: 99%
See 1 more Smart Citation
“…By [12, Theorem A.2] (see also [9]), we may replace F by the function field of the F -variety R K/F SB(D). Since now the F -algebra D ′ splits over the quadratic extension K/F , it is equivalent to a quaternion F -algebra Q.…”
Section: ])mentioning
confidence: 99%
“…There exists a finite field extension L/F of odd degree such that the algebra D ⊗ F L is split (cf. [1, 3.3.1] or [9,Page 938]). Let h be a hermitian form such that the involution σ L on the split algebra D ⊗ F L is adjoint to h. Let q be the quadratic form over L (of dimension 2m) given by h. Let q ′ be a nondegenerate subform of q of dimension 2m − 1.…”
Section: ])mentioning
confidence: 99%
“…The implication (1) ⇒ (2) is known as the Pfister Factor Conjecture, and was proven in [3]. The implication (1) ⇒ (3), and the equivalence (2) ⇔ (3), follows from the Pfister Factor Conjecture and the non-hyperbolic splitting result of [8]. The converse implication, (2) or (3) ⇒ (1), is still open in general.…”
Section: Introductionmentioning
confidence: 99%
“…In particular, whether totally decomposable involutions are exactly those involutions on algebras of two-power degree that are either anisotropic or hyperbolic after extending scalars. That totally decomposable orthogonal involutions over a field of characteristic different from two are always either anisotropic or hyperbolic can be shown (see [6, (3.2)]) using the nonhyperbolic splitting results of Karpenko, [10], and that any totally decomposable orthogonal involution on a split algebra is adjoint to a Pfister form. The later result is known as the Pfister Factor Conjecture, and was shown in [3].…”
Section: Introductionmentioning
confidence: 99%