Quadratic and Hermitian Forms over Rings

Knus, Max-Albert

doi:10.1007/978-3-642-75401-2

Cited by 361 publications

(261 citation statements)

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“…It is identified with the pointed set H 1 et (O S , O V ) simply obtained by changing the base point to (V, q) (cf. [Knu,IV,Prop. 8.2]).…”

Section: ])mentioning

confidence: 99%

“…The hyperbolic plane 1, −1 has trivial Witt invariant, and it is orthogonal to −t in V , thus w(q) = 0 (cf. [Knu,IV,Prop. 8.1.1, 1), 3)]), so [q] corresponds to the trivial element in 2 Br(O S ).…”

Section: A Splitting Hyperbolic Planementioning

confidence: 99%

“…We assume it to be O S -regular, namely, the induced homomorphism V → V ∨ := Hom(V, O S ) is an isomorphism ( [Knu,I. §3,3.2]).…”

Section: Introductionmentioning

confidence: 99%

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On the classification of quadratic forms over an integral domain of a global function field

Bitan

2017

Journal of Number Theory

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Abstract. Let C be a smooth projective curve defined over the finite field Fq (q is odd) and let K = Fq(C) be its function field. Any finite set S of closed points of C gives rise to an integral domain OS := Fq[C − S] in K. We show that given an OS-regular quadratic space (V, q) of rank n ≥ 3, the set of genera in the proper classification of quadratic OS-spaces isomorphic to (V, q) in the flat orétale topology, is in 1 : 1 correspondence with 2Br(OS ), thus there are 2 |S|−1 such. If (V, q) is isotropic, then Pic (OS)/2 classifies the forms in the genus of (V, q). For n ≥ 5 this is true for all genera, hence the full classification is via the abelian group H 2 et (OS, µ 2 ).

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“…It is identified with the pointed set H 1 et (O S , O V ) simply obtained by changing the base point to (V, q) (cf. [Knu,IV,Prop. 8.2]).…”

Section: ])mentioning

confidence: 99%

Section: A Splitting Hyperbolic Planementioning

confidence: 99%

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On the classification of quadratic forms over an integral domain of a global function field

Bitan

2017

Journal of Number Theory

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“…We refer to [11,Chapter 1] and [9, Chapters 1 and 2] for standard notation and terminology, and as general references on hermitian and quadratic forms respectively.…”

Section: Hermitian and Quadratic Formsmentioning

confidence: 99%

“…Then there exists a vector x ∈ V \ {0} such that h(x, x) = q(x) = 0. As (V, h) is nondegenerate and even, by [11,Chapter 1,(3.7.4)] there exists a vector y ∈ V \ {0} such that h(y, y) = 0 and h(x, y) = 1. Let U be the subspace of V generated by x and y.…”

Section: Proof We Clearly Have Thatmentioning

confidence: 99%

Totally decomposable symplectic and unitary involutions

Dolphin

2016

manuscripta math.

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Abstract. We study totally decomposable symplectic and unitary involutions on central simple algebras of index 2 and on split central simple algebras respectively. We show that for every field extension, these involutions are either anisotropic or hyperbolic after extending scalars, and that the converse holds if the algebras are of 2-power degree. These results are new in characteristic 2, otherwise were shown in [3] and [6] respectively.

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Quadratic descent of hermitian forms

Nokhodkar

2019

Mathematische Nachrichten

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Quadratic and Hermitian Forms over Rings

Cited by 361 publications

References 0 publications

On the classification of quadratic forms over an integral domain of a global function field

On the classification of quadratic forms over an integral domain of a global function field

Totally decomposable symplectic and unitary involutions

Quadratic descent of hermitian forms

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