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

Bitan, Rony A.

doi:10.1016/j.jnt.2017.03.007

Cited by 3 publications

(3 citation statements)

References 16 publications

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“…In order to provide a counter-example, we refer in Section 4 more concretely to the case in which (O Γ V ) 0 is the special orthogonal group of another isotropic form (V ′ , q ′ ). Based on a result established in [Bit1] and [Bit2], stating that if q is isotropic of rank ≥ 3, then c(q) ∼ = Pic (O S )/2, we show that if q ′ is isotropic of rank 2 then it may possess twisted forms which are only stably isomorphic, namely, that become isomorphic over (any) regular non-trivial extension of V ′ . So our suggested obstruction to Question 1.2 arises from the failure of the Witt cancellation theorem over O S .…”

Section: After Showing In Section 2 That [Hmentioning

confidence: 77%

“…We would like to refer to the case in which (O Γ V ) 0 is the special orthogonal group of another isotropic Γ-form. It is shown in [Bit1,Proposition 4.4] for |S| = 1 and more generally in [Bit2,Theorem 4.6] for any finite S, that if rank(V ) ≥ 3 (q is isotropic), then c(q) ∼ = Pic (O S )/2. For rank(V ) = 2, however, this genus might be larger.…”

Section: An Explicit Obstructionmentioning

confidence: 99%

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Between the genus and the $\varGamma $-genus of an integral quadratic $\varGamma $-form

Bitan¹

2017

Acta Arith.

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Let Γ be a finite group and (V, q) be a regular quadratic Γ-form defined over an integral domain O S of a global function field (of odd characteristic). We use flat cohomology to classify the quadratic Γ-forms defined over O S that are locally Γ-isomorphic for the flat topology to (V, q) and compare between the genus c(q) and the Γgenus c Γ (q) of q. We show that c Γ (q) should not inject in c(q). The suggested obstruction arises from the failure of the Witt cancellation theorem for O S .

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Section: After Showing In Section 2 That [Hmentioning

confidence: 77%

Section: An Explicit Obstructionmentioning

confidence: 99%

Between the genus and the $\varGamma $-genus of an integral quadratic $\varGamma $-form

Bitan¹

2017

Acta Arith.

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“…In case |S| = 1 and q is split by an hyperbolic plane, an algorithm producing explicitly the inner forms of q is provided in [Bit2,Algorithm1].…”

Section: Split Fundamental Groupmentioning

confidence: 99%

On the genera of semisimple groups defined over an integral domain of a global function field

Bitan¹

2018

J. Théor. Nombres Bordeaux

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Let K = Fq(C) be the global field of rational functions on a smooth and projective curve C defined over a finite field Fq. Any finite but non-empty set S of closed points on C gives rise to a Hasse integral domain OS = Fq[C − S] of K. Given an almost-simple group scheme G defined over Spec OS with a smooth fundamental group F (G), we describe the finite set of (OS-classes of) twisted-forms of G in terms of geometric invariants of F (G) and the absolute type of the Dynkin diagram of G. This turns out in most cases to biject to a disjoint union of finite abelian groups. − −− → H 1 fl (O S , Aut(R)). Let A = D(B, σ) be the discriminant algebra. If R splits, namely, R ∼ = O S ×O S , then B ∼ = A×A op and σ is the exchange involution. Then U(B, σ) ∼ = GL 1 (A) and SU(B, σ) ∼ = SL 1 (A), so we are back in the previous situation. 20 Corollary 7.3. The map U(B, σ)(O S ) Nrd − − → R × is surjective if and only if the Hasse-principle holds for SU(B, σ). Acknowledgements: The authors thank P. Gille, B. Kunyavskiȋ and A. Quéguiner-Mathieu for valuable discussions concerning the topics of the present article.

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The twisted forms of a semisimple group over an 𝔽 q -curve

Bitan¹,

Köhl²,

Schoemann³

2021

Journal De Théorie Des Nombres De Bordeaux

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The twisted forms of a semisimple group over an F q -curve Tome 33, n o 1 (2021), p. 17-38. © Société Arithmétique de Bordeaux, 2021, tous droits réservés. L'accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.centre-mersenne.org/ Journal de Théorie des Nombres de Bordeaux 33 (2021), 17-38 The twisted forms of a semisimple group over an F q -curve par Rony A. BITAN, Ralf KÖHL et Claudia SCHOEMANN Résumé. Soit C une courbe projective, lisse et connexe définie sur un corps fini F q . Étant donné un C − S-schéma en groupes semisimples où S est un ensemble fini de points fermés de C, nous décrivons l'ensemble de (O S -classes de) formes tordues de G en termes d'invariants géométriques de son groupe fondamental F (G).

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

Cited by 3 publications

References 16 publications

Between the genus and the $\varGamma $-genus of an integral quadratic $\varGamma $-form

Between the genus and the $\varGamma $-genus of an integral quadratic $\varGamma $-form

On the genera of semisimple groups defined over an integral domain of a global function field

The twisted forms of a semisimple group over an 𝔽 q -curve

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