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

JOURNAL OF NUMBER THEORY

卷:180

Article

Bitan, Rony A.^1,2 

[1] Bar Ilan Univ, Ramat Gan, Israel

[2] Tel Aviv Acad Coll Engn, Afeka, Tel Aviv, Israel

关键词: Quadratic forms;    Number theory;    Etale cohomology;    Global function fields;   

DOI  :  10.1016/j.jnt.2017.03.007

来源: Elsevier

【 摘 要 】

Let C be a smooth projective curve defined over the finite field F-q (q is odd) and let K = F-q(C) be its function field. Any finite set S of closed points of C gives rise to an integral domain O-s := Fq[C - S] in K. We show that given an O-s-regular quadratic space (V, q) of rank n >= 3, the set of genera in the proper classification of quadratic O-s-spaces isomorphic to (V, q) in the flat or etale topology, is in 1 : 1 correspondence with Br-2(O-s), thus there are 2(|s|-1) genera. If (V, q) is isotropic, then Pic (O-s)/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-et(2)(O-s,(mu) under bar (2)). (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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