The Hasse principle for bilinear symmetric forms over a ring of integers of a global function field

JOURNAL OF NUMBER THEORY

卷:168

Article

关键词: Global function field;    Bilinear forms;    Etale cohomology;    Hasse principle;   

DOI  :  10.1016/j.jnt.2016.04.004

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. Removing one closed point C-af = C - {infinity} results in an integral domain O-{infinity} = F-q [C-af] of K, over which we consider a non-degenerate bilinear and symmetric form f with orthogonal group (O) under barv. We show that the set C1(infinity) ((O) under barv) of O-{infinity} -isomorphism classes in the genus of f of rank n > 2 is bijective as a pointed set to the abelian groups H-et(2) = (O-{infinity}, (mu) under bar (2), i.e. it is an invariant C-af. We then deduce that any such f of rank n > 2 admits the local-global Hasse principal if and only if vertical bar Pic (C-af)vertical bar is odd. For rank 2 this principle holds if the integral closure of O-{infinity}. in the splitting field of (O) under bar (v)circle times O-{infinity} is a UFD. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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