TY - JOUR
T1 - On the classification of quadratic forms over an integral domain of a global function field
AU - Bitan, Rony A.
N1 - Publisher Copyright:
© 2017 Elsevier Inc.
PY - 2017/11
Y1 - 2017/11
N2 - 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 or étale topology, is in 1:1 correspondence with Br2(OS) thus there are 2|S|−1 genera. 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 Hét 2(OS,μ_2).
AB - 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 or étale topology, is in 1:1 correspondence with Br2(OS) thus there are 2|S|−1 genera. 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 Hét 2(OS,μ_2).
KW - Global function fields
KW - Number theory
KW - Quadratic forms
KW - Étale cohomology
UR - http://www.scopus.com/inward/record.url?scp=85019679181&partnerID=8YFLogxK
U2 - 10.1016/j.jnt.2017.03.007
DO - 10.1016/j.jnt.2017.03.007
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AN - SCOPUS:85019679181
SN - 0022-314X
VL - 180
SP - 26
EP - 44
JO - Journal of Number Theory
JF - Journal of Number Theory
ER -