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

Rony A. Bitan

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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 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).

Original languageEnglish
Pages (from-to)26-44
Number of pages19
JournalJournal of Number Theory
Volume180
DOIs
StatePublished - Nov 2017

Bibliographical note

Publisher Copyright:
© 2017 Elsevier Inc.

Keywords

  • Global function fields
  • Number theory
  • Quadratic forms
  • Étale cohomology

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