## Abstract

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}:=F_{q}[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 étale topology, is in 1:1 correspondence with Br2(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_{ét} ^{2}(O_{S},μ__{2}).

Original language | English |
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Pages (from-to) | 26-44 |

Number of pages | 19 |

Journal | Journal of Number Theory |

Volume | 180 |

DOIs | |

State | Published - Nov 2017 |

### Bibliographical note

Publisher Copyright:© 2017 Elsevier Inc.

## Keywords

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