## 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. Removing one closed point C^{af}=C−{∞} results in an integral domain O_{{∞}}=F_{q}[C^{af}] of K, over which we consider a non-degenerate bilinear and symmetric form f with orthogonal group O__{V}. We show that the set Cl_{∞}(O__{V}) of O_{{∞}}-isomorphism classes in the genus of f of rank n>2 is bijective as a pointed set to the abelian groups H_{ét}^{2}(O_{{∞}},μ__{2})≅Pic (C^{af})/2, i.e. it is an invariant of C^{af}. We then deduce that any such f of rank n>2 admits the local-global Hasse principal if and only if |Pic (C^{af})| is odd. For rank 2 this principle holds if the integral closure of O_{{∞}} in the splitting field of O__{V}⊗_{O{∞}}K is a UFD.

Original language | English |
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Pages (from-to) | 346-359 |

Number of pages | 14 |

Journal | Journal of Number Theory |

Volume | 168 |

DOIs | |

State | Published - 1 Nov 2016 |

### Bibliographical note

Publisher Copyright:© 2016 Elsevier Inc.

## Keywords

- Bilinear forms
- Global function field
- Hasse principle
- Étale cohomology