## Abstract

Let K = F _{q} (C) be the global function field of rational functions over a smooth and projective curve C defined over a finite field F _{q} . The ring of regular functions on C − S Where S ≠ ∅ is any finite set of closed points on C is a Dedekind domain O _{S} of K. For a semisimple O _{S} -group G With a smooth fundamental group F, We aim to describe both the set of genera of G and its principal genus (the latter if G ⊗ _{OS} K is isotropic at S) in terms of abelian groups depending on O _{S} and F only. This leads to a necessary and sufficient condition for the Hasse local-global principle to hold for certain G. We also use it to express the TamagaWa number τ(G) of a semisimple K-group G by the Euler–Poincaré invariant. This facilitates the computation of τ(G) for tWisted K-groups.

Original language | English |
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Pages (from-to) | 1037-1057 |

Number of pages | 21 |

Journal | Journal de Theorie des Nombres de Bordeaux |

Volume | 30 |

Issue number | 3 |

DOIs | |

State | Published - 2018 |

### Bibliographical note

Publisher Copyright:© Société Arithmétique de Bordeaux, 2018,.

## Keywords

- Class number
- Hasse principle
- Mots-clefs
- Tamagawa number