The starting point of this note is the observation that the local condition used in the notion of a Hilbert-symbol equivalence and a quaternion-symbol equivalence — once it is expressed in terms of the Witt invariant — admits a natural generalisation. In this paper we show that for global function fields as well as the formally real function fields over a real closed field all the resulting equivalences coincide.
Let be an integer. Let be the modular curve over , as constructed by Katz and Mazur. The minimal resolution of over is computed. Let be a prime, such that , with prime to . Let . It is shown that has stable reduction at over , and the fibre at of the stable model is computed.
Soient un fibré de rang 2 sur l’espace projectif de dimension 3 sur un corps algébriquement clos et un entier tel que et . Toute courbe schéma des zéros d’une section non nulle de est une courbe minimale dans sa classe de biliaison.