Preliminaries on Adele-Geometry

Abstract

We always denote by k a field of algebraic numbers or a field of algebraic functions of one variable over a finite constant field. We denote by k the completion of k with respect to a valuation v of k; if v is discrete, we use often the notation p and denote by $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}\to {o} _p$ the ring of p-adic integers in k_p. We denote by S any finite set of valuations which contains all the non discrete valuations (infinite places).