Méthode du cercle adélique et principe de Hasse fin pour certains systèmes de formes. (Adélic circle method and finite Hasse principle for certain systems of forms) (Q1069987)

The circle method of Hardy and Littlewood and a form of Hasse principle are placed in the setting of adèles. The development is based on four hypotheses, the first being related to Weyl's inequality and Dirichlet's theorem on simultaneous rational approximation, and the usual order of dealing with major and minor arcs and the singular series and integral is followed. A special case of the principal theorem of the paper is that when the first two hypotheses are satisfied, the version of the Hasse principle holds for a system of r integral forms of degree d in n variables. The work of Davenport, Birch and Schmidt is discussed from the point of view of this development.