Non-split reductive groups over Z

Abstract. — We study the following phenomenon: some non-split connected semisimple Q-groups G admit flat affine Z-group models G with “everywhere good reduction ” (i.e., GFp is a connected semisimple Fp-group for every prime p). Moreover, considering such G up to Z-group isomorphism, there can be more than one such G for a given G. This is seen classically for types B and D by using positive-definite quadratic lattices. The study of such Z-groups provides concrete applications of many facets of the theory of reductive groups over rings (scheme of Borel subgroups, auto-morphism scheme, relative non-abelian cohomology, etc.), and it highlights the role of number theory (class field theory, mass formulas, strong approximation, point-counting over finite fields, etc.) in analyzing the possibilities. In part, this is an expository account of [G96]. Résumé. — Nous étudions le phénomène suivant: certains Q-groupes G semi-simples connexes non déployés admettent comme modèles des Z-groupes G affines et plats avec “partout bonne réduction ” (c’est a ̀ dire, GFp est un Fp