This is my M2 thesis at Sorbonne University under the guidance of Gaëtan Chenevier. It is a survey of Gross's Paper Groups over Z.
Abstract: When we study automorphic representations of a reductive Q-group G, sometimes we need G to be the generic fiber of some reductive Z-group scheme. If this holds, we say that G admits a Z-model. In SGA3, the theory of Chevalley groups tells us any split connected reductive Q-group has a unique Z-model up to Z-group isomorphism. For semisimple groups there are also some non-split examples. However, not all non- split semisimple Q-groups have Z-models. In his famous survey paper, Gross states two necessary and suﬀicient conditions for semisimple groups to admit Z-models, which are proved by Harder, and enumerates all the possibilities via the mass formula in some cases. In this thesis, we are going to give detailed proofs of these conditions (Proposition 1.5,1.7) and the mass formula (Theorem 2.4), and then follow Gross’s route to construct Z-models for these non-split Q-groups, especially for anisotropic groups of exceptional types G2,F4.