Groups over \(\mathbb{Z}\). (English) Zbl 0846.20049

The main objects of the paper are \(\mathbb{Z}\)-models \(\mathbf G\) of connected simply connected simple algebraic \(\mathbb{Q}\)-groups \(G\), that is smooth affine group schemes of finite type over \(\mathbb{Z}\) with general fibre \(G\) such that all special fibres \({\mathbf G}\otimes\mathbb{Z}/p\mathbb{Z}\) are reductive. The author enumerates \(\mathbb{Q}\)-groups admitting \(\mathbb{Z}\)-models and proves an analog of a theorem by G. Harder [Ann. Sci. Éc. Norm. Supér., IV. Sér. 4, 409-455 (1971; Zbl 0232.20088)] on the Euler-Poincaré characteristic of \({\mathbf G}(\mathbb{Z})\). If \(G(\mathbb{R})\) is compact, this gives a mass formula allowing to compute the number of different \(\mathbb{Z}\)-models for all groups of rank at most 8 (except \(E_8\)); these models are exhibited explicitly as well as two \(\mathbb{Z}\)-models of \(E_8\) obtained via its adjoint representation.


20G30 Linear algebraic groups over global fields and their integers
14L15 Group schemes


Zbl 0232.20088
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