Birational geometry and arithmetic of linear algebraic groups. IV.
For part III of this paper see: ibid. 1998, No. 2(8), 5--54 (1998; Zbl 1038.14019).NEWLINENEWLINEThis is the fourth part of a survey concerning the geometry and the arithmetic of linear algebraic groups. This part has two chapters: \(R\)-equivalence on algebraic groups and index formulas in the arithmetic of algebraic tori.NEWLINENEWLINEThe concept of \(R\)-equivalence was introduced by \textit{Yu. I. Manin} in his book ``Cubic forms'' Moskov 1972; English translation 1974; Zbl 0277.14014). After a general discussion on the \(R\)-equivalence on algebraic varieties, the author deals with the \(R\)-equivalence on algebraic tori, with the unimodular group of simple algebras, and with algebras with involution and groups of adjoint type. Several results due to Colliot-Thélène and Sansuc on the computation of \(R\)-equivalence of algebraic tori, then some of the author's results on the \(R\)-equivalence on simply connected semisimple groups and the connection (due to Platonov) between the class group of \(R\)-equivalence with Whitehead group \(\mathrm{SK}_1(A)\) are discussed. Finally, the \(R\)-equivalence of the groups of adjoint type (including results due to Merkur'ev) are presented. In the final part the author calculates the class numbers of algebraic tori in two significant cases, which leads to some interesting formulae in the arithmetic of number fields.
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