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Arithmetic of linear algebraic groups over 2-dimensional geometric fields. (English) Zbl 1129.11014

The general theme of the paper is that most properties of linear algebraic groups and their homogeneous spaces over totally imaginary number fields have analogues for such groups over various classes of fields “of dimension 2.”
The first main result holds for all fields \(K\) of characteristic zero and cohomological dimension at most \(2\) such that the index and the exponent of every finite-dimensional simple \(K\)-algebra coincide. It states that for the standard isogeny \(1\to\mu\to G\to G^{\text{ad}}\to 1\) between a simply connected algebraic group \(G\) and its adjoint group, the boundary map \(H^1(K,G^{\text{ad}})\to H^2(K,\mu)\) is surjective. Moreover, \(G\) is isotropic if it is not purely of type \(A\). (If \(G\) has some factor of type \(E_8\), an extra hypothesis is required.)
Building on work of Chernousov, Merkurjev, Monastyrniĭ, Platonov and Yanchevskiĭ, the authors then show that the group of \(R\)-equivalence classes \(G(K)/R\) is trivial if \(G\) is a semisimple group which is either simply connected or adjoint or absolutely almost simple or an inner form of a group which is split by a metacyclic extension of \(K\).
Suppose now that \(K\) is either (i) the field of fractions of a Henselian excellent 2-dimensional local domain with algebraically closed residue field of characteristic zero, or (ii) the function field of a smooth projective connected surface over an algebraically closed field of characteristic zero. In each case, there is an associated set \(\Omega\) of discrete valuations, and the authors show that weak approximation holds for semisimple groups which are either simply connected, adjoint, absolutely almost simple or inner forms of a group split by a metacyclic extension of \(K\). (In case (ii), they assume that \(G\) has no factor of type \(E_8\).) They go on to prove the Hasse principle for projective homogeneous spaces of arbitrary connected linear groups over fields of type (i).
A substantial part of the argument relies on a careful analysis of tori, for which analogues of results of J.-L. Colliot-Thélène and J.-J. Sansuc [Ann. Sci. Éc. Norm. Supér., IV. Sér. 10, 175–229 (1977; Zbl 0356.14007)] are established: finiteness of \(R\)-equivalence on sets of rational points, computation of groups measuring the failure of weak approximation or of the Hasse principle for homogeneous spaces.
The main results of the paper were announced in [C. R. Acad. Sci., Paris, Sér. I, Math. 333, No. 9, 827–832 (2001; Zbl 1037.20050)].

MSC:

11E76 Forms of degree higher than two
14G35 Modular and Shimura varieties
20G35 Linear algebraic groups over adèles and other rings and schemes
20G15 Linear algebraic groups over arbitrary fields
14G25 Global ground fields in algebraic geometry
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