Arithmetical birational invariants of linear algebraic groups over two-dimensional geometric fields
Extending results of Colliot-Thélène, Gille and Parimala, the paper aims to compute some invariants of linear algebraic groups over a field~\(k\). These fields are mostly of cohomological dimension~\(2\): they may be global (function field of a surface), local (field of fractions of an excellent local domain) or semi-local (Laurent series over a field of cohomological dimension~\(1\). \(G\) is a connected linear \(k\)-group: in the global case it is assumed that there are no \(E_8\) factors. The invariants computed are \(G(k)/R\), the group of \(R\)-equivalence classes; \(A_\Sigma(G)\), the defect of weak approximation for a set \(\Sigma\) of places (in the global and local cases); the Galois cohomology \(H^1(k,G)\); and the Tate-Shafarevich set Ш\(^1(k,G)\) (in the local case). It is shown that \(G(k)/R\), \(A_\Sigma(G)\) and Ш\(^1(k,G)\) are stably \(k\)-birational invariants of~\(G\). Put \(\Gamma=\text{Gal}(\bar k/k)\) and let \(\pi_1(G)\) be the algebraic fundamental group of the reductive group obtained by dividing out the unipotent radical. The procedure is to look at various functors \({\mathcal H}\) from \(k\)-tori to abelian groups that are trivial on quasi-trivial tori, such as \({\mathcal H}(T)=H^1(k,T)\). Taking \(0\to Q\to P\to \pi_1(G)\to 0\), with \(P\) a permutation module and \(Q\) a coflasque \(\Gamma\)-module, we get a torus \(F_G\) whose cocharacters are~\(Q\): then \({\mathcal H}(F_G)\) is uniquely determined by~\(G\). Given a smooth rational \(k\)-variety \(X\) there is a torus \(S_X\) whose characters are the Picard group of a smooth compactification over the algebraic closure, and \({\mathcal H}(S_X)\) is uniquely determined by \(X\): moreover, it is a \(k\)-stably birational invariant. If \(X=G\), \({\mathcal H}(S_G)={\mathcal H}(T_G)\). For the fields considered, \(G(k)/R=H^1(k,F_G)\) so \(G(k)/R\) is a \(k\)-stably birational invariant. For weak approximation, it is shown that \[ A_\Sigma(G)=\text{coker} (H^1(k,S_G)\to \prod_{v\in \Sigma}H^1(k_v,S_G)). \] The case of \(H^1(k,G)\) is dealt with by using other the results of the first author, which compare it with abelian Galois cohomology. Using this last relation, it is shown here that Ш\(^1(k,G)\cong\text Ш^2(k,F_G)\) and one may take \({\mathcal H}(T)=\text Ш^2(k,T)\) and deduce the results about Ш\(^1\): in particular, \(\#\text Ш^1(k,G)=1\) if \(G\) is stably \(k\)-rational. Some extensions of these results to number fields are also given. One important technical lemma is that \(G(k)\to G^{\text{tor}}(k)=G/G^{ss}(k)\) induces an isomorphism \(G(k)/R\to G^{\text{tor}}(k)/R\). Two proofs are given: one brief by perhaps slightly mysterious methods, suggested by Colliot-Thélène, and one more self-contained and explicit, by Gille in an appendix to this paper.
scientific article; zbMATH DE number 2105417
scientific article