The main object of the paper under review is an \(X\)-torsor \(Y\) under \(G\), where \(X\) is a smooth variety over a local field \(K\) (the field of fractions of a Henselian discrete valuation ring \(R\)) and \(G\) is an algebraic \(K\)-torus. The residue field \(k\) of \(K\) is assumed perfect. The author’s goal is to show that, under certain conditions, the evaluation map \(X(K)\to H^1(K,G)\), which associates to each point \(P\) the isomorphism class of the fibre \(Y_P\), factors through reduction to the special fibre, i.e., for a suitable \(R\)-model \(\mathcal X\) of \(X\), the evaluation map comes from \(\mathcal X_s(k)\to H^1(k,\mathcal G_s)\), where \(\mathcal G\) is the Néron–Raynaud model of \(G\) and the subscript \(s\) refers to the special fibres of \(\mathcal X\) and \(\mathcal G\). This is proved when the torus \(G\) is split by a tamely ramified extension of \(K\). The author presents an example showing that one should not expect such a result in wildly ramified cases. Some intermediate results on extending torsors are formulated for more general algebraic groups \(G\) and are interesting by their own right. Finally, the author discusses some applications to the case where \(X\) is a rational surface.