Skip Garibaldi and Detlev

In a 2004 paper, Totaro asked whether a G-torsor X that has a zero-cycle of degree d > 0 will necessarily have a closed étale point of degree dividing d, where G is a connected algebraic group. This question is closely related to several conjectures regarding exceptional algebraic groups. Totaro gave a positive answer to his question in the following cases: G simple, split, and of type G2, type F4, or simply connected of type E6. We extend the list of cases where the answer is “yes” to all groups of type G2 and some nonsplit groups of type F4 and E6. No assumption on the characteristic of the base field is made. The key tool is a lemma regarding linkage of Pfister forms. For certain linear algebraic groups G over a field k and certain homogeneous G-varieties X, Totaro asked in [To]: (0.2) If X has a zero-cycle of degree d > 0, does X necessarily have a closed étale point of degree dividing d? (This question is closely related to earlier questions raised by Veisfeiler, Serre, and Colliot-Thélène.) This question can be rephrased as: (0.2) If X has a (closed) point over finite extensions K1,K2, . . . ,Kn of k, does X necessarily have a point over a separable extension of k of degree dividing every [Ki : k]? The purpose of this note is to prove the following theorem. Theorem 0.3. The answer to (0.2) is “yes” when X is a G-torsor and G is • of type G2, • reduced of type F4, or • simply connected of type 1E0 6,6 (split) or of type 1E28 6,2. A G-torsor is a principal G-bundle over Speck. Torsors are often called principal homogeneous spaces, as in [SeGC]. It is possible that the answer to (0.2) is “yes” whenever G is semisimple and X is a G-torsor (and, in particular, is affine); no counterexamples are known. In contrast, for X projective, there are examples where the answer is “no” even when d = 1, see [Fl] and [Pa]. Every group of type F4 is the group of automorphisms of some uniquely determined Albert k-algebra J . We say that the group is reduced if the 2000 Mathematics Subject Classification. 11E72 (20G15).