Formules pour l’invariant de Rost

Rost invariant of the center, revisited

“…As mentioned in [Gil00, §2.3], there are several definitions of the Rost invariant that may differ by a sign. In [GQM11], Gille and Quéguiner proved that for the definition of the Rost invariant r G they choose, in the case G = SL 1 (A) for a central simple algebra A of degree n over F , the value of r G on the image of the class 6d. The main theorem.…”

“…for every x ∈ K × . (The formula in [GQM11] contains an additional minus sign, but it does not appear here due to the adjustment in the definition of r G in §6c.) 8b.…”

“…This improves on the results of [MPT02] and [GQM07] by removing the hypothesis on the characteristic and avoiding the ad hoc typeby-type arguments used in those papers. We do rely on [GQM11] for the computation of m for type A, but nothing more. The strategy is to (1) extend the determination of invariants of quasi-trivial tori from [MPT02] to all fields (see Theorem 3.7), (2) to follow the general outline of [GQM07] to reduce to the case of type A, and (3) to avoid the ad hoc formulas used in previous work by giving a formula independent of the Killing-Cartan type of G.…”

“…for C the center of G, and under some assumptions on char F , computed the composition in terms of the value of m for type A. Eventually the value of m was determined in [GQM11]. The main result is Theorem 1.2, which gives a formula for (1.1) which does not depend on the type of G nor on char F .…”

“…(see Remark 2.11 after Proposition 2.10). The bottom arrow is given by the cup product map m Br(F ) ⊗ (F × /F ×m ) → H 3 (F, Q/Z(2)) (see [GQM11,4D]). It is shown in [GQM11, Theorem 1.1] that the p-component of the Rost invariant of G is given by the formula…”