On Serre’s injectivity question and norm principle

Abstract

Let k be a field of characteristic not 2. We give a positive answer to Serre's injectivity question for any smooth connected reductive k-group whose Dynkin diagram contains connected components only of type A n , B n or C n . We do this by relating Serre's question to the norm principles proved by Barquero and Merkurjev. We give a scalar obstruction defined up to spinor norms whose vanishing will imply the norm principle for the non-trialitarian D n case and yield a positive answer to Serre's question for connected reductive k-groups whose Dynkin diagrams contain components of non-trialitarian type D n also. We also investigate Serre's question for reductive k-groups whose derived subgroups admit quasi-split simply connected covers. arXiv:1511.00311v2 [math.AG]

Totaro’s question on zero-cycles on torsors

“…Affirmative answers are known in many other special cases (cf. [Bha16], [Bla11a], [Bla11b]), though in general, Serre's question is still open.…”

The norm principle for type $D_n$ groups over complete discretely valued fields

“…In ( [BM00]), it was established that the norm principle holds in general for all reductive groups of classical type without D n components. The D n case was investigated in ( [Bh16]) and a scalar obstruction defined up to spinor norms, whose vanishing would imply the norm principle, was given. However, the triviality of this scalar obstruction is far from clear and the question whether the norm principle holds for reductive groups with type D n components still remains open.…”