On Serre's question and norm principles

A zero cycle on a k-variety X is any element of the free abelian group of closed points of X and its degree is the sum of its coefficients, weighted by the degrees of the residue fields of the closed points involved. Any k-rational point of X is a zero cycle of degree one. In this talk, we discuss Serre’s injectivity question which asks whether the converse is true for principal homogeneous spaces X/k under connected linear algebraic groups G/k, i.e. whether such an X admitting a zero cycle of degree one in fact has a rational point. This naturally brings into the picture the so-called norm principles, which examine the behaviour of the images of group morphisms over field extensions from a linear algebraic group into a commutative one with respect to the norm map. Norm principles are interesting in their own right and have been previously studied by Merkurjev-Gille, especially in conjunction to the rationality of the algebraic group in question. However a result of Merkurjev and Barquero shows that the norm principle holds for all classical reductive groups of type A,B,C without any rationality assumptions with type D still remaining open. If time permits, we will report progress on the type D case (which part is joint work with Chernousov and Merkurjev).