On the equation N_{K/k}(\Xi)=P(t)
  For varieties given by an equation N_{K/k}(\Xi)=P(t), where N_{K/k} is the norm form attached to a field extension K/k and P(t) in k[t] is a polynomial, three topics have been investigated:   (1) computation of the unramified Brauer group of such varieties over arbitrary fields;   (2) rational points and Brauer-Manin obstruction over number fields (under Schinzel's hypothesis);   (3) zero-cycles and Brauer-Manin obstruction over number fields.   In this paper, we produce new results in each of three directions. We obtain quite general results under the assumption that K/k is abelian (as opposed to cyclic in earlier investigation). 
Dasheng Wei
math.NT