On the equation N_{K/k}(Ξ)=P(t)

Details On the equation N_{K/k}(Ξ)=P(t) On the equation N_{K/k}(Ξ)=P(t)

2012-02-18

arxiv.org/abs/1202.4115v1

Mathematics

Number Theory

51 pages

Scientific paper

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).

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