Rational maps between varieties associated to central simple algebras

In this paper we show that if two central simple $k$-algebras generate the same cyclic subgroup in $\mathrm{B}\mathrm{r}(k)$, then there are rational maps between varieties associated to these algebras, such as Brauer–Severi varieties, norm hypersurfaces and symmetric powers. In some cases we even have rational embeddings. We also relate the obtained results to the Amitsur conjecture.