Birational maps between generalized Severi–Brauer varieties

A conjecture of Amitsur states that two Severi–Brauer varieties V(A) and V(B) are birationally isomorphic if and only if the underlying algebras A and B are the same degree and generate the same cyclic subgroup of the Brauer group. We examine the question of finding birational isomorphisms between generalized Severi–Brauer varieties. As a first step, we exhibit a birational isomorphism between the generalized Severi–Brauer variety of an algebra and its opposite. We also extend a theorem of P. Roquette to generalized Severi–Brauer varieties and use this to show that one may often reduce the problem of finding birational isomorphisms to the case where each of the separable subfields of the corresponding algebras are maximal, and therefore to the case where the algebras have prime power degree. We observe that this fact allows us to verify Amitsur’s conjecture for many particular cases.