The Diophantine problem in isotropic reductive groups

Paper

Abstract

We begin to study model-theoretic properties of non-split isotropic reductive group schemes. In this paper we show that the base ring

is e-interpretable in the point group

of every sufficiently isotropic reductive group scheme

. In particular, the Diophantine problems in

and

are equivalent. We also compute the centralizer of the elementary subgroup of

and the common normalizer of all its root subgroups.