Integral Models of Algebraic Tori Over Fields of Algebraic Numbers

Abstract

Algebraic tori occupy a special place among linear algebraic groups. An algebraic torus can be defined over an arbitrary field but if the ground field is of arithmetic type, one can additionally consider schemes over the ring of integers of this field, which are related to the original tori and called their integral models. The Neron and Voskresenskiĭ models are most well known among them. There exists a broad range of problems dealing with the construction of these models and the elucidation of their properties. This paper is devoted to the study of the main integral models of algebraic tori over fields of algebraic numbers, to the comparison of their properties, and to the clarification of links between them. At the end of this paper, a special family of maximal algebraic tori unaffected inside semisimple groups of Bn type is presented as an example for realization of previously investigated constructions.