Mathematical logic, algebra and number theory

On the coincidence of standard and canonical integral models of an arbitrary algebraic torus over a number field

M. V. Grekhov

Samara National Research University, Department of Algebra and Geometry, Ac. Pavlova st., 1, 443068, Samara, Russia

Abstract: Algebraic tori occupy a special place among linear algebraic groups. An algebraic torus can be defined over an arbitrary field but if a ground field belongs to an arithmetic type one can additionally consider schemes over this field's ring of integers which are linked to the original tori and called their integral models. Néron model and Voskresenskiĭ model are most well-known among them. There exists a broad range of problems dealing with the construction of these models and the research of their properties. This paper is dedicated to the research of some important integral models of algebraic tori over number fields, namely, standard and canonical integral models. Finally, the coincidence of these two models for an arbitrary algebraic torus is proven.

Keywords: algebraic tori, integral models.

UDC: 512.761.2

MSC: 14L17

Received December 20, 2016, published October 16, 2017

DOI: 10.17377/semi.2017.14.086

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