Integral models of algebraic tori over fields of algebraic numbers
The paper is concerned with integral models of algebraic tori over algebraic number fields together with the comperison of properties of the models. It depends on the scheme which represent the (Néron, Voskresenskií) model under consideration. The author offers a setting for the Néron model of an algebraic torus, namely the standard or the canonical integral models of the torus. The basic result concerns the Néron model of a maximal algebraic torus unaffected in a semisimple algebraic group. Integral models of algebraic tori over Dedekind schemes are presented in the book by \textit{S. Bosch} et al. [Néron models. Berlin etc.: Springer-Verlag (1990; Zbl 0705.14001)], in papers by \textit{V. E. Voskresenskij} [Math. Notes 44, No. 3--4, 651--655 (1988; Zbl 0699.20037); translation from Mat. Zametki 44, No. 3, 309--318 (1988)], by \textit{S. Bosch} and \textit{Q. Liu} [Manuscr. Math. 98, No. 3, 275--293 (1999; Zbl 0934.14029)] and by \textit{S. Yu. Popov} [Vestn. Samar. Gos. Univ., Mat. Mekh. Fiz. Khim. Biol. 2001, No. 4(22), 85--108 (2001; Zbl 1072.14529)]. Let \(T\) be an algebraic torus over a field \(k\), \([L:k]\) its splitting field, \(G = {\mathcal Gal}(L:k)\). Section 1 introduces \(G\)-module \(\hat T\) of rational characters of \(T\), its coordinate ring \(k[T] = L[\hat T]^G\) and affine realization of \(T\). Section 2 concerns definitions of (Voskresenskií, canonical integral, standard integral) models of algebraic tori. Section 3 gives some indications on algorithms of constracting of Néron models of algebraic torus over global field of characteristic 0 and over its localizations. Section 4 presents properties of standard integral model over a number field and of the corresponding models over local fields. The last section contains interesting observations on a family of algebraic tori of a special structure and concludes with an explicit representation of the Néron model of a maximal algebraic torus unaffected in a semisimple algebraic group with the root system \(B_n\). It would be interesting to know how the author's work relates to \textit{A. Mézard}, \textit{M. Romagny} and \textit{D. Tossici} [Ann. Inst. Fourier 63, No. 3, 1055--1135 (2013; Zbl 1297.14051)] solution under the construction of models of \((\mathbb{G}_{m,K})^n\). (Mézard, Romagny, Tossici assume that \(K\) is the fractional field of a discrete valuation ring and note that ``we set up a precise, nontrivial dictionary that indicates how to translate the congruences in a discrete valuation ring of characteristic 0 on one side, into congruences in a discrete valuation ring of characteristic \(p\) on the other side''. They present models of \((\mathbb{G}_{m,K})^n\) constructed by successive extensions of affine, smooth, one-dimensional models of \(\mathbb{G}_{m,K}\) with connected fibers, called filtered group schemes).
scientific article; zbMATH DE number 6676853