On the birational equivalence of tori with a cyclic decomposition field. (Russian) Zbl 0358.14017

Let \(K\) be an arbitrary field, and \(L\) a cyclic finite extension of \(K\) with \(G=\)Gal\((L/K)\) of order \(n\). Let \(C(L/K)\) be the category of algebraic tori over \(K\) with decomposition field \(L\). If \(T\in C(L/K)\), then the group of characters \(\hat T\) of \(T\) is naturally a module of finite type over the group ring \(\mathbb Z[G]\cong\mathbb Z [X]/(X^n-1)\). For every polynomial \(f\in \mathbb Z[X]\) dividing \(X^n -1\), one can define a functor \(T\to T^f\) on \(C(L/K)\) as follows: \(\widehat{T^f}=\)Hom\((T,G_m)\) is nothing but Hom\(_G (\mathbb Z[X]/(f(X)), \hat T)\) \(T\) is said to be rational over \(K\) if the function field \(K(T)\) is a pure transcendental extension of \(K\), and stable rational, if \(T\times G_m^r\) is rational for a suitable natural number \(r\). Finally, let \(\Phi_d (d|n)\) denote the cyclotomic polynomial. It is known by V. E. Voskresenskiĭ [Usp. Mat. Nauk 28, No. 4 (172), 77–102 (1973; Zbl 0283.14004)] that the tori \(T\) such that \(\hat T=\mathbb Z[X]/(\Phi_d)\) are stable rational. If moreover \(n=p^t\) (\(p\) prime), then every stable rational torus over \(K\) is rational, every \(T\in C(L/K)\) is birationally equivalent over \(K\) to the product \(\prod_{d|n}T^{\Phi_d}\), and \(T\) is stable rational over \(K\) if and only if \(\widehat{T^{\Phi_d}}\) is a free \(\mathbb Z[X]/(\Phi_d )-d|n\) module for every \(d|n\). In general these statements are false for an arbitrary natural number \(n\), and the aim of this paper is to give some analogous results for arbitrary \(n\) by changing convenably the functors \(T\to T^{\Phi_d}\).


14J10 Families, moduli, classification: algebraic theory
14E05 Rational and birational maps
14J25 Special surfaces
14L10 Group varieties


Zbl 0283.14004
Full Text: EuDML