Finite-order automorphisms of a certain torus

Abstract

A classical result of Higman [H], [S, Exer. 6.3], asserts that the roots of unity in the group ring Z[Γ] of a finite commutative group Γ are the elements ±γ for γ ∈ Γ. One application of this result is the determination of the group of finite-order automorphisms of a torus T = ResS′/S(Gm) for a finite etale Galois covering f : S′ → S with S and S′ connected schemes and abelian Galois group Γ = Gal(S′/S) (e.g., an unramified extension of local fields). Indeed, the torus T has character group Z[Gal(S′/S)] with Gal(S′/S) acting through left translation, and an automorphism of T is “the same” as an automorphism of its character group X(T ) as a Galois module. Thus, to give an automorphism of T is to give a generator of Z[Gal(S′/S)] as a left module over itself. This in turn is just a unit in the group ring. Once we have the list of roots of unity, this gives the list of finite-order automorphisms. It follows that the group of finite-order automorphisms of the torus T is the finite group that is generated (as a direct product) by inversion and the action of Gal(S′/S). Now consider the problem (raised to the author by G. Prasad) of finding all finite-order automorphisms of the norm-1 subtorus T 1 in T when Γ is cyclic (e.g., the Galois group of an unramified extension of local fields). This subtorus is functorially described as the kernel of the determinant map