Heights and representations of split tori

Let Gm denote the d-dimensional split torus defined over a number field k. To each Gm-module E we associate a height function hE defined by means of the spectral height on GL(E). This gives rise to a height pairing between the monoid of irreducible Gm-modules of Gm and the group Gm ( k ) . Our main results are a characterization of those Gm-modules E for which hE satisfeis Northcott’s finiteness theorem, the determination of the kernels of the height pairing, as well as, for a few special classes of Gm-modules, of the group of automorphisms that preserve hE .