Closures of locally divergent orbits of maximal tori and values of homogeneous forms

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Let be a semisimple algebraic group over a number field K, S a finite set of places of K, Kₛ the direct product of the completions Kᵥ, v ∈ S, and the ring of S-integers of K. Let G = (Kₛ), Γ= () and π:G → G/Γ the quotient map. We describe the closures of the locally divergent orbits Tπ(g) %in G/Γ where T is a maximal Kₛ-split torus in G. If # S = 2 then the closure Tπ(g) is a finite union of T-orbits stratified in terms of parabolic subgroups of × and, consequently, Tπ(g) is homogeneous (i.e., Tπ(g)= Hπ(g) for a subgroup H of G) if and only if Tπ(g) is closed. On the other hand, if # S > 2 and K is not a CM-field then Tπ(g) is homogeneous for = SLₙ and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple K-ranks for ≠ SLₙ. As an application, we prove that f(ⁿ) = Kₛ for the class of non-rational locally K-decomposable homogeneous forms f ∈ Kₛ[x₁,, xₙ].