Finiteness of Class Numbers for Algebraic Groups

Let G be an affine group scheme of finite type over a global field F . (We do not assume G to be reductive or smooth or connected.) Let AF denote the locally compact adele ring of F , S be a finite non-empty set of places of F that contains the archimedean places, and AF the factor ring of adeles with vanishing component along S; for FS = ∏ v∈S Fv, we have AF = FS ×AF . Recall that if X is a finite type affine scheme over a topological ring R then the set X(R) inherits a natural topology that is functorial in X, and the formation of this topology is compatible with fiber products (in the categories of R-schemes and topological spaces respectively). In particular, X(R) is locally compact when R is, and X(R) is a topological group when X is an R-group scheme. We are interested in the locally compact topological group G(AF ). Since F is a discrete subring of AF , the subgroup G(F ) inside of G(AF ) is discrete (and closed). Let K be a compact open subgroup in G(AF ). Consider the double coset space ΣG,S,K = G(F )\G(AF )/G(FS)K = G(F )\G(AF )/K. Clearly for any two compact open subgroups K and K ′, K ∩K ′ is again compact open and hence of finite index in each of K and K ′. It follows that the finiteness property of ΣG,S,K is independent of the choice of K, and so is an intrinsic property of G (and S). In fact, it is equivalent to the compactness of the (typically non-Hausdorff) coset space G(F )\G(AF )/G(FS) = G(F )\G(AF ). Definition 1.1. The F -group scheme G has finite class numbers with respect to S if ΣG,S,K is finite for one (equivalently, every) compact open subgroup K ⊆ G(AF ). If this holds for all choices of S then G has finite class numbers.