In order to generalize the “modular” resolution of the K(n)-local sphere given by M. Behrens by using topological automorphic forms (M. Behrens/T. Lawson), it seems useful construct dense subgroups of stabilizer groups in terms of quasi-isogenies of suitable abelian varieties. After reviewing the motivating topological results, we state and solve the resulting arithmetic problems which evolve around approximation in algebraic groups.