There are at most finitely many singular moduli that are S-units

We show that for every finite set of prime numbers S , there are at most finitely many singular moduli that are S -units. The key new ingredient is that for every prime number p , singular moduli are p -adically disperse. We prove analogous results for the Weber modular functions, the λ -invariants and the McKay–Thompson series associated with the elements of the monster group. Finally, we also obtain that a modular function that specializes to infinitely many algebraic units at quadratic imaginary numbers must be a weak modular unit.