Decay of weighted cusp counts for congruence subgroups of $S{L}_2$ over number fields

For congruence subgroups commensurable with ${\mathrm{SL}}_2$ over number fields, we study cusp counts with certain multiplicities. We prove that the ratio of the total weighted cusp count to the group index is bounded by a negative power of the norm of the congruence level. This generalizes a theorem of Cox–Parry over $\mathbb{Q}$, and supports the heuristic that cusp terms occurring in topological, arithmetic and representation-theoretical formulas are subleading. The proof proceeds by localizing at a prime and reducing the problem to finite quotients, where it becomes a counting problem for finite groups. The main technical part is a counting problem for subgroups of ${\mathrm{SL}}_2$ over finite non-reduced principal local rings, proved by an analysis reminiscent of additive combinatorics.