Decay of weighted cusp counts for congruence subgroups of $SL_2$ over number fields

Abstract

For congruence subgroups commensurable with $\operatorname{SL}_2$ 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 $\operatorname{SL}_2$ over finite non-reduced principal local rings, proved by an analysis reminiscent of additive combinatorics.