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

For congruence subgroups commensurable with $\operatorname{SL}_2$ over number fields, we study cusp counts with arithmetic 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, with an exponent that can be chosen explicitly in terms of the degree of the number field.

This is a generalization of a theorem of Cox--Parry over rational numbers.

The main input is an explicit local orbit estimate for arbitrary subgroups of exact level in $\operatorname{SL}_2(\mathcal O_K/\mathfrak p^e)$, uniform over all number fields of fixed degree.

This estimate is covered by a general result of Finis--Lapid.

We give a new explicit proof in our setting based on an analysis reminiscent of additive combinatorics.