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

TL;DR

This paper investigates the decay of weighted cusp counts for congruence subgroups of SL_2 over number fields, establishing bounds that generalize previous results and suggest subleading behavior in various mathematical contexts.

Contribution

It extends Cox--Parry's theorem to number fields and introduces a novel counting approach for subgroups over finite non-reduced local rings.

Findings

The ratio of weighted cusp count to group index is bounded by a negative power of the level's norm.

The method reduces the problem to finite quotients and subgroup counting in finite groups.

Supports the heuristic that cusp contributions are subleading in topological, arithmetic, and representation-theoretic formulas.

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.