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

Shengyuan Zhao

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arXiv · 2605.17972

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

Shengyuan Zhao

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Abstract

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.

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Shengyuan Zhao. 2026-05-18. Decay of weighted cusp counts for congruence subgroups of $SL_2$ over number fields. https://arxiv.org/abs/2605.17972

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