Decay of weighted cusp counts for congruence subgroups of $SL_2$ over number fields
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911693829832704 |
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| author | Zhao, Shengyuan |
| author_facet | Zhao, Shengyuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17972 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Decay of weighted cusp counts for congruence subgroups of $SL_2$ over number fields Zhao, Shengyuan Number Theory Algebraic Geometry Group Theory 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. |
| title | Decay of weighted cusp counts for congruence subgroups of $SL_2$ over number fields |
| topic | Number Theory Algebraic Geometry Group Theory |
| url | https://arxiv.org/abs/2605.17972 |