{"paper":{"title":"Decay of weighted cusp counts for congruence subgroups of $SL_2$ over number fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.GR"],"primary_cat":"math.NT","authors_text":"Shengyuan Zhao","submitted_at":"2026-05-18T07:28:15Z","abstract_excerpt":"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.\n  The proof proceeds by localizing at a prime and reducing the problem to finite quotients, where it becomes a counting probl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.17972","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17972/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-19T23:33:35.574156Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"57f50e22bae83534ed27ac06e295fbb53eb2e1796f8bf419cac1bcf20300caf6"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}