{"paper":{"title":"Top-dimensional rational cohomology of the congruence subgroup $\\Gamma_{0,n}^+(p)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AT","authors_text":"Tatiana Abdelnaim","submitted_at":"2026-05-22T11:42:34Z","abstract_excerpt":"Let $\\Gamma_{0,n}^+(p)\\subset \\mathrm{SL}_n(\\mathbb{Z})$ be the congruence subgroup of level-$p$ whose first column is of the form $(*,0,\\dots,0)^t\\bmod p$. We prove that the top-dimensional cohomology group $H^{\\binom{n}{2}}(\\Gamma_{0,n}^+(p);\\mathbb{Q})$ vanishes for $p\\in\\{2,3,5,7,13\\}$ if $n \\geq 3$, as well as for $p \\leq 6n-14$.\n  Additionally, we prove a non-vanishing result, showing that this cohomology group is nonzero for $n = 2$ for every prime $p$, and for $n=3$ for all primes $p \\notin \\{2,3,5,7,13\\}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.23526","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/2605.23526/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}