{"paper":{"title":"Lifts of projective congruence groups, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.NT","authors_text":"Ian Kiming","submitted_at":"2011-12-29T08:29:07Z","abstract_excerpt":"We continue and complete our previous paper `Lifts of projective congruence groups' [2] concerning the question of whether there exist noncongruence subgroups of $\\SL_2(\\Z)$ that are projectively equivalent to one of the groups $\\Gamma_0(N)$ or $\\Gamma_1(N)$. A complete answer to this question is obtained: In case of $\\Gamma_0(N)$ such noncongruence subgroups exist precisely if $N\\not\\in {3,4,8}$ and we additionally have either that $4\\mid N$ or that $N$ is divisible by an odd prime congruent to 3 modulo 4. In case of $\\Gamma_1(N)$ these noncongruence subgroups exist precisely if $N>4$.\n  As i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.6250","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}