{"paper":{"title":"A Riccati differential equation and free subgroup numbers for lifts of $\\PSL_2(\\Z)$ modulo powers of primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"Christian Krattenthaler (Universit\\\"at Wien), Thomas W. M\\\"uller (Queen Mary)","submitted_at":"2012-11-13T10:42:00Z","abstract_excerpt":"It is shown that the number $f_\\lambda$ of free subgroups of index $6\\lambda$ in the modular group $\\PSL_2(\\Z)$, when considered modulo a prime power $p^\\al$ with $p\\ge5$, is always (ultimately) periodic. In fact, an analogous result is established for a one-parameter family of lifts of the modular group (containing $\\PSL_2(\\Z)$ as a special case), and for a one-parameter family of lifts of the Hecke group $\\mathfrak{H}(4)=C_2*C_4$. All this is achieved by explicitly determining Pad\\'e approximants to solutions of a certain multi-parameter family of Riccati differential equations. Our main res"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2947","kind":"arxiv","version":3},"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"}