{"paper":{"title":"Congruences for odd class numbers of quadratic fields with odd discriminant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jigu Kim, Yoshinori Mizuno","submitted_at":"2022-01-12T03:55:23Z","abstract_excerpt":"For any distinct two primes $p_1\\equiv p_2\\equiv 3$ $(\\text{mod }4)$, let $h(-p_1)$, $h(-p_2)$ and $h(p_1p_2)$ be the class numbers of the quadratic fields $\\mathbb{Q}(\\sqrt{-p_1})$, $\\mathbb{Q}(\\sqrt{-p_2})$ and $\\mathbb{Q}(\\sqrt{p_1p_2})$, respectively. Let $\\omega_{p_1p_2}:=(1+\\sqrt{p_1p_2})/2$ and let $\\Psi(\\omega_{p_1p_2})$ be the Hirzebruch sum of $\\omega_{p_1p_2}$. We show that $h(-p_1)h(-p_2)\\equiv h(p_1p_2)\\Psi(\\omega_{p_1p_2})/n$ $(\\text{mod }8)$, where $n=6$ (respectively, $n=2$) if $\\min\\{p_1,p_2\\}>3$ (respectively, otherwise). We also consider the real quadratic order with conduct"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.04291","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2201.04291/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"}