{"paper":{"title":"Relative $p$-class groups and $p$-Selmer groups","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Debajyoti De, Dipramit Majumdar, Sudipa Mondal","submitted_at":"2024-12-17T15:41:50Z","abstract_excerpt":"Let $E$ be an elliptic curve with $j$-invariant $0$ or $1728$ and let $\\widetilde{E}$ be a $k^{th}$ twist of $E$. We show that for any prime $p$ of good reduction of $\\widetilde{E}$, a degree $k$ relative $p$-class group and the root number of $\\widetilde{E}$ determines the dimension of the $p$-Selmer group of $\\widetilde{E}$. As a consequence, we construct families of large rank $p$-class group. We also relate congruent number and cube sum problem with relative $p$-class group."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.13022","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/2412.13022/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"}