{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:Q6HIVT75V5COMGZFVPNPSRR7ET","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"cee9d2f0dd30ab8a88621b0cfcd4ccaeea10933bd855616bcfdd89e5d5b71444","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2024-12-17T15:41:50Z","title_canon_sha256":"725653b0bc7e51b10232639699f4ffbf6aabbd516c1a281540b7f59856667aae"},"schema_version":"1.0","source":{"id":"2412.13022","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.13022","created_at":"2026-07-05T09:50:39Z"},{"alias_kind":"arxiv_version","alias_value":"2412.13022v1","created_at":"2026-07-05T09:50:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.13022","created_at":"2026-07-05T09:50:39Z"},{"alias_kind":"pith_short_12","alias_value":"Q6HIVT75V5CO","created_at":"2026-07-05T09:50:39Z"},{"alias_kind":"pith_short_16","alias_value":"Q6HIVT75V5COMGZF","created_at":"2026-07-05T09:50:39Z"},{"alias_kind":"pith_short_8","alias_value":"Q6HIVT75","created_at":"2026-07-05T09:50:39Z"}],"graph_snapshots":[{"event_id":"sha256:e92b2b7005bed32aac589a3b39902e8be302e7fccc9d6fe2260f3b67f2fab39a","target":"graph","created_at":"2026-07-05T09:50:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2412.13022/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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.","authors_text":"Debajyoti De, Dipramit Majumdar, Sudipa Mondal","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2024-12-17T15:41:50Z","title":"Relative $p$-class groups and $p$-Selmer groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.13022","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2d0cbbf74236616eeb7ac8e1fbb23b4007cef3bb8cf7db997f347e8a3011b0a6","target":"record","created_at":"2026-07-05T09:50:39Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"cee9d2f0dd30ab8a88621b0cfcd4ccaeea10933bd855616bcfdd89e5d5b71444","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2024-12-17T15:41:50Z","title_canon_sha256":"725653b0bc7e51b10232639699f4ffbf6aabbd516c1a281540b7f59856667aae"},"schema_version":"1.0","source":{"id":"2412.13022","kind":"arxiv","version":1}},"canonical_sha256":"878e8acffdaf44e61b25abdaf9463f24ed566ca7af38c4e6e82f6a97cb3046da","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"878e8acffdaf44e61b25abdaf9463f24ed566ca7af38c4e6e82f6a97cb3046da","first_computed_at":"2026-07-05T09:50:39.169992Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:50:39.169992Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kgm42Ulgb/Y0qDAZzRieJCmm8olGj43VUKr8Fbqbk0gsgnEZvIVPK1hSM67Ik9vQyAOYJhAyDD9pzHlcKslNCw==","signature_status":"signed_v1","signed_at":"2026-07-05T09:50:39.170580Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.13022","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2d0cbbf74236616eeb7ac8e1fbb23b4007cef3bb8cf7db997f347e8a3011b0a6","sha256:e92b2b7005bed32aac589a3b39902e8be302e7fccc9d6fe2260f3b67f2fab39a"],"state_sha256":"33a40c9584c2a035d33b595609bb7795bd7af02eefb11b6a54168bc2d345ce18"}