{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:Q4YGXPTMMRJ6YSHMQJS4KCJ6PQ","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":"40209c2350528cc939eb2dfa8dcbafecd891fa776609f5a87b59afba408d9863","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-07-25T19:26:28Z","title_canon_sha256":"746a270ded4cb1fbeef0d4f321cd54c08ee74826b0bed3b474f6f6c76cc1d5e7"},"schema_version":"1.0","source":{"id":"2207.12487","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2207.12487","created_at":"2026-07-05T10:03:22Z"},{"alias_kind":"arxiv_version","alias_value":"2207.12487v2","created_at":"2026-07-05T10:03:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2207.12487","created_at":"2026-07-05T10:03:22Z"},{"alias_kind":"pith_short_12","alias_value":"Q4YGXPTMMRJ6","created_at":"2026-07-05T10:03:22Z"},{"alias_kind":"pith_short_16","alias_value":"Q4YGXPTMMRJ6YSHM","created_at":"2026-07-05T10:03:22Z"},{"alias_kind":"pith_short_8","alias_value":"Q4YGXPTM","created_at":"2026-07-05T10:03:22Z"}],"graph_snapshots":[{"event_id":"sha256:730cf4497933f480f684e093b0732ac0ee82eaf781b94a367dee1a9adca3423e","target":"graph","created_at":"2026-07-05T10:03:22Z","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/2207.12487/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Consider a Mordell curve $E_a:y^2=x^3+a$ with $a \\in \\mathbb Z$. These curves have a rational $3$-isogeny, say $\\varphi$. We give an upper and a lower bound on the rank of the $\\varphi$-Selmer group of $E_a$ over $\\mathbb Q(\\zeta_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $\\mathbb Q(\\zeta_3)$. Using our bounds on the Selmer groups, we prove some cases of the rational cube sum problem. Further, using these bounds, we give explicit families of the Mordell curves to show that for a positive proportion of $E_a$, ${\\rm Sel}^3(E_{a}/\\mathbb Q)=0$ (respect","authors_text":"Dipramit Majumdar, Pratiksha Shingavekar, Somnath Jha","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-07-25T19:26:28Z","title":"$3$-Selmer group, ideal class groups and cube sum problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.12487","kind":"arxiv","version":2},"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:d35db3d9b2bfc6c2a3ed80072cf2381372712061319a3a17a373066644bcb50e","target":"record","created_at":"2026-07-05T10:03:22Z","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":"40209c2350528cc939eb2dfa8dcbafecd891fa776609f5a87b59afba408d9863","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2022-07-25T19:26:28Z","title_canon_sha256":"746a270ded4cb1fbeef0d4f321cd54c08ee74826b0bed3b474f6f6c76cc1d5e7"},"schema_version":"1.0","source":{"id":"2207.12487","kind":"arxiv","version":2}},"canonical_sha256":"87306bbe6c6453ec48ec8265c5093e7c31cb84365fd846699fedd910b2d9d7c8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"87306bbe6c6453ec48ec8265c5093e7c31cb84365fd846699fedd910b2d9d7c8","first_computed_at":"2026-07-05T10:03:22.283422Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:03:22.283422Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UhV2TFLOh7Hj1XKwAovlR/EheoXig+2UDVSiNUWyDVYSWxkLep+mHqiqCLU8ITRq962Cv5R71vXd8dt+TmEZDA==","signature_status":"signed_v1","signed_at":"2026-07-05T10:03:22.283765Z","signed_message":"canonical_sha256_bytes"},"source_id":"2207.12487","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d35db3d9b2bfc6c2a3ed80072cf2381372712061319a3a17a373066644bcb50e","sha256:730cf4497933f480f684e093b0732ac0ee82eaf781b94a367dee1a9adca3423e"],"state_sha256":"14588933a7af6ecfaedcc1ca889bd1dc3c2dd4ef0e5d6975c11e60c6718bd0c0"}