{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:T3QBYGXE6DLC7BAVGY7HYDKJ54","short_pith_number":"pith:T3QBYGXE","schema_version":"1.0","canonical_sha256":"9ee01c1ae4f0d62f8415363e7c0d49ef1b8fb3765db8e10d7c9ef51127a2a46a","source":{"kind":"arxiv","id":"2502.01069","version":1},"attestation_state":"computed","paper":{"title":"$\\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dipramit Majumdar, Pratiksha Shingavekar, Somnath Jha","submitted_at":"2025-02-03T05:25:34Z","abstract_excerpt":"We consider the family of elliptic curves $E_{a,b}:y^2=x^3+a(x-b)^2$ with $a,b \\in \\mathbb{Z}$. These elliptic 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 $E_{a,b}$ over $K:=\\mathbb{Q}(\\zeta_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $K$. Using our bounds on the Selmer groups, we construct infinitely many curves in this family with arbitrary large $3$-Selmer rank over $K$ and no non-trivial $K$-rational point of order $3$. We also show that for a positive proporti"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2502.01069","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.NT","submitted_at":"2025-02-03T05:25:34Z","cross_cats_sorted":[],"title_canon_sha256":"7b3f1d9490e438ee50415eb9c91316188679b36ba97f0305109fb132617511f3","abstract_canon_sha256":"01994aa04b5713d80ac7ca162fca18ce004bdc58060f5b031c771de7d0e0093d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:08:42.522268Z","signature_b64":"0G860jvE7t0rPBbKuZb+3YExhP1dy4CN5a5ijCAmxqSQQ/Z/WxdmfBsYycikIXayUGa3Tr7SnnX7MIfyq4KOCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ee01c1ae4f0d62f8415363e7c0d49ef1b8fb3765db8e10d7c9ef51127a2a46a","last_reissued_at":"2026-07-05T10:08:42.521808Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:08:42.521808Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$\\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dipramit Majumdar, Pratiksha Shingavekar, Somnath Jha","submitted_at":"2025-02-03T05:25:34Z","abstract_excerpt":"We consider the family of elliptic curves $E_{a,b}:y^2=x^3+a(x-b)^2$ with $a,b \\in \\mathbb{Z}$. These elliptic 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 $E_{a,b}$ over $K:=\\mathbb{Q}(\\zeta_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $K$. Using our bounds on the Selmer groups, we construct infinitely many curves in this family with arbitrary large $3$-Selmer rank over $K$ and no non-trivial $K$-rational point of order $3$. We also show that for a positive proporti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.01069","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/2502.01069/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2502.01069","created_at":"2026-07-05T10:08:42.521866+00:00"},{"alias_kind":"arxiv_version","alias_value":"2502.01069v1","created_at":"2026-07-05T10:08:42.521866+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2502.01069","created_at":"2026-07-05T10:08:42.521866+00:00"},{"alias_kind":"pith_short_12","alias_value":"T3QBYGXE6DLC","created_at":"2026-07-05T10:08:42.521866+00:00"},{"alias_kind":"pith_short_16","alias_value":"T3QBYGXE6DLC7BAV","created_at":"2026-07-05T10:08:42.521866+00:00"},{"alias_kind":"pith_short_8","alias_value":"T3QBYGXE","created_at":"2026-07-05T10:08:42.521866+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/T3QBYGXE6DLC7BAVGY7HYDKJ54","json":"https://pith.science/pith/T3QBYGXE6DLC7BAVGY7HYDKJ54.json","graph_json":"https://pith.science/api/pith-number/T3QBYGXE6DLC7BAVGY7HYDKJ54/graph.json","events_json":"https://pith.science/api/pith-number/T3QBYGXE6DLC7BAVGY7HYDKJ54/events.json","paper":"https://pith.science/paper/T3QBYGXE"},"agent_actions":{"view_html":"https://pith.science/pith/T3QBYGXE6DLC7BAVGY7HYDKJ54","download_json":"https://pith.science/pith/T3QBYGXE6DLC7BAVGY7HYDKJ54.json","view_paper":"https://pith.science/paper/T3QBYGXE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2502.01069&json=true","fetch_graph":"https://pith.science/api/pith-number/T3QBYGXE6DLC7BAVGY7HYDKJ54/graph.json","fetch_events":"https://pith.science/api/pith-number/T3QBYGXE6DLC7BAVGY7HYDKJ54/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/T3QBYGXE6DLC7BAVGY7HYDKJ54/action/timestamp_anchor","attest_storage":"https://pith.science/pith/T3QBYGXE6DLC7BAVGY7HYDKJ54/action/storage_attestation","attest_author":"https://pith.science/pith/T3QBYGXE6DLC7BAVGY7HYDKJ54/action/author_attestation","sign_citation":"https://pith.science/pith/T3QBYGXE6DLC7BAVGY7HYDKJ54/action/citation_signature","submit_replication":"https://pith.science/pith/T3QBYGXE6DLC7BAVGY7HYDKJ54/action/replication_record"}},"created_at":"2026-07-05T10:08:42.521866+00:00","updated_at":"2026-07-05T10:08:42.521866+00:00"}