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We show that (under mild conditions on $E$) for every $r>0$, there are infinitely many quadratic twists $E^d/\\mathbf{Q}$ of $E/\\mathbf{Q}$ such that $\\mathrm{dim}_{\\mathbf{F}_2}(\\mathrm{ST}(E^d/K)[2]) > r$"},"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":"1801.10536","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-01-31T16:42:59Z","cross_cats_sorted":[],"title_canon_sha256":"0b54cf36d980fecbdec1d938ffa7917ec2c4f475fbba8ad2d1a46dd17250755c","abstract_canon_sha256":"13a856b5c252c6bb790939b9648f7420b12422f55f8b4b3c652dc05aaa9f830a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:40.576849Z","signature_b64":"fiqfwGuHKXUfXKoEbfkyC00DMk2uaHnhN5J0WfZfp/KGOhUNdchlRqMx0N/CMvrx7TX4/ZmDfYrZtG6UHYTeCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a4e16b0f168ede2d7a599055c21b37f3d597520aa904f7e5bb4f7b29e71188e","last_reissued_at":"2026-05-18T00:24:40.576292Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:40.576292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Large Shafarevich-Tate groups over quadratic number fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Myungjun Yu","submitted_at":"2018-01-31T16:42:59Z","abstract_excerpt":"Let $E$ be an elliptic curve over the rational field $\\mathbf{Q}$. 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We show that (under mild conditions on $E$) for every $r>0$, there are infinitely many quadratic twists $E^d/\\mathbf{Q}$ of $E/\\mathbf{Q}$ such that $\\mathrm{dim}_{\\mathbf{F}_2}(\\mathrm{ST}(E^d/K)[2]) > r$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.10536","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":""},"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":"1801.10536","created_at":"2026-05-18T00:24:40.576403+00:00"},{"alias_kind":"arxiv_version","alias_value":"1801.10536v1","created_at":"2026-05-18T00:24:40.576403+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1801.10536","created_at":"2026-05-18T00:24:40.576403+00:00"},{"alias_kind":"pith_short_12","alias_value":"JJHBNMHRNDW6","created_at":"2026-05-18T12:32:31.084164+00:00"},{"alias_kind":"pith_short_16","alias_value":"JJHBNMHRNDW6FV5F","created_at":"2026-05-18T12:32:31.084164+00:00"},{"alias_kind":"pith_short_8","alias_value":"JJHBNMHR","created_at":"2026-05-18T12:32:31.084164+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/JJHBNMHRNDW6FV5FTECVYINTP4","json":"https://pith.science/pith/JJHBNMHRNDW6FV5FTECVYINTP4.json","graph_json":"https://pith.science/api/pith-number/JJHBNMHRNDW6FV5FTECVYINTP4/graph.json","events_json":"https://pith.science/api/pith-number/JJHBNMHRNDW6FV5FTECVYINTP4/events.json","paper":"https://pith.science/paper/JJHBNMHR"},"agent_actions":{"view_html":"https://pith.science/pith/JJHBNMHRNDW6FV5FTECVYINTP4","download_json":"https://pith.science/pith/JJHBNMHRNDW6FV5FTECVYINTP4.json","view_paper":"https://pith.science/paper/JJHBNMHR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1801.10536&json=true","fetch_graph":"https://pith.science/api/pith-number/JJHBNMHRNDW6FV5FTECVYINTP4/graph.json","fetch_events":"https://pith.science/api/pith-number/JJHBNMHRNDW6FV5FTECVYINTP4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JJHBNMHRNDW6FV5FTECVYINTP4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JJHBNMHRNDW6FV5FTECVYINTP4/action/storage_attestation","attest_author":"https://pith.science/pith/JJHBNMHRNDW6FV5FTECVYINTP4/action/author_attestation","sign_citation":"https://pith.science/pith/JJHBNMHRNDW6FV5FTECVYINTP4/action/citation_signature","submit_replication":"https://pith.science/pith/JJHBNMHRNDW6FV5FTECVYINTP4/action/replication_record"}},"created_at":"2026-05-18T00:24:40.576403+00:00","updated_at":"2026-05-18T00:24:40.576403+00:00"}