{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:MLRJUCGBLBEWCPQUVXS6KTE5AD","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":"4b08c9ccec7d378e706d41b1d2fb5af9bb1cc3705213b814114742b9a5fa68ef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-23T15:26:49Z","title_canon_sha256":"293ddd9d96163dfe9b8eb73b4a8b60ba71913a2dec91d114fb6a9088f1ef4d52"},"schema_version":"1.0","source":{"id":"1611.07840","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.07840","created_at":"2026-05-18T00:56:59Z"},{"alias_kind":"arxiv_version","alias_value":"1611.07840v1","created_at":"2026-05-18T00:56:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.07840","created_at":"2026-05-18T00:56:59Z"},{"alias_kind":"pith_short_12","alias_value":"MLRJUCGBLBEW","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MLRJUCGBLBEWCPQU","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MLRJUCGB","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:fdf90384e756c98939442c6ba7f3b213d6825c998093474cd39e3c555db6747d","target":"graph","created_at":"2026-05-18T00:56:59Z","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"},"paper":{"abstract_excerpt":"We present the results of our search for the orders of Tate-Shafarevich groups for the quadratic twists of elliptic curves. We formulate a general conjecture, giving for a fixed elliptic curve $E$ over $\\Bbb Q$ and positive integer $k$, an asymptotic formula for the number of quadratic twists $E_d$, $d$ positive square-free integers less than $X$, with finite group $E_d(\\Bbb Q)$ and $|\\Sha(E_d(\\Bbb Q))| = k^2$. This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of the curve $X_0(49)$. In section 8 we exhibit $88$ examples o","authors_text":"Andrzej D\\k{a}browski, Lucjan Szymaszkiewicz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-23T15:26:49Z","title":"Behaviour of the order of Tate-Shafarevich groups for the quadratic twists of elliptic curves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.07840","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":"sha256:0208abee4729030bd0028f510365fbbf7b22e5e54762c0cfd69ab9fa8df31571","target":"record","created_at":"2026-05-18T00:56:59Z","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":"4b08c9ccec7d378e706d41b1d2fb5af9bb1cc3705213b814114742b9a5fa68ef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-23T15:26:49Z","title_canon_sha256":"293ddd9d96163dfe9b8eb73b4a8b60ba71913a2dec91d114fb6a9088f1ef4d52"},"schema_version":"1.0","source":{"id":"1611.07840","kind":"arxiv","version":1}},"canonical_sha256":"62e29a08c15849613e14ade5e54c9d00d295214918e2705b1938e7f2d7676af7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62e29a08c15849613e14ade5e54c9d00d295214918e2705b1938e7f2d7676af7","first_computed_at":"2026-05-18T00:56:59.688563Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:56:59.688563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6RbKeBEARtxlfGj8nHG9FdknStUHigFSX/8Nhi/J9Sg1n1yW942gkLjUJMXOJZxQuEUHRQZ6QhYrgo8dJVfSDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:56:59.689098Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.07840","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0208abee4729030bd0028f510365fbbf7b22e5e54762c0cfd69ab9fa8df31571","sha256:fdf90384e756c98939442c6ba7f3b213d6825c998093474cd39e3c555db6747d"],"state_sha256":"a52a6e322a686e0794ae2aff35c7a0f7c25c6cb8a2dc3b8f07e275879003e03f"}