{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:JEODALI4AIZ6M3J4OF6VKL5QNF","short_pith_number":"pith:JEODALI4","canonical_record":{"source":{"id":"1705.01880","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-04T15:30:44Z","cross_cats_sorted":[],"title_canon_sha256":"f6253fd5deb224ccc2cf3bade4294103f759e13f5835cbfe1648da00674d78c8","abstract_canon_sha256":"e632bc305787410f2fe46235a460b785423677bf953b9cc1da053bf97d8faa8d"},"schema_version":"1.0"},"canonical_sha256":"491c302d1c0233e66d3c717d552fb0696baedc32382bce2ea1de26c4d188bf8b","source":{"kind":"arxiv","id":"1705.01880","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.01880","created_at":"2026-05-18T00:45:02Z"},{"alias_kind":"arxiv_version","alias_value":"1705.01880v1","created_at":"2026-05-18T00:45:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.01880","created_at":"2026-05-18T00:45:02Z"},{"alias_kind":"pith_short_12","alias_value":"JEODALI4AIZ6","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_16","alias_value":"JEODALI4AIZ6M3J4","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_8","alias_value":"JEODALI4","created_at":"2026-05-18T12:31:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:JEODALI4AIZ6M3J4OF6VKL5QNF","target":"record","payload":{"canonical_record":{"source":{"id":"1705.01880","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-04T15:30:44Z","cross_cats_sorted":[],"title_canon_sha256":"f6253fd5deb224ccc2cf3bade4294103f759e13f5835cbfe1648da00674d78c8","abstract_canon_sha256":"e632bc305787410f2fe46235a460b785423677bf953b9cc1da053bf97d8faa8d"},"schema_version":"1.0"},"canonical_sha256":"491c302d1c0233e66d3c717d552fb0696baedc32382bce2ea1de26c4d188bf8b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:02.121459Z","signature_b64":"RSvvdVj/pq07mYqdm5ujmDhGlg6acvfH31d7oRIQJuqaGuZC5jLjHiUa2xJWbNhYenJjaQAROFVq8+1f1YI5DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"491c302d1c0233e66d3c717d552fb0696baedc32382bce2ea1de26c4d188bf8b","last_reissued_at":"2026-05-18T00:45:02.121057Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:02.121057Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1705.01880","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:45:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sOmFRyNMkFOdtJah2jeZijk9otEWzVgOGbmYi6r+fBUCBS8jr1N2ru8RdEA5u6GWPZf8awyg6BIePRpC6QbCCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T10:34:42.597049Z"},"content_sha256":"dbdc7a81592ed0d0e26a90ca9c5359a86f36e42e7474a1ff631f332855d03466","schema_version":"1.0","event_id":"sha256:dbdc7a81592ed0d0e26a90ca9c5359a86f36e42e7474a1ff631f332855d03466"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:JEODALI4AIZ6M3J4OF6VKL5QNF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Counterexamples to the local-global divisibility over elliptic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Gabriele Ranieri","submitted_at":"2017-05-04T15:30:44Z","abstract_excerpt":"Let $p \\geq 5$ be a prime number. We find all the possible subgroups $G$ of ${\\rm GL}_2 ( \\mathbb{Z} / p \\mathbb{Z} )$ such that there exists a number field $k$ and an elliptic curve ${\\mathcal{E}}$ defined over $k$ such that the ${\\rm Gal} ( k ( {\\mathcal{E}}[p] ) / k )$-module ${\\mathcal{E}}[p]$ is isomorphic to the $G$-module $( \\mathbb{Z} / p \\mathbb{Z} )^2$ and there exists $n \\in \\mathbb{N}$ such that the local-global divisibility by $p^n$ does not hold over ${\\mathcal{E}} ( k )$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.01880","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:45:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dsKzASRvMIx5hgrBtQAEq1eT0U1Wr8XuuSLmAX2oowniWfwUtojiEByNAbJjIY9NCrk6t0Zbf4RVME2oPGs9Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T10:34:42.597779Z"},"content_sha256":"a11e54e459f08faad79cdb52bd4e405fa38e13ee50ddff050fd019c8260cd04c","schema_version":"1.0","event_id":"sha256:a11e54e459f08faad79cdb52bd4e405fa38e13ee50ddff050fd019c8260cd04c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JEODALI4AIZ6M3J4OF6VKL5QNF/bundle.json","state_url":"https://pith.science/pith/JEODALI4AIZ6M3J4OF6VKL5QNF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JEODALI4AIZ6M3J4OF6VKL5QNF/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-11T10:34:42Z","links":{"resolver":"https://pith.science/pith/JEODALI4AIZ6M3J4OF6VKL5QNF","bundle":"https://pith.science/pith/JEODALI4AIZ6M3J4OF6VKL5QNF/bundle.json","state":"https://pith.science/pith/JEODALI4AIZ6M3J4OF6VKL5QNF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JEODALI4AIZ6M3J4OF6VKL5QNF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:JEODALI4AIZ6M3J4OF6VKL5QNF","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":"e632bc305787410f2fe46235a460b785423677bf953b9cc1da053bf97d8faa8d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-04T15:30:44Z","title_canon_sha256":"f6253fd5deb224ccc2cf3bade4294103f759e13f5835cbfe1648da00674d78c8"},"schema_version":"1.0","source":{"id":"1705.01880","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1705.01880","created_at":"2026-05-18T00:45:02Z"},{"alias_kind":"arxiv_version","alias_value":"1705.01880v1","created_at":"2026-05-18T00:45:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.01880","created_at":"2026-05-18T00:45:02Z"},{"alias_kind":"pith_short_12","alias_value":"JEODALI4AIZ6","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_16","alias_value":"JEODALI4AIZ6M3J4","created_at":"2026-05-18T12:31:24Z"},{"alias_kind":"pith_short_8","alias_value":"JEODALI4","created_at":"2026-05-18T12:31:24Z"}],"graph_snapshots":[{"event_id":"sha256:a11e54e459f08faad79cdb52bd4e405fa38e13ee50ddff050fd019c8260cd04c","target":"graph","created_at":"2026-05-18T00:45:02Z","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":"Let $p \\geq 5$ be a prime number. We find all the possible subgroups $G$ of ${\\rm GL}_2 ( \\mathbb{Z} / p \\mathbb{Z} )$ such that there exists a number field $k$ and an elliptic curve ${\\mathcal{E}}$ defined over $k$ such that the ${\\rm Gal} ( k ( {\\mathcal{E}}[p] ) / k )$-module ${\\mathcal{E}}[p]$ is isomorphic to the $G$-module $( \\mathbb{Z} / p \\mathbb{Z} )^2$ and there exists $n \\in \\mathbb{N}$ such that the local-global divisibility by $p^n$ does not hold over ${\\mathcal{E}} ( k )$.","authors_text":"Gabriele Ranieri","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-04T15:30:44Z","title":"Counterexamples to the local-global divisibility over elliptic curves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.01880","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:dbdc7a81592ed0d0e26a90ca9c5359a86f36e42e7474a1ff631f332855d03466","target":"record","created_at":"2026-05-18T00:45:02Z","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":"e632bc305787410f2fe46235a460b785423677bf953b9cc1da053bf97d8faa8d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-05-04T15:30:44Z","title_canon_sha256":"f6253fd5deb224ccc2cf3bade4294103f759e13f5835cbfe1648da00674d78c8"},"schema_version":"1.0","source":{"id":"1705.01880","kind":"arxiv","version":1}},"canonical_sha256":"491c302d1c0233e66d3c717d552fb0696baedc32382bce2ea1de26c4d188bf8b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"491c302d1c0233e66d3c717d552fb0696baedc32382bce2ea1de26c4d188bf8b","first_computed_at":"2026-05-18T00:45:02.121057Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:45:02.121057Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RSvvdVj/pq07mYqdm5ujmDhGlg6acvfH31d7oRIQJuqaGuZC5jLjHiUa2xJWbNhYenJjaQAROFVq8+1f1YI5DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:45:02.121459Z","signed_message":"canonical_sha256_bytes"},"source_id":"1705.01880","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dbdc7a81592ed0d0e26a90ca9c5359a86f36e42e7474a1ff631f332855d03466","sha256:a11e54e459f08faad79cdb52bd4e405fa38e13ee50ddff050fd019c8260cd04c"],"state_sha256":"f2770639e1e8d88c48e1a9ea694ca4b87d38d429e111738fe7bb4bb4e7d528f7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FRp4BEYKn2TccHftIRrOgZK2nim7AOscdz9H6zX+cymoOaENLT4wJqI8scB8HZi2rvQA5G/MRpip7FeCz274Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T10:34:42.602362Z","bundle_sha256":"ff44519cac869fada65d3eca3507dae1d8d4c13d094ee39b2e29a92e36fed486"}}