{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:PNK3KWPQFHRFRLOFUTXZQWYVLI","short_pith_number":"pith:PNK3KWPQ","canonical_record":{"source":{"id":"1712.02148","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-12-06T12:04:10Z","cross_cats_sorted":[],"title_canon_sha256":"b704efb21ae66045c0f2837c80c9b558e21a1bead6ae39f398045b0ee9da52b7","abstract_canon_sha256":"97ea1d1f4655564051b7db34cdd611b92d194c2181ee515a3df8dddb624e912e"},"schema_version":"1.0"},"canonical_sha256":"7b55b559f029e258adc5a4ef985b155a14a020482286040445a11ac42e237588","source":{"kind":"arxiv","id":"1712.02148","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.02148","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"arxiv_version","alias_value":"1712.02148v2","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.02148","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"pith_short_12","alias_value":"PNK3KWPQFHRF","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_16","alias_value":"PNK3KWPQFHRFRLOF","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_8","alias_value":"PNK3KWPQ","created_at":"2026-05-18T12:31:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:PNK3KWPQFHRFRLOFUTXZQWYVLI","target":"record","payload":{"canonical_record":{"source":{"id":"1712.02148","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-12-06T12:04:10Z","cross_cats_sorted":[],"title_canon_sha256":"b704efb21ae66045c0f2837c80c9b558e21a1bead6ae39f398045b0ee9da52b7","abstract_canon_sha256":"97ea1d1f4655564051b7db34cdd611b92d194c2181ee515a3df8dddb624e912e"},"schema_version":"1.0"},"canonical_sha256":"7b55b559f029e258adc5a4ef985b155a14a020482286040445a11ac42e237588","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:26:59.950177Z","signature_b64":"9Gsnpaheb2FMTnT9mzisF9DGSZdbVrzr7+iBG7B+6KhTaWnZrJ0ZLtXPZMtVoXFAqBfrb9s4yggfeljGqUpdBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b55b559f029e258adc5a4ef985b155a14a020482286040445a11ac42e237588","last_reissued_at":"2026-05-18T00:26:59.949384Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:26:59.949384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1712.02148","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:26:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H9hvRD6Kx/mjd00l9m8oNti+R+FnqTNviODmNruAkGMsLBu2xzfC+RvA245d+cE9fbx2CF4Eo/71Mj7Ah1KQCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T15:22:01.362673Z"},"content_sha256":"b0a51ee3804b663c523a798afa8cc1befb76feebe203fe942f3c0b31695a4373","schema_version":"1.0","event_id":"sha256:b0a51ee3804b663c523a798afa8cc1befb76feebe203fe942f3c0b31695a4373"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:PNK3KWPQFHRFRLOFUTXZQWYVLI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Horizontal variation of Tate--Shafarevich groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ashay A. Burungale, Haruzo Hida, Ye Tian","submitted_at":"2017-12-06T12:04:10Z","abstract_excerpt":"Let $E$ be an elliptic curve over $\\mathbb{Q}$. Let $p$ be an odd prime and $\\iota: \\overline{\\mathbb{Q}}\\hookrightarrow \\mathbb{C}_p$ an embedding. Let $K$ be an imaginary quadratic field and $H_{K}$ the corresponding Hilbert class field. For a class group character $\\chi$ over $K$, let $\\mathbb{Q}(\\chi)$ be the field generated by the image of $\\chi$ and $\\mathfrak{p}_{\\chi}$ the prime of $\\mathbb{Q}(\\chi)$ above $p$ determined via $\\iota_p$. Under mild hypotheses, we show that the number of class group characters $\\chi$ such that the $\\chi$-isotypic Tate--Shafarevich group of $E$ over $H_{K}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.02148","kind":"arxiv","version":2},"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":"2026-05-18T00:26:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5p7XGWy4eVKE7nJvD6lWvqvsgUJvmXH3nKeylwr5EDtkOnaSqac3x5X+cZlGGRN5ueN6I+GXz6d+g+mmm2mPAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T15:22:01.363389Z"},"content_sha256":"4155d2c6d7ed038c7c6ead42f42ee2c0f9715ea99aafcca62afb600409392984","schema_version":"1.0","event_id":"sha256:4155d2c6d7ed038c7c6ead42f42ee2c0f9715ea99aafcca62afb600409392984"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PNK3KWPQFHRFRLOFUTXZQWYVLI/bundle.json","state_url":"https://pith.science/pith/PNK3KWPQFHRFRLOFUTXZQWYVLI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PNK3KWPQFHRFRLOFUTXZQWYVLI/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":"2026-05-25T15:22:01Z","links":{"resolver":"https://pith.science/pith/PNK3KWPQFHRFRLOFUTXZQWYVLI","bundle":"https://pith.science/pith/PNK3KWPQFHRFRLOFUTXZQWYVLI/bundle.json","state":"https://pith.science/pith/PNK3KWPQFHRFRLOFUTXZQWYVLI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PNK3KWPQFHRFRLOFUTXZQWYVLI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:PNK3KWPQFHRFRLOFUTXZQWYVLI","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":"97ea1d1f4655564051b7db34cdd611b92d194c2181ee515a3df8dddb624e912e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-12-06T12:04:10Z","title_canon_sha256":"b704efb21ae66045c0f2837c80c9b558e21a1bead6ae39f398045b0ee9da52b7"},"schema_version":"1.0","source":{"id":"1712.02148","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.02148","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"arxiv_version","alias_value":"1712.02148v2","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.02148","created_at":"2026-05-18T00:26:59Z"},{"alias_kind":"pith_short_12","alias_value":"PNK3KWPQFHRF","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_16","alias_value":"PNK3KWPQFHRFRLOF","created_at":"2026-05-18T12:31:37Z"},{"alias_kind":"pith_short_8","alias_value":"PNK3KWPQ","created_at":"2026-05-18T12:31:37Z"}],"graph_snapshots":[{"event_id":"sha256:4155d2c6d7ed038c7c6ead42f42ee2c0f9715ea99aafcca62afb600409392984","target":"graph","created_at":"2026-05-18T00:26:59Z","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":"Let $E$ be an elliptic curve over $\\mathbb{Q}$. Let $p$ be an odd prime and $\\iota: \\overline{\\mathbb{Q}}\\hookrightarrow \\mathbb{C}_p$ an embedding. Let $K$ be an imaginary quadratic field and $H_{K}$ the corresponding Hilbert class field. For a class group character $\\chi$ over $K$, let $\\mathbb{Q}(\\chi)$ be the field generated by the image of $\\chi$ and $\\mathfrak{p}_{\\chi}$ the prime of $\\mathbb{Q}(\\chi)$ above $p$ determined via $\\iota_p$. Under mild hypotheses, we show that the number of class group characters $\\chi$ such that the $\\chi$-isotypic Tate--Shafarevich group of $E$ over $H_{K}","authors_text":"Ashay A. Burungale, Haruzo Hida, Ye Tian","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-12-06T12:04:10Z","title":"Horizontal variation of Tate--Shafarevich groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.02148","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:b0a51ee3804b663c523a798afa8cc1befb76feebe203fe942f3c0b31695a4373","target":"record","created_at":"2026-05-18T00:26:59Z","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":"97ea1d1f4655564051b7db34cdd611b92d194c2181ee515a3df8dddb624e912e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-12-06T12:04:10Z","title_canon_sha256":"b704efb21ae66045c0f2837c80c9b558e21a1bead6ae39f398045b0ee9da52b7"},"schema_version":"1.0","source":{"id":"1712.02148","kind":"arxiv","version":2}},"canonical_sha256":"7b55b559f029e258adc5a4ef985b155a14a020482286040445a11ac42e237588","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7b55b559f029e258adc5a4ef985b155a14a020482286040445a11ac42e237588","first_computed_at":"2026-05-18T00:26:59.949384Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:26:59.949384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9Gsnpaheb2FMTnT9mzisF9DGSZdbVrzr7+iBG7B+6KhTaWnZrJ0ZLtXPZMtVoXFAqBfrb9s4yggfeljGqUpdBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:26:59.950177Z","signed_message":"canonical_sha256_bytes"},"source_id":"1712.02148","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b0a51ee3804b663c523a798afa8cc1befb76feebe203fe942f3c0b31695a4373","sha256:4155d2c6d7ed038c7c6ead42f42ee2c0f9715ea99aafcca62afb600409392984"],"state_sha256":"372e003bfb52295a778bf8d47f0cf3167bec8f1789f6306322b6d7306408a8e7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HHglIYD6jPAffGDJD75ifkngYanXwnhinf8XqxM14ag2ncf0MpqRF8+wXlmQcelRFL9iqp36hIFBs7H/0kshCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T15:22:01.367368Z","bundle_sha256":"def33230fe68ff56eabfca00e757b809e9c608d7dc1713d3e86fb1a06ca659b1"}}