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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}"},"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":"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1712.02148","created_at":"2026-05-18T00:26:59.949521+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.02148v2","created_at":"2026-05-18T00:26:59.949521+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.02148","created_at":"2026-05-18T00:26:59.949521+00:00"},{"alias_kind":"pith_short_12","alias_value":"PNK3KWPQFHRF","created_at":"2026-05-18T12:31:37.085036+00:00"},{"alias_kind":"pith_short_16","alias_value":"PNK3KWPQFHRFRLOF","created_at":"2026-05-18T12:31:37.085036+00:00"},{"alias_kind":"pith_short_8","alias_value":"PNK3KWPQ","created_at":"2026-05-18T12:31:37.085036+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/PNK3KWPQFHRFRLOFUTXZQWYVLI","json":"https://pith.science/pith/PNK3KWPQFHRFRLOFUTXZQWYVLI.json","graph_json":"https://pith.science/api/pith-number/PNK3KWPQFHRFRLOFUTXZQWYVLI/graph.json","events_json":"https://pith.science/api/pith-number/PNK3KWPQFHRFRLOFUTXZQWYVLI/events.json","paper":"https://pith.science/paper/PNK3KWPQ"},"agent_actions":{"view_html":"https://pith.science/pith/PNK3KWPQFHRFRLOFUTXZQWYVLI","download_json":"https://pith.science/pith/PNK3KWPQFHRFRLOFUTXZQWYVLI.json","view_paper":"https://pith.science/paper/PNK3KWPQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.02148&json=true","fetch_graph":"https://pith.science/api/pith-number/PNK3KWPQFHRFRLOFUTXZQWYVLI/graph.json","fetch_events":"https://pith.science/api/pith-number/PNK3KWPQFHRFRLOFUTXZQWYVLI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PNK3KWPQFHRFRLOFUTXZQWYVLI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PNK3KWPQFHRFRLOFUTXZQWYVLI/action/storage_attestation","attest_author":"https://pith.science/pith/PNK3KWPQFHRFRLOFUTXZQWYVLI/action/author_attestation","sign_citation":"https://pith.science/pith/PNK3KWPQFHRFRLOFUTXZQWYVLI/action/citation_signature","submit_replication":"https://pith.science/pith/PNK3KWPQFHRFRLOFUTXZQWYVLI/action/replication_record"}},"created_at":"2026-05-18T00:26:59.949521+00:00","updated_at":"2026-05-18T00:26:59.949521+00:00"}