{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:JIJO4PMSOOL7QM2L3UWHRJRWUQ","short_pith_number":"pith:JIJO4PMS","schema_version":"1.0","canonical_sha256":"4a12ee3d927397f8334bdd2c78a636a430eb2b600e394ea1750a5242f93c3aef","source":{"kind":"arxiv","id":"2412.07716","version":3},"attestation_state":"computed","paper":{"title":"On The Telescopic Picard Group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.AT","authors_text":"Shai Keidar","submitted_at":"2024-12-10T18:09:27Z","abstract_excerpt":"We prove that for any prime $p$ and height $n \\ge 1$, the telescopic Picard group $\\mathrm{Pic}(\\mathrm{Sp}_{Tn})$ contains a subgroup of the form $\\mathbb{Z}_p \\times \\mathbb{Z}/a_p(p^n-1)$, where $a_p = 1$ if $p = 2$ and $a_p = 2$ if $p$ is odd. Using Kummer theory, we obtain an $(\\mathbb{F}_{p^n}^\\times \\rtimes \\mathbb{Z}/n)$-Galois extension of $\\mathbb{S}_{T(n)}$, obtaining the first example of a lift of a non-Abelian Galois extension of the $K(n)$-local sphere to the telescopic world, at arbitrary positive height and prime. Our proof proceeds by setting up a higher categorical framework "},"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":"2412.07716","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2024-12-10T18:09:27Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"7b998f3e22223ca76bb66e89263c1f1c4267379a4e6cc0837da6e16d37edf414","abstract_canon_sha256":"17dedefe4334d55eddc073a7dbfcec33f302dd38596df3a6a4b190792abcd5b0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:48:38.171906Z","signature_b64":"ujirKBnlcBSsVKgr5vjREAJ1R5ebT+4NVhai7u/Q4kFqYnmigCnMB3Ja09XQACPB0h5oZyFRlH4BWIm2S6V1Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a12ee3d927397f8334bdd2c78a636a430eb2b600e394ea1750a5242f93c3aef","last_reissued_at":"2026-07-05T09:48:38.171440Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:48:38.171440Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On The Telescopic Picard Group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.AT","authors_text":"Shai Keidar","submitted_at":"2024-12-10T18:09:27Z","abstract_excerpt":"We prove that for any prime $p$ and height $n \\ge 1$, the telescopic Picard group $\\mathrm{Pic}(\\mathrm{Sp}_{Tn})$ contains a subgroup of the form $\\mathbb{Z}_p \\times \\mathbb{Z}/a_p(p^n-1)$, where $a_p = 1$ if $p = 2$ and $a_p = 2$ if $p$ is odd. Using Kummer theory, we obtain an $(\\mathbb{F}_{p^n}^\\times \\rtimes \\mathbb{Z}/n)$-Galois extension of $\\mathbb{S}_{T(n)}$, obtaining the first example of a lift of a non-Abelian Galois extension of the $K(n)$-local sphere to the telescopic world, at arbitrary positive height and prime. Our proof proceeds by setting up a higher categorical framework "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.07716","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.07716/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2412.07716","created_at":"2026-07-05T09:48:38.171497+00:00"},{"alias_kind":"arxiv_version","alias_value":"2412.07716v3","created_at":"2026-07-05T09:48:38.171497+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.07716","created_at":"2026-07-05T09:48:38.171497+00:00"},{"alias_kind":"pith_short_12","alias_value":"JIJO4PMSOOL7","created_at":"2026-07-05T09:48:38.171497+00:00"},{"alias_kind":"pith_short_16","alias_value":"JIJO4PMSOOL7QM2L","created_at":"2026-07-05T09:48:38.171497+00:00"},{"alias_kind":"pith_short_8","alias_value":"JIJO4PMS","created_at":"2026-07-05T09:48:38.171497+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/JIJO4PMSOOL7QM2L3UWHRJRWUQ","json":"https://pith.science/pith/JIJO4PMSOOL7QM2L3UWHRJRWUQ.json","graph_json":"https://pith.science/api/pith-number/JIJO4PMSOOL7QM2L3UWHRJRWUQ/graph.json","events_json":"https://pith.science/api/pith-number/JIJO4PMSOOL7QM2L3UWHRJRWUQ/events.json","paper":"https://pith.science/paper/JIJO4PMS"},"agent_actions":{"view_html":"https://pith.science/pith/JIJO4PMSOOL7QM2L3UWHRJRWUQ","download_json":"https://pith.science/pith/JIJO4PMSOOL7QM2L3UWHRJRWUQ.json","view_paper":"https://pith.science/paper/JIJO4PMS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2412.07716&json=true","fetch_graph":"https://pith.science/api/pith-number/JIJO4PMSOOL7QM2L3UWHRJRWUQ/graph.json","fetch_events":"https://pith.science/api/pith-number/JIJO4PMSOOL7QM2L3UWHRJRWUQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JIJO4PMSOOL7QM2L3UWHRJRWUQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JIJO4PMSOOL7QM2L3UWHRJRWUQ/action/storage_attestation","attest_author":"https://pith.science/pith/JIJO4PMSOOL7QM2L3UWHRJRWUQ/action/author_attestation","sign_citation":"https://pith.science/pith/JIJO4PMSOOL7QM2L3UWHRJRWUQ/action/citation_signature","submit_replication":"https://pith.science/pith/JIJO4PMSOOL7QM2L3UWHRJRWUQ/action/replication_record"}},"created_at":"2026-07-05T09:48:38.171497+00:00","updated_at":"2026-07-05T09:48:38.171497+00:00"}