{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:UCSH2BZFRLH27HQS37JOWE3R5O","short_pith_number":"pith:UCSH2BZF","canonical_record":{"source":{"id":"1409.7702","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2014-09-26T20:01:06Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"c433b7c53fdf6e25f5771e79be835c185d5b5cf0354cba23304dfc35980d931b","abstract_canon_sha256":"40964424b26d71ad7d05f6aa5984790f1a1f02e5f1ec7dd0d0a18e83c6681997"},"schema_version":"1.0"},"canonical_sha256":"a0a47d07258acfaf9e12dfd2eb1371ebb7b3e733257f49071eddd077d4523c5f","source":{"kind":"arxiv","id":"1409.7702","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.7702","created_at":"2026-05-18T00:53:26Z"},{"alias_kind":"arxiv_version","alias_value":"1409.7702v4","created_at":"2026-05-18T00:53:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.7702","created_at":"2026-05-18T00:53:26Z"},{"alias_kind":"pith_short_12","alias_value":"UCSH2BZFRLH2","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_16","alias_value":"UCSH2BZFRLH27HQS","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_8","alias_value":"UCSH2BZF","created_at":"2026-05-18T12:28:52Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:UCSH2BZFRLH27HQS37JOWE3R5O","target":"record","payload":{"canonical_record":{"source":{"id":"1409.7702","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2014-09-26T20:01:06Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"c433b7c53fdf6e25f5771e79be835c185d5b5cf0354cba23304dfc35980d931b","abstract_canon_sha256":"40964424b26d71ad7d05f6aa5984790f1a1f02e5f1ec7dd0d0a18e83c6681997"},"schema_version":"1.0"},"canonical_sha256":"a0a47d07258acfaf9e12dfd2eb1371ebb7b3e733257f49071eddd077d4523c5f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:26.392773Z","signature_b64":"jFr0Van7HPgX7J2YsqwxDD0tsjAZ11WaO5bW2Hn2+yF3CTmsrnQN5N2eDi8jcaM9whTmUwWpvmPlOqfCz7m8CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a0a47d07258acfaf9e12dfd2eb1371ebb7b3e733257f49071eddd077d4523c5f","last_reissued_at":"2026-05-18T00:53:26.392212Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:26.392212Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1409.7702","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:53:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8S4JS82qpypPCGk2fsMWTpEIfgAZc1XydhYGCqSvggIHGk9qRYmimB9XDIQP8DP8tAquOLSHybq21/7stPUKAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T16:35:03.440658Z"},"content_sha256":"95f431ef21e2df5cf2330926acbf9b42cd27d84aa4318861d82f27033e1270a8","schema_version":"1.0","event_id":"sha256:95f431ef21e2df5cf2330926acbf9b42cd27d84aa4318861d82f27033e1270a8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:UCSH2BZFRLH27HQS37JOWE3R5O","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Picard group of topological modular forms via descent theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.AT","authors_text":"Akhil Mathew, Vesna Stojanoska","submitted_at":"2014-09-26T20:01:06Z","abstract_excerpt":"This paper starts with an exposition of descent-theoretic techniques in the study of Picard groups of $\\mathbf{E}_{\\infty}$-ring spectra, which naturally lead to the study of Picard spectra. We then develop tools for the efficient and explicit determination of differentials in the associated descent spectral sequences for the Picard spectra thus obtained. As a major application, we calculate the Picard groups of the periodic spectrum of topological modular forms $TMF$ and the non-periodic and non-connective $Tmf$. We find that $\\mathrm{Pic} (TMF)$ is cyclic of order 576, generated by the suspe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.7702","kind":"arxiv","version":4},"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":"2026-05-18T00:53:26Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5RFp3HiPS4FZn1hQd/oy31XIIhAvxk4TjDSokBfJL5swi0aabZ14FjuWJxN4m4f1duA5qPvBiqbzHvxp3vV+Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T16:35:03.441372Z"},"content_sha256":"b79444e527ab754ce1a2a5b119cdff382b1138b02bac6ab925e671b8a22a94e3","schema_version":"1.0","event_id":"sha256:b79444e527ab754ce1a2a5b119cdff382b1138b02bac6ab925e671b8a22a94e3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UCSH2BZFRLH27HQS37JOWE3R5O/bundle.json","state_url":"https://pith.science/pith/UCSH2BZFRLH27HQS37JOWE3R5O/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UCSH2BZFRLH27HQS37JOWE3R5O/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":"2026-05-31T16:35:03Z","links":{"resolver":"https://pith.science/pith/UCSH2BZFRLH27HQS37JOWE3R5O","bundle":"https://pith.science/pith/UCSH2BZFRLH27HQS37JOWE3R5O/bundle.json","state":"https://pith.science/pith/UCSH2BZFRLH27HQS37JOWE3R5O/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UCSH2BZFRLH27HQS37JOWE3R5O/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:UCSH2BZFRLH27HQS37JOWE3R5O","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":"40964424b26d71ad7d05f6aa5984790f1a1f02e5f1ec7dd0d0a18e83c6681997","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2014-09-26T20:01:06Z","title_canon_sha256":"c433b7c53fdf6e25f5771e79be835c185d5b5cf0354cba23304dfc35980d931b"},"schema_version":"1.0","source":{"id":"1409.7702","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.7702","created_at":"2026-05-18T00:53:26Z"},{"alias_kind":"arxiv_version","alias_value":"1409.7702v4","created_at":"2026-05-18T00:53:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.7702","created_at":"2026-05-18T00:53:26Z"},{"alias_kind":"pith_short_12","alias_value":"UCSH2BZFRLH2","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_16","alias_value":"UCSH2BZFRLH27HQS","created_at":"2026-05-18T12:28:52Z"},{"alias_kind":"pith_short_8","alias_value":"UCSH2BZF","created_at":"2026-05-18T12:28:52Z"}],"graph_snapshots":[{"event_id":"sha256:b79444e527ab754ce1a2a5b119cdff382b1138b02bac6ab925e671b8a22a94e3","target":"graph","created_at":"2026-05-18T00:53:26Z","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":"This paper starts with an exposition of descent-theoretic techniques in the study of Picard groups of $\\mathbf{E}_{\\infty}$-ring spectra, which naturally lead to the study of Picard spectra. We then develop tools for the efficient and explicit determination of differentials in the associated descent spectral sequences for the Picard spectra thus obtained. As a major application, we calculate the Picard groups of the periodic spectrum of topological modular forms $TMF$ and the non-periodic and non-connective $Tmf$. We find that $\\mathrm{Pic} (TMF)$ is cyclic of order 576, generated by the suspe","authors_text":"Akhil Mathew, Vesna Stojanoska","cross_cats":["math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2014-09-26T20:01:06Z","title":"The Picard group of topological modular forms via descent theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.7702","kind":"arxiv","version":4},"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":"sha256:95f431ef21e2df5cf2330926acbf9b42cd27d84aa4318861d82f27033e1270a8","target":"record","created_at":"2026-05-18T00:53:26Z","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":"40964424b26d71ad7d05f6aa5984790f1a1f02e5f1ec7dd0d0a18e83c6681997","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AT","submitted_at":"2014-09-26T20:01:06Z","title_canon_sha256":"c433b7c53fdf6e25f5771e79be835c185d5b5cf0354cba23304dfc35980d931b"},"schema_version":"1.0","source":{"id":"1409.7702","kind":"arxiv","version":4}},"canonical_sha256":"a0a47d07258acfaf9e12dfd2eb1371ebb7b3e733257f49071eddd077d4523c5f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a0a47d07258acfaf9e12dfd2eb1371ebb7b3e733257f49071eddd077d4523c5f","first_computed_at":"2026-05-18T00:53:26.392212Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:26.392212Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jFr0Van7HPgX7J2YsqwxDD0tsjAZ11WaO5bW2Hn2+yF3CTmsrnQN5N2eDi8jcaM9whTmUwWpvmPlOqfCz7m8CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:26.392773Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.7702","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:95f431ef21e2df5cf2330926acbf9b42cd27d84aa4318861d82f27033e1270a8","sha256:b79444e527ab754ce1a2a5b119cdff382b1138b02bac6ab925e671b8a22a94e3"],"state_sha256":"671a848c88d05cc714ea72bc508fb452d167deffd8a37570ebc80ff5dfd9bfde"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UOFqSxC4WYQT9DPqhx6WwIGAwpeLIYl1ECBDx1GNJTZjSWCaCRWytcpwZNDqrmG0wkSUzRLFDgyjStIHjYVKBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T16:35:03.445394Z","bundle_sha256":"3e4d7d13ad46f1066eaeb56ca2917a1d49d334221bd464172330baa650d0dfd9"}}