{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:QBP6F5WS7RH55MOXZGW7I3PR7J","short_pith_number":"pith:QBP6F5WS","canonical_record":{"source":{"id":"1007.2213","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-13T21:56:12Z","cross_cats_sorted":[],"title_canon_sha256":"97bfbeac7b8ecc613dc5cafa0c88b521177eeef223237620a51c3ad993ce99a1","abstract_canon_sha256":"5cd127801ecdd534cd3bb5d3bdbfb88faf127ff9b206044003da48a08bea6e84"},"schema_version":"1.0"},"canonical_sha256":"805fe2f6d2fc4fdeb1d7c9adf46df1fa556a1f28267af9a7829912e3e34a459d","source":{"kind":"arxiv","id":"1007.2213","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.2213","created_at":"2026-05-17T23:53:19Z"},{"alias_kind":"arxiv_version","alias_value":"1007.2213v1","created_at":"2026-05-17T23:53:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.2213","created_at":"2026-05-17T23:53:19Z"},{"alias_kind":"pith_short_12","alias_value":"QBP6F5WS7RH5","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"QBP6F5WS7RH55MOX","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"QBP6F5WS","created_at":"2026-05-18T12:26:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:QBP6F5WS7RH55MOXZGW7I3PR7J","target":"record","payload":{"canonical_record":{"source":{"id":"1007.2213","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-13T21:56:12Z","cross_cats_sorted":[],"title_canon_sha256":"97bfbeac7b8ecc613dc5cafa0c88b521177eeef223237620a51c3ad993ce99a1","abstract_canon_sha256":"5cd127801ecdd534cd3bb5d3bdbfb88faf127ff9b206044003da48a08bea6e84"},"schema_version":"1.0"},"canonical_sha256":"805fe2f6d2fc4fdeb1d7c9adf46df1fa556a1f28267af9a7829912e3e34a459d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:19.686658Z","signature_b64":"+xXpGKKQZS/rLzxjO7WakW9SeMSAaqiIeK38voLRM0VYk/R+nJVe2GsBregRJlBIGGUWpCR1AK+K/0TAnnNgCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"805fe2f6d2fc4fdeb1d7c9adf46df1fa556a1f28267af9a7829912e3e34a459d","last_reissued_at":"2026-05-17T23:53:19.686025Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:19.686025Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1007.2213","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":"2026-05-17T23:53:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"n4q+ZqfxteP0gG0XhJbCbELSwl+rjAickpBEXXOTiSgnYtsrGm0DHhcaBAlKgqlIS14I9McJmaKmGpt4MiOaBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T14:35:44.390514Z"},"content_sha256":"3809aecda8467483f15ef7dcc591d857775c47fbe391283d1d3920530e8bcf3b","schema_version":"1.0","event_id":"sha256:3809aecda8467483f15ef7dcc591d857775c47fbe391283d1d3920530e8bcf3b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:QBP6F5WS7RH55MOXZGW7I3PR7J","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Greenberg's $L$-invariant of the symmetric sixth power of an ordinary cusp form","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Robert Harron","submitted_at":"2010-07-13T21:56:12Z","abstract_excerpt":"We derive a formula for Greenberg's $L$-invariant of Tate twists of the symmetric sixth power of an ordinary non-CM cuspidal newform of weight $\\geq4$, under some technical assumptions. This requires a \"sufficiently rich\" Galois deformation of the symmetric cube which we obtain from the symmetric cube lift to $\\GSp(4)_{/\\QQ}$ of Ramakrishnan--Shahidi and the Hida theory of this group developed by Tilouine--Urban. The $L$-invariant is expressed in terms of derivatives of Frobenius eigenvalues varying in the Hida family. Our result suggests that one could compute Greenberg's $L$-invariant of all"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2213","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":"2026-05-17T23:53:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"K1XcUtqjP5S5OJ78ony8LTVgsO8f8nlvnvnYe99lnxJYIfQ2CMQ3LO6gR6UvBD4whp/86+vW4HZcYXQ9V8BrDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T14:35:44.390855Z"},"content_sha256":"062fcbcbc4b0f42eda8936c7db2dfdfa1acc25208b9993d89474eef9b4c9f2f0","schema_version":"1.0","event_id":"sha256:062fcbcbc4b0f42eda8936c7db2dfdfa1acc25208b9993d89474eef9b4c9f2f0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QBP6F5WS7RH55MOXZGW7I3PR7J/bundle.json","state_url":"https://pith.science/pith/QBP6F5WS7RH55MOXZGW7I3PR7J/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QBP6F5WS7RH55MOXZGW7I3PR7J/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":"2026-06-23T14:35:44Z","links":{"resolver":"https://pith.science/pith/QBP6F5WS7RH55MOXZGW7I3PR7J","bundle":"https://pith.science/pith/QBP6F5WS7RH55MOXZGW7I3PR7J/bundle.json","state":"https://pith.science/pith/QBP6F5WS7RH55MOXZGW7I3PR7J/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QBP6F5WS7RH55MOXZGW7I3PR7J/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:QBP6F5WS7RH55MOXZGW7I3PR7J","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":"5cd127801ecdd534cd3bb5d3bdbfb88faf127ff9b206044003da48a08bea6e84","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-13T21:56:12Z","title_canon_sha256":"97bfbeac7b8ecc613dc5cafa0c88b521177eeef223237620a51c3ad993ce99a1"},"schema_version":"1.0","source":{"id":"1007.2213","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1007.2213","created_at":"2026-05-17T23:53:19Z"},{"alias_kind":"arxiv_version","alias_value":"1007.2213v1","created_at":"2026-05-17T23:53:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.2213","created_at":"2026-05-17T23:53:19Z"},{"alias_kind":"pith_short_12","alias_value":"QBP6F5WS7RH5","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"QBP6F5WS7RH55MOX","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"QBP6F5WS","created_at":"2026-05-18T12:26:12Z"}],"graph_snapshots":[{"event_id":"sha256:062fcbcbc4b0f42eda8936c7db2dfdfa1acc25208b9993d89474eef9b4c9f2f0","target":"graph","created_at":"2026-05-17T23:53:19Z","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":"We derive a formula for Greenberg's $L$-invariant of Tate twists of the symmetric sixth power of an ordinary non-CM cuspidal newform of weight $\\geq4$, under some technical assumptions. This requires a \"sufficiently rich\" Galois deformation of the symmetric cube which we obtain from the symmetric cube lift to $\\GSp(4)_{/\\QQ}$ of Ramakrishnan--Shahidi and the Hida theory of this group developed by Tilouine--Urban. The $L$-invariant is expressed in terms of derivatives of Frobenius eigenvalues varying in the Hida family. Our result suggests that one could compute Greenberg's $L$-invariant of all","authors_text":"Robert Harron","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-13T21:56:12Z","title":"On Greenberg's $L$-invariant of the symmetric sixth power of an ordinary cusp form"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.2213","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":"sha256:3809aecda8467483f15ef7dcc591d857775c47fbe391283d1d3920530e8bcf3b","target":"record","created_at":"2026-05-17T23:53:19Z","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":"5cd127801ecdd534cd3bb5d3bdbfb88faf127ff9b206044003da48a08bea6e84","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-07-13T21:56:12Z","title_canon_sha256":"97bfbeac7b8ecc613dc5cafa0c88b521177eeef223237620a51c3ad993ce99a1"},"schema_version":"1.0","source":{"id":"1007.2213","kind":"arxiv","version":1}},"canonical_sha256":"805fe2f6d2fc4fdeb1d7c9adf46df1fa556a1f28267af9a7829912e3e34a459d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"805fe2f6d2fc4fdeb1d7c9adf46df1fa556a1f28267af9a7829912e3e34a459d","first_computed_at":"2026-05-17T23:53:19.686025Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:53:19.686025Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+xXpGKKQZS/rLzxjO7WakW9SeMSAaqiIeK38voLRM0VYk/R+nJVe2GsBregRJlBIGGUWpCR1AK+K/0TAnnNgCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:53:19.686658Z","signed_message":"canonical_sha256_bytes"},"source_id":"1007.2213","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3809aecda8467483f15ef7dcc591d857775c47fbe391283d1d3920530e8bcf3b","sha256:062fcbcbc4b0f42eda8936c7db2dfdfa1acc25208b9993d89474eef9b4c9f2f0"],"state_sha256":"22203e77a3beb63816343178abbcb36f4716dba321956064a03b3bbdda436e8b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aaO1557hDbkYndUTfYW1R5TQkZO9SlobC26QSD29Mwa6PGEqA2CR4AmjEPenp8JzslzfRidDP1CYizZxJ6N6BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T14:35:44.392751Z","bundle_sha256":"b9c9dbe26b1e9d50f3aa7edee52b14a3df8d41793e39451951b6861677f32b9f"}}