{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:6WL4INALYFXVJC5WA2FP7EU35A","short_pith_number":"pith:6WL4INAL","canonical_record":{"source":{"id":"1012.0175","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-01T12:24:14Z","cross_cats_sorted":[],"title_canon_sha256":"e87a5e09a3db0aab612288e505c3407b54beb53c47472e3417f0112eed6dfdf3","abstract_canon_sha256":"4f82e096e1271f38c01d01ed5d0454bab5d2290bf5277996a8e90bd67700980e"},"schema_version":"1.0"},"canonical_sha256":"f597c4340bc16f548bb6068aff929be83091d3a732e6476f16a03aab5506742f","source":{"kind":"arxiv","id":"1012.0175","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.0175","created_at":"2026-05-18T03:20:59Z"},{"alias_kind":"arxiv_version","alias_value":"1012.0175v1","created_at":"2026-05-18T03:20:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.0175","created_at":"2026-05-18T03:20:59Z"},{"alias_kind":"pith_short_12","alias_value":"6WL4INALYFXV","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"6WL4INALYFXVJC5W","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"6WL4INAL","created_at":"2026-05-18T12:26:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:6WL4INALYFXVJC5WA2FP7EU35A","target":"record","payload":{"canonical_record":{"source":{"id":"1012.0175","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-01T12:24:14Z","cross_cats_sorted":[],"title_canon_sha256":"e87a5e09a3db0aab612288e505c3407b54beb53c47472e3417f0112eed6dfdf3","abstract_canon_sha256":"4f82e096e1271f38c01d01ed5d0454bab5d2290bf5277996a8e90bd67700980e"},"schema_version":"1.0"},"canonical_sha256":"f597c4340bc16f548bb6068aff929be83091d3a732e6476f16a03aab5506742f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:20:59.049331Z","signature_b64":"BtM8SExqilJxSVjqaUwXLc9c93RIVjTY6VTlK3WSQuQagbQA9xLmxY+rzWIJEye/AJQcjPzFEolqmCUZ8XW5Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f597c4340bc16f548bb6068aff929be83091d3a732e6476f16a03aab5506742f","last_reissued_at":"2026-05-18T03:20:59.048723Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:20:59.048723Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1012.0175","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":"2026-05-18T03:20:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Db/oQba6GWq5OxZf7dzFoD7PiiWZlftFIqwdl4Lzj6Lcw7qiGeShsg+loNQUEXtuBiG1Y/Km7hKOPlGcAvD9Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T21:26:48.904768Z"},"content_sha256":"0d066d1004c61f8bc15673a9099fcfef127f7c26ac398ec30794fe544cd1b284","schema_version":"1.0","event_id":"sha256:0d066d1004c61f8bc15673a9099fcfef127f7c26ac398ec30794fe544cd1b284"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:6WL4INALYFXVJC5WA2FP7EU35A","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Wach modules and critical slope p-adic L-functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"David Loeffler, Sarah Livia Zerbes","submitted_at":"2010-12-01T12:24:14Z","abstract_excerpt":"We study Kato and Perrin-Riou's critical slope p-adic L-function attached to an ordinary modular form, using the methods of our earlier work with Lei. We show that it may be decomposed as a sum of two bounded measures multiplied by explicit distributions depending only on the local properties of the modular form at p. We use this decomposition to prove results on the zeros of the p-adic L-function, and we show that our results match the behaviour observed in examples calculated by Pollack and Stevens."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.0175","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":"2026-05-18T03:20:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ebgxqfxjS2kjXuJxgOELh87+pxRXq3N4t7pCTlHoTVFlNU9Wt0a7tyQDlLqtaC8WPepT1AUqw5ZkEFI+DbCgAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T21:26:48.905372Z"},"content_sha256":"03728f7a7028775b6b7ac2abe88ae3b292b02e0db05c5237f1e9d999313c6582","schema_version":"1.0","event_id":"sha256:03728f7a7028775b6b7ac2abe88ae3b292b02e0db05c5237f1e9d999313c6582"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6WL4INALYFXVJC5WA2FP7EU35A/bundle.json","state_url":"https://pith.science/pith/6WL4INALYFXVJC5WA2FP7EU35A/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6WL4INALYFXVJC5WA2FP7EU35A/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":"2026-06-22T21:26:48Z","links":{"resolver":"https://pith.science/pith/6WL4INALYFXVJC5WA2FP7EU35A","bundle":"https://pith.science/pith/6WL4INALYFXVJC5WA2FP7EU35A/bundle.json","state":"https://pith.science/pith/6WL4INALYFXVJC5WA2FP7EU35A/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6WL4INALYFXVJC5WA2FP7EU35A/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:6WL4INALYFXVJC5WA2FP7EU35A","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":"4f82e096e1271f38c01d01ed5d0454bab5d2290bf5277996a8e90bd67700980e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-01T12:24:14Z","title_canon_sha256":"e87a5e09a3db0aab612288e505c3407b54beb53c47472e3417f0112eed6dfdf3"},"schema_version":"1.0","source":{"id":"1012.0175","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.0175","created_at":"2026-05-18T03:20:59Z"},{"alias_kind":"arxiv_version","alias_value":"1012.0175v1","created_at":"2026-05-18T03:20:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.0175","created_at":"2026-05-18T03:20:59Z"},{"alias_kind":"pith_short_12","alias_value":"6WL4INALYFXV","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"6WL4INALYFXVJC5W","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"6WL4INAL","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:03728f7a7028775b6b7ac2abe88ae3b292b02e0db05c5237f1e9d999313c6582","target":"graph","created_at":"2026-05-18T03:20:59Z","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":"We study Kato and Perrin-Riou's critical slope p-adic L-function attached to an ordinary modular form, using the methods of our earlier work with Lei. We show that it may be decomposed as a sum of two bounded measures multiplied by explicit distributions depending only on the local properties of the modular form at p. We use this decomposition to prove results on the zeros of the p-adic L-function, and we show that our results match the behaviour observed in examples calculated by Pollack and Stevens.","authors_text":"David Loeffler, Sarah Livia Zerbes","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-01T12:24:14Z","title":"Wach modules and critical slope p-adic L-functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.0175","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":"sha256:0d066d1004c61f8bc15673a9099fcfef127f7c26ac398ec30794fe544cd1b284","target":"record","created_at":"2026-05-18T03:20:59Z","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":"4f82e096e1271f38c01d01ed5d0454bab5d2290bf5277996a8e90bd67700980e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-01T12:24:14Z","title_canon_sha256":"e87a5e09a3db0aab612288e505c3407b54beb53c47472e3417f0112eed6dfdf3"},"schema_version":"1.0","source":{"id":"1012.0175","kind":"arxiv","version":1}},"canonical_sha256":"f597c4340bc16f548bb6068aff929be83091d3a732e6476f16a03aab5506742f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f597c4340bc16f548bb6068aff929be83091d3a732e6476f16a03aab5506742f","first_computed_at":"2026-05-18T03:20:59.048723Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:20:59.048723Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BtM8SExqilJxSVjqaUwXLc9c93RIVjTY6VTlK3WSQuQagbQA9xLmxY+rzWIJEye/AJQcjPzFEolqmCUZ8XW5Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T03:20:59.049331Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.0175","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0d066d1004c61f8bc15673a9099fcfef127f7c26ac398ec30794fe544cd1b284","sha256:03728f7a7028775b6b7ac2abe88ae3b292b02e0db05c5237f1e9d999313c6582"],"state_sha256":"de5be12088c6a1f11fe97d8a09d9fbc927f759ae059d7f242c7b20a3570754e9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fwwlWMezjB1SG5l1+dJlE9L5ni7LenvVIfaXIRDf5/+2vDAZl0bMlgKRDtL1u+QE3GIAh5rQReTP9en5VTB0Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T21:26:48.908704Z","bundle_sha256":"7f6a90eef825e7324770ca03fe46498b5dce474cb039e4c54ed6bb137acd0d0a"}}