{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:WSF52C2FO7JZDOSW7IGOBC3YTI","short_pith_number":"pith:WSF52C2F","canonical_record":{"source":{"id":"0909.0799","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2009-09-04T02:56:27Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"6d65c30e74d97141485ff806c39afb63edc75953fd832edb11567b30436acc9b","abstract_canon_sha256":"e25450f7a10a03ca2299d3924025a7b5c47642d1cb082409f94e72914f749a7c"},"schema_version":"1.0"},"canonical_sha256":"b48bdd0b4577d391ba56fa0ce08b789a0fe16a67f930dbd807518455199f7ff7","source":{"kind":"arxiv","id":"0909.0799","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.0799","created_at":"2026-05-18T01:03:12Z"},{"alias_kind":"arxiv_version","alias_value":"0909.0799v2","created_at":"2026-05-18T01:03:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.0799","created_at":"2026-05-18T01:03:12Z"},{"alias_kind":"pith_short_12","alias_value":"WSF52C2FO7JZ","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"WSF52C2FO7JZDOSW","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"WSF52C2F","created_at":"2026-05-18T12:26:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:WSF52C2FO7JZDOSW7IGOBC3YTI","target":"record","payload":{"canonical_record":{"source":{"id":"0909.0799","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2009-09-04T02:56:27Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"6d65c30e74d97141485ff806c39afb63edc75953fd832edb11567b30436acc9b","abstract_canon_sha256":"e25450f7a10a03ca2299d3924025a7b5c47642d1cb082409f94e72914f749a7c"},"schema_version":"1.0"},"canonical_sha256":"b48bdd0b4577d391ba56fa0ce08b789a0fe16a67f930dbd807518455199f7ff7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:12.296170Z","signature_b64":"uaU3koduXDYfNOsInQWSryUGiMFT3Ax2KvC6bCuQQSNUWoSIMg5LN9qirqkLWTV61POuPvHRossm+F5z+37cDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b48bdd0b4577d391ba56fa0ce08b789a0fe16a67f930dbd807518455199f7ff7","last_reissued_at":"2026-05-18T01:03:12.295686Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:12.295686Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0909.0799","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:03:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2JxlT8eDxj8ka7u87FQp6CgyC/MvrBaSLiFuAOTcY7GD6f4pkECgHcdArk9jhfStMCN+9hrXFTrJzon2e1uICg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T08:38:40.999093Z"},"content_sha256":"70409e9bd6f2f0bd3c95ee41d207bf6fadd445f5056dab0180761cd8e944deca","schema_version":"1.0","event_id":"sha256:70409e9bd6f2f0bd3c95ee41d207bf6fadd445f5056dab0180761cd8e944deca"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:WSF52C2FO7JZDOSW7IGOBC3YTI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The cusp amplitudes and quasi-level of a congruence subgroup of SL2 over any Dedekind domain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.GR","authors_text":"Andreas Schweizer, A. W. Mason","submitted_at":"2009-09-04T02:56:27Z","abstract_excerpt":"We extend some algebraic properties of the classical modular group SL_2(Z) to equivalent groups in the theory of Drinfeld modules, in particular properties which are important in the theory of modular curves.\n  We study cusp amplitudes and the level of a (congruence) subgroup of SL_2(D) for any Dedekind domain D, as ideals of D. In particular, we extend a remarkable result of Larcher.\n  We introduce finer notions of quasi-amplitude and quasi-level, which are not required to be ideals and encode more information about the subgroup.\n  Our results also provide several new necessary conditions for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.0799","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:03:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gfyiS5XYioe9PoFUs+WIasqMzZ3g/qsJ1fGd8QbCcp5sHw6Cj9+YApBv14ijkRH++oOEzCz9rGnBJSLesMR4AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T08:38:40.999759Z"},"content_sha256":"0a73f59c84539e86331aae29e00fb344b7ce4cdb76fd045277728da15df42bd6","schema_version":"1.0","event_id":"sha256:0a73f59c84539e86331aae29e00fb344b7ce4cdb76fd045277728da15df42bd6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WSF52C2FO7JZDOSW7IGOBC3YTI/bundle.json","state_url":"https://pith.science/pith/WSF52C2FO7JZDOSW7IGOBC3YTI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WSF52C2FO7JZDOSW7IGOBC3YTI/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":"2026-05-30T08:38:41Z","links":{"resolver":"https://pith.science/pith/WSF52C2FO7JZDOSW7IGOBC3YTI","bundle":"https://pith.science/pith/WSF52C2FO7JZDOSW7IGOBC3YTI/bundle.json","state":"https://pith.science/pith/WSF52C2FO7JZDOSW7IGOBC3YTI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WSF52C2FO7JZDOSW7IGOBC3YTI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:WSF52C2FO7JZDOSW7IGOBC3YTI","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":"e25450f7a10a03ca2299d3924025a7b5c47642d1cb082409f94e72914f749a7c","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2009-09-04T02:56:27Z","title_canon_sha256":"6d65c30e74d97141485ff806c39afb63edc75953fd832edb11567b30436acc9b"},"schema_version":"1.0","source":{"id":"0909.0799","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0909.0799","created_at":"2026-05-18T01:03:12Z"},{"alias_kind":"arxiv_version","alias_value":"0909.0799v2","created_at":"2026-05-18T01:03:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.0799","created_at":"2026-05-18T01:03:12Z"},{"alias_kind":"pith_short_12","alias_value":"WSF52C2FO7JZ","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"WSF52C2FO7JZDOSW","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"WSF52C2F","created_at":"2026-05-18T12:26:02Z"}],"graph_snapshots":[{"event_id":"sha256:0a73f59c84539e86331aae29e00fb344b7ce4cdb76fd045277728da15df42bd6","target":"graph","created_at":"2026-05-18T01:03:12Z","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 extend some algebraic properties of the classical modular group SL_2(Z) to equivalent groups in the theory of Drinfeld modules, in particular properties which are important in the theory of modular curves.\n  We study cusp amplitudes and the level of a (congruence) subgroup of SL_2(D) for any Dedekind domain D, as ideals of D. In particular, we extend a remarkable result of Larcher.\n  We introduce finer notions of quasi-amplitude and quasi-level, which are not required to be ideals and encode more information about the subgroup.\n  Our results also provide several new necessary conditions for","authors_text":"Andreas Schweizer, A. W. Mason","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2009-09-04T02:56:27Z","title":"The cusp amplitudes and quasi-level of a congruence subgroup of SL2 over any Dedekind domain"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.0799","kind":"arxiv","version":2},"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":"sha256:70409e9bd6f2f0bd3c95ee41d207bf6fadd445f5056dab0180761cd8e944deca","target":"record","created_at":"2026-05-18T01:03:12Z","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":"e25450f7a10a03ca2299d3924025a7b5c47642d1cb082409f94e72914f749a7c","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2009-09-04T02:56:27Z","title_canon_sha256":"6d65c30e74d97141485ff806c39afb63edc75953fd832edb11567b30436acc9b"},"schema_version":"1.0","source":{"id":"0909.0799","kind":"arxiv","version":2}},"canonical_sha256":"b48bdd0b4577d391ba56fa0ce08b789a0fe16a67f930dbd807518455199f7ff7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b48bdd0b4577d391ba56fa0ce08b789a0fe16a67f930dbd807518455199f7ff7","first_computed_at":"2026-05-18T01:03:12.295686Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:03:12.295686Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uaU3koduXDYfNOsInQWSryUGiMFT3Ax2KvC6bCuQQSNUWoSIMg5LN9qirqkLWTV61POuPvHRossm+F5z+37cDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:03:12.296170Z","signed_message":"canonical_sha256_bytes"},"source_id":"0909.0799","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:70409e9bd6f2f0bd3c95ee41d207bf6fadd445f5056dab0180761cd8e944deca","sha256:0a73f59c84539e86331aae29e00fb344b7ce4cdb76fd045277728da15df42bd6"],"state_sha256":"9b914d987888d97e61f68390cdaa18e53f01e877dfe6752189d32bcc95a946a2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ChaAjV9EagSsJ4H93Tr1pDRYKzgfYImxtVUhFZzDzapQJ7+olo+eXBQxBBAOtadlBIoajbSBjRUzSOKOgwksDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T08:38:41.003658Z","bundle_sha256":"4295216ccde14fe9f64360a5f05acfee4b045acd651f27db3f608d2ac7c8f3ce"}}