{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:MYUI2KLQUAQHQMKX6OY3PKDSXV","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":"f5250ae1cfb22d42b3d99c436db1a5ba5db6b5f7b1bd3d388a7a5ce1abf96a18","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2019-02-11T23:32:30Z","title_canon_sha256":"47f1f89b52110ff660163f45c5465276e12f9979c1d9bae67789efe2d69549c4"},"schema_version":"1.0","source":{"id":"1902.04180","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1902.04180","created_at":"2026-07-05T02:13:10Z"},{"alias_kind":"arxiv_version","alias_value":"1902.04180v3","created_at":"2026-07-05T02:13:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1902.04180","created_at":"2026-07-05T02:13:10Z"},{"alias_kind":"pith_short_12","alias_value":"MYUI2KLQUAQH","created_at":"2026-07-05T02:13:10Z"},{"alias_kind":"pith_short_16","alias_value":"MYUI2KLQUAQHQMKX","created_at":"2026-07-05T02:13:10Z"},{"alias_kind":"pith_short_8","alias_value":"MYUI2KLQ","created_at":"2026-07-05T02:13:10Z"}],"graph_snapshots":[{"event_id":"sha256:0d5d218b07dab09757d25c05237536e1b2cd734e42725712ae39262e9f5f9c47","target":"graph","created_at":"2026-07-05T02:13:10Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1902.04180/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Modular graph functions are $SL(2,{\\mathbb Z})$-invariant functions associated with Feynman graphs of a two-dimensional conformal field theory on a torus of modulus $\\tau$. For one-loop graphs they reduce to real analytic Eisenstein series. We obtain the Fourier series, including the constant and non-constant Fourier modes, of all two-loop modular graph functions, as well as their Poincar\\'e series with respect to $\\Gamma_\\infty \\backslash PSL(2,{\\mathbb Z})$. The Fourier and Poincar\\'e series provide the tools to compute the Petersson inner product of two-loop modular graph functions using Ra","authors_text":"Eric D'Hoker, Justin Kaidi","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2019-02-11T23:32:30Z","title":"Modular graph functions and odd cuspidal functions -- Fourier and Poincar\\'e series"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.04180","kind":"arxiv","version":3},"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":"sha256:25653d0d6bd3dd08873a866b4e3c35b7c49abeb661881d68bcf8aae9ad8837c6","target":"record","created_at":"2026-07-05T02:13:10Z","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":"f5250ae1cfb22d42b3d99c436db1a5ba5db6b5f7b1bd3d388a7a5ce1abf96a18","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2019-02-11T23:32:30Z","title_canon_sha256":"47f1f89b52110ff660163f45c5465276e12f9979c1d9bae67789efe2d69549c4"},"schema_version":"1.0","source":{"id":"1902.04180","kind":"arxiv","version":3}},"canonical_sha256":"66288d2970a020783157f3b1b7a872bd44032585694b26b319aee9e0eeb28dfd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"66288d2970a020783157f3b1b7a872bd44032585694b26b319aee9e0eeb28dfd","first_computed_at":"2026-07-05T02:13:10.268223Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:13:10.268223Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jcf0GDdRwaZK17zLrW5XtnkHXKM5kxcb62O+b/xQuuRW7qPoTZ3p9shIinKOnaBLciRW5SZsEvAX6aulMd0DDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T02:13:10.268660Z","signed_message":"canonical_sha256_bytes"},"source_id":"1902.04180","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:25653d0d6bd3dd08873a866b4e3c35b7c49abeb661881d68bcf8aae9ad8837c6","sha256:0d5d218b07dab09757d25c05237536e1b2cd734e42725712ae39262e9f5f9c47"],"state_sha256":"72a8c1537db8641e29f7d246e81214da8355f7b6bb5c88850a24e54550e4ac08"}