{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:M5MKMWZAJGEF7KBIB3BFOONSJA","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":"7d92427ac8e3e6caade8ec0eb006e67211ea371fef81024a0e8ee0a7a697de67","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-09T20:12:07Z","title_canon_sha256":"1eacf234ef12820a6a236e97b185a22796f5fb2ae77fe63efa76ae1fe90f6e95"},"schema_version":"1.0","source":{"id":"1908.03616","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.03616","created_at":"2026-07-04T23:52:42Z"},{"alias_kind":"arxiv_version","alias_value":"1908.03616v1","created_at":"2026-07-04T23:52:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.03616","created_at":"2026-07-04T23:52:42Z"},{"alias_kind":"pith_short_12","alias_value":"M5MKMWZAJGEF","created_at":"2026-07-04T23:52:42Z"},{"alias_kind":"pith_short_16","alias_value":"M5MKMWZAJGEF7KBI","created_at":"2026-07-04T23:52:42Z"},{"alias_kind":"pith_short_8","alias_value":"M5MKMWZA","created_at":"2026-07-04T23:52:42Z"}],"graph_snapshots":[{"event_id":"sha256:90e8709395b34eb3573c137f4076bfe2a6c918682a8348477094e408c7ff1912","target":"graph","created_at":"2026-07-04T23:52:42Z","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/1908.03616/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show that every elliptic modular form of integral weight greater than $1$ can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central $\\mathrm{L}$-values present in all previous work. For weights greater than $2$, we refine our result further, showing that linear combinations of products of exactly two cusp expansions of Eisenstein series suffice.","authors_text":"Jiacheng Xia, Martin Raum","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-09T20:12:07Z","title":"All modular forms of weight 2 can be expressed by Eisenstein series"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03616","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":"sha256:4fcaae0c7caf569408d1685d1957682391a7d97f19408e6da4364c36f1786ff0","target":"record","created_at":"2026-07-04T23:52:42Z","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":"7d92427ac8e3e6caade8ec0eb006e67211ea371fef81024a0e8ee0a7a697de67","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-09T20:12:07Z","title_canon_sha256":"1eacf234ef12820a6a236e97b185a22796f5fb2ae77fe63efa76ae1fe90f6e95"},"schema_version":"1.0","source":{"id":"1908.03616","kind":"arxiv","version":1}},"canonical_sha256":"6758a65b2049885fa8280ec25739b248209e2a2fc85d26c42a0cfa6cf7960046","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6758a65b2049885fa8280ec25739b248209e2a2fc85d26c42a0cfa6cf7960046","first_computed_at":"2026-07-04T23:52:42.791457Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T23:52:42.791457Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/qTjLIkNeXcbv0F/NKe1TiQ7CHfrg3vgic3ZLN8HOgDvb6UYEXnUrEc1MTMDibK3UDpJfUx7B8MMI/d9o4FDAA==","signature_status":"signed_v1","signed_at":"2026-07-04T23:52:42.791951Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.03616","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4fcaae0c7caf569408d1685d1957682391a7d97f19408e6da4364c36f1786ff0","sha256:90e8709395b34eb3573c137f4076bfe2a6c918682a8348477094e408c7ff1912"],"state_sha256":"4ff98d8b645ebde1d42eb7930112227b1ee1f491d1b3b4ad5074d31f9ece0433"}