{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:ZD7FHNX2ZBRPONUBFCPHOKVM2F","short_pith_number":"pith:ZD7FHNX2","canonical_record":{"source":{"id":"2407.01205","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-07-01T11:51:28Z","cross_cats_sorted":[],"title_canon_sha256":"4c20b0bc202a672c7f77cfdb9d5f3006c3a7f06532213925f8138af55e94841f","abstract_canon_sha256":"b5e6aec744dc901ff43849dc7dd210263f33b5b262d8bc3b2f2ea49352a3c3e9"},"schema_version":"1.0"},"canonical_sha256":"c8fe53b6fac862f73681289e772aacd167cc440b4ca73b3778486887c17a6db4","source":{"kind":"arxiv","id":"2407.01205","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.01205","created_at":"2026-07-05T09:23:07Z"},{"alias_kind":"arxiv_version","alias_value":"2407.01205v2","created_at":"2026-07-05T09:23:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.01205","created_at":"2026-07-05T09:23:07Z"},{"alias_kind":"pith_short_12","alias_value":"ZD7FHNX2ZBRP","created_at":"2026-07-05T09:23:07Z"},{"alias_kind":"pith_short_16","alias_value":"ZD7FHNX2ZBRPONUB","created_at":"2026-07-05T09:23:07Z"},{"alias_kind":"pith_short_8","alias_value":"ZD7FHNX2","created_at":"2026-07-05T09:23:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:ZD7FHNX2ZBRPONUBFCPHOKVM2F","target":"record","payload":{"canonical_record":{"source":{"id":"2407.01205","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-07-01T11:51:28Z","cross_cats_sorted":[],"title_canon_sha256":"4c20b0bc202a672c7f77cfdb9d5f3006c3a7f06532213925f8138af55e94841f","abstract_canon_sha256":"b5e6aec744dc901ff43849dc7dd210263f33b5b262d8bc3b2f2ea49352a3c3e9"},"schema_version":"1.0"},"canonical_sha256":"c8fe53b6fac862f73681289e772aacd167cc440b4ca73b3778486887c17a6db4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:23:07.744918Z","signature_b64":"JyInil0f3NJ0bFVQVXE7iYJhE5UFRxDL+GfweYDBOXxgod6+pt8JIpk/s1kWqRL91psfVGUQ0JpYeFJHz2YkDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c8fe53b6fac862f73681289e772aacd167cc440b4ca73b3778486887c17a6db4","last_reissued_at":"2026-07-05T09:23:07.744444Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:23:07.744444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2407.01205","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":"2026-07-05T09:23:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6MbKDtaXylnoq7bolDcCad3KS7cwyHb/PKYsbb+uOjPN/RDTvVR1qQIUdXUeMoQRYXbbitFqEICO6AcDoGkjBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T15:45:59.685506Z"},"content_sha256":"582b85fb83786d6284432ac4377f6a460a8d1c02c2be1c2a6a6ac38eb23cfcd6","schema_version":"1.0","event_id":"sha256:582b85fb83786d6284432ac4377f6a460a8d1c02c2be1c2a6a6ac38eb23cfcd6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:ZD7FHNX2ZBRPONUBFCPHOKVM2F","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The basis problem for modular forms for the Weil representation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Manuel K.-H. M\\\"uller","submitted_at":"2024-07-01T11:51:28Z","abstract_excerpt":"The vector valued theta series of a positive-definite even lattice is a modular form for the Weil representation of $\\mathrm{SL}_2(\\mathbb{Z})$. We show that the space of cusp forms for the Weil representation is generated by such functions. This gives a positive answer to Eichler's basis problem in this case. As applications we derive Waldspurger's result on the basis problem for scalar valued modular forms and give a new proof of the surjectivity of the Borcherds lift based on the analysis of local Picard groups."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.01205","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2407.01205/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T09:23:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nBx7iGixUe9f0G3zKnn1GCWkct3rLNn0Dr7+2Nov82Le1EaalNl1DQxJLwJzJn1nM92piMXIUlfnogT6NOPyCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T15:45:59.686012Z"},"content_sha256":"1017bca71eed0f0a6313247461052897019ededd69c8bb647d2656010c748d6c","schema_version":"1.0","event_id":"sha256:1017bca71eed0f0a6313247461052897019ededd69c8bb647d2656010c748d6c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZD7FHNX2ZBRPONUBFCPHOKVM2F/bundle.json","state_url":"https://pith.science/pith/ZD7FHNX2ZBRPONUBFCPHOKVM2F/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZD7FHNX2ZBRPONUBFCPHOKVM2F/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":"2026-08-08T15:45:59Z","links":{"resolver":"https://pith.science/pith/ZD7FHNX2ZBRPONUBFCPHOKVM2F","bundle":"https://pith.science/pith/ZD7FHNX2ZBRPONUBFCPHOKVM2F/bundle.json","state":"https://pith.science/pith/ZD7FHNX2ZBRPONUBFCPHOKVM2F/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZD7FHNX2ZBRPONUBFCPHOKVM2F/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:ZD7FHNX2ZBRPONUBFCPHOKVM2F","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":"b5e6aec744dc901ff43849dc7dd210263f33b5b262d8bc3b2f2ea49352a3c3e9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-07-01T11:51:28Z","title_canon_sha256":"4c20b0bc202a672c7f77cfdb9d5f3006c3a7f06532213925f8138af55e94841f"},"schema_version":"1.0","source":{"id":"2407.01205","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2407.01205","created_at":"2026-07-05T09:23:07Z"},{"alias_kind":"arxiv_version","alias_value":"2407.01205v2","created_at":"2026-07-05T09:23:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2407.01205","created_at":"2026-07-05T09:23:07Z"},{"alias_kind":"pith_short_12","alias_value":"ZD7FHNX2ZBRP","created_at":"2026-07-05T09:23:07Z"},{"alias_kind":"pith_short_16","alias_value":"ZD7FHNX2ZBRPONUB","created_at":"2026-07-05T09:23:07Z"},{"alias_kind":"pith_short_8","alias_value":"ZD7FHNX2","created_at":"2026-07-05T09:23:07Z"}],"graph_snapshots":[{"event_id":"sha256:1017bca71eed0f0a6313247461052897019ededd69c8bb647d2656010c748d6c","target":"graph","created_at":"2026-07-05T09:23:07Z","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/2407.01205/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The vector valued theta series of a positive-definite even lattice is a modular form for the Weil representation of $\\mathrm{SL}_2(\\mathbb{Z})$. We show that the space of cusp forms for the Weil representation is generated by such functions. This gives a positive answer to Eichler's basis problem in this case. As applications we derive Waldspurger's result on the basis problem for scalar valued modular forms and give a new proof of the surjectivity of the Borcherds lift based on the analysis of local Picard groups.","authors_text":"Manuel K.-H. M\\\"uller","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-07-01T11:51:28Z","title":"The basis problem for modular forms for the Weil representation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.01205","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":"sha256:582b85fb83786d6284432ac4377f6a460a8d1c02c2be1c2a6a6ac38eb23cfcd6","target":"record","created_at":"2026-07-05T09:23:07Z","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":"b5e6aec744dc901ff43849dc7dd210263f33b5b262d8bc3b2f2ea49352a3c3e9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-07-01T11:51:28Z","title_canon_sha256":"4c20b0bc202a672c7f77cfdb9d5f3006c3a7f06532213925f8138af55e94841f"},"schema_version":"1.0","source":{"id":"2407.01205","kind":"arxiv","version":2}},"canonical_sha256":"c8fe53b6fac862f73681289e772aacd167cc440b4ca73b3778486887c17a6db4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c8fe53b6fac862f73681289e772aacd167cc440b4ca73b3778486887c17a6db4","first_computed_at":"2026-07-05T09:23:07.744444Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:23:07.744444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JyInil0f3NJ0bFVQVXE7iYJhE5UFRxDL+GfweYDBOXxgod6+pt8JIpk/s1kWqRL91psfVGUQ0JpYeFJHz2YkDg==","signature_status":"signed_v1","signed_at":"2026-07-05T09:23:07.744918Z","signed_message":"canonical_sha256_bytes"},"source_id":"2407.01205","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:582b85fb83786d6284432ac4377f6a460a8d1c02c2be1c2a6a6ac38eb23cfcd6","sha256:1017bca71eed0f0a6313247461052897019ededd69c8bb647d2656010c748d6c"],"state_sha256":"7bbe63c0a4505d2c2cccdb150981a3066df67f5497d69a6dc96303283d23a338"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fpbqd+qW9z0T/gF92XGN454dxN/bQ5AKHsfmeU9PTMCNC4VeP/ebN4wGFTPrhoxYh3SGm93Z2bi5iqGdN0hZBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T15:45:59.689617Z","bundle_sha256":"4bdb5104c4d927530f86c77af91e5033227eb227224310ec366c8adcab430926"}}