{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:DC5V6CWUBVMZCDEHDJO3NSW3ET","short_pith_number":"pith:DC5V6CWU","canonical_record":{"source":{"id":"2108.09311","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-08-20T18:00:01Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"d1bab22236ddcd78e6405cc3b9abcb74360117c0350814429314fdfc4c9b46b1","abstract_canon_sha256":"97af5a5c5bbf52bc131f4b472c9dfa970bb2faf57a8ff660f946bc2c7ef57c50"},"schema_version":"1.0"},"canonical_sha256":"18bb5f0ad40d59910c871a5db6cadb24d39866db70e1bfe453d524d9f1c7d3dc","source":{"kind":"arxiv","id":"2108.09311","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2108.09311","created_at":"2026-07-05T03:57:11Z"},{"alias_kind":"arxiv_version","alias_value":"2108.09311v1","created_at":"2026-07-05T03:57:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.09311","created_at":"2026-07-05T03:57:11Z"},{"alias_kind":"pith_short_12","alias_value":"DC5V6CWUBVMZ","created_at":"2026-07-05T03:57:11Z"},{"alias_kind":"pith_short_16","alias_value":"DC5V6CWUBVMZCDEH","created_at":"2026-07-05T03:57:11Z"},{"alias_kind":"pith_short_8","alias_value":"DC5V6CWU","created_at":"2026-07-05T03:57:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:DC5V6CWUBVMZCDEHDJO3NSW3ET","target":"record","payload":{"canonical_record":{"source":{"id":"2108.09311","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-08-20T18:00:01Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"d1bab22236ddcd78e6405cc3b9abcb74360117c0350814429314fdfc4c9b46b1","abstract_canon_sha256":"97af5a5c5bbf52bc131f4b472c9dfa970bb2faf57a8ff660f946bc2c7ef57c50"},"schema_version":"1.0"},"canonical_sha256":"18bb5f0ad40d59910c871a5db6cadb24d39866db70e1bfe453d524d9f1c7d3dc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:57:11.064140Z","signature_b64":"1IyKKr+qA3EbGHSTHnByQbb4vfrcs46sAzzd9gnZEKY53BDZukBkaUNPd8nOfJkVUAcX9v31YjPuJ163U1pNCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"18bb5f0ad40d59910c871a5db6cadb24d39866db70e1bfe453d524d9f1c7d3dc","last_reissued_at":"2026-07-05T03:57:11.063734Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:57:11.063734Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2108.09311","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":"2026-07-05T03:57:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nwFBthNCz5Duf9nbjd1ywyI+flxI5iwe6iUsXgBtQhEa3bXE3i2SxX+W3vmzLslMzZXIWu8waBbEE/nWA2f5BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T15:07:38.019477Z"},"content_sha256":"05152b076594474ac990e32a6e84423dc045b5f1e8fa8800c52fc48dcdda8e4f","schema_version":"1.0","event_id":"sha256:05152b076594474ac990e32a6e84423dc045b5f1e8fa8800c52fc48dcdda8e4f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:DC5V6CWUBVMZCDEHDJO3NSW3ET","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"hep-th","authors_text":"Thorsten Schimannek","submitted_at":"2021-08-20T18:00:01Z","abstract_excerpt":"We show that the stringy K\\\"ahler moduli space of a generic genus one curve of degree $N$, for $N\\le 5$, is the $\\Gamma_1(N)$ modular curve $X_1(N)$. This implies a correspondence between the cusps of the modular curves and certain large volume limits in the stringy K\\\"ahler moduli spaces of genus one fibered Calabi-Yau manifolds with $N$-sections. Using Higgs transitions in M-theory and F-theory as well as modular properties of the topological string partition function, we identify these large volume limits with elements of the Tate-Shafarevich group of the genus one fibration. Singular eleme"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.09311","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2108.09311/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":"2026-07-05T03:57:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/oCQ55E5R0LueMRR/ZOaX3edliQ8VE2IJxJeKxQXFUb0j6215aJaK60zIid094Mddjew0o7PrOUERgreTUfGCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T15:07:38.019978Z"},"content_sha256":"d3fc0b311df90dfda5362d22da8400a4a009236ba9bdfe17dd52934ac3986651","schema_version":"1.0","event_id":"sha256:d3fc0b311df90dfda5362d22da8400a4a009236ba9bdfe17dd52934ac3986651"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DC5V6CWUBVMZCDEHDJO3NSW3ET/bundle.json","state_url":"https://pith.science/pith/DC5V6CWUBVMZCDEHDJO3NSW3ET/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DC5V6CWUBVMZCDEHDJO3NSW3ET/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":"2026-08-16T15:07:38Z","links":{"resolver":"https://pith.science/pith/DC5V6CWUBVMZCDEHDJO3NSW3ET","bundle":"https://pith.science/pith/DC5V6CWUBVMZCDEHDJO3NSW3ET/bundle.json","state":"https://pith.science/pith/DC5V6CWUBVMZCDEHDJO3NSW3ET/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DC5V6CWUBVMZCDEHDJO3NSW3ET/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:DC5V6CWUBVMZCDEHDJO3NSW3ET","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":"97af5a5c5bbf52bc131f4b472c9dfa970bb2faf57a8ff660f946bc2c7ef57c50","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-08-20T18:00:01Z","title_canon_sha256":"d1bab22236ddcd78e6405cc3b9abcb74360117c0350814429314fdfc4c9b46b1"},"schema_version":"1.0","source":{"id":"2108.09311","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2108.09311","created_at":"2026-07-05T03:57:11Z"},{"alias_kind":"arxiv_version","alias_value":"2108.09311v1","created_at":"2026-07-05T03:57:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.09311","created_at":"2026-07-05T03:57:11Z"},{"alias_kind":"pith_short_12","alias_value":"DC5V6CWUBVMZ","created_at":"2026-07-05T03:57:11Z"},{"alias_kind":"pith_short_16","alias_value":"DC5V6CWUBVMZCDEH","created_at":"2026-07-05T03:57:11Z"},{"alias_kind":"pith_short_8","alias_value":"DC5V6CWU","created_at":"2026-07-05T03:57:11Z"}],"graph_snapshots":[{"event_id":"sha256:d3fc0b311df90dfda5362d22da8400a4a009236ba9bdfe17dd52934ac3986651","target":"graph","created_at":"2026-07-05T03:57:11Z","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/2108.09311/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show that the stringy K\\\"ahler moduli space of a generic genus one curve of degree $N$, for $N\\le 5$, is the $\\Gamma_1(N)$ modular curve $X_1(N)$. This implies a correspondence between the cusps of the modular curves and certain large volume limits in the stringy K\\\"ahler moduli spaces of genus one fibered Calabi-Yau manifolds with $N$-sections. Using Higgs transitions in M-theory and F-theory as well as modular properties of the topological string partition function, we identify these large volume limits with elements of the Tate-Shafarevich group of the genus one fibration. Singular eleme","authors_text":"Thorsten Schimannek","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-08-20T18:00:01Z","title":"Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.09311","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":"sha256:05152b076594474ac990e32a6e84423dc045b5f1e8fa8800c52fc48dcdda8e4f","target":"record","created_at":"2026-07-05T03:57:11Z","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":"97af5a5c5bbf52bc131f4b472c9dfa970bb2faf57a8ff660f946bc2c7ef57c50","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"hep-th","submitted_at":"2021-08-20T18:00:01Z","title_canon_sha256":"d1bab22236ddcd78e6405cc3b9abcb74360117c0350814429314fdfc4c9b46b1"},"schema_version":"1.0","source":{"id":"2108.09311","kind":"arxiv","version":1}},"canonical_sha256":"18bb5f0ad40d59910c871a5db6cadb24d39866db70e1bfe453d524d9f1c7d3dc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"18bb5f0ad40d59910c871a5db6cadb24d39866db70e1bfe453d524d9f1c7d3dc","first_computed_at":"2026-07-05T03:57:11.063734Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:57:11.063734Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1IyKKr+qA3EbGHSTHnByQbb4vfrcs46sAzzd9gnZEKY53BDZukBkaUNPd8nOfJkVUAcX9v31YjPuJ163U1pNCA==","signature_status":"signed_v1","signed_at":"2026-07-05T03:57:11.064140Z","signed_message":"canonical_sha256_bytes"},"source_id":"2108.09311","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:05152b076594474ac990e32a6e84423dc045b5f1e8fa8800c52fc48dcdda8e4f","sha256:d3fc0b311df90dfda5362d22da8400a4a009236ba9bdfe17dd52934ac3986651"],"state_sha256":"e0dd0d4af214357a3888460c47d0b6836b90b12c3bbc6e47089ddce7d3f750aa"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"osLWgpHnb0FtSDYMfBvYJd0THSH0losTJ37uK0Ka1uj4igCZMkKT22wo0br1NtOvXJ1cio6twKRT2M5Hdoa9AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T15:07:38.023933Z","bundle_sha256":"c44be5839942009149d0dd5d7b2d3a359a246cb249952000a2dd64764828cf66"}}