{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:R36WZ7CT3NTZOJJPDAWUS6BFH2","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":"52c8c2205d048ddfcaff7b7c43fc1a1c5ef85476bde8fb0ea7eeed04217f1819","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-10T09:41:42Z","title_canon_sha256":"0984900556a52e8ccca9d34e0df3c4520547ef91d4cf6f46f1343a89752074e8"},"schema_version":"1.0","source":{"id":"1109.2207","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.2207","created_at":"2026-05-18T03:54:47Z"},{"alias_kind":"arxiv_version","alias_value":"1109.2207v2","created_at":"2026-05-18T03:54:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.2207","created_at":"2026-05-18T03:54:47Z"},{"alias_kind":"pith_short_12","alias_value":"R36WZ7CT3NTZ","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_16","alias_value":"R36WZ7CT3NTZOJJP","created_at":"2026-05-18T12:26:39Z"},{"alias_kind":"pith_short_8","alias_value":"R36WZ7CT","created_at":"2026-05-18T12:26:39Z"}],"graph_snapshots":[{"event_id":"sha256:0ceb127ac08b22fcf4fd2d7dc6e5dda3d918fb02cf43d4b27bb78c8ec949b664","target":"graph","created_at":"2026-05-18T03:54:47Z","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 study the number of elliptic curves, up to isomorphism, over a fixed quartic field $K$ having a prescribed torsion group $T$ as a subgroup. Let $T=\\Z/m\\Z \\oplus \\Z/n\\Z$, where $m|n$, be a torsion group such that the modular curve $X_1(m,n)$ is an elliptic curve. Let $K$ be a number field such that there is a positive and finite number of elliptic curves $E_T$ over $K$ having $T$ as a subgroup. We call such pairs $(E_T, K)$ \\emph{exceptional}. It is known that there are only finitely many exceptional pairs when $K$ varies through all quadratic or cubic fields. We prove that when $K$ varies t","authors_text":"Filip Najman","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-10T09:41:42Z","title":"Exceptional elliptic curves over quartic fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.2207","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:d3910d32cd42ad760412cba4c8c1583220cbc077b1f6b65175d7abd9abded9cf","target":"record","created_at":"2026-05-18T03:54:47Z","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":"52c8c2205d048ddfcaff7b7c43fc1a1c5ef85476bde8fb0ea7eeed04217f1819","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-10T09:41:42Z","title_canon_sha256":"0984900556a52e8ccca9d34e0df3c4520547ef91d4cf6f46f1343a89752074e8"},"schema_version":"1.0","source":{"id":"1109.2207","kind":"arxiv","version":2}},"canonical_sha256":"8efd6cfc53db6797252f182d4978253ea24f5759d10e3620ac617f9c46bf3d74","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8efd6cfc53db6797252f182d4978253ea24f5759d10e3620ac617f9c46bf3d74","first_computed_at":"2026-05-18T03:54:47.878812Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:54:47.878812Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4rlSqQrLQhFfg2AKb04nYjUqS3KsRb534gDrdcqXK6wungjlRpfu1zbPW4OY6bZWgYTlVwpieX0L5hSHpipOAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:54:47.879527Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.2207","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d3910d32cd42ad760412cba4c8c1583220cbc077b1f6b65175d7abd9abded9cf","sha256:0ceb127ac08b22fcf4fd2d7dc6e5dda3d918fb02cf43d4b27bb78c8ec949b664"],"state_sha256":"36c7b21161003dd21b90ae2fff8171ce7c2e5c9405727a04dbc365b0218b135e"}