{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:IJILZJWM2XSZLBFPG3R57EIJKW","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":"7dbb9c9f007f2b4b7e4d67a1d0d4fb3a6bf012e0655b7797006cf6c5bfcb29ea","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-02-23T12:23:39Z","title_canon_sha256":"60847f3e7036029554f833555aace945ab5d293f3d7e01829db0e40e9f7061ae"},"schema_version":"1.0","source":{"id":"2602.19771","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2602.19771","created_at":"2026-05-17T23:39:15Z"},{"alias_kind":"arxiv_version","alias_value":"2602.19771v3","created_at":"2026-05-17T23:39:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.19771","created_at":"2026-05-17T23:39:15Z"},{"alias_kind":"pith_short_12","alias_value":"IJILZJWM2XSZ","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"IJILZJWM2XSZLBFP","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"IJILZJWM","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:73a790f212da29fed2b5f787de6446abfdf351bfaa00f7ee07222562d81429eb","target":"graph","created_at":"2026-05-17T23:39:15Z","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":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"We show that the stacky Batyrev--Manin conjecture [DY24] holds for the naive height on X_0(N) when F=Q."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The listed values of N are precisely those for which the coarse moduli space of X_0(N) is isomorphic to P^1, and that the height in question is the naive height on the stack."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"The stacky Batyrev-Manin conjecture holds for naive heights on the modular stacks X_0(N) for N in {1,2,3,4,5,6,7,8,9,10,12,13,16,18,25} over Q."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"The stacky Batyrev-Manin conjecture holds for the naive height on the modular stacks X_0(N) over the rationals for N where the coarse space is the projective line."}],"snapshot_sha256":"d3e56cd5e82616afe48e40933b861a8cbfc6670c4c4cab6e8e560e53d7fc1d3b"},"formal_canon":{"evidence_count":2,"snapshot_sha256":"d973f1e40c12a12cef0a3f035a559f75bb21a0c05a89e202220aff2dbd5cd397"},"paper":{"abstract_excerpt":"Let $\\mathcal{X}_0(N)$ be the Deligne--Rapoport modular stack of elliptic curves endowed with a cyclic rational $N$-isogeny over a number field $F$. Let $N\\in\\{1,2,3,4,5,6,7,8,9,10,12,13,16,18,25\\},$ which are precisely the values for which the coarse moduli space of $\\mathcal{X}_0(N)$ is isomorphic to $\\mathbb{P}^1$. We show that the stacky Batyrev--Manin conjecture [DY24] holds for the naive height on $\\mathcal{X}_0(N)$ when $F=\\mathbb{Q}$. In the process, we give a concrete description of $\\mathcal{X}_0(N)$ as a square root stack over a stacky curve.","authors_text":"Changho Han, Ratko Darda","cross_cats":["math.AG"],"headline":"The stacky Batyrev-Manin conjecture holds for the naive height on the modular stacks X_0(N) over the rationals for N where the coarse space is the projective line.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-02-23T12:23:39Z","title":"The stacky Batyrev-Manin conjecture and modular curves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2602.19771","kind":"arxiv","version":3},"verdict":{"created_at":"2026-05-15T20:25:02.111981Z","id":"f3dbe265-af60-4777-9697-c94804a604e5","model_set":{"reader":"grok-4.3"},"one_line_summary":"The stacky Batyrev-Manin conjecture holds for naive heights on the modular stacks X_0(N) for N in {1,2,3,4,5,6,7,8,9,10,12,13,16,18,25} over Q.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"The stacky Batyrev-Manin conjecture holds for the naive height on the modular stacks X_0(N) over the rationals for N where the coarse space is the projective line.","strongest_claim":"We show that the stacky Batyrev--Manin conjecture [DY24] holds for the naive height on X_0(N) when F=Q.","weakest_assumption":"The listed values of N are precisely those for which the coarse moduli space of X_0(N) is isomorphic to P^1, and that the height in question is the naive height on the stack."}},"verdict_id":"f3dbe265-af60-4777-9697-c94804a604e5"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:bb5c23e838c4718813bb124de00dc2c56629c6414779e0b09411992cf7fb39cc","target":"record","created_at":"2026-05-17T23:39:15Z","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":"7dbb9c9f007f2b4b7e4d67a1d0d4fb3a6bf012e0655b7797006cf6c5bfcb29ea","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-02-23T12:23:39Z","title_canon_sha256":"60847f3e7036029554f833555aace945ab5d293f3d7e01829db0e40e9f7061ae"},"schema_version":"1.0","source":{"id":"2602.19771","kind":"arxiv","version":3}},"canonical_sha256":"4250bca6ccd5e59584af36e3df910955a02689ff972a9e733391a8f0be747975","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4250bca6ccd5e59584af36e3df910955a02689ff972a9e733391a8f0be747975","first_computed_at":"2026-05-17T23:39:15.987697Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:15.987697Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zu2tSZTfZobqaYu4CUInLkEvBTbngb60MEP/M+U6RONDAo000fCZX00x92nvvLHrJjv+8fn1CN7OqhPMflgLBQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:15.988288Z","signed_message":"canonical_sha256_bytes"},"source_id":"2602.19771","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bb5c23e838c4718813bb124de00dc2c56629c6414779e0b09411992cf7fb39cc","sha256:73a790f212da29fed2b5f787de6446abfdf351bfaa00f7ee07222562d81429eb"],"state_sha256":"8c9912b00f96a87db69c3c78d1714b7d51008034c850491b08e0a17520b63b6a"}