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Given a superelliptic curve $C/k$ of $\\ell$-power degree, we describe the field generated by the $\\ell$-power torsion of the Jacobian variety in terms of the branch set and reduction type of $C$ (and hence, in terms of data determined by a suitable affine model of $C$). If the Jacobian is good away from $\\ell$ and the branch set is defined over a pro-$\\ell$ extension of $k(\\mu_{\\ell^\\infty})$ unramified away from $\\ell$, then the $\\ell$-power torsion of the Jacobian is rational over the maximal such extension.\n  By decomposing the covering"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1803.08524","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2018-03-22T18:07:09Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"68e57283ef9fa9c86978639619ab1033f2ce4f0c22183231fd779ab3fd29e69a","abstract_canon_sha256":"7fae382ec4b7ed0eeb590a454901401ec27dd33beb65e1f4c0b08b24e2cb4877"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:21.437140Z","signature_b64":"r4ePYYAIoaoP+YWvGreGMUATufej7kqhRF+0sEsAO73Qup+79GK2zu8pHS/eET/eA/F7d5kgiB+KgSWcLmmrBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cdffad7c4744b54f5fe590048073d416a03506720dae4e57421a12732d939693","last_reissued_at":"2026-05-18T00:20:21.436595Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:21.436595Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cyclic covers and Ihara's Question","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Akio Tamagawa, Christopher Rasmussen","submitted_at":"2018-03-22T18:07:09Z","abstract_excerpt":"Let $\\ell$ be a rational prime and $k$ a number field. Given a superelliptic curve $C/k$ of $\\ell$-power degree, we describe the field generated by the $\\ell$-power torsion of the Jacobian variety in terms of the branch set and reduction type of $C$ (and hence, in terms of data determined by a suitable affine model of $C$). If the Jacobian is good away from $\\ell$ and the branch set is defined over a pro-$\\ell$ extension of $k(\\mu_{\\ell^\\infty})$ unramified away from $\\ell$, then the $\\ell$-power torsion of the Jacobian is rational over the maximal such extension.\n  By decomposing the covering"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.08524","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1803.08524","created_at":"2026-05-18T00:20:21.436673+00:00"},{"alias_kind":"arxiv_version","alias_value":"1803.08524v1","created_at":"2026-05-18T00:20:21.436673+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1803.08524","created_at":"2026-05-18T00:20:21.436673+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZX7227CHIS2U","created_at":"2026-05-18T12:33:07.085635+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZX7227CHIS2U6X7F","created_at":"2026-05-18T12:33:07.085635+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZX7227CH","created_at":"2026-05-18T12:33:07.085635+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZX7227CHIS2U6X7FSACIA46UC2","json":"https://pith.science/pith/ZX7227CHIS2U6X7FSACIA46UC2.json","graph_json":"https://pith.science/api/pith-number/ZX7227CHIS2U6X7FSACIA46UC2/graph.json","events_json":"https://pith.science/api/pith-number/ZX7227CHIS2U6X7FSACIA46UC2/events.json","paper":"https://pith.science/paper/ZX7227CH"},"agent_actions":{"view_html":"https://pith.science/pith/ZX7227CHIS2U6X7FSACIA46UC2","download_json":"https://pith.science/pith/ZX7227CHIS2U6X7FSACIA46UC2.json","view_paper":"https://pith.science/paper/ZX7227CH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1803.08524&json=true","fetch_graph":"https://pith.science/api/pith-number/ZX7227CHIS2U6X7FSACIA46UC2/graph.json","fetch_events":"https://pith.science/api/pith-number/ZX7227CHIS2U6X7FSACIA46UC2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZX7227CHIS2U6X7FSACIA46UC2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZX7227CHIS2U6X7FSACIA46UC2/action/storage_attestation","attest_author":"https://pith.science/pith/ZX7227CHIS2U6X7FSACIA46UC2/action/author_attestation","sign_citation":"https://pith.science/pith/ZX7227CHIS2U6X7FSACIA46UC2/action/citation_signature","submit_replication":"https://pith.science/pith/ZX7227CHIS2U6X7FSACIA46UC2/action/replication_record"}},"created_at":"2026-05-18T00:20:21.436673+00:00","updated_at":"2026-05-18T00:20:21.436673+00:00"}