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We determine the group of fixed points of the cyclic group $\\mathbf{mu}_N\\cong \\mathbb{Z}/N\\mathbb{Z}$ acting on the Jacobian $J_N:=\\Jac(C_{f,N})$. For each $\\ell$ distinct from the characteristic of the base field, the Tate module $T_\\ell J_N$ is shown to be a free module over the ring $\\mathbb{Z}_\\ell[T]/(\\sum_{i=0}^{N-1}T^i)$. We also calculate the degree of the induced polarization on the new part $J_N^{new}$ of the Jacobian."},"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":"1309.0295","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-09-02T03:22:08Z","cross_cats_sorted":[],"title_canon_sha256":"afbdc510a9beb85244cf016ecdbcd54153d1e86f581d7cc4769915b941938988","abstract_canon_sha256":"3762ae0965ea1ab9cb3d8a94167ee060e80c9460b055c1693018ef1f561c930b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:35.384441Z","signature_b64":"aVBhlaiVxlyUGwNEE43sFaB/8GoAlNBnBlUDXlTZAxNGAfkLeEHhopiy6DN2u6zEtv0cu9r9AGZoZKis74ajBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9f2daf0cfd7553a91214c3308659195db32923d542e1be7e93507a3af59f9e66","last_reissued_at":"2026-05-18T03:13:35.383757Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:35.383757Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fixed points and homology of superelliptic Jacobians","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Chia-Fu Yu, Haining Wang, Jiangwei Xue","submitted_at":"2013-09-02T03:22:08Z","abstract_excerpt":"Let $\\eta: C_{f,N}\\to \\mathbb{P}^1$ be a cyclic cover of $\\mathbb{P}^1$ of degree $N$ which is totally and tamely ramified for all the ramification points. We determine the group of fixed points of the cyclic group $\\mathbf{mu}_N\\cong \\mathbb{Z}/N\\mathbb{Z}$ acting on the Jacobian $J_N:=\\Jac(C_{f,N})$. For each $\\ell$ distinct from the characteristic of the base field, the Tate module $T_\\ell J_N$ is shown to be a free module over the ring $\\mathbb{Z}_\\ell[T]/(\\sum_{i=0}^{N-1}T^i)$. We also calculate the degree of the induced polarization on the new part $J_N^{new}$ of the Jacobian."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0295","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":""},"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":"1309.0295","created_at":"2026-05-18T03:13:35.383867+00:00"},{"alias_kind":"arxiv_version","alias_value":"1309.0295v2","created_at":"2026-05-18T03:13:35.383867+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.0295","created_at":"2026-05-18T03:13:35.383867+00:00"},{"alias_kind":"pith_short_12","alias_value":"T4W26DH5OVJ2","created_at":"2026-05-18T12:27:59.945178+00:00"},{"alias_kind":"pith_short_16","alias_value":"T4W26DH5OVJ2SEQU","created_at":"2026-05-18T12:27:59.945178+00:00"},{"alias_kind":"pith_short_8","alias_value":"T4W26DH5","created_at":"2026-05-18T12:27:59.945178+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/T4W26DH5OVJ2SEQUYMYIMWIZLW","json":"https://pith.science/pith/T4W26DH5OVJ2SEQUYMYIMWIZLW.json","graph_json":"https://pith.science/api/pith-number/T4W26DH5OVJ2SEQUYMYIMWIZLW/graph.json","events_json":"https://pith.science/api/pith-number/T4W26DH5OVJ2SEQUYMYIMWIZLW/events.json","paper":"https://pith.science/paper/T4W26DH5"},"agent_actions":{"view_html":"https://pith.science/pith/T4W26DH5OVJ2SEQUYMYIMWIZLW","download_json":"https://pith.science/pith/T4W26DH5OVJ2SEQUYMYIMWIZLW.json","view_paper":"https://pith.science/paper/T4W26DH5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1309.0295&json=true","fetch_graph":"https://pith.science/api/pith-number/T4W26DH5OVJ2SEQUYMYIMWIZLW/graph.json","fetch_events":"https://pith.science/api/pith-number/T4W26DH5OVJ2SEQUYMYIMWIZLW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/T4W26DH5OVJ2SEQUYMYIMWIZLW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/T4W26DH5OVJ2SEQUYMYIMWIZLW/action/storage_attestation","attest_author":"https://pith.science/pith/T4W26DH5OVJ2SEQUYMYIMWIZLW/action/author_attestation","sign_citation":"https://pith.science/pith/T4W26DH5OVJ2SEQUYMYIMWIZLW/action/citation_signature","submit_replication":"https://pith.science/pith/T4W26DH5OVJ2SEQUYMYIMWIZLW/action/replication_record"}},"created_at":"2026-05-18T03:13:35.383867+00:00","updated_at":"2026-05-18T03:13:35.383867+00:00"}