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In this paper we classify the $\\text{PGL}_3(K)$-conjugacy classes of semi-stable dominant degree $2$ rational maps $f:{\\mathbb P}^2_K\\dashrightarrow{\\mathbb P}^2_K$ whose automorphism group $$\\text{Aut}(f):=\\{\\phi\\in\\text{PGL}_3(K): \\phi^{-1}\\circ f\\circ\\phi=f\\}$$ is finite and of order at least $3$. In particular, we prove that $\\#\\text{Aut}(f)\\le24$ in general, that $\\#\\text{Aut}(f)\\le21$ for morphisms, and that $\\#\\text{Aut}(f)\\le6$ for all but finitely many conjugacy classes of $f$."},"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":"1607.05772","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2016-07-19T22:17:53Z","cross_cats_sorted":["math.DS","math.NT"],"title_canon_sha256":"87fcc268dda5aafc1ed4bf51604c789ecf9fe9e8c2ac574bb7b2bb79831a0309","abstract_canon_sha256":"3b3bad7bded84293410b06412c6e126206fabf0ee62c027e64161f8a42a2a870"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:41.261159Z","signature_b64":"pPIfOshTusrhk1ul+R9YZPj3xDjJmlQa0Snzq5FlKBk6wf3tXAu0Mb26JQMw9RmwXa1uOG5MJqqJs7wOBg48BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b653fbdc0e44d732eb17468dc6fb7e8ba8f23d12066bac7ad7d40e87f8c14e80","last_reissued_at":"2026-05-18T00:36:41.260488Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:41.260488Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A classification of degree $2$ semi-stable rational maps $\\mathbb{P}^2\\to\\mathbb{P}^2$ with large finite dynamical automorphism group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.NT"],"primary_cat":"math.AG","authors_text":"Joseph H. Silverman, Michelle Manes","submitted_at":"2016-07-19T22:17:53Z","abstract_excerpt":"Let $K$ be an algebraically closed field of characteristic $0$. In this paper we classify the $\\text{PGL}_3(K)$-conjugacy classes of semi-stable dominant degree $2$ rational maps $f:{\\mathbb P}^2_K\\dashrightarrow{\\mathbb P}^2_K$ whose automorphism group $$\\text{Aut}(f):=\\{\\phi\\in\\text{PGL}_3(K): \\phi^{-1}\\circ f\\circ\\phi=f\\}$$ is finite and of order at least $3$. In particular, we prove that $\\#\\text{Aut}(f)\\le24$ in general, that $\\#\\text{Aut}(f)\\le21$ for morphisms, and that $\\#\\text{Aut}(f)\\le6$ for all but finitely many conjugacy classes of $f$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.05772","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":"1607.05772","created_at":"2026-05-18T00:36:41.260596+00:00"},{"alias_kind":"arxiv_version","alias_value":"1607.05772v2","created_at":"2026-05-18T00:36:41.260596+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.05772","created_at":"2026-05-18T00:36:41.260596+00:00"},{"alias_kind":"pith_short_12","alias_value":"WZJ7XXAOITLT","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_16","alias_value":"WZJ7XXAOITLTF2YX","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_8","alias_value":"WZJ7XXAO","created_at":"2026-05-18T12:30:51.357362+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/WZJ7XXAOITLTF2YXI2G4N636RO","json":"https://pith.science/pith/WZJ7XXAOITLTF2YXI2G4N636RO.json","graph_json":"https://pith.science/api/pith-number/WZJ7XXAOITLTF2YXI2G4N636RO/graph.json","events_json":"https://pith.science/api/pith-number/WZJ7XXAOITLTF2YXI2G4N636RO/events.json","paper":"https://pith.science/paper/WZJ7XXAO"},"agent_actions":{"view_html":"https://pith.science/pith/WZJ7XXAOITLTF2YXI2G4N636RO","download_json":"https://pith.science/pith/WZJ7XXAOITLTF2YXI2G4N636RO.json","view_paper":"https://pith.science/paper/WZJ7XXAO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1607.05772&json=true","fetch_graph":"https://pith.science/api/pith-number/WZJ7XXAOITLTF2YXI2G4N636RO/graph.json","fetch_events":"https://pith.science/api/pith-number/WZJ7XXAOITLTF2YXI2G4N636RO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WZJ7XXAOITLTF2YXI2G4N636RO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WZJ7XXAOITLTF2YXI2G4N636RO/action/storage_attestation","attest_author":"https://pith.science/pith/WZJ7XXAOITLTF2YXI2G4N636RO/action/author_attestation","sign_citation":"https://pith.science/pith/WZJ7XXAOITLTF2YXI2G4N636RO/action/citation_signature","submit_replication":"https://pith.science/pith/WZJ7XXAOITLTF2YXI2G4N636RO/action/replication_record"}},"created_at":"2026-05-18T00:36:41.260596+00:00","updated_at":"2026-05-18T00:36:41.260596+00:00"}