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These maps arise by Hilbert-90 descent from the trace-zero maps $X^{dq}-X^d$ on $\\ker\\operatorname{Tr}_{\\mathbb{F}_{q^3}/\\mathbb{F}_q}$, but the principal object is the resulting $\\tau$-equivariant quotient-map family; nonconstant separable members are viewed as covers.\n  We prove that cancellation is exactly a torsion-defect phenomenon. If $\\ell(-)$ denotes scheme-theoretic length and $\\boldsymbol{\\mu}_d=\\ker([d]:\\m"},"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":"2605.25291","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-24T23:12:06Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"48c7fae777444dd875f48a76e4418633084ff2536d8e6f84d3f6f5390a0922c6","abstract_canon_sha256":"b8d19c0a621642b6938f26bb4425f4fd5ac9d5230933e3019fcb4327d449b772"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T02:04:27.522570Z","signature_b64":"MxIOiP9P0XX8KWX20XgIum5FBNG8zKub0iNK1CnLEk65N1bNApB8/8iAbrUEuNkp/CVqtA05C3uF7vynvnwKAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5241c8029f831105de2a9270bd5b7c882d55db1169abde10316dcbf617604ce9","last_reissued_at":"2026-05-26T02:04:27.521836Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T02:04:27.521836Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hilbert-90 quotient maps, torsion defects, and symmetric monodromy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Henry Shin","submitted_at":"2026-05-24T23:12:06Z","abstract_excerpt":"Let $\\tau(z)=-1-z^{-1}$. We study the reduced rational maps $h_d:\\mathbb{P}^1\\to\\mathbb{P}^1$ obtained by cancelling common factors in $H_d^{\\rm raw}(z)=z^d(\\tau(z)^d-1)/(z^d-1)$. These maps arise by Hilbert-90 descent from the trace-zero maps $X^{dq}-X^d$ on $\\ker\\operatorname{Tr}_{\\mathbb{F}_{q^3}/\\mathbb{F}_q}$, but the principal object is the resulting $\\tau$-equivariant quotient-map family; nonconstant separable members are viewed as covers.\n  We prove that cancellation is exactly a torsion-defect phenomenon. 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