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For $m\\geq 1,$ let $X_m$ be the curve $X$ over the finite field $\\mathbb{F}_m$, the $m$-th extension of $\\mathbb{F}.$ Let $K_n(m)$ be the $K$-group $K_n(X_m)$ of the smooth projective curve $X_m.$\n  In this paper, we study the structure of the groups $K_n(m).$ If $l$ is a prime, we establish an analogue of Iwasawa theorem in algebraic number theory for the orders of the $l$-primary part $K_n(l^m)\\{l\\}$ of $K_n(l^m)$. In particular, when $X$ is an elliptic curve $E$ defined over $\\mathbb{F},$ our method det"},"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":"1112.5920","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-12-27T04:24:22Z","cross_cats_sorted":[],"title_canon_sha256":"c7d6a12aafeacf108c1f026c8153de5b6689e0ed3fb14decb4bdca05a7bf44bc","abstract_canon_sha256":"a391f7fe759ada3d1730c72644ef1fc9639d75f0880a59b47ebee8212af9106a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:37.259596Z","signature_b64":"vfO6UXCglx2CcH0AaqRGOS5ZQKIxhpKOlYQnToeQuQ+79dLfuCktAr9RvDQ6r59QmUwnAqKh6ckyVEtCRZn0DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9e1a9362a89ac350b6d8ced576c76d5ab719dd9c0d54cf0b64aea0cb3e50b650","last_reissued_at":"2026-05-18T04:05:37.259015Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:37.259015Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Higher $K$-Groups of Smooth Projective Curves Over Finite Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hourong Qin, Qingzhong Ji","submitted_at":"2011-12-27T04:24:22Z","abstract_excerpt":"Let $X$ be a smooth projective curve over a finite field $\\mathbb{F}$ with $q$ elements. 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