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We prove that there are only finitely many isomorphism classes of continuous geometric semisimple representations $\\rho:G_{K}\\to \\mathrm{GL}_{n}(\\overline{\\mathbb{F}}_{p})$ such that their Artin conductors are bounded. It is worth emphasizing that we do not need to assume that $p$ does not divide $n$."},"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":"2606.29277","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-28T08:49:03Z","cross_cats_sorted":[],"title_canon_sha256":"681f4eb47e270e66beca13590163ccb377af29602f7a34c3b32bd4b55742b40c","abstract_canon_sha256":"f8d3357e9fefe47b92a0597bcb823f8e0ee8797627b97f752bdba2a985050f07"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-30T01:17:59.992978Z","signature_b64":"BnYiKeNpWw7rC6Kq+JQjXu3MTMQZQuEdZB/HetfBkecYbsDqeiy4St9vOb0x/zHs2Zai0o/6xza6xg73XfLmCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b1f8f22ff34d233d3a32650bbc469fb286069595e6ff8c63f0ae599d65331125","last_reissued_at":"2026-06-30T01:17:59.992546Z","signature_status":"signed_v1","first_computed_at":"2026-06-30T01:17:59.992546Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A finiteness theorem for mod $p$ Galois representations over global function fields","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Yufan Luo","submitted_at":"2026-06-28T08:49:03Z","abstract_excerpt":"Let $p$ be an odd prime number and let $\\overline{\\mathbb{F}}_p$ be a fixed algebraic closure of the finite field of order $p$. Let $K$ be a global function field of characteristic different from $p$ and let $G_{K}$ be the absolute Galois group of $K$. We prove that there are only finitely many isomorphism classes of continuous geometric semisimple representations $\\rho:G_{K}\\to \\mathrm{GL}_{n}(\\overline{\\mathbb{F}}_{p})$ such that their Artin conductors are bounded. 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