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We prove that #X(K)<=#\\scrX(F_{\\pp})+2r, extending the refined version of the Chabauty-Coleman bound to the case of bad reduction. The new technical insight is to isolate variants of the classical rank of a divisor on a curve which are better suited for singular curves and which satisfy Clifford's theorem."},"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":"1204.3335","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-04-15T23:15:16Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"79c3ac3f360f768685a4cb4afa3f3628d13584d5e475a79e0ed1c1ddd158319b","abstract_canon_sha256":"d9bd32de183979148caa8299a84367ea2115e39b998d211dee2c2a6caed01257"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:35:28.247527Z","signature_b64":"yk8jThuzEVBCFfaAs7Aw4HPu4s6qkne1SAke9SwgQAKSn921tTww9eUeNLxTMTTkzk41rdASrj3SkDs5/AihBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c4e960019fa2bd069272519989c7330b7cdf5dec5bac8e957c3eebc237f7a595","last_reissued_at":"2026-05-18T03:35:28.246920Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:35:28.246920Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Chabauty-Coleman bound at a prime of bad reduction and clifford bounds for geometric rank functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"David Zureick-Brown, Eric Katz","submitted_at":"2012-04-15T23:15:16Z","abstract_excerpt":"Let X be a curve over a number field K with genus g>=2, $\\pp$ a prime of O_K over an unramified rational prime p>2r, J the Jacobian of X, r=rank J(K), and $\\scrX$ a regular proper model of X at $\\pp$. Suppose r<g. We prove that #X(K)<=#\\scrX(F_{\\pp})+2r, extending the refined version of the Chabauty-Coleman bound to the case of bad reduction. 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