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In case $r \\neq a-1$ and $b \\neq a+1$ we prove $C$ has no other point $Q \\neq P$ having Weierstrass semigroup equal to $<a;b>$. We say the Weierstrass semigroup $<a;b>$ occurs at most once. The curve $C_{a;b}$ has genus $(a-1)(b-1)/2$ and the result is generalized to genus $g<(a-1)(b-1)/2$. We obtain a lower bound on $g$ (sharp in many cases) such that all Weierstrass semigrou"},"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":"1708.04271","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-08-14T18:47:00Z","cross_cats_sorted":[],"title_canon_sha256":"1e6ee507f3ec40a06c23618e7a3e3c35ca622c85dcec11efd3c25eb08031a3a5","abstract_canon_sha256":"3c99fdeaf65b7d2f887247e77edf69193bd53f7e362c67423552bc19d681ee76"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:37:59.825164Z","signature_b64":"yJ9aITjGeIO/0HzJJhJvlNhpiVWibamFx4U+lYXzD8wbC9OwRbZCdO4cc/pPX80PAdDpGpsiiY36qkaq/thQAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"72e433b1735151c0581e83faeb52054bbac5c9c41c018d029aa5bae7c04e41a0","last_reissued_at":"2026-05-18T00:37:59.824568Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:37:59.824568Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The uniqueness of Weierstrass points with semigroup <a;b> and related subgroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Marc Coppens","submitted_at":"2017-08-14T18:47:00Z","abstract_excerpt":"Assume $a$ and $b=na+r$ with $n \\geq 1$ and $0<r<a$ are relatively prime integers. In case $C$ is a smooth curve and $P$ is a point on $C$ with Weierstrass semigroup equal to $<a;b>$ then $C$ is called a $C_{a;b}$-curve. In case $r \\neq a-1$ and $b \\neq a+1$ we prove $C$ has no other point $Q \\neq P$ having Weierstrass semigroup equal to $<a;b>$. We say the Weierstrass semigroup $<a;b>$ occurs at most once. The curve $C_{a;b}$ has genus $(a-1)(b-1)/2$ and the result is generalized to genus $g<(a-1)(b-1)/2$. We obtain a lower bound on $g$ (sharp in many cases) such that all Weierstrass semigrou"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.04271","kind":"arxiv","version":1},"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":"1708.04271","created_at":"2026-05-18T00:37:59.824658+00:00"},{"alias_kind":"arxiv_version","alias_value":"1708.04271v1","created_at":"2026-05-18T00:37:59.824658+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.04271","created_at":"2026-05-18T00:37:59.824658+00:00"},{"alias_kind":"pith_short_12","alias_value":"OLSDHMLTKFI4","created_at":"2026-05-18T12:31:34.259226+00:00"},{"alias_kind":"pith_short_16","alias_value":"OLSDHMLTKFI4AWA6","created_at":"2026-05-18T12:31:34.259226+00:00"},{"alias_kind":"pith_short_8","alias_value":"OLSDHMLT","created_at":"2026-05-18T12:31:34.259226+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/OLSDHMLTKFI4AWA6QP5OWUQFJO","json":"https://pith.science/pith/OLSDHMLTKFI4AWA6QP5OWUQFJO.json","graph_json":"https://pith.science/api/pith-number/OLSDHMLTKFI4AWA6QP5OWUQFJO/graph.json","events_json":"https://pith.science/api/pith-number/OLSDHMLTKFI4AWA6QP5OWUQFJO/events.json","paper":"https://pith.science/paper/OLSDHMLT"},"agent_actions":{"view_html":"https://pith.science/pith/OLSDHMLTKFI4AWA6QP5OWUQFJO","download_json":"https://pith.science/pith/OLSDHMLTKFI4AWA6QP5OWUQFJO.json","view_paper":"https://pith.science/paper/OLSDHMLT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1708.04271&json=true","fetch_graph":"https://pith.science/api/pith-number/OLSDHMLTKFI4AWA6QP5OWUQFJO/graph.json","fetch_events":"https://pith.science/api/pith-number/OLSDHMLTKFI4AWA6QP5OWUQFJO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OLSDHMLTKFI4AWA6QP5OWUQFJO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OLSDHMLTKFI4AWA6QP5OWUQFJO/action/storage_attestation","attest_author":"https://pith.science/pith/OLSDHMLTKFI4AWA6QP5OWUQFJO/action/author_attestation","sign_citation":"https://pith.science/pith/OLSDHMLTKFI4AWA6QP5OWUQFJO/action/citation_signature","submit_replication":"https://pith.science/pith/OLSDHMLTKFI4AWA6QP5OWUQFJO/action/replication_record"}},"created_at":"2026-05-18T00:37:59.824658+00:00","updated_at":"2026-05-18T00:37:59.824658+00:00"}