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In one of our algorithms, the group of $\\F_{p^2}$-valued points of the Jacobian has prime order, while another allows for a prescribed embedding degree with respect to a subgroup of prescribed order. The curves are defined over $\\F_{p^2}$ out of necessity: we show that curves of $p$-rank 1 over $\\F_p$ for large $p$ cannot be efficiently constructed using explicit CM constructions."},"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":"0811.3434","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2008-11-20T22:14:24Z","cross_cats_sorted":[],"title_canon_sha256":"01d8f3479108fe768ef22870280d195708a8546b1a58b478e533528bae4d1bc6","abstract_canon_sha256":"8c2f992f971d0b0945f0d14854fab36107f6f7b37fbca07f6b8a80ab4a74920f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:27.534458Z","signature_b64":"QI2i254mkVGCVWWZlaQKp/9WC8aPuWo+A/S8IcQQxEPIjCXNYR35MQMVLcP1XlGy2hd+zg3OWRp1LHT8yBA7Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2a7d84c93e4896ab3407c2a9b7e5a5fd3856303c52417f9add27ce26b90a003f","last_reissued_at":"2026-05-17T23:54:27.533765Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:27.533765Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A CM construction for curves of genus 2 with p-rank 1","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Gary McGuire, Laura Hitt O'Connor, Marco Streng, Michael Naehrig","submitted_at":"2008-11-20T22:14:24Z","abstract_excerpt":"We construct Weil numbers corresponding to genus-2 curves with $p$-rank 1 over the finite field $\\F_{p^2}$ of $p^2$ elements. 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