{"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. The corresponding curves can be constructed using explicit CM constructions. 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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0811.3434","kind":"arxiv","version":3},"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"}