{"paper":{"title":"Prime form and sigma function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["nlin.SI"],"primary_cat":"math.AG","authors_text":"John Gibbons, Shigeki Matsutani, Yoshihiro Onishi","submitted_at":"2012-04-17T09:53:14Z","abstract_excerpt":"In this article, we study some cyclic $(r,s)$ curves $X$ given by $y^r =x^s + \\lambda_{1} x^{s-1} +...+ \\lambda_{s-1} x + \\lambda_s$. We give an expression for the prime form $\\cE(P,Q)$, where $(P, Q \\in X)$, in terms of the sigma function for some such curves, specifically any hyperelliptic curve $(r,s) = (2, 2g+1)$ as well as the cyclic trigonal curve $(r,s) = (3,4)$, $$ \\cE(P,Q) =\\frac{\\sigma_{\\natural_{r}}(u - v)}{\\sqrt{du_1}\\sqrt{d v_1}}, $$ where $\\natural_r$ is a certain index of differentials. Here $u_1$ and $v_1$ are respectively the first components of $u = w(P)$ and $v = w(Q)$ which"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.3747","kind":"arxiv","version":2},"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"}