{"paper":{"title":"On superspecial hyperelliptic curves of Rosenhain forms","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Ryo Ohashi","submitted_at":"2026-06-09T04:40:33Z","abstract_excerpt":"Any genus-$g$ hyperelliptic curve $C$ defined over an algebraically closed field of characteristic $p \\geq 3$ can be written in a Rosenhain form as $y^2 = x(x-1)\\prod_{i=1}^{2g-1}(x-\\lambda_i)$. In this paper, we first show that, if $C$ is superspecial, then each of $\\lambda_i,1-\\lambda_i$, and $\\lambda_i-\\lambda_j$ is a square in $\\mathbb{F}_{p^2}$. As an application, we propose a new algorithm for enumerating superspecial hyperelliptic curves in small characteristic. By implementing our algorithm, we successfully computed the number of isomorphism classes of such curves of genera $4$ and $5$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.10414","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.10414/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}