{"paper":{"title":"Generalized Howe curves of genus 4, 5, and 6 with completely decomposable Jacobians","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Generalized Howe curves with Jacobians splitting into four elliptic curves are superspecial for genus 4 in every prime between 20001 and 999999.","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Ryo Ohashi","submitted_at":"2026-04-20T10:42:50Z","abstract_excerpt":"Superspecial curves are important objects in number theory and algebraic geometry, and the existence in genus $g \\geq 4$ remains an open problem for all but finitely many characteristics $p > 0$. As a computational approach to this problem, Kudo-Harashita-Howe (2020) showed that a superspecial curve of genus 4 exists in each characteristic $p$ with $7 < p < 20000$. Their method restricted attention to a specific class of curves, known as Howe curves, for which superspeciality is reduced to those of curves of genus at most 2. In this paper, we focus on a more specific class of curves, namely Ho"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we confirmed by computer the existence of such superspecial curves of genus 4 in characteristics p with 20000 < p < 10^6. Using a similar approach, we also propose constructions of superspecial curves of genera 5 and 6 from only supersingular elliptic curves.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The superspeciality of the generalized Howe curve reduces exactly to the supersingularity of its four elliptic curve factors, and the computer enumeration correctly finds all such curves without implementation errors or missed cases.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Generalized Howe curves with completely decomposable Jacobians yield superspecial genus-4 curves for all primes 20000 < p < 10^6, plus constructions and existence checks for genera 5 and 6 up to p=10^5.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Generalized Howe curves with Jacobians splitting into four elliptic curves are superspecial for genus 4 in every prime between 20001 and 999999.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"00c72c982fa5c0978f820f65fa3dd52ce70818e0b13a7f7cf0045f788558ab0a"},"source":{"id":"2604.18074","kind":"arxiv","version":3},"verdict":{"id":"c5d4ad39-48d4-4d59-b081-04a61440d675","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T04:10:18.960411Z","strongest_claim":"we confirmed by computer the existence of such superspecial curves of genus 4 in characteristics p with 20000 < p < 10^6. Using a similar approach, we also propose constructions of superspecial curves of genera 5 and 6 from only supersingular elliptic curves.","one_line_summary":"Generalized Howe curves with completely decomposable Jacobians yield superspecial genus-4 curves for all primes 20000 < p < 10^6, plus constructions and existence checks for genera 5 and 6 up to p=10^5.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The superspeciality of the generalized Howe curve reduces exactly to the supersingularity of its four elliptic curve factors, and the computer enumeration correctly finds all such curves without implementation errors or missed cases.","pith_extraction_headline":"Generalized Howe curves with Jacobians splitting into four elliptic curves are superspecial for genus 4 in every prime between 20001 and 999999."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.18074/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-20T04:27:42.761012Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"ba27b9f7b01d323f6c595d311cf36529372128cc30f90b6406ca015e4b1d8e9d"},"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"}