{"paper":{"title":"Hilbert's 10th Problem via Mordell curves","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Debanjana Kundu, Dipramit Majumdar, Somnath Jha","submitted_at":"2024-12-05T15:33:54Z","abstract_excerpt":"We show that for $5/6$-th of all primes $p$, Hilbert's 10-th Problem is unsolvable for $\\mathbb{Q}(\\zeta_3, \\sqrt[3]{p})$. We also show that there is an infinite set $S$ of square free integers such tha Hilbert's 10-th Problem is unsolvable over the number fields $\\mathbb{Q}(\\zeta_3, \\sqrt{D}, \\sqrt[3]{p})$ for every $D \\in S$ and every prime $p \\equiv 2,5 \\pmod{9}$. We use the CM elliptic curves $Y^2=X^3-432D^2$ associated to the cube sum problem, with $D$ varying in suitable congruence class, in our proof."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.04253","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2412.04253/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"}