{"paper":{"title":"Sums of rational cubes and the $3$-Selmer group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexander Smith, Peter Koymans","submitted_at":"2024-05-15T13:11:13Z","abstract_excerpt":"Recently, Alp\\\"oge-Bhargava-Shnidman determined the average size of the $2$-Selmer group in the cubic twist family of any elliptic curve over $\\mathbb{Q}$ with $j$-invariant $0$. We obtain the distribution of the $3$-Selmer groups in the same family. As a consequence, we improve their upper bound on the density of integers expressible as a sum of two rational cubes. Assuming a $3$-converse theorem, we also improve their lower bound on this density.\n  The $\\sqrt{-3}$-Selmer group in this cubic twist family is well-known to be large, which poses significant challenges to the methods previously d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.09311","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/2405.09311/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"}