{"paper":{"title":"Composite images of Galois for elliptic curves over $\\mathbf{Q}$ & Entanglement fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jackson S. Morrow","submitted_at":"2017-07-14T21:38:03Z","abstract_excerpt":"Let $E$ be an elliptic curve defined over $\\mathbf{Q}$ without complex multiplication. For each prime $\\ell$, there is a representation $\\rho_{E,\\ell}\\colon \\text{Gal}(\\overline{\\mathbf{Q}}/\\mathbf{Q}) \\to \\text{GL}_2(\\mathbf{F}_{\\ell})$ that describes the Galois action on the $\\ell$-torsion points of $E$. Building on recent work of Rouse--Zureick-Brown and Zywina, we find models for composite level modular curves whose rational points classify elliptic curves over $\\mathbf{Q}$ with simultaneously non-surjective, composite image of Galois. We also provably determine the rational points on almo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.04646","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"}