{"paper":{"title":"On the possible images of the mod ell representations associated to elliptic curves over Q","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"David Zywina","submitted_at":"2015-08-31T01:28:58Z","abstract_excerpt":"Consider a non-CM elliptic curve $E$ defined over $\\mathbb{Q}$. For each prime $\\ell$, there is a representation $\\rho_{E,\\ell}: G \\to GL_2(\\mathbb{F}_\\ell)$ that describes the Galois action on the $\\ell$-torsion points of $E$, where $G$ is the absolute Galois group of $\\mathbb{Q}$. A famous theorem of Serre says that $\\rho_{E,\\ell}$ is surjective for all large enough $\\ell$. We will describe all known, and conjecturally all, pairs $(E,\\ell)$ such that $\\rho_{E,\\ell}$ is not surjective. Together with another paper, this produces an algorithm that given an elliptic curve $E/\\mathbb{Q}$, outputs"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.07660","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":""},"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"}