{"paper":{"title":"On the Surjectivity of Galois Representations Associated to Elliptic Curves over Number Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dmitry Vaintrob, Eric Larson","submitted_at":"2012-03-30T23:28:16Z","abstract_excerpt":"Given an elliptic curve $E$ over a number field $K$, the $\\ell$-torsion points $E[\\ell]$ of $E$ define a Galois representation $\\gal(\\bar{K}/K) \\to \\gl_2(\\ff_\\ell)$. A famous theorem of Serre states that as long as $E$ has no Complex Multiplication (CM), the map $\\gal(\\bar{K}/K) \\to \\gl_2(\\ff_\\ell)$ is surjective for all but finitely many $\\ell$.\n  We say that a prime number $\\ell$ is exceptional (relative to the pair $(E,K)$) if this map is not surjective. Here we give a new bound on the largest exceptional prime, as well as on the product of all exceptional primes of $E$. We show in particul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.0046","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"}