{"paper":{"title":"Effective bounds for adelic Galois representations attached to elliptic curves over the rationals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Lorenzo Furio","submitted_at":"2024-12-13T18:37:45Z","abstract_excerpt":"Given an elliptic curve $E$ defined over $\\mathbb{Q}$ without complex multiplication, we provide an explicit sharp bound on the index of the image of the adelic representation $\\rho_E$. In particular, if $\\operatorname{h}_{\\mathcal{F}}(E)$ is the stable Faltings height of $E$, we show that $[\\operatorname{GL}_2(\\widehat{\\mathbb{Z}}) : \\operatorname{Im}\\rho_E]$ is bounded above by $10^{21} (\\operatorname{h}_{\\mathcal{F}}(E)+40)^{4.42}$, and, for $\\operatorname{h}_{\\mathcal{F}}(E)$ tending to infinity, by $\\operatorname{h}_{\\mathcal{F}}(E)^{3+o(1)}$. We also classify the possible (conjecturally "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.10340","kind":"arxiv","version":4},"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/2412.10340/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"}