{"paper":{"title":"The density of odd order reductions for elliptic curves with a rational point of order 2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jeremy Rouse, Ke Liang","submitted_at":"2018-10-24T19:22:35Z","abstract_excerpt":"Suppose that $E/\\mathbb{Q}$ is an elliptic curve with a rational point $T$ of order $2$ and $\\alpha \\in E(\\mathbb{Q})$ is a point of infinite order. We consider the problem of determining the density of primes $p$ for which $\\alpha \\in E(\\mathbb{F}_{p})$ has odd order. This density is determined by the image of the arboreal Galois representation $\\tau_{E,2^{k}} : {\\rm Gal}(\\overline{\\mathbb{Q}}/\\mathbb{Q}) \\to {\\rm AGL}_{2}(\\mathbb{Z}/2^{k}\\mathbb{Z})$. Assuming that $\\alpha$ is primitive (that is, neither $\\alpha$ nor $\\alpha + T$ is twice a point over $\\mathbb{Q}$) and that the image of the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.10583","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"}