{"paper":{"title":"Divisibility Biases in the Orders of Elliptic Curve Reductions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Nara Sheen, Sung Min Lee","submitted_at":"2026-06-23T18:20:14Z","abstract_excerpt":"Let $E$ be an elliptic curve over the rationals. In 2004, Cojocaru proved, using the Chebotarev density theorem, that the set of primes $p \\leq x$ for which $m$ divides $\\#E_p(\\mathbb{F}_p)$ has a natural density. In 2009, Banks and Shparlinski proved an averaged version of this result over families of elliptic curves. In this article, we give a more explicit analysis of these densities. In particular, we show that, for Serre curves, the density of primes $p$ for which $m \\mid \\#E_p(\\mathbb{F}_p)$ is approximately $1/\\varphi(m)$, and is always greater than $1/m$ for every $m \\geq 2$. Thus, the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.25067","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.25067/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"}