{"paper":{"title":"Using Indices of Points on an Elliptic Curve to Construct A Diophantine Model of $\\Z$ and Define $\\Z$ Using One Universal Quantifier in Very Large Subrings of Number Fields, Including $\\Q$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.LO"],"primary_cat":"math.NT","authors_text":"Alexandra Shlapentokh","submitted_at":"2009-01-27T03:53:06Z","abstract_excerpt":"Let $K$ be a number field and let $E$ be an elliptic curve defined and of rank one over $K$. For a set $\\calW_K$ of primes of $K$, let $O_{K,\\calW_K}=\\{x\\in K: \\ord_{\\pp}x \\geq 0, \\forall \\pp \\not \\in \\calW_K\\}$. Let $P \\in E(K)$ be a generator of $E(K)$ modulo the torsion subgroup. Let $(x_n(P),y_n(P))$ be the affine coordinates of $[n]P$ with respect to a fixed Weierstrass equation of $E$. We show that there exists a set $\\calW_K$ of primes of $K$ of natural density one such that in $O_{K,\\calW_K}$ multiplication of indices (with respect to some fixed multiple of $P$) is existentially defina"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.4168","kind":"arxiv","version":3},"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"}