{"paper":{"title":"Patterns on elliptic curves beyond Bremner's conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hector Pasten, Natalia Garcia-Fritz","submitted_at":"2026-05-14T15:29:23Z","abstract_excerpt":"In the late 1990's, Bremner conjectured that long arithmetic progressions among the $x$-coordinates of rational points of an elliptic curve $E$ over $\\mathbb{Q}$ should force the rank of $E$ to be large. This conjecture (and a broad generalization of it) was proved by the authors two decades later, by combining Nevanlinna theory and the Uniform Mordell--Lang theorem of Gao--Ge--K\\\"uhne. The proof inspired subsequent work by the authors where a generalization of the Bogomolov--Fu--Tschinkel conjecture was proved by similar means. In this note we isolate a flexible pattern principle implicit in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.14962","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"}