{"paper":{"title":"A note on Bremner's conjecture and uniformity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"If ranks of elliptic curves over the rationals are uniformly bounded, then arithmetic progressions of their rational x-coordinates have uniformly bounded lengths.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Hector Pasten, Natalia Garcia-Fritz","submitted_at":"2026-04-06T16:52:27Z","abstract_excerpt":"In 1998, Bremner conjectured that elliptic curves over the rationals having long sequences of distinct rational points whose $x$-coordinates are in arithmetic progression, have large rank. This was proved some years ago in a strong form as a consequence of previous work by the authors, by a combination of Nevanlinna theory and the uniform Mordell--Lang theorem of Gao--Ge--K\\\"uhne. Thus, if the ranks of elliptic curves over the rationals are uniformly bounded, then so are the lengths of the aforementioned arithmetic progressions. In this note we give a much more direct proof of this last statem"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"if the ranks of elliptic curves over the rationals are uniformly bounded, then so are the lengths of the aforementioned arithmetic progressions.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The uniform Mordell--Lang conjecture for curves due to Dimitrov--Gao--Habegger, invoked to replace the earlier Nevanlinna-theoretic argument.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A more direct proof establishes that uniform boundedness of ranks of rational elliptic curves implies uniform boundedness of lengths of arithmetic progressions in x-coordinates of rational points, relying only on the uniform Mordell-Lang conjecture for curves.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"If ranks of elliptic curves over the rationals are uniformly bounded, then arithmetic progressions of their rational x-coordinates have uniformly bounded lengths.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"78155a50d74fc1b9cd4f565ee4e4c3b1fe6a197c30a5153453dab90226a0bc48"},"source":{"id":"2604.04850","kind":"arxiv","version":2},"verdict":{"id":"e5f4d50f-18cd-4f61-88fa-86db15053376","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T18:58:58.798079Z","strongest_claim":"if the ranks of elliptic curves over the rationals are uniformly bounded, then so are the lengths of the aforementioned arithmetic progressions.","one_line_summary":"A more direct proof establishes that uniform boundedness of ranks of rational elliptic curves implies uniform boundedness of lengths of arithmetic progressions in x-coordinates of rational points, relying only on the uniform Mordell-Lang conjecture for curves.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The uniform Mordell--Lang conjecture for curves due to Dimitrov--Gao--Habegger, invoked to replace the earlier Nevanlinna-theoretic argument.","pith_extraction_headline":"If ranks of elliptic curves over the rationals are uniformly bounded, then arithmetic progressions of their rational x-coordinates have uniformly bounded lengths."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.04850/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":1,"snapshot_sha256":"46d1795f6522b6386abf701080cd9ca31e49fd3604ad72ebe4567e2c1a20dbbd"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}