{"abstract":[{"text":"For a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists E \r\nD have no integral points, as D ranges over square-free integers ordered by size. Our result is conditional on a weak form of the Hall–Lang conjecture in the case that E has partial 2-torsion. The proof uses a correspondence of Mordell and the reduction theory of binary quartic forms in order to transfer the problem to counting rational points of bounded height on a certain singular cubic surface, together with extensive use of cancellation in character sum estimates, drawn from Heath-Brown’s analysis of Selmer group statistics for the congruent number curve.","lang":"eng"}],"OA_type":"diamond","OA_place":"publisher","status":"public","date_published":"2025-09-17T00:00:00Z","acknowledgement":"The authors are grateful to Roger Heath-Brown and to the anonymous referees for useful comments. The first author was supported by an FWF grant (DOI 10.55776/P36278).","quality_controlled":"1","oa":1,"title":"Almost all quadratic twists of an elliptic curve have no integral points","month":"09","language":[{"iso":"eng"}],"_id":"21266","arxiv":1,"main_file_link":[{"open_access":"1","url":"https://doi.org/10.4171/JEMS/1704"}],"project":[{"_id":"bd8a4fdc-d553-11ed-ba76-80a0167441a3","name":"Rational curves via function field analytic number theory","grant_number":"P36278"}],"citation":{"ama":"Browning TD, Chan S. Almost all quadratic twists of an elliptic curve have no integral points. <i>Journal of the European Mathematical Society</i>. 2025. doi:<a href=\"https://doi.org/10.4171/jems/1704\">10.4171/jems/1704</a>","chicago":"Browning, Timothy D, and Stephanie Chan. “Almost All Quadratic Twists of an Elliptic Curve Have No Integral Points.” <i>Journal of the European Mathematical Society</i>. European Mathematical Society Press, 2025. <a href=\"https://doi.org/10.4171/jems/1704\">https://doi.org/10.4171/jems/1704</a>.","short":"T.D. Browning, S. Chan, Journal of the European Mathematical Society (2025).","mla":"Browning, Timothy D., and Stephanie Chan. “Almost All Quadratic Twists of an Elliptic Curve Have No Integral Points.” <i>Journal of the European Mathematical Society</i>, European Mathematical Society Press, 2025, doi:<a href=\"https://doi.org/10.4171/jems/1704\">10.4171/jems/1704</a>.","apa":"Browning, T. D., &#38; Chan, S. (2025). Almost all quadratic twists of an elliptic curve have no integral points. <i>Journal of the European Mathematical Society</i>. European Mathematical Society Press. <a href=\"https://doi.org/10.4171/jems/1704\">https://doi.org/10.4171/jems/1704</a>","ieee":"T. D. Browning and S. Chan, “Almost all quadratic twists of an elliptic curve have no integral points,” <i>Journal of the European Mathematical Society</i>. European Mathematical Society Press, 2025.","ista":"Browning TD, Chan S. 2025. Almost all quadratic twists of an elliptic curve have no integral points. Journal of the European Mathematical Society."},"external_id":{"arxiv":["2401.04375"]},"DOAJ_listed":"1","oa_version":"Published Version","publisher":"European Mathematical Society Press","corr_author":"1","year":"2025","article_processing_charge":"No","type":"journal_article","doi":"10.4171/jems/1704","publication":"Journal of the European Mathematical Society","day":"17","author":[{"first_name":"Timothy D","last_name":"Browning","orcid":"0000-0002-8314-0177","id":"35827D50-F248-11E8-B48F-1D18A9856A87","full_name":"Browning, Timothy D"},{"orcid":"0000-0001-8467-4106","id":"c4c0afc8-9262-11ed-9231-d8b0bc743af1","first_name":"Yik Tung","last_name":"Chan","full_name":"Chan, Yik Tung"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_identifier":{"issn":["1435-9855"],"eissn":["1435-9863"]},"article_type":"original","publication_status":"epub_ahead","date_updated":"2026-02-23T10:54:40Z","department":[{"_id":"TiBr"}],"date_created":"2026-02-17T07:46:26Z"}