{"title":"Non-thin rank jumps for double elliptic K3 surfaces","authors":"Hector Pasten, Cecília Salgado","doi":"10.1007/s00229-024-01554-2","DOIUrl":null,"url":null,"abstract":"<p>For an elliptic surface <span>\\(\\pi :X\\rightarrow \\mathbb {P}^1\\)</span> defined over a number field <i>K</i>, a theorem of Silverman shows that for all but finitely many fibres above <i>K</i>-rational points, the resulting elliptic curve over <i>K</i> has Mordell-Weil rank at least as large as the rank of the group of sections of <span>\\(\\pi \\)</span>. When <i>X</i> is a <i>K</i>3 surface with two distinct elliptic fibrations, we show that the set of <i>K</i>-rational points of <span>\\(\\mathbb {P}^1\\)</span> for which this rank inequality is strict, is not a thin set, under certain hypothesis on the fibrations. Our results provide one of the first cases of this phenomenon beyond that of rational elliptic surfaces.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00229-024-01554-2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
For an elliptic surface \(\pi :X\rightarrow \mathbb {P}^1\) defined over a number field K, a theorem of Silverman shows that for all but finitely many fibres above K-rational points, the resulting elliptic curve over K has Mordell-Weil rank at least as large as the rank of the group of sections of \(\pi \). When X is a K3 surface with two distinct elliptic fibrations, we show that the set of K-rational points of \(\mathbb {P}^1\) for which this rank inequality is strict, is not a thin set, under certain hypothesis on the fibrations. Our results provide one of the first cases of this phenomenon beyond that of rational elliptic surfaces.