{"title":"Counting rational points on elliptic curves with a rational 2-torsion point","authors":"F. Naccarato","doi":"10.4171/rlm/945","DOIUrl":null,"url":null,"abstract":"Let $E/\\mathbb{Q}$ be an elliptic curve over the rational numbers. It is known, by the work of Bombieri and Zannier, that if $E$ has full rational $2$-torsion, the number $N_E(B)$ of rational points with Weil height bounded by $B$ is $\\exp\\left(O\\left(\\frac{\\log B}{\\sqrt{\\log\\log B}}\\right)\\right)$. In this paper we exploit the method of descent via $2$-isogeny to extend this result to elliptic curves with just one nontrivial rational $2$-torsion point. Moreover, we make use of a result of Petsche to derive the stronger upper bound $N_{E}(B) = \\exp\\left(O\\left(\\frac{\\log B}{\\log\\log B}\\right)\\right)$ for these curves and to remove a deep transcendence theory ingredient from the proof.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2021-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Rendiconti Lincei-Matematica e Applicazioni","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4171/rlm/945","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
Let $E/\mathbb{Q}$ be an elliptic curve over the rational numbers. It is known, by the work of Bombieri and Zannier, that if $E$ has full rational $2$-torsion, the number $N_E(B)$ of rational points with Weil height bounded by $B$ is $\exp\left(O\left(\frac{\log B}{\sqrt{\log\log B}}\right)\right)$. In this paper we exploit the method of descent via $2$-isogeny to extend this result to elliptic curves with just one nontrivial rational $2$-torsion point. Moreover, we make use of a result of Petsche to derive the stronger upper bound $N_{E}(B) = \exp\left(O\left(\frac{\log B}{\log\log B}\right)\right)$ for these curves and to remove a deep transcendence theory ingredient from the proof.
期刊介绍:
The journal is dedicated to the publication of high-quality peer-reviewed surveys, research papers and preliminary announcements of important results from all fields of mathematics and its applications.