Uniform Manin-Mumford for a family of genus 2 curves

• 成果类型: Article
• 署名作者: DeMarco, Laura; Krieger, Holly; Ye, Hexi
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2020.191.3.5
• 发表日期: 2020
• 页码: 949-1001
• 关键词: canonical heights; torsion points; equidistribution; VARIETIES

摘要

We introduce a general strategy for proving quantitative and uniform bounds on the number of common points of height zero for a pair of inequivalent height functions on P-1((Q) over bar). We apply this strategy to prove a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds on the number of common torsion points of elliptic curves in the case of two Legendre curves over C. As a consequence, we obtain two uniform bounds for a two-dimensional family of genus 2 curves: a uniform Manin-Mumford bound for the family over C, and a uniform Bogomolov bound for the family over (Q) over bar.