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Uniform Manin-Mumford for a family of genus 2 curves
Annals of Mathematics ( IF 5.7 ) Pub Date : 2020-01-01 , DOI: 10.4007/annals.2020.191.3.5
Laura DeMarco 1 , Holly Krieger 2 , Hexi Ye 3
Affiliation  

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 $\mathbb{P}^1(\overline{\mathbb{Q}}).$ 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 $\mathbb{C}$. 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 $\mathbb{C}$, and a uniform Bogomolov bound for the family over $\overline{\mathbb{Q}}.$

中文翻译:

属 2 曲线家族的 Uniform Manin-Mumford

我们介绍了一种通用策略,用于证明 $\mathbb{P}^1(\overline{\mathbb{Q}}) 上一对不等价高度函数的高度为零的公共点数量的定量和统一边界。应用此策略来证明 Bogomolov、Fu 和 Tschinkel 的猜想,在 $\mathbb{C}$ 上的两条勒让德曲线的情况下,椭圆曲线的公共扭转点数断言一致边界。因此,我们获得了属 2 曲线的二维族的两个统一边界:$\mathbb{C}$ 上的家庭的统一 Manin-Mumford 界,以及 $\overline 上的家庭的统一 Bogomolov 界{\mathbb{Q}}.$
更新日期:2020-01-01
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