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arXiv · 1901.09945

Uniform Manin-Mumford for a family of genus 2 curves

Laura DeMarco·Holly Krieger·Hexi Ye
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Abstract

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 $\overline{\mathbb{Q}}.$

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Laura DeMarco, Holly Krieger, Hexi Ye. 2019-01-28. Uniform Manin-Mumford for a family of genus 2 curves. https://arxiv.org/abs/1901.09945

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