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May 2020 Uniform Manin-Mumford for a family of genus 2 curves
Laura DeMarco, Holly Krieger, Hexi Ye
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Ann. of Math. (2) 191(3): 949-1001 (May 2020). DOI: 10.4007/annals.2020.191.3.5

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. "Uniform Manin-Mumford for a family of genus 2 curves." Ann. of Math. (2) 191 (3) 949 - 1001, May 2020. https://doi.org/10.4007/annals.2020.191.3.5

Information

Published: May 2020
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2020.191.3.5

Subjects:
Primary: 14H40
Secondary: 11G50 , 37F45 , 37P50

Keywords: arithmetic intersection , Elliptic curves , Lattès maps , Manin-Mumford , non-archimedean potential theory , preperiodic points , torsion points

Rights: Copyright © 2020 Department of Mathematics, Princeton University

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Vol.191 • No. 3 • May 2020
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