The Manin–Mumford conjecture and the Tate–Voloch conjecture for a product of Siegel moduli spaces

Qiu, Congling

doi:10.2140/ant.2023.17.981

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The Manin–Mumford conjecture and the Tate–Voloch conjecture for a product of Siegel moduli spaces

Congling Qiu

Vol. 17 (2023), No. 5, 981–1016

DOI: 10.2140/ant.2023.17.981

Abstract

We use perfectoid spaces associated to abelian varieties and Siegel moduli spaces to study torsion points and ordinary CM points. We reprove the Manin–Mumford conjecture, i.e., Raynaud’s theorem. We also prove the Tate–Voloch conjecture for a product of Siegel moduli spaces, namely ordinary CM points outside a closed subvariety can not be $p$ -adically too close to it.

Keywords

Manin–Mumford conjecture, Tate–Voloch conjecture, CM points, Siegel moduli spaces, perfectoid spaces

Mathematical Subject Classification

Primary: 11G10, 11G18, 14G35, 14G45, 14K12

Milestones

Received: 12 February 2021

Revised: 27 March 2022

Accepted: 6 July 2022

Published: 9 May 2023

Authors

Congling Qiu
Department of Mathematics Yale University New Haven, CT United States

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Vol. 17, No. 5, 2023