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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
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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