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

## TL;DR

This paper employs perfectoid spaces to reprove the Manin-Mumford conjecture and establish the Tate-Voloch conjecture for products of Siegel moduli spaces, advancing understanding of torsion and CM points in arithmetic geometry.

## Contribution

It introduces a novel approach using perfectoid spaces to prove longstanding conjectures related to torsion and CM points in Siegel moduli spaces.

## Key findings

- Reproved the Manin-Mumford conjecture using perfectoid spaces.
- Proved the Tate-Voloch conjecture for products of Siegel moduli spaces.
- Showed ordinary CM points cannot be p-adically too close to a subvariety outside it.

## 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.