A note on the Manin-Mumford conjecture

TL;DR

This paper proves a special case of the André-Oort conjecture for Kuga varieties, showing that the Zariski closure of certain special subvarieties is a finite union of special subvarieties, based on a variant of the Manin-Mumford conjecture.

Contribution

It establishes a specific case of the André-Oort conjecture for Kuga varieties by linking it to a Manin-Mumford type statement for abelian schemes.

Findings

Zariski closure of special subvarieties is a finite union of special subvarieties.

Reduces the problem to a Manin-Mumford conjecture variant.

Provides a proof for a special case of the André-Oort conjecture.

Abstract

In this paper we prove a special case of the Andr\'e-Oort conjecture for Kuga varieties. If $M$ is a Kuga variety fibred over a pure Shimura variety $S$ as an abelian scheme, and $(M_n)$ is a sequence of special subvarieties in $M$ which are faithfully flat over $S$, then the Zariski closure of the union of the $M_n$'s is a finite union of special subvarieties faithfully flat over $S$. The proof is reduced to a variant of the Manin-Mumford conjecture for abelian schemes.