On Special Subvarieties of the Universal Semi-abelian Scheme and Pink Conjectures

TL;DR

This paper classifies special subvarieties of the universal semi-abelian scheme, connects the Zilber-Pink conjecture for mixed Shimura varieties to semi-abelian varieties, and reformulates the Relative Manin-Mumford Conjecture.

Contribution

It provides a classification of special subvarieties of the universal semi-abelian scheme and links conjectures across different contexts, correcting previous errors.

Findings

Classification of special subvarieties for arbitrary toric rank.

Implication of Zilber-Pink conjecture for mixed Shimura varieties on semi-abelian varieties.

A reformulation of the Relative Manin-Mumford Conjecture for semi-abelian schemes.

Abstract

The universal Poincar\'e torsor, or more generally the universal semi-abelian scheme, can be viewed as a mixed Shimura variety. We give a classification of special subvarieties of the universal semi-abelian scheme of arbitrary toric rank. Given this classification, we show that the Zilber-Pink conjecture for mixed Shimura varieties implies the Zilber-Pink conjecture for semi-abelian varieties, correcting an error in an unpublished manuscript of Pink. Moreover, we give a more reasonable reformulation of the Relative Manin-Mumford Conjecture for semi-abelian schemes.