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

2024/10/23 by Kangsheng Gu, Gu, Kaiyuan, Huang, Chenxin
Engineering · Mathematics · #11G15 #14G35 #14K05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #Rings, Modules, and Algebras #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2410.17755

openalex publication_date 2024/10/23 · openalex created_date 2024/11/13 · openalex updated_date 2026/07/28

Abstract

The universal Poincaré 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for mixed Shimura varieties implies the Zilber-Pink conjecture for semi-abelian varieties, correcting an error in an unpublished manuscript of Pink.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.

Related

• The Zilber-Pink conjecture for varieties not defined over \mathbb Q
• Generalised André-Pink-Zannier Conjecture for Shimura varieties of abelian type
• A special point problem of André-Pink-Zannier in the universal family of abelian varieties
• On the Zilber-Pink conjecture for complex abelian varieties
• Relative Manin-Mumford for semi-abelian surfaces