Distinguished categories and the Zilber-Pink conjecture

authored by

Fabrizio Barroero, Gabriel Andreas Dill

Abstract

We propose an axiomatic approach towards studying unlikely intersections by introducing the framework of distinguished categories. This includes commutative algebraic groups and mixed Shimura varieties. It allows us to define all basic concepts of the field and prove some fundamental facts about them, e.g. the defect condition. In some categories that we call very distinguished, we are able to show some implications between Zilber-Pink statements with respect to base change. This yields unconditional results, i.e. the Zilber-Pink conjecture for a complex curve in \(\mathcal{A}_2\) that cannot be defined over \(\bar{\mathbb{Q}}\), a complex curve in the \(g\)-th fibered power of the Legendre family, and a complex curve in the base change of a semiabelian variety over \(\bar{\mathbb{Q}}\).

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Article

Journal

American Journal of Mathematics

Volume

147

Pages

715-778

No. of pages

64

ISSN

0002-9327

Publication date

06.2025

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

General Mathematics

Electronic version(s)

https://doi.org/10.1353/ajm.2025.a961345 (Access: Closed)
https://doi.org/10.48550/arXiv.2103.07422 (Access: Open)