G-functions, motives, and unlikely intersections -- old and new
Number Theory
2025-01-20 v1 Algebraic Geometry
Abstract
In this survey, we outline the role of G-functions in arithmetic geometry, notably their link with Picard-Fuchs differential equations and periods. We explain how polynomial relations between special values of G-functions arising from a pencil of algebraic varieties may occur at a parameter where the fiber has more ``motivic" symmetries; and how Bombieri's principle of global relations can be used to control the height of such parameters (which was also one of the origins of the Andr\'e-Oort conjecture). At the end, we sketch the recent revival of the G-function method in the context of unlikely intersections and the Zilber-Pink conjecture.
Cite
@article{arxiv.2501.09867,
title = {G-functions, motives, and unlikely intersections -- old and new},
author = {Yves André},
journal= {arXiv preprint arXiv:2501.09867},
year = {2025}
}