$G$-functions, motives, and unlikely intersections – old and new

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 André–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.