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

TL;DR

This survey explores the role of G-functions in arithmetic geometry, their connection with differential equations and periods, and recent advances in unlikely intersections and the Zilber-Pink conjecture.

Contribution

It provides a comprehensive overview of G-functions' applications in controlling special values and their recent role in unlikely intersection problems.

Findings

G-functions relate to Picard-Fuchs equations and periods.

Polynomial relations between G-functions can indicate enhanced symmetries.

The G-function method is being revitalized in the context of unlikely intersections.

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.