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

TL;DR

This survey traces how G-functions and G-operators illuminate the link between periods, Picard–Fuchs equations, and arithmetic geometry, highlighting Bombieri's global-relations principle as a tool to bound exceptional values and connect to the André-Oort and Zilber-Pink conjectures. It explains how functional periods of Picard–Fuchs families translate into G-functions and how period relations at points of enhanced symmetry yield polynomial relations among G-values, with special $M_0$- setups clarifying $2\pi i$-factors. The work surveys top-down results in finiteness and height bounds for special points, and discusses how combining G-function methods with Pila–Zannier’s counting yields advances on unlikely intersections in moduli spaces, includingCM and Hodge-generic scenarios, while extending to $p$-adic contexts. The approach provides a unified framework tying differential equations, periods, and Diophantine geometry to modern conjectures like André-Oort and Zilber–Pink, offering both conceptual insight and concrete finiteness results.

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.