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arXiv · 2501.09867

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

Yves André
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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.

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Yves André. 2025-01-16. G-functions, motives, and unlikely intersections -- old and new. https://arxiv.org/abs/2501.09867

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