Geometric G-functions and Atypicality

Abstract

We describe a general method for giving $p$-adic interpretations of $G$-functions arising from degenerating periods of smooth projective algebraic varieties. Using this, we are able to implement a strategy due to Andr\'e for bounding heights of moduli points where period functions acquire unusual algebraic relations. This leads to new results on Galois lower bounds for special moduli, and new cases of the Zilber-Pink conjecture. In particular, we establish the first Galois-orbit lower bounds on CM moduli in non-Shimura settings. As a more technical contribution, we introduce a refinement of the Pila-Zannier strategy capable of handling Zilber-Pink-type atypical intersection problems in arbitrary dimension and for arbitrary smooth projective families.