Geometric G-functions and Atypicality

Abstract

In a seminal research manuscript, André showed how the theory of G-functions could be used to give height bounds on the moduli of smooth projective algebraic varieties acquiring non-generic algebraic cycles. The method was limited by the lack of a suitable cohomological interpretation for these G-functions at finite places. In this paper we use recent developments in p-adic Hodge theory to remove this constraint.With respect to André's strategy for producing height bounds from algebraic relations on G-function values at special points, we give a general method for producing relations that hold at all finite places, and show that producing relations at the infinite places is the essential difficulty. This leads to new cases of the Zilber-Pink conjecture, as well as new height bounds on special moduli, including the first unconditional finiteness results for 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.