$G$-functions and quaternionic multiplication in ${\mathcal{A}}_2$

Abstract

We consider $G$-functions associated with one--parameter families of principally polarized abelian surfaces. We construct relations among their values at QM points, under the assumption that the family is Hodge generic and has a distinguished QM point. The proof uses archimedean and $p$-adic period matrices of QM abelian surfaces, via the de Rham--Betti and de Rham--crystalline comparison isomorphisms. As an application of our relations and the André--Bombieri Hasse principle, we obtain a height bound for QM points and discuss consequences for the Zilber--Pink conjecture in ${\mathcal{A}}_2$.

arXiv commentSubmitted version. Accompanying Mathematica code: https://doi.org/10.5281/zenodo.23453560