On the $v$-adic values of G-functions III

TL;DR

This paper investigates the relationships among G-function values at special points related to abelian surfaces with Quaternionic multiplication, exploring both archimedean and p-adic contexts, and discusses implications for the Zilber-Pink conjecture.

Contribution

It extends previous work on G-functions by analyzing their values at special points in abelian surfaces with Quaternionic multiplication, linking p-adic and archimedean properties.

Findings

Established relations among G-function values at special points.

Connected G-function values to Quaternionic multiplication structures.

Discussed implications for the Zilber-Pink conjecture in moduli space.

Abstract

In this third part in this series we continue from \cite{papaspadicpart1}, the study of relations among values of G-functions associated to a $1$-parameter family of principally polarized abelian surfaces. In particular, we establish relations among the values of these G-functions, in both the archimedean and $p$-adic setting, at points corresponding to abelian surfaces with Quaternionic multiplication. We also discuss applications to the Zilber-Pink conjecture in $\mathcal{A}_2$ that naturally follow from our discussion.