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

Abstract

This is the second paper in a series by the author, centered on the study of values of G-functions associated to a $1$-parameter family of abelian varieties $f:\CX\rightarrow S$ and a point $s_0\in S(K)$ over some number field $K$. Here we study the case where $f:\CX\rightarrow S$ is a family of elliptic curves. Extending work of Andr\'e and Beukers, we construct relations among the values of G-functions in this setting at points whose fibers are CM elliptic curves.