On the $v$-adic values of G-functions II
Georgios Papas
TL;DR
This work develops an André–Beukers–Beukers-inspired framework to study the $v$-adic values of G-functions attached to a 1-parameter family of CM elliptic curves. It derives explicit relations at ordinary and supersingular places, including archimedean analogues, and uses them to prove effective height bounds for CM fibers, connecting to an effective Siegel-type problem for imaginary quadratic fields. The results reduce the Siegel-type question to bounding a set of primes tied to CM discriminants, and they articulate how Beukers-type archimedean relations and André–Bombieri-type Hasse principles drive the conclusions. Collectively, the paper advances a path toward effective class-number bounds via the G-functions method in the CM elliptic setting, with explicit, computable constants and a clear roadmap for handling ramified and supersingular phenomena.
Abstract
This is the second in a series of papers by the author centered around the study of values of G-functions associated to $1$-parameter families of abelian varieties $f:\CX\rightarrow S$ and a point $s_0\in S(K)$ with smooth fiber over some number field $K$. Here we study the case where $f:\CX\rightarrow S$ is a family of elliptic curves. We construct relations among the values of G-functions in this setting at points whose fiber is a CM elliptic curve. These lead to bounds for the height of such points, via André's G-functions method. We also discuss implications of our height bounds to the search for an effective version of Siegel's lower bounds for class numbers of imaginary quadratic number fields.