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

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 André and Beukers, we construct relations among the values of G-functions in this setting at points whose fibers are CM elliptic curves.

💡 Research Summary

The paper “v‑adic G‑function values and CM elliptic curve relations” is the second installment in a series devoted to the arithmetic of G‑functions attached to one‑parameter families of abelian varieties, focusing here on families of elliptic curves. The author studies a smooth irreducible base curve S defined over a number field K, a family f : E → S of elliptic curves, and a distinguished K‑rational point s₀ ∈ S(K). By André’s construction (And89) one can associate to the pair (E → S, s₀) a finite collection of G‑functions Y_G centered at s₀. The main goal is to exhibit explicit polynomial relations among the values of these G‑functions at points s ∈ S(K) whose fibers E_s are CM elliptic curves and which are v‑adically close to s₀ for a given place v of the field K(s).

Main Result (Theorem 1.1).
Assume that the central fiber E_{s₀} has complex multiplication (CM). For any place v of K(s) such that s lies within the v‑adic radius of convergence of the power series defining Y_G, there exists a non‑zero polynomial R_{s,v} ∈ \overline{Q}