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

Abstract

We study $G$-functions centered at smooth fibers of one--parameter families of principally polarized abelian varieties. Building on work of Yves André, we associate to such families a collection of $G$-functions whose local values admit period interpretations at archimedean places and at finite places of good reduction of the distinguished fiber, and whose radii of convergence control reduction-theoretic proximity at the latter places. We also discuss a conjectural extension to places of bad semistable reduction using Hyodo--Kato cohomology. Specializing to ${\mathcal{A}}_2$, we construct non-trivial algebraic relations among the values of these $G$-functions at $E^2$- and $E\times \mathrm{CM}$-points in both the archimedean and $p$-adic settings. At non-supersingular finite places of good reduction, we construct relations independent of the place. Finally, we obtain height bounds via André-Bombieri's Hasse principle and discuss applications to large Galois orbit estimates and Zilber--Pink finiteness results for curves in ${\mathcal{A}}_2$.

arXiv commentSubmitted version. Accompanying Mathematica code available at https://doi.org/10.5281/zenodo.23453166