Bogomolov property for Galois representations with big local image

Abstract

An algebraic extension of the rational numbers is said to have the $\textit{Bogomolov property}$ (B) if the absolute logarithmic Weil height of its non-torsion elements is uniformly bounded from below. Given a continuous representation $\rho$ of the absolute Galois group $G_{\mathbb{K}}$ of a number field ${\mathbb{K}}$, one says that $\rho$ has (B) if the subfield of $\overline{\mathbb{Q}}$ fixed by $\mathrm{ker}\,\rho$ has (B). We prove that, if $\rho:G_{\mathbb{K}} \to \mathrm{GL}_d({\mathbb{Z}}_p)$ maps an inertia subgroup at a prime above $p$ surjectively onto an open subgroup of $\mathrm{GL}_d({\mathbb{Z}}_p)$, then $\rho$ has (B). More generally, we show that if the image of inertia is open in the image of the decomposition group, the normal closure of the local image is sufficiently large in the global one, and a certain condition on the center of $\rho(G_{\mathbb{K}})$ satisfied, then $\rho$ has (B). In particular, no assumption on the modularity of $\rho$ is needed, contrary to previous work of Habegger and Amoroso--Terracini. We provide several examples both in modular and non-modular cases. Our methods rely on a result of Sen comparing the ramification and Lie filtrations on the $p$-adic Lie group $\rho(G_{\mathbb{K}})$.