Bogomolov property for Galois representations with big local image
Andrea Conti, Lea Terracini
TL;DR
The paper proves that a wide class of Galois representations with big local image satisfy the Bogomolov property (B) for the fixed field, without assuming modularity. The core method combines Sen's comparison between ramification and Lie filtrations with metric inequalities at a fixed prime to produce lower bounds for Weil heights; this yields a robust criterion (BforLie) under local potential total ramification, a central element, and a large-normal-closure condition. The results translate into concrete criteria for representations, including inertia-full 2D cases and modular forms, and provide numerous modular and non-modular examples such as $p$-adic families on eigencurves, elliptic curves over number fields, and GL$_2$-type abelian varieties. This broad framework extends Bogomolov-type phenomena beyond abelian or modular settings and offers practical criteria for verifying (B) in diverse arithmetic contexts. The work highlights the pivotal role of local image size, HT–Sen weights, and deformation theory in establishing height lower bounds with wide applicability.
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 $ρ$ of the absolute Galois group $G_{\mathbb{K}}$ of a number field ${\mathbb{K}}$, one says that $ρ$ has (B) if the subfield of $\overline{\mathbb{Q}}$ fixed by $\mathrm{ker}\,ρ$ has (B). We prove that, if $ρ:G_{\mathbb{K}} \to \mathrm{GL}_d({\mathbb{Z}}_p)$ maps an inertia subgroup at a prime above $p$ surjectively onto an open subgroup of $\mathrm{GL}_d({\mathbb{Z}}_p)$, then $ρ$ 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 $ρ(G_{\mathbb{K}})$ satisfied, then $ρ$ has (B). In particular, no assumption on the modularity of $ρ$ 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 $ρ(G_{\mathbb{K}})$.