Andrea Conti (Heidelberg)

An algebraic extension of the rational numbers is said to have the Bogomolov property 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 $G_{\mathbb Q}$ of $\mathbb Q$, one can ask whether the field fixed by $\mathrm{ker}(\rho)$ has the Bogomolov property (in short, we say that $\rho$ has (B)). In a joint work with Lea Terracini, we prove that, if $\rho\colon G_{\mathbb Q}\to\mathrm{GL}_N(\mathbb Z_p)$ maps an inertia subgroup at $p$ surjectively onto an open subgroup of $\mathrm{GL}_N(\mathbb Z_p)$, then $\rho$ has (B). More generally, we show that if the image of a decomposition group at $p$ is open in the image of $G_\Q$, plus a certain condition on the center of the image is 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.