STNB2025 (38th edition)
Bogomolov property for Galois representations with big local image
Presenters
Andrea Conti
Abstract
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 to 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 $G_{\mathbb Q}$, 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.
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