Seminar des SFB/TRR 326 GAUS Bogomolov property for Galois representations with big local image

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 ρ of the absolute Galois group G_ℚ of ℚ, one can ask whether the field fixed by ker(ρ) has the Bogomolov property (in short, we say that ρ has (B)). In a joint work with Lea Terracini, we prove that, if ρ: G_ℚ → GL_N(ℤ_p) maps an inertia subgroup at p surjectively onto an open subgroup of GL_N(ℤ_p), then ρ 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 ρ has (B). In particular, no assumption on the modularity of ρ is needed, contrary to previous work of Habegger and Amoroso—Terracini.