The Bogomolov Property through Galois Representations

Abstract

The Bogomolov property \B for an algebraic extension of \(\QQ\) asserts the existence of a uniform positive lower bound for the absolute logarithmic Weil height outside the group of roots of unity. Originally introduced as a weakening of Northcott's property and closely related to Lehmer's conjecture, it has been established for several natural classes of infinite extensions, including abelian extensions and fields generated by torsion points of elliptic curves defined over the rationals. Given a Galois representation of an absolute Galois group, one can associate with it the algebraic extension fixed by its kernel and ask whether this extension has property \B. This point of view allows one to reinterpret classical results, and to generalize them to other Galois representations, both of geometric and non-geometric origin. This expository paper gives a survey of the techniques and results in this framework. We present some results on modular representations, with a particular focus on the role of local \(p\)-adic information. We explain how Sen's theorem on totally ramified \(p\)-adic Lie extensions enters the proof of new criteria for the Bogomolov property, and how these criteria apply to representations with large local image. This contribution is based on joint work with Francesco Amoroso, Andrea Conti, and Pietro Piras.