Bogomolov property for modular Galois representations with nontrivial nebentypus

TL;DR

This paper extends the Bogomolov property to fields associated with modular Galois representations of eigenforms with nontrivial nebentypus, and introduces ADZ fields where this property is stable under composition.

Contribution

It generalizes previous results on the Bogomolov property to a broader class of modular forms and introduces ADZ fields with stable property (B).

Findings

Proves property (B) for fields from eigenforms with nontrivial nebentypus.

Introduces ADZ fields where property (B) is preserved under composition.

Extends earlier work on elliptic curves and modular forms.

Abstract

A field in which the (logarithmic) Weil height is bounded from below by a strictly positive constant is said to have the Bogomolov property (property (B)). Given a normalized eigenform $f\in S_k(\Gamma_0(N))$ Amoroso and Terracini proved (B) for the field "cut out" by the adelic representation associated to $f$ under some assumptions on $f$, generalizing the earlier work of Habegger on elliptic curves. In this paper we extend this result to the case of normalized eigenforms with nontrivial nebentypus character. We also introduce the notion of ADZ field, inspired by earlier work of Amoroso, David and Zannier, exhibiting a class of fields in which property (B) is preserved under (arbitrary) composition.