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Bogomolov property for modular Galois representations with nontrivial nebentypus

Pietro Piras

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 $f\in S_k(Γ_0(N))$ 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.

Bogomolov property for modular Galois representations with nontrivial nebentypus

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 Amoroso and Terracini proved (B) for the field "cut out" by the adelic representation associated to under some assumptions on , 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.
Paper Structure (11 sections, 20 theorems, 100 equations, 3 figures)

This paper contains 11 sections, 20 theorems, 100 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Deligne-Shimura construction
  3. Property (B) for morphisms
  4. Main result
  5. Proof of the main result
  6. Hypotheses and proof of the merging theorem
  7. Outside of $p$
  8. Crystalline representations with character
  9. The proof of theorem \ref{['main:thm']}
  10. The proof of the normal closure lemma
  11. Examples

Key Result

Proposition 3.2

For $x_1,\ldots,x_n\in\overline{\mathbb{Q}}$ the Weil height satisfies the following prpoerties

Figures (3)

  • Figure 1: Field extensions considered in the proof of proposition \ref{['prop:cep']}.
  • Figure :
  • Figure :

Theorems & Definitions (47)

  • Conjecture : Rem17, Conjecture 3.4
  • Definition 3.1
  • Proposition 3.2
  • Definition 3.3: Definition 1.1, AT25
  • Remark 4.1
  • Remark 4.2
  • Theorem 4.3
  • Lemma 5.1
  • proof
  • Lemma 5.2: Lemma 2.1, ADZ14
  • ...and 37 more