Bogomolov property for modular Galois representations with nontrivial nebentypus

1. Problem Statement and Context

The Bogomolov Property (Property B):
A field $K \subseteq \overline{\mathbb{Q}}$ is said to have the Bogomolov property if there exists a constant $C > 0$ such that for all non-zero elements $\alpha \in K$ that are not roots of unity, the logarithmic Weil height satisfies $h(\alpha) \geq C$. This property implies that the set of algebraic numbers in the field with small height is finite (excluding roots of unity).

Previous Work:

• Habegger (2013): Proved that the field generated by the torsion points of an elliptic curve over $\mathbb{Q}$, denoted $\mathbb{Q}(E_{\text{tors}})$, has Property (B).
• Amoroso and Terracini (2025): Generalized Habegger's result to modular forms. They proved that for a normalized eigenform $f \in S_k(\Gamma_0(N))$ (with trivial nebentypus) satisfying specific conditions, the field cut out by the adelic Galois representation associated to $f$ has Property (B).

The Gap:
The result by Amoroso and Terracini relied heavily on the assumption that the nebentypus character $\chi$ is trivial. When $\chi$ is nontrivial, the Galois representation can possess inner twists (automorphisms of the coefficient field that twist the form by a Dirichlet character). These inner twists complicate the structure of the image of the Galois representation, making the direct application of previous methods impossible.

Objective:
Piras aims to extend the Amoroso-Terracini result to the case of normalized eigenforms $f \in S_k(\Gamma_0(N), \chi)$ where the nebentypus $\chi$ is nontrivial.

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2. Methodology

The proof follows a strategy of decomposing the Galois representation into "outside $p$" and "inside $p$" components, analyzing them separately, and then merging the results using a generalized "merging theorem."

A. Handling Inner Twists and Image Control

The primary difficulty with nontrivial nebentypus is the existence of inner twists. If $\gamma$ is an automorphism of the coefficient field $E_f$, there may exist a Dirichlet character $\chi_\gamma$ such that $f^\gamma = f \otimes \chi_\gamma$.

• Strategy: The author restricts the Galois group to a finite extension $K$ of $\mathbb{Q}$ where all such twisting characters become trivial. On this field, the image of the representation is better controlled.
• Assumptions: The paper imposes specific conditions on the form $f$ and a prime $p$ (Assumption 1):
  • $p \nmid N$ (level).
  • $a_p(f) = 0$ (supersingular condition).
  • The image of the representation modulo $p$ contains $SL_2(\mathcal{O}_f/p\mathcal{O}_f)$.
  • $p \geq 5$ and specific non-divisibility conditions on $k-1$.

B. The ADZ Field Concept

To handle the "outside $p$" part of the representation, the author introduces the notion of an ADZ field (inspired by Amoroso, David, and Zannier).

• Definition: A Galois extension $F/K$ is ADZ at a place $v$ if the fixed field of the center of $\text{Gal}(F/K)$ has uniformly bounded local degrees over $v$.
• Key Contribution: The paper proves that the class of ADZ fields is closed under compositum. This is a significant technical advancement because Property (B) is generally not preserved under compositum of fields.
• Application: The author shows that the field cut out by the representation outside of $p$ is an ADZ field (specifically, the square of Frobenius elements becomes central), thereby inheriting Property (B).

C. Crystalline Representations and Ramification

For the "inside $p$" part (the $p$-adic representation):

• Classification: Using the condition $a_p(f)=0$, the representation restricted to $G_{\mathbb{Q}_p}$ is identified as a crystalline representation of type $V_{k,0,\omega}$ (following Breuil's classification).
• Structure: The author analyzes the ramification groups of the field extensions generated by the mod $p^n$ reductions of this representation.
• Normal Closure Lemma: A critical technical lemma (Proposition 5.19) is proven, showing that the normal closure of the Galois group of the extension at level $n$ (relative to $n-1$) inside the full Galois group is the entire group at level $n$. This ensures the "merging" of the two parts works.

D. The Merging Theorem

The core of the proof relies on Theorem 5.3, a generalization of a result by Amoroso-Terracini. It states that if:

1. Field $F$ is unramified at $p$ and has Property (B).
2. Field $L$ satisfies specific ramification and centrality assumptions (Assumptions 2 and 3).
   Then the compositum $FL$ has Property (B).

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3. Key Contributions

1. Extension to Nontrivial Nebentypus: The paper successfully generalizes the Bogomolov property result from trivial to nontrivial nebentypus characters, overcoming the obstruction posed by inner twists.
2. Introduction of ADZ Fields: The author formalizes the concept of ADZ fields and proves their stability under arbitrary compositum (Theorem 5.8). This provides a robust new class of fields known to satisfy Property (B).
3. Normal Closure Lemma for Modular Forms: The paper provides a rigorous proof of the normal closure lemma for modular forms with nontrivial nebentypus, adapting group-theoretic arguments from the elliptic curve and trivial nebentypus cases to the more complex setting of modular forms.
4. Explicit Examples: The paper constructs explicit examples of modular forms (specifically in weight $k=2$) satisfying the necessary assumptions, utilizing a theorem of Diamond to ensure the representations are not "old" at $p$.

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4. Main Results

Theorem 4.3 (Main Theorem):
Let $f \in S_k(\Gamma_0(N), \chi)$ be a normalized eigenform with $k \geq 2$ and nontrivial nebentypus $\chi$. Let $p$ be a prime satisfying Assumption 1 ($p \nmid N$, $a_p(f)=0$, large image mod $p$, etc.). Then the field $K(\hat{\rho}_f)$ cut out by the adelic representation associated to $f$ (restricted to the field $K$ fixed by the inner twists) has the Bogomolov property.

Corollary:
Since the field cut out by the representation over $\mathbb{Q}$ is contained in the field cut out over $K$, the result implies that the field generated by the torsion points of the modular abelian variety (or the representation itself) has Property (B) under these conditions.

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5. Significance

• Theoretical Advancement: This work bridges a gap in the arithmetic of modular forms, showing that the "small height" phenomenon (Bogomolov property) is robust even when the underlying modular form has a nontrivial character.
• Methodological Innovation: The introduction of ADZ fields and the proof of their closure under compositum offer a powerful new tool for studying height bounds in infinite Galois extensions. This is particularly relevant because the preservation of Property (B) under compositum is a notoriously difficult problem.
• Connection to Conjectures: The results support the broader landscape of conjectures related to Lehmer's problem and the distribution of algebraic numbers of small height. By establishing Property (B) for these specific infinite extensions, the paper provides concrete evidence for the behavior of heights in fields generated by automorphic representations.
• Computational Relevance: By providing a list of explicit modular forms (via LMFDB labels) that satisfy the conditions, the paper moves the theory from abstract existence to concrete, verifiable examples, facilitating future computational investigations.

In summary, Pietro Piras's paper successfully generalizes a major result in the arithmetic of modular forms by developing new group-theoretic tools (ADZ fields) and overcoming the technical hurdles introduced by inner twists in the presence of nontrivial nebentypus.