On the Divisibility Properties of the Fourier Coefficients of Meromorphic Hilbert Modular Forms (Associated with Real Quadratic Fields)

Core Concepts

This article investigates the rationality and divisibility properties of Fourier coefficients of meromorphic Hilbert modular forms associated with real quadratic fields, revealing specific divisibility patterns under certain conditions.

Abstract

Bibliographic Information:

Depouilly, B. (2024). On the Divisibility Properties of the Fourier Coefficients of Meromorphic Hilbert Modular Forms. arXiv. https://arxiv.org/abs/2411.00701v1

Research Objective:

This paper investigates the divisibility and rationality properties of the Fourier coefficients of meromorphic Hilbert modular forms associated with real quadratic fields, expanding on the work of Zagier who examined similar properties for cusp forms.

Methodology:

The author utilizes the framework of Doi-Naganuma theta lifts to establish a connection between meromorphic Hilbert modular forms and vector-valued Maass-Poincaré series. By analyzing the Fourier expansions of these series and leveraging properties of harmonic Maass forms, the divisibility of the coefficients is studied.

Key Findings:

• The paper establishes conditions under which the Fourier coefficients of the studied meromorphic Hilbert modular forms are rational with bounded denominators.
• It demonstrates that for a negative integer m and weight k ≥ 4, if the space of cusp forms of weight k is trivial, the ν-th coefficient of the form ωm is divisible by (DℓνN(ν0))^(k−1), where D is the discriminant of the real quadratic field, ν0 is the primitive part of ν, and ℓν is the integer such that ν = ℓνν0.
• In cases where the space of cusp forms is non-trivial, the paper shows that similar divisibility properties hold for specific linear combinations of the forms ωm.

Main Conclusions:

The research reveals inherent divisibility patterns within the Fourier coefficients of a specific class of meromorphic Hilbert modular forms. These findings contribute to a deeper understanding of the arithmetic properties of these forms and their connection to other modular objects through theta lifts.

Significance:

This work enhances the understanding of the arithmetic nature of Hilbert modular forms, particularly their connection to real quadratic fields. The divisibility properties uncovered could potentially have implications for further research in areas such as number theory and algebraic geometry.

Limitations and Future Research:

The study focuses on meromorphic Hilbert modular forms associated with real quadratic fields. Exploring similar divisibility properties for forms associated with other number fields or in higher dimensions could be a potential avenue for future research. Additionally, investigating the implications of these divisibility patterns for related mathematical objects could yield further insights.

Stats

The article examines Hilbert modular forms of even weight k ≥ 4.
Numerical examples are provided for the real quadratic field F = Q(√5), m = -1/5, and weight 4.

Quotes

"In this article, we exhibit rationality and divisibility properties for the Fourier coefficients cν when m < 0."
"For m < 0 and an even integer k ≥ 4, if the space of cusp forms Sk,ρL is trivial, then, up to multiplication by an integer that doesn’t depend on ν, the ν-th coefficient of ωm is divisible by (DℓνN(ν0))^(k−1) for all ν ∈ ∂^(−1)_F."

Key Insights Distilled From

On the Divisibility Properties of the Fourier Coefficients of Meromorphic Hilbert Modular Forms

by Baptiste Dep... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00701.pdf

Deeper Inquiries

Can the divisibility properties of Fourier coefficients be extended to Hilbert modular forms associated with number fields beyond real quadratic fields?

Extending the divisibility properties of Fourier coefficients to Hilbert modular forms associated with number fields beyond real quadratic fields is a natural and challenging question. While the paper focuses on real quadratic fields, several of the tools and concepts used have analogues in higher degree number fields. However, several significant obstacles arise:

Complexity of the Weil representation: The Weil representation, crucial for constructing the theta lift, becomes significantly more intricate for number fields of higher degree. The structure of the dual lattice and the discriminant group, which directly impact the representation, become more complex.
Geometry of the associated spaces: The paper utilizes the connection between Hilbert modular forms and orthogonal modular forms in signature (2,2). For higher degree number fields, the corresponding orthogonal groups have higher rank and signature, leading to more complicated symmetric spaces. The geometry of these spaces and the associated Hirzebruch-Zagier divisors are less understood.
Explicit Formulas: Deriving explicit formulas for the Fourier coefficients of Hilbert modular forms, like those in Theorem 3.2, becomes significantly harder in higher dimensions. These formulas are crucial for directly analyzing divisibility properties.
Despite these challenges, some avenues for generalization exist:

Restricting to specific classes of number fields: One could focus on specific classes of number fields, such as totally real cubic fields, where the complexity might be more manageable.
Exploring alternative approaches: Instead of directly generalizing the techniques, exploring alternative approaches to studying Fourier coefficients, such as p-adic methods or representation-theoretic techniques, might be fruitful.
Therefore, while extending the divisibility properties presents significant challenges, it is a promising area for future research. New techniques and a deeper understanding of the underlying structures associated with higher degree number fields are likely required.

How do these findings about the divisibility of Fourier coefficients impact the understanding of the algebraic structure of meromorphic Hilbert modular forms?

The divisibility properties of Fourier coefficients provide valuable insights into the algebraic structure of meromorphic Hilbert modular forms. Here's how:

Integrality properties: The existence of bounded denominators for linear combinations of the ωβ,m, as shown in Theorem 5.5, suggests a hidden integral structure within the space of meromorphic Hilbert modular forms. This hints at the possibility of defining a natural integral basis for certain subspaces of these forms.
Connections to special values: The divisibility by (DℓνN(ν0))^(k−1) relates the Fourier coefficients to invariants of the underlying number field and the lattice L. This suggests potential connections between these coefficients and special values of L-functions, similar to the theory of modular forms for GL(2).
Congruences and Hecke operators: Divisibility properties often foreshadow deeper congruences between modular forms. These findings could pave the way for studying the action of Hecke operators on meromorphic Hilbert modular forms and establishing congruence relations among them.
In essence, these divisibility properties provide a glimpse into the arithmetic nature of meromorphic Hilbert modular forms. They suggest a rich algebraic structure governed by the properties of the underlying number field and hint at deeper connections with other arithmetic objects. Further investigation of these properties could lead to a more profound understanding of the algebraic framework of these modular forms.

If we consider the Fourier coefficients as a sequence, what can be said about the properties of this sequence and its potential applications in other mathematical domains?

Considering the Fourier coefficients of meromorphic Hilbert modular forms as sequences opens up intriguing possibilities for exploring their properties and potential applications in other mathematical domains. Here are some key aspects:

Growth properties: Understanding the growth behavior of these sequences is crucial. Unlike holomorphic cusp forms, which have exponentially decaying Fourier coefficients, meromorphic forms can exhibit more complicated growth patterns due to the presence of poles. Analyzing these growth properties could provide insights into the distribution of poles and the asymptotic behavior of these forms.
Recurrence relations: The intricate formulas for Fourier coefficients, like those in Theorem 3.2, often lead to recurrence relations among them. These relations can be studied using techniques from difference equations and combinatorial number theory, potentially revealing hidden combinatorial interpretations or connections to special functions.
Connections to other areas: The divisibility properties and potential congruences among Fourier coefficients suggest connections to:

Representation theory: The sequences could encode information about representations of certain groups, potentially leading to new insights into the Langlands program.
Coding theory: Sequences with good distribution and correlation properties are valuable in coding theory. The special arithmetic properties of these Fourier coefficients might be beneficial in constructing codes with desirable error-correction capabilities.
Dynamical systems: The iterative nature of Fourier coefficient formulas could be relevant to the study of certain dynamical systems, where the coefficients might describe the evolution of a system over time.
In conclusion, viewing the Fourier coefficients as sequences provides a rich framework for investigating their properties and potential applications. Their connections to various mathematical areas make them a promising object of study, potentially leading to new insights and applications in diverse fields.