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Modular Forms Related to Real Quadratic Fields as Traces of Cycle Integrals

This paper establishes that the representations of Zagier's $f_{k,D}$ and Mono's $g_{k+1,D}$ functions as traces of cycle integrals are not unique, contributing to the development of a unifying theory for locally harmonic Maaß forms and their applications to twisted central $L$ -values.

Original authors: Johann Stumpenhusen

Published 2026-08-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Johann Stumpenhusen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematics, there exists a realm where numbers and geometry intertwine in patterns so precise they seem to hum with a hidden music. This is the world of modular forms, functions that behave with a rigid, rhythmic symmetry when the space they inhabit is twisted or stretched in specific ways. For decades, mathematicians have used these forms to unlock secrets of prime numbers and the deep structure of the universe. A particularly fascinating branch of this field involves real quadratic fields, which are number systems built from square roots of positive integers that are not perfect squares. These systems create a unique geometric landscape filled with curves and paths that behave differently than the flat planes we are used to. Within this landscape, researchers have long searched for specific functions that can act as bridges, connecting the chaotic behavior of these number systems to the orderly world of modular forms. The challenge has been that while some examples of these bridges were known, they appeared as isolated islands, lacking a single theory to explain how they all fit together.

Enter a new study that seeks to map the connections between these isolated islands. The researcher, Johann Stumpenhausen, focuses on two specific types of mathematical objects that had been discovered in the past decade. One type, introduced by a mathematician named Zagier, acts like a sum of many small, sharp peaks scattered across the geometric landscape. The other, developed by a researcher named Mono, is a slightly different kind of object that behaves harmonically in most places but has specific lines where it changes its nature. For years, these two objects were understood through different lenses, and while they were known to be related, the exact nature of their relationship remained somewhat mysterious. The prevailing view was that the way these objects could be represented as sums of integrals along specific paths was unique, like a fingerprint that could only be formed in one way.

Stumpenhausen's work challenges this assumption of uniqueness. By constructing a new, more flexible type of mathematical tool—a generalized series that can be tuned with different parameters—the author demonstrates that these two objects can be represented in multiple, distinct ways. Imagine trying to describe a complex shape by tracing its outline; previously, it was believed there was only one correct way to trace that outline to capture the shape's essence. This new research shows that there are actually many different ways to trace the same outline, each using a slightly different set of rules, yet all arriving at the exact same result. The author proves that the functions Zagier and Mono studied can be expressed as a collection of integrals along curved paths, known as cycle integrals, but that this collection is not limited to a single, fixed formula. Instead, there is a whole family of formulas that work, depending on how the underlying mathematical tool is adjusted.

The significance of this finding goes beyond mere variety. The study reveals that these modular forms are deeply connected to the geometry of the quadratic fields they inhabit. The paths along which the integrals are taken are not random; they are the natural curves, or geodesics, that connect the roots of the quadratic equations defining the number system. The research shows that by summing the values of a specific series along these natural curves, one can reconstruct the complex modular forms. Furthermore, the paper investigates the behavior of these forms near the edges of the number system, specifically at rational numbers. It confirms that while these forms are perfectly smooth and symmetric in the middle of the space, they exhibit a fascinating, slightly broken symmetry when approached from the rational numbers. This broken symmetry is what mathematicians call a quantum modular form, a concept that extends the idea of perfect symmetry to the messy, real-world boundary of the number system.

The study also clarifies the relationship between these modular forms and their "shadows," which are simpler functions that capture the essential growth and decay of the original forms. The author shows that the complex objects studied by Zagier and Mono are not just random collections of numbers but are built from the shadows of these simpler functions, woven together with specific mathematical operators. While the paper does not provide a single unifying theory that explains every example under one roof, it successfully demonstrates that the representations of these functions are not unique. This finding removes the barrier of uniqueness, showing that the mathematical landscape is richer and more interconnected than previously thought. By demonstrating that these representations are not unique, the paper opens the door for mathematicians to choose the most convenient representation for their specific needs, potentially leading to new discoveries in how these forms relate to central values of L-functions, which are crucial for understanding the distribution of prime numbers.

Ultimately, this research serves as a bridge between the abstract world of locally harmonic forms and the concrete world of cycle integrals. It takes objects that were once seen as isolated curiosities and places them firmly within a broader, more flexible framework. The author proves that the representations of these functions are not fixed in stone but can be varied, offering a new perspective on how modular forms related to real quadratic fields are constructed. This flexibility suggests that the underlying structure of these number systems is more adaptable than previously believed, allowing for a variety of mathematical expressions to describe the same fundamental reality. The work stands as a testament to the power of looking at familiar problems through a new, more generalized lens, revealing that what seemed like a single, rigid path was actually part of a vast network of possibilities.

Technical Summary: Modular Forms Related to Real Quadratic Fields as Traces of Cycle Integrals

Problem Statement
The paper addresses the representation of specific modular forms associated with real quadratic fields as traces of cycle integrals. Historically, Zagier's functions $f_{k,D}$ and Mono's functions $g_{k+1,D}$ have been studied in the context of locally harmonic Maaß forms and twisted central $L$ -values. While Löbrich and Schwagenscheidt established that the locally harmonic Maaß form $F_{1-k,D}$ can be expressed as a trace of cycle integrals of Petersson's Poincaré series, and Mono provided an analogous representation for $g_{k+1,D}$ , these representations were viewed as somewhat isolated examples. The paper notes that there is currently "no unifying theory behind these examples" and that the known representations are not unique. The primary objective is to demonstrate that the representation of $f_{k,D}$ and $g_{k+1,D}$ as traces of cycle integrals is not unique by utilizing a broader class of Poincaré series, moving towards a potential unifying direction without claiming a complete theory has been established.

Methodology
The author employs a combination of classical modular form theory, differential operators, and the theory of cycle integrals. The methodology proceeds through the following steps:

Generalization of Poincaré Series: The paper introduces a generalized Poincaré series $H_{k_1, k_2}(z, \tau)$ , which transforms as a modular form of weight $2k_1$ in $z$ and weight $-2k_2$ in $\tau$ . This series generalizes the standard Petersson Poincaré series used in previous works.
Differential Operators: The study utilizes the Maaß raising ( $R_k$ ) and lowering ( $L_k$ ) operators, as well as the Bol operator ( $D^{k-1}$ ) and the shadow operator ( $\xi_{2-k}$ ). Lemma 4.1 establishes how these operators act on $H_{k_1, k_2}$ , allowing the author to relate different weights and indices within the family of series.
Trace of Cycle Integrals: The core technique involves expressing modular forms as sums of cycle integrals over Heegner geodesics ( $S_Q$ ) associated with binary integral quadratic forms of discriminant $D$ . The author leverages a theorem by Alfes and Schwagenscheidt (Theorem 4.3) which relates cycle integrals of functions transformed by raising/lowering operators.
Inductive Relations: By combining the action of differential operators on $H_{k_1, k_2}$ with the trace formulas, the author derives inductive relations (Lemma 4.4) that connect cycle integrals of $H_{k+1, k-m}$ and $H_{k, -k-m}$ to the base cases corresponding to known representations of $g_{k+1,D}$ and $f_{k,D}$ .
Quantum Modular Forms: The paper investigates the decomposition of $g_{k+1,D}$ into a modular integral and a rational function, analyzing the transformation behavior of the rational part to establish its status as a weak quantum modular form.

Key Contributions and Results

Non-Uniqueness of Representations: The central result (Theorem 1.2) proves that the representation of $g_{k+1,D}$ $g_{k + 1, D}$ and $f_{k,D}$ $f_{k, D}$ as traces of cycle integrals is not unique. Specifically, for any even integer $m$ $m$ within certain bounds, these functions can be written as traces involving the generalized series $H_{k_1, k_2}$ $H_{k_{1}, k_{2}}$ with varying parameters.
- For $g_{k+1,D}$ , the representation involves $H_{k+1, k-m}$ .
- For $f_{k,D}$ , the representation involves $H_{k, -(k+m)}$ .
Extension of Previous Work: The paper recovers the known results of Löbrich/Schwagenscheidt (for $F_{1-k,D}$ ) and Mono (for $g_{k+1,D}$ ) as specific cases ( $m=0$ ) of the new, more general formulas, thereby expanding the known landscape of representations rather than providing a full unifying theory.
New Representation for $f_{k,D}$ : Theorem 3.7 provides a new representation of Zagier's $f_{k,D}$ as a trace of cycle integrals of the Poincaré series $P_{2k}(z, \tau)$ , which is identified as a special case of $H_{k, -k}$ .
Quantum Modular Form Connection: Proposition 1.3 demonstrates that the rational part of Mono's decomposition of $g_{k+1,D}$ , denoted $P g_{k+1,D}$ , gives rise to a weak quantum modular form of weight $2k+2$ with Haupttypus for the full modular group.
Fourier Coefficients: The paper establishes that the Fourier coefficients of $g_{k+1,D}$ are real in every connected component of the upper half-plane excluding the exceptional set. It also discusses the relationship between the Fourier coefficients of $G_{-k,D}$ and $g_{k+1,D}$ via Eichler integrals, though it notes that deriving explicit formulas for the local part of $G_{-k,D}$ via cycle integrals remains an open challenge.

Significance and Claims
The paper contributes to the direction of a "unifying theory" for the examples of locally harmonic Maaß forms and their cycle integral representations by demonstrating that the representations of $f_{k,D}$ and $g_{k+1,D}$ are not unique and depend on a parameter $m$ . This provides a broader structural context for these objects, though the author explicitly acknowledges that a complete unifying theory is not yet established.

The significance is framed as follows:

It resolves the question of uniqueness regarding the trace of cycle integral representations for these specific modular forms by showing multiple valid representations exist.
It connects the theory of locally harmonic Maaß forms with quantum modular forms by explicitly identifying the non-modular part of $g_{k+1,D}$ as a quantum modular form.
It offers a potential pathway for finding preimages of seed functions under the Bol and shadow operators, although the author modestly notes that finding a representation for $G_{-k,D}$ as a trace of cycle integrals (analogous to the other forms) has "remained fruitless so far" for the base case $m=0$ , suggesting that other cases might yield new perspectives.

The paper does not claim to solve the problem of finding a representation for $G_{-k,D}$ generally, nor does it propose new experimental applications. Instead, it focuses on the theoretical expansion of existing examples and the exploration of the non-uniqueness of their integral representations.

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