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An Analogue of the Dedekind Eta Function for Hecke Groups $H(\sqrt{D})$

This paper constructs an analogue of the Dedekind eta function for Hecke groups $H(\sqrt{D})$ associated with real quadratic fields, establishing new holomorphic modular functions and analyzing the asymptotic growth and sign patterns of their Fourier coefficients.

Original authors: Debmalya Basak, Dorian Goldfeld, Winston Heap, Nicolas Robles, Alexandru Zaharescu

Published 2026-03-17

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CC BY 4.0

Original authors: Debmalya Basak, Dorian Goldfeld, Winston Heap, Nicolas Robles, Alexandru Zaharescu

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

Imagine you are a musician trying to compose a new symphony. You have a famous, perfect piece of music from the past called the Dedekind Eta function. It's like a "Golden Standard" melody that has been studied for over a century. This melody has a special property: if you play it in a certain way (transforming the notes), it changes slightly but keeps its soul intact. Mathematicians call this modularity.

For a long time, this "Golden Standard" only worked for one specific type of musical group (the Modular Group, related to the number 1). But what if you wanted to create a similar masterpiece for a different, more complex group of numbers? Specifically, groups related to the square roots of numbers like 5, 13, or 17?

This is exactly what the authors of this paper did. They built a new musical instrument (a new mathematical function) that acts like the Dedekind Eta function but for these new, more complex groups.

Here is a breakdown of their journey, using simple analogies:

1. The New Instrument (The Analogue)

The authors created a function they call $\eta_D(z)$ .

The Old Way: The original Dedekind function is built by multiplying an infinite string of numbers together, like a chain: $(1-q)(1-q^2)(1-q^3)...$
The New Way: To make their new function work for the "Hecke Groups" (the new musical groups), they had to twist this chain. They added a special "filter" based on the properties of the number $D$ (like 5 or 13). Some links in the chain get a positive sign, some get a negative sign, and some get multiplied by complex "roots of unity" (think of these as rotating the notes on a wheel).
The Result: This new chain, $\eta_D(z)$ , behaves beautifully. If you apply the rules of the new group to it, it transforms perfectly, just like the original did. They even created a "super-version" called $\Delta_D(z)$ (by raising $\eta_D$ to a power) that acts exactly like the famous Ramanujan function, but for this new world.

2. The Mystery of the Notes (The Coefficients)

Every musical piece can be broken down into individual notes. In math, these notes are called Fourier coefficients (let's call them $a_D(N)$ ).

The Question: When you play this new symphony, what do the notes look like? Do they get louder and louder in a predictable way? Do they jump up and down randomly?
The Discovery: The authors found that these notes grow incredibly fast—exponentially fast, like a population of bacteria. But it's not just random noise. The notes have a very specific rhythm.
- The Growth: The size of the note $N$ is roughly determined by a formula involving $\sqrt{N}$ . It's like a wave that gets taller and taller as you go further out.
- The Sign Pattern: The notes don't just get big; they flip between positive and negative. This is the "sign pattern." The authors discovered that this flipping isn't random chaos. It follows a strict, repeating cycle determined by the smallest number that cannot be a square in that specific group (called the "least quadratic non-residue").
- The Metaphor: Imagine a pendulum swinging. For some numbers (like $D=17$ ), the pendulum swings in a simple "Left, Right, Right" pattern. For others (like $D=5$ ), the swing is much more jagged and complex because there are "secondary waves" interfering with the main swing.

3. The "Secondary Waves" (Why $D=5$ is special)

The authors noticed something fascinating about the case where $D=5$ .

In most cases, there is one giant "main wave" that dictates the size of the notes.
But for $D=5$ , there is a second wave that is almost as big as the first one. It's like having two drummers playing at nearly the same volume. They interfere with each other, creating a "jagged" or "bumpy" sound (mathematically, a jagged graph).
The authors wrote a detailed "two-term" formula to explain this bumpy behavior, showing exactly how these two waves interact. This is a rare and detailed look at the "noise" in the music.

4. Connecting to Partitions (The Puzzle Pieces)

Finally, the authors found a surprising link between their new musical notes and partitions.

A "partition" is just a way of breaking a number into a sum of smaller numbers (e.g., 4 can be 1+1+1+1, or 2+2, or 3+1, etc.).
The authors showed that their complex new notes ( $a_D(N)$ ) can be calculated by counting specific types of these partitions, but with a twist: they only count partitions where the pieces follow specific rules (like only using numbers that are "non-residues" modulo $D$ ).
The Analogy: It's like saying, "The sound of this new symphony is actually just the total number of ways you can build a tower out of Lego bricks, but you can only use red and blue bricks, and you can't use any bricks that are multiples of 5."

Summary

In short, this paper is about building a new, complex musical instrument based on an old, famous one. The authors:

Constructed the instrument (the new function).
Proved it plays the right notes (modularity).
Analyzed the sound waves (the growth and sign patterns of the coefficients).
Discovered that the sound is a mix of a giant wave and a slightly smaller interfering wave (especially for $D=5$ ).
Connected the sound to the art of counting puzzle pieces (partitions).

They have given mathematicians a new tool to understand the hidden rhythms of numbers in these complex groups, revealing that even in the most chaotic-looking sequences of numbers, there is a beautiful, predictable structure waiting to be heard.

1. Problem Statement and Motivation

The paper addresses the construction of holomorphic modular forms for Hecke groups $H(\lambda)$ , specifically where $\lambda = \sqrt{D}$ and $D \equiv 1 \pmod 4$ is a fundamental discriminant of a real quadratic field.

Context: While the classical Dedekind eta function $\eta(z)$ and the Ramanujan cusp form $\Delta(z)$ are fundamental to the theory of the modular group $SL(2, \mathbb{Z})$ , analogous constructions for Hecke groups with $\lambda > 2$ (which are Fuchsian groups of the second kind with infinite area fundamental domains) have been elusive.
Challenge: Standard square-integrable cusp forms do not exist for these groups in the holomorphic category due to the infinite area of the fundamental domain. The authors aim to construct a new family of holomorphic modular functions that vanish at the cusp at infinity but are not square-integrable, serving as analogues to $\eta(z)$ and $\Delta(z)$ .
Specific Gap: Previous work focused on non-holomorphic Eisenstein series and Maass cusp forms. This paper seeks to fill the gap in the holomorphic theory for these specific groups.

2. Methodology

The authors employ a multi-faceted approach combining analytic number theory, complex analysis, and the theory of modular forms:

Construction of the Eta Analogue:
They define a function $\eta_D(z)$ for $D \equiv 1 \pmod 4$ using an infinite product twisted by the primitive real Dirichlet character $\chi_D$ modulo $D$ .
$\eta_D(z) := q^{m_D} \prod_{n=1}^\infty \left( (1-q^n)^{\chi_D(n)} \prod_{a=1}^D (1 - e^{2\pi i a/D} q^n)^{\chi_D(a)} \right)$
where $q = e^{2\pi i z / \sqrt{D}}$ and $m_D$ is derived from $L(-1, \chi_D)$ .
- For $D=5$ , they define a specific analogue $\eta_5(z)$ and its fifth power $\Delta_5(z) = \eta_5(z)^5$ .
Modularity Proofs (Weil's Method):
To prove modularity, the authors generalize Weil's proof for the classical eta function. They introduce a "dual" product function $\Phi_D^\#(z)$ and establish a modular relation between $\Phi_D(z)$ and $\Phi_D^\#(z)$ under the inversion $z \mapsto -1/z$ . This involves:
- Defining auxiliary functions $F_D(z)$ and $F_D^\#(z)$ as logarithms of the products.
- Utilizing Mellin transforms to relate these functions to Dirichlet $L$ -functions and the Riemann zeta function.
- Shifting contours in the complex plane to derive functional equations, ultimately showing that $\eta_D(z)$ transforms with a specific multiplier system (a root of unity) under the generators of $H(\sqrt{D})$ .
Asymptotic Analysis (Circle Method):
To study the Fourier coefficients $a_D(N)$ of $\eta_D(z)$ , the authors apply a variation of the Hardy-Littlewood Circle Method (specifically Cauchy's integral formula).
- They analyze the generating function $G(q)$ near its dominant singularities on the unit circle.
- They identify the singularities at roots of unity $e(u/v)$ , determining that the dominant singularity occurs at $v = n_D$ , the least quadratic non-residue modulo $D$ .
- They use stationary phase approximation to evaluate the integrals, separating the leading exponential growth from oscillatory terms.
Connection to Partitions:
The paper establishes exact identities linking $a_D(N)$ to twisted partition functions and classical partition theory, utilizing Euler's Pentagonal Number Theorem and properties of quadratic residues/non-residues.

3. Key Contributions and Results

A. Construction of Modular Forms

Theorem 1.6 & 1.10: The authors prove that $\Delta_5(z)$ (for $D=5$ ) and $\eta_D(z)$ (for $D > 5$ ) are holomorphic modular forms for the Hecke group $H(\sqrt{D})$ .
They explicitly calculate the multiplier system $u_\gamma$ (a root of unity) such that $\eta_D(\gamma z) = u_\gamma \eta_D(z)$ .
They show these functions vanish at the cusp $\infty$ .

B. Asymptotic Growth of Fourier Coefficients

The paper provides a precise asymptotic formula for the Fourier coefficients $a_D(N)$ of $\eta_D(z)$ (Theorem 1.17).

Growth Rate: The coefficients exhibit exponential growth of the form $\exp(2\sqrt{c_D N})$ , where $c_D = \frac{\sqrt{D} L(2, \chi_D)}{n_D^2}$ . This places them in an intermediate regime between the polynomial growth of $\Delta(z)$ and the exponential growth of the partition function $p(N)$ .
Sign Patterns: The sign of $a_D(N)$ $a_{D} (N)$ is governed by a singular series $S_D(N)$ $S_{D} (N)$ , which is periodic modulo $n_D$ $n_{D}$ .
- If $n_D = 2$ , signs alternate: $(-1)^N$ .
- If $n_D = 3$ , signs follow a pattern of $2, -1, -1$ (or similar) based on $N \pmod 3$ .
- The paper provides explicit formulas for the phase shifts $\beta_D(u, n_D)$ involving $L$ -values.

C. Two-Term Asymptotic Expansion (Case $D=5$ )

For the specific case $D=5$ , the authors derive a two-term asymptotic expansion (Theorem 1.25).

Unlike $D=13$ where the secondary term is negligible, for $D=5$ , the secondary exponential term is significant and close in magnitude to the leading term.
This explains the "jagged" oscillation observed in the plots of $a_5(N)$ compared to $a_{13}(N)$ .
The expansion includes a cosine factor that is out of phase with the leading sign factor, accounting for complex sign changes.

D. Exact Identities

The paper derives an exact combinatorial formula (Equation 7.5) expressing $a_D(N)$ as a sum involving:

Partitions into quadratic non-residues ( $p_{NR}$ ).
Generalized partition functions with roots of unity weights ( $p_{ord}$ ).
Coefficients from Euler's Pentagonal Number Theorem.

4. Significance and Implications

New Class of Modular Forms: The paper successfully constructs the first known family of holomorphic modular forms for Hecke groups $H(\sqrt{D})$ with $D > 5$ that vanish at the cusp, expanding the landscape of automorphic forms beyond $SL(2, \mathbb{Z})$ and congruence subgroups.
Arithmetic Insight: The results link the analytic behavior of these forms (growth and signs) directly to deep arithmetic invariants: the least quadratic non-residue $n_D$ , class numbers of imaginary quadratic fields, and values of Dirichlet $L$ -functions.
Bridging Fields: By connecting the coefficients to partition theory, the work opens new avenues for studying recurrence relations and congruences for these coefficients, similar to Ramanujan's work on $p(n)$ .
Methodological Advancement: The use of Weil-style arguments combined with detailed stationary phase analysis for non-standard modular groups provides a template for analyzing other Hecke groups and similar infinite-area domains.

5. Conclusion

The authors have successfully generalized the Dedekind eta function to Hecke groups $H(\sqrt{D})$ . They established the modularity of these functions, derived precise asymptotic formulas for their Fourier coefficients (revealing a unique interplay between exponential growth and periodic sign patterns), and connected these coefficients to the theory of partitions. The work resolves the existence of such holomorphic forms for these groups and provides a rich framework for future research into their arithmetic properties.