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The Eichler--Selberg trace formula for Hilbert cusp forms, the class numbers of quartic CM fields, and their distributions

Motivated by Su's work on Cohen-type Eisenstein series, this paper introduces generalized Hurwitz class numbers defined via quartic CM fields to establish an Eichler--Selberg trace formula for Hilbert cusp forms over real quadratic fields of narrow class number one, subsequently applying this framework to study distributions, prove class number relations, and compute Hecke operator traces.

Original authors: Seiji Kuga, Andrei Seymour-Howell, Satoshi Wakatsuki

Published 2026-07-03

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Seiji Kuga, Andrei Seymour-Howell, Satoshi Wakatsuki

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 master architect trying to understand the hidden blueprints of a very complex, multi-dimensional building. In the world of mathematics, this building is made of "Hilbert cusp forms"—intricate, high-dimensional shapes that follow strict rules. The architects (mathematicians) want to know: How many rooms are in this building? and What are the specific patterns inside?

This paper by Kuga, Seymour-Howell, and Wakatsuki provides a new, powerful tool to answer these questions. Here is the breakdown of their work using everyday analogies:

1. The Old Blueprint vs. The New Tool

For a long time, mathematicians had a famous tool called the Eichler–Selberg Trace Formula. Think of this as a master key that could count the rooms in a standard, two-dimensional building (related to regular numbers). This key worked by counting something called "Hurwitz class numbers."

The Analogy: Imagine "Hurwitz class numbers" are like counting the different types of unique tiles used to build a floor. In the old world, these tiles were based on simple, imaginary quadratic fields (a specific type of number system).

The authors realized that to build their new, more complex "Hilbert" building (which exists over real quadratic fields, like $\mathbb{Q}(\sqrt{5})$ or $\mathbb{Q}(\sqrt{29})$ ), the old tiles wouldn't fit. They needed a new set of tiles.

2. The New Tiles: Generalized Hurwitz Class Numbers

The authors invented a new type of tile called Generalized Hurwitz Class Numbers.

The Analogy: If the old tiles were based on simple, imaginary gardens, these new tiles are based on Quartic CM fields. Think of these as four-dimensional gardens with complex, symmetrical structures.
The Discovery: The authors showed that the "weight" or "value" of these new tiles is determined by the class numbers of these four-dimensional gardens. In simple terms, they found a way to count the unique symmetries of these complex 4D gardens and use that count to solve problems about their high-dimensional building.

3. The Master Key: The New Trace Formula

The core achievement of the paper is a new version of the Trace Formula.

The Analogy: Imagine you want to know the total number of bricks in a massive wall, but you can't count them one by one. Instead, you have a special scanner (the Trace Formula) that tells you the total count by looking at the shadows cast by the wall.
How it works: The authors built a scanner specifically for their Hilbert building. This scanner says: "To count the rooms (the space of cusp forms), you don't need to look at the rooms directly. Instead, look at the shadows cast by the new 4D garden tiles (the Generalized Hurwitz Class Numbers)."
The Result: They proved that if you sum up these new class numbers in a specific way, you get the exact count of the mathematical objects they are studying.

4. What They Found (The Corollaries)

Once they had this new scanner, they could see things they couldn't see before:

Counting Empty Rooms: They proved that for certain specific weights (like weight 2,2), the building is completely empty for most types of fields, but has exactly one room for specific fields like $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(\sqrt{29})$ .
Class Number Relations: They found a new rule connecting the counts of these 4D gardens. It's like discovering that if you have a certain number of red tiles, you must have a specific number of blue tiles to keep the floor balanced.
Distribution Patterns: They studied how these new class numbers are spread out. They proved that if you look at a huge number of these 4D gardens, the patterns of their symmetries settle into a predictable, smooth curve (similar to how a bell curve appears in statistics). This is called an equidistribution theorem.

5. The Computer Test (The Proof of Concept)

Mathematicians often build theoretical tools and then test them on a computer to make sure they work.

The Experiment: The authors wrote a computer program to calculate these new class numbers for the fields $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(\sqrt{29})$ .
The Check: They used their new formula to predict the "trace" (the count of rooms) for these fields.
- For $\mathbb{Q}(\sqrt{5})$ , the formula predicted the building should be empty. The computer confirmed: Zero rooms.
- For $\mathbb{Q}(\sqrt{29})$ , the formula predicted a specific pattern. The computer confirmed this matched perfectly with known data from elliptic curves (another type of mathematical object).
The Scale: They did this for over 148,000 different prime numbers, calculating millions of class numbers. The fact that the numbers matched perfectly proved their new "scanner" works.

Summary

In short, this paper:

Invented a new way to count complex mathematical objects (Generalized Hurwitz Class Numbers) based on four-dimensional number fields.
Built a new formula (Trace Formula) that uses these counts to determine the size of spaces of Hilbert cusp forms.
Proved that these new counts follow predictable statistical patterns.
Verified everything with massive computer calculations, showing that their new mathematical "blueprint" is accurate.

They didn't just find a new number; they found a new language to describe the hidden architecture of these complex mathematical worlds.

Technical Summary: The Eichler–Selberg Trace Formula for Hilbert Cusp Forms, Class Numbers of Quartic CM Fields, and Their Distributions

Problem Statement
The paper addresses the need for an explicit Eichler–Selberg trace formula for the space of holomorphic Hilbert cusp forms over real quadratic fields of narrow class number one. While classical trace formulas for modular forms over $\mathbb{Q}$ relate traces of Hecke operators to Hurwitz class numbers (associated with imaginary quadratic fields), a direct analogue for Hilbert modular forms requires a generalization of these class numbers. Specifically, the authors seek to express the traces of Hecke operators acting on Hilbert cusp forms in terms of arithmetic invariants of quartic CM fields (specifically, extensions $F(\sqrt{-n})$ where $F$ is a real quadratic field). Additionally, the paper investigates the distribution and asymptotic behavior of these generalized class numbers.

Methodology
The authors employ a combination of adelic automorphic forms theory, orbital integral computations, and algebraic number theory.

Generalized Hurwitz Class Numbers: Motivated by Su's construction of Cohen-type Eisenstein series of half-integral weight over totally real fields, the authors define generalized Hurwitz class numbers $H_F(n)$ . Unlike the classical case where $H(n)$ relates to imaginary quadratic fields, $H_F(n)$ is defined via the class numbers of quartic CM fields $F(\sqrt{-n})$ . The definition involves the class number of the extension, the index of unit groups, and divisor sums involving Artin symbols.
Trace Formula Derivation: The core of the paper is the derivation of the Eichler–Selberg trace formula (Theorem 1.1).
- The authors utilize the adelic formulation of Hecke operators and the trace formula for $GL(2)$ over a totally real field.
- They avoid the complex orbital integral computations over individual $p$ -adic fields found in previous general formulas (e.g., Saito [Sai84]) by adapting the method of Koyama and Lim [KL06]. This approach allows for a direct relation between global orbital integrals of elliptic conjugacy classes and the generalized Hurwitz class numbers.
- The proof relies on the assumption that the narrow class number of the real quadratic field $F$ is 1, which simplifies the structure of the arithmetic quotient and the unit group actions.
Resolvent and Distribution Analysis: To study the distribution of the class numbers, the authors derive a resolvent trace formula for Hecke operators. This involves analyzing the generating function of the traces and applying Tauberian theorems and properties of Chebyshev polynomials to establish equidistribution results.
Numerical Verification: The theoretical results are implemented numerically for the fields $F = \mathbb{Q}(\sqrt{5})$ and $F = \mathbb{Q}(\sqrt{29})$ . The authors compute class numbers unconditionally (without assuming the Generalized Riemann Hypothesis) using PARI/GP and verify the trace formula against known dimensions of cusp form spaces and Sato–Tate distributions.

Key Contributions and Results

The Eichler–Selberg Trace Formula (Theorem 1.1): The authors establish a trace formula for the space $S_\kappa(SL_2(\mathfrak{o}_F))$ where $\kappa = (\kappa_1, \kappa_2)$ are even integers $>1$ . The trace of the Hecke operator $T_{n,F}$ is expressed as a sum over $t \in \mathfrak{o}_F$ of the product of generalized Hurwitz class numbers $H_F(4n - t^2)$ and Chebyshev polynomials.
$\text{tr } T_{n,F} | S_\kappa = \frac{1}{2} \sum_{t \in \mathfrak{o}_F} H_F(4n - t^2) P_{\kappa_1}(t, n) P_{\kappa_2}(\sigma(t), \sigma(n)) + \text{correction terms}$
This formula explicitly links the spectral side (traces) to the geometric side (class numbers of quartic CM fields).
Dimension Formulas and Class Number Relations: As corollaries, the paper derives dimension formulas for spaces of Hilbert cusp forms (Corollary 1.3). In cases where the dimension is zero (e.g., weight $(2,2)$ for specific discriminants), the trace formula yields new class number relations (Corollary 1.5), analogous to the classical relations of Hurwitz and Kronecker.
Distribution and Equidistribution: The paper proves an optimal estimate for weighted averages of the generalized Hurwitz class numbers (Corollary 1.11) and establishes an equidistribution theorem (Theorem 1.12). It shows that as the norm of a prime $p$ grows, the values of the normalized traces (and thus the associated class number sums) become equidistributed with respect to a specific measure on $[-1, 1] \times [-1, 1]$ related to the Sato–Tate distribution.
Numerical Computations: The authors provide extensive numerical data for $\mathbb{Q}(\sqrt{5})$ and $\mathbb{Q}(\sqrt{29})$ .
- For $\mathbb{Q}(\sqrt{5})$ , they verify that the trace is zero for weight $(2,2)$ (as the space is empty) and compute Hecke eigenvalues for weights $(8,8)$ and $(4,8)$ , confirming the Sato–Tate conjecture.
- For $\mathbb{Q}(\sqrt{29})$ , they verify the trace against the modularity of a specific elliptic curve for weight $(2,2)$ and compute eigenvalues for weight $(2,6)$ .
- The computations involved calculating millions of generalized Hurwitz class numbers, demonstrating the feasibility of the formula for practical number-theoretic computations.

Significance
The paper claims significance in providing a "transparent alternative proof" of the trace formula that directly relates global orbital integrals to generalized Hurwitz class numbers, avoiding the technical complexity of local orbital integrals found in earlier works. By expressing the trace in terms of class numbers of quartic CM fields, the authors provide a tool to study the distribution of these class numbers, which are otherwise difficult to analyze directly. The work also offers a computational framework for verifying class number computations and studying Hecke eigenvalues for Hilbert modular forms, potentially serving as a method to generate class number data without relying on the Generalized Riemann Hypothesis. The authors note that while the formula itself is not essentially new in its existence (following from Saito's work), the specific form and its application to the distribution of quartic CM class numbers are novel contributions.