Explicit Hecke eigenform product identities for Hilbert modular forms

This paper characterizes product identities of the form $g=f\cdot h$ among Hecke eigenforms over totally real number fields, proving that such identities occur exclusively for the field $\mathbb{Q}(\sqrt{5})$ with exactly two instances, while also demonstrating their non-existence when both factors are Eisenstein series of distinct weights.

The Gist

Imagine you are a master chef in a vast, infinite kitchen. In this kitchen, you have special ingredients called Hilbert Modular Forms. These aren't your average flour or sugar; they are complex mathematical recipes that follow very strict rules.

Some of these recipes are "eigenforms." Think of an eigenform as a perfectly balanced flavor profile. If you apply a specific "taste test" (called a Hecke operator) to it, the flavor doesn't change its character; it just gets scaled up or down. It's like a pure note on a piano that stays pure no matter how loud you play it.

The Big Question: Can You Mix Two Perfect Flavors?

About 30 years ago, a mathematician named William Duke asked a delicious question: "If I take two of these perfect, balanced recipes (let's call them $f$ and $h$) and mix them together to make a new dish ($g = f \times h$), will the new dish also be a perfect, balanced flavor?"

Usually, the answer is no. Mixing two perfect things often creates something messy. But sometimes, by sheer luck or mathematical magic, the mix does stay perfect. Duke found 16 such "magic mixes" in the simplest kitchen (the rational numbers, $\mathbb{Q}$).

The New Discovery: The "Grand Riemann Hypothesis" Kitchen

The authors of this paper (Hao, Qin, and Zhou) decided to explore this question in a much bigger, more complex kitchen: Totally Real Number Fields.

Think of these fields as different "dimensions" or "universes" of numbers.

• The Quadratic Fields: These are like 2D universes (specifically, fields like $\mathbb{Q}(\sqrt{5})$).
• The Higher-Degree Fields: These are 3D, 4D, or even higher-dimensional universes.

They wanted to know: In these complex universes, can we find any new "magic mixes" where two perfect recipes combine to make a third perfect recipe?

The Two Main Ingredients

To solve this, they looked at two types of ingredients:

1. Eisenstein Series: These are the "basic building blocks" of the kitchen. They are like the flour and water—simple, predictable, and always present.
2. Cusp Forms: These are the "secret spices." They are rare, complex, and vanish at the edges of the kitchen (mathematically speaking).

The Findings: A Tale of Two Cases

Case 1: Mixing Two "Basic Building Blocks" (Eisenstein Series)

The authors asked: If I mix two basic recipes, do I get a perfect new one?

• The Result: They found that for almost all universes, the answer is NO.
• The Exception: There is exactly one universe where it works: $\mathbb{Q}(\sqrt{5})$ (a specific 2D universe).
• The "Why": In this specific universe, the "room" (mathematically called the dimension of the space) is just the right size. It's like a small closet where two coats can hang perfectly side-by-side without touching. In any larger universe (with a larger "discriminant"), the room gets too big, and the coats (the mathematical identities) can't fit together perfectly anymore.
• The Magic: They found exactly two specific recipes in this universe that work:
  1. A weight-4 recipe is exactly 60 times the square of a weight-2 recipe.
  2. A weight-8 recipe is exactly 120 times the product of a weight-2 and a weight-6 recipe.

Case 2: Mixing a "Basic Block" with a "Secret Spice"

The authors asked: If I mix a basic recipe with a secret spice, do I get a perfect new one?

• The Result: NO. Never.
• The Catch: To prove this for the most complex scenarios, they had to assume the Grand Riemann Hypothesis is true.
  • Analogy: Imagine the Grand Riemann Hypothesis is a "Law of Physics" for this kitchen. It says that all the "ghosts" (zeros of certain functions) in the kitchen are hiding on a specific invisible line. If we assume this law holds, then the math proves that mixing a basic block with a secret spice never creates a perfect new dish. The flavors simply don't align.

Case 3: The High-Dimensional Universes (Degree 3 and up)

Finally, they looked at the 3D, 4D, and higher kitchens.

• The Result: NO magic mixes exist here.
• The Reason: As the universe gets bigger (higher degree), the "room" for these recipes grows so fast that the mathematical equations required for a "perfect mix" become impossible to satisfy. It's like trying to fit a square peg in a round hole that keeps getting rounder and rounder. The numbers just don't add up.

The "Grand Riemann Hypothesis" Safety Net

The paper relies heavily on a famous, unproven conjecture called the Grand Riemann Hypothesis.

• Simple Explanation: Think of it as a guarantee that the "noise" in the kitchen is perfectly organized. Without this guarantee, the authors can't be 100% sure about the most complex cases (mixing a weight-2 basic block with a secret spice).
• The Takeaway: If the Grand Riemann Hypothesis is true (which most mathematicians believe it is), then the list of "magic mixes" is complete and finite.

Summary for the General Audience

This paper is a mathematical census. The authors went through every possible "universe" of numbers (specifically those with simple structures) and asked, "Can two perfect mathematical recipes mix to make a third perfect one?"

1. In the complex, high-dimensional universes: The answer is a hard NO. The math is too crowded.
2. In the 2D universes: The answer is almost always NO, except for one special universe ($\mathbb{Q}(\sqrt{5})$).
3. In that one special universe: There are exactly two magical combinations that work.

They used powerful computers to check the numbers and deep theoretical logic (assuming the Grand Riemann Hypothesis) to prove that no other hidden combinations exist. It's a definitive "closed case" for this specific type of mathematical puzzle.

Technical Summary

Here is a detailed technical summary of the paper "Explicit Hecke Eigenform Product Identities for Hilbert Modular Forms" by Zeping Hao, Chao Qin, and Yang Zhou.

1. Problem Statement

The paper addresses a fundamental question in the theory of automorphic forms, originally posed by William Duke: Under what conditions is the product of two Hecke eigenforms also a Hecke eigenform?

Specifically, the authors investigate the equation $g = f \cdot h$, where $f, h,$ and $g$ are Hilbert modular forms over a totally real number field $F$. The forms are required to be Hecke eigenforms. The paper focuses on two specific scenarios:

1. Real Quadratic Fields: Enumerating all such product identities over real quadratic fields ($n=2$) with narrow class number one ($h^+=1$).
2. Higher Degree Totally Real Fields: Investigating identities where $f$ and $h$ are Eisenstein series of distinct weights over totally real fields of degree $n \geq 3$.

The study distinguishes between cases where the forms are Eisenstein series (non-cuspidal) and cusp forms. It is noted that the product of two cuspidal eigenforms cannot be an eigenform because the Fourier coefficient at the unit ideal vanishes. Thus, the analysis focuses on:

• The product of two Eisenstein series.
• The product of an Eisenstein series and a cusp form.

2. Methodology

The authors employ a combination of analytic number theory, explicit dimension formulas, and computational verification. The methodology is structured as follows:

A. Analytic Bounds via Dedekind Zeta Functions

For identities involving Eisenstein series, the constant terms of the Fourier expansions provide a necessary condition. If $g = f \cdot h$, the constant terms must satisfy a specific relation involving the Dedekind zeta function $\zeta_F(s)$.

• The authors derive an equation of the form:
  $$1 = \frac{\zeta_F(1-k_1) + \zeta_F(1-k_2)}{\zeta_F(1-k_1-k_2)} \cdot \frac{\zeta_F(1-k_1-k_2)}{\zeta_F(1-k_1)\zeta_F(1-k_2)}$$
• They utilize known bounds for $|\zeta_F(1-k)|$ involving the discriminant $D$ and the Gamma function $\Gamma(k)$. By establishing lower bounds for the right-hand side of the derived inequality, they prove that for sufficiently large $D$ or specific weight combinations, the identity cannot hold.

B. Fourier Coefficient Comparisons at Small Primes

When analytic bounds on constant terms are insufficient (e.g., in the equal-weight Eisenstein case), the authors compare Fourier coefficients at small prime ideals (specifically at primes dividing 2).

• They analyze the Hecke recurrence relations: $c(p^{j+1}) = c(p^j)c(p) - N(p)^{k-1}c(p^{j-1})$.
• By comparing coefficients of $g$ and $f \cdot h$ at small primes, they derive contradictions or strict constraints on the discriminant $D$ and weights $k$.

C. Dimension Formulas and the Grand Riemann Hypothesis (GRH)

For cases involving cusp forms (specifically $g = E_k \cdot h$ where $h$ is cuspidal), the authors rely on the dimension of the space of cusp forms $S_k(\Gamma_F)$.

• Key Theorem (from [16]): If $\dim S_{k_1+k_2}(\Gamma_F) > 1$, then under the Grand Riemann Hypothesis (GRH), the product of a normalized Eisenstein series and a normalized cusp eigenform cannot be an eigenform.
• The authors use explicit dimension formulas for Hilbert cusp forms (involving $\zeta_F(-1)$ and class numbers) to show that for the relevant ranges of $D$ and weights, the dimension is strictly greater than 1. This allows them to rule out non-trivial identities without computing high-dimensional Fourier coefficients.

D. Computational Verification

The theoretical bounds reduce the search space to a finite set of discriminants and weights. The authors use SageMath and Magma to:

• Compute generalized Bernoulli numbers for explicit verification of zeta relations.
• Calculate dimensions of cusp form spaces.
• Verify the non-existence of identities for the remaining small cases (e.g., $D=8, 13, 17, \dots$).

3. Key Contributions and Results

Theorem 1: Real Quadratic Fields ($n=2, h^+=1$)

The paper provides a complete classification of eigenform product identities over real quadratic fields with narrow class number one.

• Existence: Under the GRH, such identities exist only for the field $F = \mathbb{Q}(\sqrt{5})$.
• Explicit Identities: There are exactly two such identities in this field:
  1. $E_4 = 60 E_2^2$ (Product of two weight-2 Eisenstein series).
  2. $h_8 = 120 E_2 \cdot h_6$ (Product of a weight-2 Eisenstein series and a weight-6 cusp form, resulting in a weight-8 cusp form).
     Note: $E_k$ denotes the normalized Eisenstein series, and $h_k$ denotes the unique normalized cusp eigenform of weight $k$.
• Non-existence:
  • For any real quadratic field with discriminant $D > 5$, no identity exists between two Eisenstein series of weight $\geq 2$.
  • No identity exists between an Eisenstein series (weight $\geq 4$) and a cusp form.
  • Under GRH, this non-existence extends to the case where the Eisenstein series has weight 2.

Theorem 2: Totally Real Fields of Degree $n \geq 3$

The authors generalize the results to totally real number fields of degree $n > 2$.

• Result: There are no eigenform product identities $g = f \cdot h$ where $f$ and $h$ are Eisenstein series of distinct weights.
• Method: They prove that the analytic estimates derived from the Hecke L-series and Odlyzko's bounds on discriminants lead to a contradiction with the necessary constant term relations for any $n > 2$.

4. Significance

• Completeness: This work exhaustively classifies eigenform product identities for Hilbert modular forms over all real quadratic fields of narrow class number one, extending previous partial results by Joshi-Zhang and You-Zhang.
• Role of GRH: The paper highlights the critical role of the Grand Riemann Hypothesis in ruling out non-trivial identities involving cusp forms. Without GRH, the dimension argument alone is insufficient for certain edge cases (specifically weight 2 Eisenstein series).
• Dimensional Constraints: The results reinforce the intuition that "dimension reasons" are the primary obstruction to such identities. As the discriminant $D$ increases, the dimension of the relevant modular form spaces grows, making it impossible for the product of two forms to land in a 1-dimensional eigenspace (unless the product is trivial).
• Uniqueness of $\mathbb{Q}(\sqrt{5})$: The paper confirms that $\mathbb{Q}(\sqrt{5})$ is unique in its ability to support these identities due to its minimal discriminant, which keeps the dimension of the cusp form spaces low enough to allow trivial identities to occur.

5. Conclusion

The paper concludes that Hecke eigenform product identities are extremely rare. For real quadratic fields, they occur exclusively in $\mathbb{Q}(\sqrt{5})$ and only for specific low-weight combinations. For higher-degree totally real fields, no such identities exist for Eisenstein series of distinct weights. The authors suggest that their methods can be extended to other cases (e.g., equal-weight Eisenstein series or cusp-Eisenstein products) once sufficient data on special values of L-series and dimension formulas becomes available.