Zeros of Hecke polynomials arising from weak eigenforms

1. Problem Statement and Context

The paper addresses the distribution of zeros of modular forms and associated polynomials, a topic initiated by the seminal work of Rankin and Swinnerton-Dyer (1976). They proved that the zeros of holomorphic Eisenstein series $E_k(\tau)$ on the full modular group $SL_2(\mathbb{Z})$ lie on the lower boundary of the standard fundamental domain $\mathcal{F}$, specifically on the arc of the unit circle $A = \{ \tau \in \mathbb{H} : |\tau|=1, -1/2 \leq \text{Re}(\tau) \leq 0 \}$.

Subsequent research extended this phenomenon to:

• Divisor polynomials of modular forms (zeros in $[0, 1728]$ via the $j$-invariant).
• Hecke polynomials derived from the $j$-function (Asai, Kaneko, Ninomiya).
• Specific harmonic Maass forms (Ono and Gomez, 2024).

The Gap: While these results exist for holomorphic forms and specific examples of harmonic Maass forms, there was no general theory establishing that the zeros of Hecke polynomials arising from a broad class of weak Hecke eigenforms (specifically harmonic Maass forms of weight $2-k$) lie on the arc $A$ (or map to $[0, 1728]$) and are simple.

The Objective: Gomez generalizes previous results to show that for any weak Hecke eigenform $F$ of weight $2-k$ with real Fourier coefficients, the associated Hecke polynomials $P_n(F; x)$ have all their zeros in the interval $[0, 1728]$ for sufficiently large $n$, and these zeros are simple.

2. Methodology

The proof combines the theory of harmonic Maass forms, Hecke operators, and asymptotic analysis of Maass-Poincaré series.

A. Construction of Hecke Polynomials

Let $F(\tau)$ be a weight $2-k$ harmonic Maass weak eigenform with real Fourier coefficients and shadow $G(\tau) = \xi_{2-k}(F)$. The author constructs a function $H_n(F; \tau)$ by applying the Hecke operator $T_{2-k}(n)$ to $F$ and subtracting the eigenvalue term:
$$H_n(F; \tau) := \Delta(\tau)^{b(k-2)} \tilde{E}_{k-2}(\tau) \cdot \left( n^{k-1} F(\tau) | T_{2-k}(n) - a(n)F(\tau) \right)$$
where $\Delta$ is the modular discriminant, $\tilde{E}_{k-2}$ is a specific Eisenstein series (or product of $E_4, E_6$) chosen to cancel trivial zeros, and $b(k)$ is a floor function related to the weight.

• Key Insight: Because $F$ is a weak eigenform, the non-holomorphic part of $F$ is an eigenfunction of the Hecke operator. The subtraction $n^{k-1} F | T - a(n)F$ eliminates the non-holomorphic part, leaving a weakly holomorphic modular function.
• Since $H_n(F; \tau)$ is a modular function of weight 0, it can be expressed as a polynomial in the $j$-invariant: $P_n(F; j(\tau)) = H_n(F; \tau)$.

B. Decomposition via Maass-Poincaré Series

To analyze the zeros, the author expresses the weak eigenform $F$ as a linear combination of Maass-Poincaré series $F_{2-k, -l}(\tau)$.
$$F(\tau) = \sum_{l=1}^m c_F^+(-l) F_{2-k, -l}(\tau)$$
This allows the Hecke action to be tracked explicitly. The action of $T_{2-k}(n)$ on these series is known, allowing the author to write $H_n(F; \tau)$ as a sum of transformed Poincaré series.

C. Asymptotic Analysis on the Arc $A$

The core of the proof involves analyzing the behavior of $H_n(F; \tau)$ on the arc $A$ (where $\tau = e^{i\theta}, \theta \in [\pi/3, \pi/2]$).

1. Dominant Term Isolation: The author isolates the "main term" of the Poincaré series expansion, which corresponds to the identity coset and the inversion coset. This term behaves like a damped cosine wave:
   $$f_{mn}(\theta) \approx 2 \cos\left( \frac{k-2}{2}\theta + 2\pi mn \cos \theta \right)$$
2. Tail Control: The remaining terms (the "tail" of the series and the error from the Hecke action on lower indices) are bounded using:
   • Whittaker/Bessel function bounds: Specifically bounds on the $M$-Whittaker function and $I$-Bessel functions appearing in the Fourier coefficients.
   • Kloosterman sums: Estimating the growth of these sums.
   • Triangle Inequality: Showing that for large $n$, the magnitude of the error terms is strictly less than the amplitude of the dominant cosine term.

3. Key Contributions and Results

Theorem 1.1: Existence and Degree

• Establishes that for any weak Hecke eigenform $F$ of weight $2-k$, the constructed function $H_n(F; \tau)$ is a weakly holomorphic modular function.
• Proves that $H_n(F; \tau) = P_n(F; j(\tau))$ where $P_n(F; x)$ is a monic polynomial of degree $mn - b(k-2)$ (where $m$ is the depth of the pole of $F$).

Theorem 1.2: Location and Simplicity of Zeros

• Main Result: If $n$ is sufficiently large (specifically satisfying $C_F n^{k-1} e^{-\pi n \sqrt{3}/2} < 1$), then the polynomial $P_n(F; x)$ has $mn - b(k-2)$ distinct (simple) zeros, all lying in the interval $[0, 1728]$.
• Mechanism: The proof shows that the function $H_n(F; e^{i\theta})$ oscillates sufficiently to change sign exactly $mn - b(k-2)$ times on the arc $A$. By the Intermediate Value Theorem, there is a zero in each subinterval. Since the map $j: A \to [0, 1728]$ is a bijection, these correspond to real zeros in the specified interval.
• Special Cases: The paper provides criteria for when $0$ or $1728$ are zeros, depending on $k \pmod{6}$ or $k \pmod{4}$.

Technical Bounds

The paper derives explicit constants ($C_F$) and bounds for the error terms involving the coefficients of $F$, ensuring the "damped cosine" approximation holds for large $n$.

4. Significance

1. Generalization: This work significantly broadens the scope of the Rankin-Swinnerton-Dyer phenomenon. It moves from holomorphic forms and specific examples to a general class of harmonic Maass forms (weak eigenforms).
2. Unification: It unifies the study of zeros for holomorphic modular forms, meromorphic Poincaré series, and harmonic Maass forms under a single framework of Hecke polynomials.
3. Methodological Advancement: The technique of isolating a dominant cosine term via Maass-Poincaré series and controlling the tail using Whittaker/Bessel bounds provides a robust toolkit for analyzing the asymptotic distribution of zeros in non-holomorphic settings.
4. Applications: Understanding the zeros of these polynomials is crucial for the theory of mock modular forms, partition congruences, and the arithmetic properties of modular forms. The result confirms that the "real" nature of the zeros is a stable property even when moving from holomorphic to harmonic Maass forms.

In summary, Gomez proves that the geometric distribution of zeros (on the unit circle arc) and the arithmetic distribution of their images (in $[0, 1728]$) are intrinsic properties of weak Hecke eigenforms, extending a classical theme in modular form theory to the realm of harmonic Maass forms.