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The variation of zeros of the Miller basis

This paper establishes a connection between the distribution of zeros in the Miller basis of modular forms and a logarithmic version of the Szegő curve, demonstrating that these zeros lie on specific arcs or curves depending on the ratio $\delta=m/\ell$ and providing conjectural thresholds and enumerations for algebraic zeros.

Original authors: Liubomir Chiriac, Andrei Jorza

Published 2026-05-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Liubomir Chiriac, Andrei Jorza

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 cartographer trying to map the location of hidden treasures. In the world of mathematics, these "treasures" are zeros (the points where a function equals zero) of special mathematical objects called modular forms. These forms are like complex, vibrating strings that exist in a specific geometric landscape known as the "upper half-plane."

For decades, mathematicians knew that for a specific, famous type of these forms (called Eisenstein series), all the treasures were hidden along a single, elegant curve: a slice of the unit circle. It was like finding all the gold coins scattered neatly along a single arc of a beach.

This paper, by Liubomir Chiriac and Andrei Jorza, investigates a different, more diverse family of these forms called the Miller basis. They ask: If we change the "recipe" for these forms, do the treasures stay on that same beach, or do they scatter to new, stranger locations?

Here is the breakdown of their findings using everyday analogies:

1. The Two Main Characters: The Beach and the Logarithmic Curve

The authors discover that the location of these zeros depends entirely on a ratio, which they call $\delta$ (delta). Think of $\delta$ as a "dial" that you can turn to change the shape of the form.

The "Beach" (The Unit Arc): This is the familiar, curved shoreline where the zeros used to live.
The "Logarithmic Szegő Curve": This is a new, more exotic shape the zeros migrate to. Imagine a curve that looks like a stretched-out, logarithmic spiral or a specific "S" shape that appears when you look at how certain mathematical polynomials behave when they get very large.

2. The Dial of Fate ( $\delta$ )

The paper maps out exactly where the zeros go based on how you turn the dial ( $\delta$ ):

When the dial is low ( $\delta < 0.62$ ): The zeros behave like the old-fashioned forms. They stay entirely on the Beach (the unit arc). It's a safe, predictable zone.
When the dial is high ( $\delta$ close to 1): The zeros abandon the beach entirely. They migrate to the Logarithmic Szegő Curve. It's as if the tide has gone out, and the coins have washed up onto a new, distant shore.
The Middle Ground: In the middle, the zeros split up. Some stay on the Beach, and some move to the new Curve. The authors propose a "Conjecture" (a strong mathematical guess) that the zeros will always be found on the upper outline formed by combining both the Beach and the Curve.

3. The "Thresholds" (The Tipping Points)

The authors didn't just guess; they calculated specific tipping points where the behavior changes.

If your ratio is below 61.94%, you can be 100% sure all zeros are on the Beach.
If your ratio is above 95.46%, you can be sure none of the zeros are on the Beach; they are all on the new Curve.
Between these numbers, it's a mixed bag, but the authors provide a formula to estimate exactly how many stay on the Beach versus how many move.

4. The "Faber Polynomial" Connection

To understand why the zeros move, the authors look at the "DNA" of these forms, which they call Faber polynomials.

Think of the Miller form as a complex machine. The Faber polynomial is the blueprint.
When the "dial" is set to certain values, this blueprint looks like a truncated exponential function (a mathematical function that grows very fast).
It is a known mathematical fact that the zeros of these truncated exponential functions naturally settle on that "Logarithmic Szegő Curve." The authors proved that as the forms get larger (higher weight), the Miller forms start to look more and more like these exponential functions, dragging their zeros along with them to the new curve.

5. The "Algebraic" Treasure Hunt

Finally, the paper looks at the nature of the zeros.

For the old, famous forms, the zeros were "transcendental" (mathematically complex numbers that can't be written as simple fractions or roots of simple equations).
The authors found that for the Miller forms, there are rare, special cases where the zeros are "algebraic" (simpler, more "constructible" numbers).
They acted like detectives, checking every possible form up to a certain size. They found a specific list of forms where the zeros are these special algebraic numbers. Interestingly, these special zeros only appear when the form is very close to the "edge" of its family (specifically when the difference between the total size and the starting point is very small).

Summary

In short, this paper draws a new map for a family of mathematical objects. It shows that their "zeros" (the points where they vanish) don't just stay in one place. Depending on a specific ratio:

They might stay on the familiar circular arc.
They might migrate to a new, logarithmic curve.
Or they might split between the two.

The authors provide the exact mathematical "compass" to predict where these zeros will be, proving that the behavior of these complex forms is governed by a beautiful, predictable transition between two distinct geometric shapes.

Technical Summary: The Variation of Zeros of the Miller Basis

Problem Statement
The paper investigates the asymptotic distribution of zeros of the Miller basis elements $g_{k,m}$ within the space of holomorphic modular forms $M_k$ of even weight $k \ge 4$ and level one. These forms are defined by their Fourier expansions $g_{k,m} = q^m + O(q^{\ell+1})$ , where $\ell = \dim M_k - 1$ and $0 \le m \le \ell$ .

Previous work has established specific behaviors for subsets of this family:

Rankin and Swinnerton-Dyer proved that zeros of Eisenstein series lie on the unit arc $A$ in the standard fundamental domain $\mathcal{F}$ .
Duke and Jenkins showed that certain weakly holomorphic forms also have zeros on $A$ .
Raveh demonstrated that for fixed $m$ , if $\ell$ is sufficiently large relative to $m$ , the zeros of $g_{k,m}$ lie on $A$ .
Rudnick showed that if the difference $D = \ell - m$ is fixed while $k \to \infty$ , zeros cluster near vertical lines.

The central problem addressed is to unify these phenomena by analyzing the variation of zeros across the entire family of Miller forms as the ratio $\delta = m/\ell$ varies, particularly when both $k$ and $D$ grow.

Methodology
The authors employ a combination of analytic number theory, complex analysis, and computational verification:

Integral Representation and Circle Method: The authors approximate $g_{k,m}(e^{i\theta})$ on the unit arc using an integral representation derived from the Duke-Jenkins and Raveh frameworks. This involves integrating a kernel function $G(z, \tau)$ over a specific contour and estimating residues at poles corresponding to the identity and the transformation $S = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ .
Error Estimation: By bounding the error terms in the integral approximation, they determine conditions under which $g_{k,m}(e^{i\theta})$ behaves asymptotically like $2\cos(2\pi m \cos \theta + k\theta/2)$ . This allows for the counting of zeros on the arc $A$ based on the oscillation of the cosine term.
Faber Polynomials: The forms $g_{k,m}$ $g_{k, m}$ are related to Faber polynomials $P(j)$ $P (j)$ via the relation $g_{k,m}/(\Delta^\ell E_{k'}) = P(j)$ $g_{k, m} / (Δ^{ℓ} E_{k^{'}}) = P (j)$ . The authors analyze the coefficients of these polynomials as the degree $D = \ell - m$ $D = ℓ - m$ grows.
- For small $D$ (relative to $k$ ), they use Stirling's approximation and properties of Laguerre polynomials to show coefficients behave like those of truncated exponential polynomials.
- For larger $D$ , they derive explicit bounds on the coefficients using combinatorial estimates and properties of the $j$ -invariant expansion.
Szeg˝o Curve Analysis: The asymptotic distribution of zeros of truncated exponential polynomials is known to converge to the Szeg˝o curve $S = \{z \in \mathbb{C} : |z e^{1-z}| = 1\}$ . The authors map this behavior back to the modular domain via the $j$ -invariant, defining a "logarithmic Szeg˝o curve" $L_\pm$ and a shifted curve $S_\delta$ .
Computational Verification: The authors utilize SageMath to numerically compute thresholds for the ratio $\delta$ where zeros transition from the arc $A$ to the curve $S_\delta$ . They also perform exhaustive searches for algebraic zeros for small $D$ .

Key Contributions and Results

Theorem A (Zeros on the Arc): The authors establish a lower bound for the proportion of non-elliptic zeros lying on the unit arc $A$ as a function of $\delta = m/\ell$ .
- If $\delta < 0.6194$ , all zeros lie on $A$ for sufficiently large $k$ .
- If $0.6194 \le \delta < 0.9546$ , a decreasing proportion of zeros lies on $A$ .
- If $\delta \ge 0.9546$ , the lower bound for zeros on $A$ drops to zero.
- These theoretical thresholds align closely with numerical experiments up to $k=600,000$ .
Theorem B (Convergence to Logarithmic Szeg˝o Curve): When $D = \ell - m \to \infty$ but grows slowly relative to $k$ (specifically $D < \alpha \log |k| / \log \log |k|$ ), the zeros of $g_{k,m}$ asymptotically approach a shifted logarithmic Szeg˝o curve $S_\delta$ . This curve is defined as $L_\pm - \frac{1}{2\pi} \log |1-\delta|$ , where $L_\pm$ is the graph of a function derived from the classical Szeg˝o curve.
Conjecture C: The authors propose that for any $\delta > 0$ , the zeros asymptotically approach the curve $C_\delta$ , defined as the upper hull of the union of the unit arc $A$ and the curve $S_\delta$ . Theorems A and B prove parts of this conjecture at the extremes of $\delta$ .
Weakly Holomorphic Forms: Parallel results are derived for weakly holomorphic modular forms (Theorem 9), identifying different threshold values for $\delta$ (e.g., $\delta > 1.1598$ for all zeros on $A$ when $k \ll 0$ ).
Algebraic Zeros (Section 7): Contrasting with Eisenstein series (where Kohnen proved non-elliptic zeros are transcendental), the authors exhibit Miller forms with non-elliptic algebraic zeros.
- They completely classify all such forms for $\ell - m \le 25$ .
- They find that non-elliptic algebraic zeros occur only when $m = \ell - 1$ and the zero corresponds to specific quadratic imaginary fields with class number 1 (discriminants $-19, -27, -43, -67, -163$).
- For $25 < \ell - m \le 100$ , they provide a probabilistic result stating that non-elliptic algebraic zeros are absent for almost all weights, based on Galois group analysis of the Faber polynomials.

Significance
The paper provides a comprehensive framework for understanding the geometric distribution of zeros of the Miller basis, bridging the gap between the "arc" phenomenon (dominant for small $\delta$ ) and the "vertical line" or "Szeg˝o curve" phenomenon (dominant for large $\delta$ or fixed $D$ ).

The authors claim that their work:

Generalizes previous results by Duke, Jenkins, Raveh, and Rudnick into a single continuous parameter family.
Identifies a new asymptotic locus (the logarithmic Szeg˝o curve) for modular form zeros, linking modular forms to the theory of truncated exponential polynomials.
Challenges the universality of the "zeros on the arc" property by demonstrating that for large $\delta$ , zeros migrate to a different curve.
Highlights the arithmetic complexity of these zeros by showing the existence of algebraic zeros, a feature absent in the classical Eisenstein series case, and provides a systematic classification for low-degree cases.

The paper concludes by positing that the transition between these regimes is governed by the ratio $\delta$ , with specific conjectural cutoffs derived from the geometry of the Szeg˝o curve and the unit arc.