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Interlacing for zeros of the Serre derivative of Eisenstein series

This paper extends the classical Rankin-Swinnerton-Dyer result on the location of zeros of Eisenstein series by proving that the zeros of their Serre derivatives interlace with each other and with the zeros of higher-weight Eisenstein series on the lower arc of the fundamental domain, thereby establishing that the zeros of the cuspidal projection of these derivatives also lie on this arc.

Original authors: Maggie Bohanek, Owen McGinty, Erick Ross, Yanhui Su, Hui Xue

Published 2026-08-18

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CC BY 4.0

Original authors: Maggie Bohanek, Owen McGinty, Erick Ross, Yanhui Su, Hui Xue

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

In the vast landscape of number theory, mathematicians often study objects called modular forms. These are highly symmetric functions that behave in very specific ways when you transform the space they live in, much like how a snowflake looks the same after you rotate it. Among the most famous of these are the Eisenstein series, a family of functions that have been studied for over a century. A key question about these functions is where their zeros lie. A zero is simply a point where the function's value drops to nothing. For decades, mathematicians have known that for the standard Eisenstein series, almost all of these zeros cluster along a specific curved path in the complex plane, a path that looks like the bottom edge of a slice of a circle. This discovery, made in 1970, revealed a hidden order within these abstract functions.

Building on this foundation, a new study by Maggie Bohanek, Owen McGinty, Erick Ross, Yanhui Su, and Hui Xue investigates a slightly different version of these functions. They looked at what happens when you apply a special mathematical operation, known as the Serre derivative, to the Eisenstein series. This operation creates a new function that is closely related to the original but has its own unique properties. While a recent result confirmed that the zeros of this new function also stay on that same curved path, the exact location of each zero remained a mystery. The researchers wanted to know not just that the zeros were there, but precisely where they sat relative to one another and how they moved as the functions changed.

The team began by developing a set of extremely precise estimates to map the location of these zeros. They treated the problem like a surveyor charting a coastline, first identifying rough positions and then refining them with increasing accuracy until they could pinpoint the zeros to a very high degree of certainty. With these maps in hand, they were able to prove four major results about how these zeros behave. First, they showed that if you take two different functions from this family, the zeros of the one with the higher weight are always nestled between the zeros of the one with the lower weight. This is a property known as interlacing, similar to how the teeth of two combs might fit together if you slide them past each other, though here the "teeth" are points on a curve.

Second, the researchers classified exactly when this interlacing happens. They found that for most pairs of these functions, the zeros fit together perfectly in an alternating pattern. However, they also identified a specific list of exceptions where this perfect pattern breaks down, providing a complete and precise rule for when the zeros will and will not interlace. Third, they discovered that the zeros of this new derivative function always interlace with the zeros of the next standard Eisenstein series in the sequence. This connection links the new derivative functions directly back to the original, well-understood family of functions.

Finally, the team used this last finding to solve a related problem about the "cuspidal projection" of these functions. In simple terms, this is a way of stripping away certain parts of the function to leave behind a core component that mathematicians are particularly interested in. The researchers proved that the zeros of this stripped-down version also lie on that same curved path. This extends a previous result and confirms that the hidden order of the zeros is robust, persisting even when the functions are modified or simplified. By combining rigorous calculation with a new, uniform approach to estimating these locations, the authors have provided a clear and complete picture of how these zeros are arranged, turning a vague understanding of their location into a precise mathematical map.

Technical Summary: Interlacing for Zeros of the Serre Derivative of Eisenstein Series

Problem Statement
The paper investigates the location and interlacing properties of the zeros of the Serre derivative of Eisenstein series, denoted $\vartheta_k(E_k)$ . While Rankin and Swinnerton-Dyer (1970) established that the non-elliptic zeros of the Eisenstein series $E_k$ lie on the lower arc of the fundamental domain, and Sugibayashi recently proved that the zeros of $\vartheta_k(E_k)$ also lie on this arc, precise quantitative estimates for their locations were lacking. Furthermore, the interlacing behavior of these zeros—specifically how the zeros of $\vartheta_\ell(E_\ell)$ relate to $\vartheta_k(E_k)$ , and how $\vartheta_k(E_k)$ relates to $E_{k+2}$ —remained unproven. The authors frame this work within the philosophy that $\vartheta_k(E_k)$ acts as a modular analog to the product $E_2 \cdot E_k$ , extending known interlacing results for products $E_w E_k$ (where $w \ge 4$ ) to the case $w=2$ .

Methodology
The authors employ a strategy of constructing increasingly precise approximations for the zeros of the function $F_k(\theta)$ , which is a real-valued transformation of $\vartheta_k(E_k)$ defined on the lower arc $A = \{e^{i\theta} : \pi/2 < \theta < 2\pi/3\}$ . The methodology proceeds in four stages of approximation:

Presample Zeros ( $\theta^{**}_{k,i}$ ): Defined as the zeros of the simplest approximation $F^{**}_k(\theta) = \sin(k\theta/2)$ . These are explicitly given by integer multiples of $2\pi/k$ .
Sample Zeros ( $\theta^*_{k,i}$ ): Defined as the zeros of the next approximation $F^*_k(\theta)$ , which includes a correction term involving $B_0(\theta)$ . The authors derive an asymptotic expansion for these zeros in terms of the presample zeros, accurate to order $O(1/k^3)$ .
Postsample Zeros ( $\hat{\theta}_{k,i}$ ): Defined as the zeros of $\hat{F}_k(\theta)$ , which adds a term involving $d(\theta)^{-k}$ . A key technical innovation is the introduction of a variable $W$ dependent on $s = k(2\pi/3 - \theta)$ , allowing for a highly accurate approximation near the endpoint $\theta = 2\pi/3$ . This avoids the need for case-by-case analysis of zero "movement" near the boundary, a difficulty in previous works.
Actual Zeros ( $\theta_{k,i}$ ): The authors prove that the actual zeros of $F_k(\theta)$ are exponentially close to the postsample zeros (specifically, within $O(1/k^2)$ ).

Using these precise location estimates, the authors analyze the distances between consecutive zeros and the relative positions of zeros for different weights to establish interlacing properties.

Key Contributions and Results
The paper establishes four main theorems regarding the zeros of $\vartheta_k(E_k)$ :

Stieltjes Interlacing: For all even integers $\ell > k \ge 4$ , the zeros of $\vartheta_\ell(E_\ell)$ Stieltjes interlace with the zeros of $\vartheta_k(E_k)$ on the lower arc. This means that between any two consecutive zeros of the lower weight series, there is at least one zero of the higher weight series.
Standard Interlacing Classification: The paper provides a precise classification of when the zeros of $\vartheta_\ell(E_\ell)$ standardly interlace with those of $\vartheta_k(E_k)$ (where "standard" implies exactly one zero of one sequence lies between any two consecutive zeros of the other). This occurs if and only if $\ell - k \in \{2, 4, 6, 8, 12\}$ or if $(k, \ell)$ belongs to a specific finite list of exceptional pairs (e.g., $(4, 14), (6, 16)$ , etc.).
Interlacing with $E_{k+2}$ : For all even integers $k \ge 4$ , the zeros of $\vartheta_k(E_k)$ interlace with the zeros of $E_{k+2}$ on the lower arc. Specifically, each zero of $E_{k+2}$ lies just to the right of a corresponding zero of $\vartheta_k(E_k)$ .
Zeros of the Cuspidal Projection: As an application of the third result, the authors show that for $k \notin \{4, 6, 8, 12\}$ , all non-cuspidal, non-elliptic zeros of the cuspidal projection $\text{cuspproj}(\vartheta_k(E_k)) = \vartheta_k(E_k) + \frac{k}{12}E_{k+2}$ lie on the lower arc. This extends a previous result by Xue and Zhu.

Significance and Claims
The authors claim that their approach offers a "uniform" method that is conceptually simpler than previous techniques used for Eisenstein series zeros. By utilizing the postsample approximation term, they avoid the complex case analysis required near the endpoint $\theta = 2\pi/3$ in earlier studies (such as those by Nozaki, Griffin et al., and Frendreiss et al.). This uniformity allows for cleaner proofs of interlacing properties and suggests the method may be amenable to generalization in other settings where zero movement near boundaries is a technical hurdle.

The paper also notes a connection to Markov's theorem on the interlacing of polynomial derivatives, observing that their results for weights that are multiples of 12 can be viewed as a modular analog of Markov's result. The work confirms that the "modified product" $\vartheta_k(E_k)$ behaves analogously to the product $E_w E_k$ (for $w \ge 4$ ) regarding the location and interlacing of its zeros, thereby validating the philosophical framework of treating the Serre derivative as a natural generalization of the product $E_2 \cdot E_k$ .

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