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On the Real Zeroes of Half-integral Weight Hecke Cusp Forms, II

This paper demonstrates that for a significant proportion of half-integral weight Hecke cusp forms in Kohnen plus subspaces with weight up to $K$ , the number of real zeroes grows at the expected rate, a result achieved by establishing sharp bounds for the mollified first and second moments of quadratic twists of modular $L$ -functions.

Original authors: Jesse Jääsaari

Published 2026-07-30

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CC BY 4.0

Original authors: Jesse Jääsaari

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 standing in a vast, infinite ocean of numbers, where the waves are not made of water, but of invisible patterns called "cusp forms." These aren't just any patterns; they are the mathematical DNA of symmetry, appearing in everything from the geometry of space to the secrets of prime numbers. For decades, mathematicians have been trying to understand where the "zeroes" of these patterns hide. A zero is simply a spot where the wave flattens out completely, touching the sea level.

In the world of these special waves, there are two main types: "integral weight" and "half-integral weight." Think of integral weight waves as smooth, predictable surfers who follow strict rules. Half-integral weight waves, however, are the rebellious cousins. They are trickier, more chaotic, and their internal numbers (called Fourier coefficients) don't play by the usual multiplication rules. Because of this, figuring out where their zeroes land on specific "real" lines (imaginary roads running through the complex number system) has been a massive headache. The big question is: as these waves get taller and more complex (their "weight" increases), do their zeroes spread out evenly, or do they clump up in weird places? If they spread out evenly, it tells us that the universe of numbers is surprisingly orderly, even in its most chaotic corners.

This paper, written by Jesse Jäätääri, tackles the rebellious half-integral waves. Previous attempts to count their zeroes on these real lines were like trying to count raindrops in a storm using a sieve with holes too big; the results were there, but they were fuzzy and missed a lot of the drops. The author wanted to know if, for a huge number of these waves, the zeroes actually appear at the rate we expect them to—roughly proportional to the size of the wave.

The paper's main finding is a resounding "yes." The author proves that for a vast majority of these half-integral weight waves (specifically, for a number of forms proportional to the square of the weight parameter, $K^2$ ), the zeroes do indeed grow at the expected rate. In plain terms, if you look at a large family of these waves, almost all of them have their zeroes scattered along the real lines exactly as predicted by the theory.

To get this result, the author had to invent a new tool called a "mollifier." Imagine trying to listen to a single violin in a room full of screaming fans. The "noise" of the mathematical fluctuations makes it hard to hear the signal. A mollifier is like a noise-canceling headphone specifically tuned to the math of these waves. It smooths out the wild fluctuations in the numbers, allowing the author to see the underlying pattern clearly. By using this tool, the author could calculate the "moments" (statistical averages) of these waves with much sharper precision than before.

The paper explicitly rules out the idea that the zeroes are sparse or irregular for most of these forms. While earlier work showed that zeroes exist, it couldn't guarantee they appeared frequently enough for a "positive proportion" of the forms without losing accuracy. This paper removes that uncertainty. It doesn't just suggest the zeroes are there; it proves that for a huge chunk of the family of forms, the zeroes are abundant and well-distributed. The author doesn't claim to have solved the problem for every single wave (mathematics rarely works that way), but they have shown that the "typical" behavior is exactly what the best theories predicted, finally bringing the rebellious half-integral waves into line with their orderly integral cousins.

Problem Statement
This paper addresses the distribution of "real" zeroes of half-integral weight Hecke cusp forms within the fundamental domain. Specifically, it focuses on zeroes located on the two geodesic segments $\delta_1 := \{s \in \mathbb{C} : \text{Re}(s) = -1/2\}$ and $\delta_2 := \{s \in \mathbb{C} : \text{Re}(s) = 0\}$ . While the holomorphic Quantum Unique Ergodicity (QUE) conjecture implies that zeroes equidistribute in the bulk of the fundamental domain as the weight $k \to \infty$ , the behavior of zeroes on these specific geodesics is more subtle.

Previous work by the author [14] established that for a positive proportion of forms, the number of real zeroes grows at a rate of $K/Y$ (where $K$ is the weight parameter and $Y$ is the height in the Siegel set), but with a significant loss of logarithmic factors, specifically $K/((\log K)^{3/2+\epsilon} Y)$ . This loss was attributed to the fluctuation in the size of the Fourier coefficients $c_g(|d|)$ at fundamental discriminants $d$ , which prevented the second and fourth moments of these coefficients from being of the same order of magnitude. The goal of this work is to remove these logarithmic losses and prove that for $\gg K^2$ forms in the Kohnen plus subspace, the number of real zeroes grows at the expected rate $\gg K/Y$ .

Methodology
The core strategy follows the approach of the author's previous work [14], which relies on detecting sign changes in the Fourier coefficients $c_g(|d|)$ along short intervals of fundamental discriminants. However, this paper introduces a crucial technical advancement: the use of a mollifier $M_g(d)$ to smooth out the fluctuations in the size of the coefficients.

Mollification: The authors construct a mollifier $M_g(d)$ designed such that the product $|c_g(|d|)|M_g(d)$ is typically of constant size. This is achieved by modeling the central $L$ -values $L(1/2, f \otimes \chi_d)$ (via Waldspurger's formula) using an iterative scheme inspired by Radziwiłł and Soundararajan [26]. The mollifier is a short Dirichlet polynomial constructed from disjoint sets of primes to approximate $L(1/2, f \otimes \chi_d)^{-1/2}$ .
Moment Estimation: The proof hinges on obtaining sharp bounds for the mollified second and fourth moments of the Fourier coefficients.
- Second Moment: Evaluated using Waldspurger's formula, the approximate functional equation, the Petersson trace formula for integral weight forms, and Poisson summation.
- Fourth Moment: This is the primary technical novelty. The authors relate the mollified fourth moment to a random model where Hecke eigenvalues are replaced by independent random variables distributed according to the Sato-Tate measure. By expressing the moment as an expectation value of a random $L$ -function and a random mollifier, they utilize the independence of these variables at primes to simplify the computation.
Random Model and Contour Integration: The authors employ a random Dirichlet series $L(X; d, k)$ and a random mollifier $M_2(X; d)$ . They evaluate the expectation of the product $L(X; d, k)M_2(X; d)$ by shifting contours in the complex plane. A key observation is that the contribution from the pole at $s_2 = -s_1$ in the double integral cancels the main term arising from the off-diagonal part of the Petersson trace formula, a phenomenon similar to that observed in Khan [17] but arising here through different mechanisms.
Sign Change Detection: Once the sharp bounds for the mollified moments are established (specifically, that the fourth moment is bounded by $XK$, matching the order of the second moment), the authors apply Markov's inequality and Cauchy-Schwarz arguments to show that for a positive proportion of forms, the mollified coefficients exhibit sign changes in short intervals.

Key Contributions and Results
The main result is Theorem 1.2, which states that for $\gg K^2$ of the forms $g$ in the union of Kohnen plus subspaces with weights up to $K$ , the number of real zeroes on $\delta_j \cap F_Y$ (for $j \in \{1, 2\}$ ) satisfies:
$\#\{z \in Z(g) \cap \delta_j \cap F_Y\} \gg \frac{K}{Y}$
for $\sqrt{K} \log K \le Y \le K^{1-\vartheta}$ .

This result unifies and improves upon the previous findings in [14] by:

Removing the logarithmic loss $(\log K)^{-3/2-\epsilon}$ .
Achieving the conjectured optimal rate of growth for the number of zeroes.
Establishing the result for a positive proportion of forms in a family of weights, rather than a single weight.

The paper also provides Proposition 4.1 and Proposition 4.2, which establish the sharp bounds for the mollified second and fourth moments, respectively. These propositions are the technical engines driving the main theorem.

Significance
The paper claims significance in the context of the distribution of zeroes of automorphic forms. It demonstrates that the methods used for integral weight forms (specifically those relying on the multiplicativity of coefficients, such as in Lester–Matomäki–Radziwiłł [21]) do not directly apply to half-integral weight forms due to the lack of multiplicativity in the Fourier coefficients. Instead, this work shows that by introducing a suitable mollifier and utilizing random models to handle the non-multiplicative fluctuations, one can recover the expected density of zeroes.

The authors note that while the Generalised Lindelöf Hypothesis suggests a similar bound for integral weight forms, the unconditional result for half-integral weight forms presented here is a significant step forward, as it overcomes the specific analytic difficulties posed by the non-multiplicative nature of the coefficients in the half-integral weight setting. The work verifies a prediction made in [14] that a mollifier could be used to eliminate the logarithmic losses in the count of real zeroes.

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