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

This paper investigates the distribution of real zeroes of half-integral weight Hecke cusp forms near the cusp at infinity, demonstrating that for a large subset of these forms, the number of such zeroes approaches the expected rate predicted by an analogue of the Ghosh-Sarnak conjecture, a result achieved through the asymptotic evaluation of averaged moments of quadratic twists of modular $L$ -functions.

Original authors: Jesse Jääsaari

Published 2026-02-09

📖 5 min read🧠 Deep dive

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 on a vast, curved landscape called the "upper half-plane." On this landscape, there are special mathematical waves called cusp forms. These waves aren't made of water or sound, but of pure numbers and geometry. They have a specific "weight" (think of this as their size or complexity), and as this weight gets huge, the waves become incredibly intricate.

The author of this paper, Jesse Jääsaari, is interested in a very specific question: Where do these waves touch the ground (where they become zero)?

The Big Picture: The "Real" Zeroes

Mathematicians have long known that these waves have zeroes scattered all over the landscape. However, there are two special, straight vertical lines on this map (called geodesics) where the waves behave in a very particular way: they stay "real" (no imaginary parts).

A famous guess (the Ghosh–Sarnak conjecture) suggests that as these waves get heavier and heavier, almost all of their zeroes that are close to the "top" of the map (near infinity) will line up perfectly on these two vertical lines. It's like predicting that if you drop a million pebbles into a specific canyon, they will all land on two specific narrow ridges rather than scattering randomly across the floor.

The Challenge: A New Type of Wave

For a long time, this was studied for "standard" waves (integral weight). But this paper looks at a trickier, half-integer version of these waves.

Think of the standard waves like a choir where every singer follows a strict, predictable rulebook (multiplicativity). If you know how one singer sounds, you can predict the others. The half-integer waves, however, are like a choir where the singers only follow the rules when they are singing specific notes (squares), but go rogue otherwise. This makes them much harder to predict using the old rulebooks.

The Strategy: Listening for Sign Changes

To find where the wave hits zero, the author uses a clever trick. Instead of looking for the zero directly, he looks for a sign change.

Imagine a wave going up and down. If it goes from positive (above the ground) to negative (below the ground), it must have crossed the ground (zero) somewhere in between.

The Goal: Prove that for many of these heavy waves, the numbers describing them flip from positive to negative very frequently along those two special vertical lines.
The Method: The author connects these waves to a different set of numbers (Fourier coefficients) and uses a statistical approach. He doesn't try to prove it for every single wave (which is too hard), but rather proves it for a massive number of them.

The Results: Two Main Findings

1. The "Almost All" Result (The Stronger Finding)
The author proves that for a huge collection of these waves (specifically, about $K^2$ of them, where $K$ is the size of the weight), the number of zeroes on those two vertical lines is almost exactly what the famous conjecture predicted.

Analogy: Imagine you have a bag of 1,000,000 dice. You can't predict exactly how many 6s each die will roll, but you can prove that for the vast majority of the dice, the number of 6s will be very close to the expected average. This paper shows that for most of these mathematical waves, the zeroes line up exactly as the experts hoped.

2. The "Positive Proportion" Result (The Broader Finding)
Even if we can't prove it for almost all waves, the author shows that it is definitely true for a significant chunk (at least half) of them.

Analogy: Even if we can't say every person in a city likes pizza, we can prove that at least 50% of them definitely do. This paper guarantees that for a large, positive group of these waves, the zeroes are indeed clustering on those two lines.

How Did They Do It?

The author had to invent new tools because the old ones (which relied on the waves following strict rules) didn't work for these "rogue" half-integer waves.

The New Tool: He used a technique involving "averaging." Instead of looking at one wave at a time, he looked at the average behavior of thousands of waves at once.
The "Mollifier": He used a mathematical "filter" (called a mollifier) to smooth out the noise and see the underlying patterns in the numbers. This is like using noise-canceling headphones to hear a specific voice in a crowded room.

Summary

In simple terms, this paper confirms that a famous guess about where mathematical waves touch the ground is correct for a massive number of these waves, even though these waves are much more chaotic and harder to study than the ones mathematicians used to look at. It's a major step forward in understanding the hidden order within these complex mathematical structures.

Technical Summary: On the Real Zeroes of Half-Integral Weight Hecke Cusp Forms

Problem Statement
This paper investigates the distribution of zeroes of half-integral weight Hecke cusp forms on the manifold $\Gamma_0(4)\backslash\mathbb{H}$ near the cusp at infinity. Specifically, it addresses the analogue of the Ghosh–Sarnak conjecture, which posits that as the weight $k \to \infty$ , the zeroes of such forms located in shrinking Siegel sets $F_Y = \{z \in F : \text{Im}(z) \ge Y\}$ (where $Y \to \infty$ ) concentrate on two specific vertical geodesics: $\delta_1 = \{s \in \mathbb{C} : \text{Re}(s) = -1/2\}$ and $\delta_2 = \{s \in \mathbb{C} : \text{Re}(s) = 0\}$ . These are termed "real" zeroes because the cusp forms, under a specific normalization, take real values on these lines.

While the equidistribution of zeroes in the fundamental domain for half-integral weight forms has been established conditionally on the Generalized Riemann Hypothesis (GRH) by Lester and Radziwiłł, the finer distributional behavior on small scales (near the cusp) had not been explored for this setting. A primary obstacle is that the Fourier coefficients of half-integral weight forms lack the full multiplicativity possessed by integral weight forms, rendering standard techniques from the integral weight case (e.g., those by Ghosh–Sarnak, Matomäki, and Lester–Matomäki–Radziwiłł) inapplicable.

Methodology
The author employs a strategy that reduces the detection of real zeroes to the study of sign changes in the Fourier coefficients of the cusp forms. The core technical approach involves two distinct pathways to establish these sign changes, utilizing the Shimura correspondence which links half-integral weight forms $g$ to integral weight forms $f$ of weight $2k$ .

Reduction to Sign Changes:
Using a steep descent argument (Proposition 6.1), the paper establishes that the value of the cusp form $g$ on the lines $\delta_j$ is essentially determined by a single Fourier coefficient $c_g(\ell)$ for specific $\ell$ . Consequently, finding a zero on these lines corresponds to detecting sign changes in the sequence of Fourier coefficients $c_g(\ell)$ as $\ell$ varies.
Approach for Theorem 1.1 (Many Forms, Optimal Count):
To prove that a large number of forms (specifically $\gg_\varepsilon K^2/(\log K)^{3/2+\varepsilon}$ ) possess the expected number of real zeroes, the author analyzes the Fourier coefficients $c_g(|d|)$ where $d$ ranges over fundamental discriminants.
- Moments of L-functions: The proof relies on estimating the first and second moments of quadratic twists of modular L-functions, $L(1/2, f \otimes \chi_d)$ . By Waldspurger's formula, $|c_g(|d|)|^2$ is proportional to these L-values.
- Averaging: The author averages over the family of forms $g$ and the weight $k$ . This averaging is crucial because the ratio of the logarithm of the conductor to the logarithm of the family size is too large for standard asymptotic formulas without additional averaging.
- Short Intervals: The argument involves showing that for many forms, the sum of coefficients over short intervals $[x, x+H]$ exhibits sign changes. This is achieved by bounding the first moment (Lemma 2.7) and the fourth moment (Lemma 2.8) of the coefficients, utilizing the Petersson formula for half-integral weight forms and the large sieve inequality of Deshouillers and Iwaniec.
Approach for Theorem 1.3 (Positive Proportion of Forms):
To establish a result for a positive proportion of forms, the author exploits the multiplicativity of Fourier coefficients at square arguments.
- Multiplicative Structure: For a fixed fundamental discriminant $d$ , the sequence $c_g(|d|m^2)$ behaves multiplicatively in $m$ .
- Mollification: The proof utilizes Iwaniec–Sarnak mollifiers to handle the size of $c_g(|d|)$ . Proposition 2.1 establishes that for a positive proportion of forms, $|c_g(|d|)|$ is not too small.
- Sign Changes: By applying results on the sign changes of multiplicative functions (specifically adapting arguments from Lester, Matomäki, and Radziwiłł), the paper demonstrates that the sequence $c_g(|d|m^2)$ undergoes sign changes for a positive proportion of forms.

Key Contributions and Results

Theorem 1.1: For $\gg_\varepsilon K^2/(\log K)^{3/2+\varepsilon}$ of the half-integral weight Hecke cusp forms in the Kohnen plus subspace with weight bounded by $K$ , the number of real zeroes in the region $F_Y$ (where $\sqrt{K}\log K \le Y \le K^{1-\delta}$ ) on each geodesic $\delta_j$ is $\gg K/Y (\log K)^{-23/2}$ . This count is optimal up to powers of logarithms, matching the expected rate derived from the area of the domain.
Theorem 1.3: For at least $(1/2 - \varepsilon)$ of the forms in the same family, the number of real zeroes on each geodesic $\delta_j$ in the same range of $Y$ is $\gg \sqrt{K/Y}$ . This provides a lower bound holding for a positive proportion of forms, albeit with a weaker growth rate than the "many forms" result.
Technical Innovation: The paper introduces methods to handle the lack of multiplicativity in half-integral weight coefficients by combining averaging over weights with the Shimura correspondence. It provides new estimates for the first and second moments of quadratic twists of L-functions averaged over the Kohnen plus subspace.

Significance and Claims
The author claims these are the first results concerning the small-scale distribution of zeroes for half-integral weight Hecke cusp forms. The work demonstrates that despite the structural differences between integral and half-integral weight forms (specifically the lack of full multiplicativity), the phenomenon of zeroes concentrating on specific geodesics near the cusp persists.

The paper explicitly notes that the techniques differ fundamentally from those used in the integral weight setting. While the integral weight case relies heavily on the multiplicativity of coefficients to study sign changes directly, the half-integral weight setting requires a more complex interplay of L-function moment estimates and the specific arithmetic properties of the Shimura lift. The author modestly states that the results for Theorem 1.1 are optimal up to logarithmic factors and that the methods used are distinct from previous works, demonstrating that non-trivial information about these zeroes can be deduced without the strong results available in the integral weight case.

The paper does not claim to prove the full Ghosh–Sarnak conjecture (that all zeroes concentrate on these lines) nor does it claim to remove the logarithmic losses in the count for Theorem 1.1. It also notes that extending the result of Theorem 1.1 to a positive proportion of forms using mollifiers remains a challenging open problem, though the author mentions in a footnote that a subsequent result using an Euler product mollifier has been obtained.