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One-level densities of large even and odd orthogonal families of automorphic L-functions

Conditional on the Generalized Riemann Hypothesis, this paper establishes one-level density results for even and odd orthogonal families of automorphic L-functions with support extended to (-3,3), yielding the strongest known non-vanishing results for these families and their derivatives at the central point.

Original authors: Vorrapan Chandee, Xiannan Li, Micah B. Milinovich

Published 2026-05-19

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CC BY 4.0

Original authors: Vorrapan Chandee, Xiannan Li, Micah B. Milinovich

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 a vast, cosmic library filled with infinite books. Each book represents a specific mathematical object called an L-function. These aren't stories about dragons or space travel; they are complex formulas that encode deep secrets about numbers, much like a DNA sequence encodes the blueprint of a living organism.

The authors of this paper—Vorrapan Chandee, Xiannan Li, and Micah B. Milinovich—are like detectives trying to understand the "genetic makeup" of these books. Specifically, they are looking for zeros. In the world of L-functions, a "zero" is a special point where the formula equals zero. Finding these zeros is crucial because their location holds the key to some of the hardest puzzles in mathematics.

Here is a breakdown of what they did, using simple analogies:

1. The Two Families: The Even and The Odd

The authors focused on two specific groups of these L-functions, which they call the Even and Odd families.

The Analogy: Think of these families as two different species of birds. One species (the Even family) has a specific trait that makes it behave one way, while the other species (the Odd family) has a trait that forces it to behave differently.
The Discovery: For the "Odd" family, the rules of the universe (specifically the "functional equation") dictate that the bird must land on a specific branch (the central point) and stay there. In math terms, the L-function is guaranteed to be zero at the center.
The Twist: For the "Even" family, the bird might land on that branch, but it doesn't have to. The big question is: How often does it actually land there? If it lands there, the function is "zero" (vanishing). If it doesn't, the function is "non-zero" (non-vanishing).

2. The Map of the Zeros: One-Level Density

To answer how often the Even family avoids the center, the authors created a new, more detailed map.

The Old Map: Previous researchers had a map that could only see zeros within a certain distance from the center. It was like having a flashlight that only lit up a small circle around your feet.
The New Map: The authors built a flashlight with a much wider beam. They extended the "support" of their map from a range of (-2, 2) to (-3, 3).
Why it matters: By seeing a wider area, they could count the zeros more accurately. It's like zooming out on a camera; you can see more of the landscape and get a better sense of the terrain.

3. The Big Achievement: Proving Non-Vanishing

The ultimate goal of their detective work was to prove that these L-functions are alive (non-zero) at the center point.

The Even Family: Using their new, wider map, they proved that at least 69.8% of the Even family members do not land on the forbidden branch. They are "non-zero." This is a significant improvement over previous estimates, which were much lower.
The Odd Family: Since we know the Odd family always lands on the branch (making the function zero), the authors looked at the derivative (the speed at which the function is moving when it hits the branch). They proved that at least 97.8% of the Odd family members are "moving" when they hit the branch, meaning the derivative is not zero.

4. The Rules of the Game (Assumptions)

It is important to note that the authors played by a specific set of rules. They assumed the Generalized Riemann Hypothesis (GRH) is true.

The Analogy: Imagine they are trying to solve a maze. They are assuming that the maze has a specific, predictable structure (the GRH). If that structure is real, their map and their counts are 100% correct. If the structure turns out to be different, their specific numbers might need adjustment, but their method of mapping remains valid.

Summary

In short, this paper is about refining our tools to look deeper into the "DNA" of a specific group of mathematical formulas. By separating the formulas into "Even" and "Odd" groups and using a wider viewing lens, the authors proved that:

Most of the "Even" formulas are not zero at their most critical point.
Almost all of the "Odd" formulas are active (their derivative is not zero) at that same point.

They achieved this by pushing the boundaries of what we can see in the mathematical landscape, conditional on a major hypothesis in mathematics being true. They did not apply this to medicine or engineering; they simply solved a pure math puzzle about how often these special numbers are zero or not.

Technical Summary: One-Level Densities of Large Even and Odd Orthogonal Families of Automorphic L-Functions

Problem Statement
This paper investigates the distribution of low-lying zeros of $L$ -functions attached to primitive cusp forms of level $q$ and fixed weight $k \geq 4$ . Specifically, the authors study the "one-level density" of these zeros, averaged over square-free levels $q$ . The study distinguishes between two sub-families based on the sign of the functional equation: the even family ( $\varepsilon_{f,q} = 1$ ) and the odd family ( $\varepsilon_{f,q} = -1$ ).

The central heuristic guiding this work is the Katz-Sarnak philosophy, which predicts that the statistical distribution of zeros of $L$ -functions in a family mirrors the eigenvalue distribution of random matrices from a specific symmetry group (in this case, the orthogonal group). While previous results established this correspondence for fixed levels or averaged over all levels, this paper addresses the specific challenge of averaging over square-free levels while separating the even and odd families. This separation is crucial because the functional equation sign dictates the vanishing behavior at the central point $s=1/2$ : odd forms vanish at $s=1/2$ , while even forms are conjectured to be non-vanishing.

Methodology
The authors employ analytic number theory techniques, primarily relying on the explicit formula relating sums over zeros to sums over prime powers, combined with the Kuznetsov trace formula.

Explicit Formula and Root Numbers: The core of the analysis involves averaging the explicit formula over the family. A critical technical step is the inclusion of the root number $\varepsilon_{f,q}$ in the average. For square-free $q$ , the root number is explicitly given by $\varepsilon_{f,q} = i^k \mu(q) \lambda_f(q) \sqrt{q}$ . The authors treat the sums with and without the root number separately to isolate the even and odd contributions.
Trace Formulas: To evaluate the sums over the Fourier coefficients $\lambda_f(n)$ , the authors utilize the Petersson trace formula for newforms (specifically the version by Iwaniec, Luo, and Sarnak) and the Kuznetsov trace formula. These tools allow the conversion of spectral sums into arithmetic sums involving Kloosterman sums and Bessel functions.
GRH Assumption: The Generalized Riemann Hypothesis (GRH) is assumed throughout. This assumption is used to bound sums over primes and to ensure the convergence and error term estimates required for the trace formula applications.
Fourier Analysis: The support of the Fourier transform of the test function, $\hat{\Phi}$ , is a key parameter. The authors extend the allowable support of $\hat{\Phi}$ to the interval $(-3, 3)$ . This extension requires delicate handling of the error terms arising from the trace formula, particularly when the root number is involved.
Optimization for Non-Vanishing: To derive non-vanishing results, the authors formulate an optimization problem involving the ratio of integrals of the test function against the symmetry kernels. They utilize methods from reproducing kernel Hilbert spaces (referencing Freeman and Miller) to determine the optimal constants for the proportion of non-vanishing forms.

Key Contributions and Results

Extended Support for One-Level Density: The primary result (Theorem 1.1) proves the one-level density conjecture for the even and odd orthogonal families of $GL(2)$ automorphic $L$ -functions, averaged over square-free levels. The authors establish that the conjecture holds when the Fourier transform of the test function, $\hat{\Phi}$ , is compactly supported in $(-3, 3)$ .
- For the even family, the density kernel is $W_{\text{even}}(x) = 1 + \frac{\sin(2\pi x)}{2\pi x}$ .
- For the odd family, the density kernel is $W_{\text{odd}}(x) = 1 - \frac{\sin(2\pi x)}{2\pi x} + \delta_0(x)$ .
- The authors note that while a previous result by the first two authors with Baluyot achieved support in $(-4, 4)$ for the full orthogonal family, the inclusion of the root number to separate even and odd families currently presents a barrier, limiting the support to $(-3, 3)$ .
Improved Non-Vanishing Proportions: Leveraging the extended support of $(-3, 3)$ , the authors derive stronger lower bounds for the proportion of non-vanishing $L$ -values and derivatives at the central point (Corollary 1.3).
- Even Family: At least 69.8% of the forms in the even family have $L(1/2, f) \neq 0$ . This improves upon the $(50-\epsilon)\%$ bound derived from results with support limited to $(-2, 2)$ or the non-optimized application of the $(-4, 4)$ result.
- Odd Family: Since $L(1/2, f) = 0$ for odd forms, the focus is on the derivative. The authors show that at least 97.8% of the forms in the odd family have $L'(1/2, f) \neq 0$ .
Technical Barrier Identification: The paper explicitly identifies the difficulty in extending the support beyond $(-3, 3)$ for the separated families. The authors suggest that this limitation arises fundamentally because their current methods do not yield a reduction in the conductor when the support exceeds this range. They posit that further improvements might require a more powerful version of the Kuznetsov trace formula on $GL(3)$.

Significance
The paper claims significance primarily in the context of understanding the non-vanishing of central $L$ -values, a central problem in number theory with implications for the Birch and Swinnerton-Dyer conjecture and other arithmetic questions. By separating the even and odd families and extending the support of the test function to $(-3, 3)$ , the authors achieve the strongest known non-vanishing results for these specific families conditional on GRH.

The work refines the understanding of the Katz-Sarnak philosophy in the number field setting, demonstrating that the statistical behavior of zeros aligns with random matrix theory even when averaging over square-free levels and separating by root number. The authors remain modest regarding the $(-3, 3)$ limit, acknowledging it as a "natural barrier" for their current methodology, rather than a definitive limit of the conjecture itself.

1. The Two Families: The Even and The Odd

2. The Map of the Zeros: One-Level Density

3. The Big Achievement: Proving Non-Vanishing

4. The Rules of the Game (Assumptions)

Summary

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