Centered Moments of Weighted One-Level Densities of $GL(2)$ $L$-Functions

This paper extends Knightly and Reno's 2018 findings on weight-dependent symmetry in one-level densities to the $n^{\text{th}}$ centered moments of $GL(2)$ $L$-functions by generalizing the weighted trace formula to arbitrary integers, analyzing resulting cross terms, and employing a combinatorial technique to facilitate comparison with random matrix theory.

The Gist

Imagine you are a detective trying to understand the hidden rhythm of a massive, chaotic orchestra. This orchestra isn't made of violins and drums, but of mathematical numbers called "L-functions." These numbers are the secret code of the universe, governing everything from prime numbers to the shapes of elliptical curves.

For decades, mathematicians have suspected that this chaotic orchestra doesn't just play random noise. Instead, they believe it follows a specific, elegant pattern, much like the sound waves produced by a Gaussian Unitary Ensemble (GUE)—a fancy term for a specific type of random matrix (think of it as a giant, complex drum machine).

The Big Mystery: The "Symmetry" of the Music

In the late 90s, two giants in the field, Katz and Sarnak, proposed a theory: The "low notes" of this mathematical orchestra (the zeros near the center of the music) should sound exactly like the vibrations of one of five specific types of symmetrical shapes (like a sphere, a cube, or a twisted loop).

However, there was a twist. In 2018, researchers Knightly and Reno discovered something strange: The type of symmetry depends on how you listen.

Imagine you are at a concert.

• If you listen to the whole orchestra equally, you hear Symmetry A (Orthogonal).
• But, if you put on special headphones that amplify the sound of the violin section while muting the drums, the music suddenly sounds like Symmetry B (Symplectic).

The "weight" (the headphones) changed the perceived symmetry of the music.

What This Paper Does: The "Centered Moments"

The authors of this paper (Lawrence Dillon and his team) decided to go a step further. Instead of just listening to the music once (the "one-level density"), they wanted to analyze the moments of the music.

Think of it this way:

• The Average: What is the average pitch of the orchestra?
• The Variance: How much does the pitch wobble around that average?
• The "Centered Moments": If you take a snapshot of the orchestra's pitch, how does it wiggle, bounce, and fluctuate around the average?

The authors asked: "If we change the headphones (the weights), does the wiggling pattern (the moments) also change, or does it stay the same?"

The Challenge: The "Cross-Talk" Problem

Calculating these wiggles for a single note is easy. But calculating them for a whole symphony (the $n$-th moment) is a nightmare.

When you multiply the sounds of $n$ different notes together, you get "cross-terms."

• Analogy: Imagine you are mixing $n$ different smoothies. If you just mix one fruit at a time, it's simple. But if you blend them all together, the flavors interact in complex ways. The "flavor of the strawberry" might change when mixed with the "flavor of the banana."

In math, these interactions are called cross terms. Previous methods could handle single notes (primes), but they broke down when trying to handle the complex mix of $n$ notes multiplied together.

The Solution: A New "Combinatorial Trick"

The team had to build a new mathematical tool to untangle this flavor-mixing mess.

1. The Generalized Formula: They took an existing formula (the "Trace Formula") that usually only worked for prime numbers and expanded it to work for any number. This allowed them to track the "cross-talk" between all the different notes.
2. The Elementary Trick: Usually, mathematicians use a very complex, high-level magic trick (involving generating series and deep combinatorics) to simplify these messy equations. The authors, however, used a clever, elementary combinatorial trick.
   • The Metaphor: Instead of using a giant, complicated machine to sort the laundry, they found a simple folding technique that made the clothes stack themselves perfectly. They rewrote the messy math terms into a form that looked exactly like the statistics of a Gaussian distribution (the famous "Bell Curve").

The Result: The "Bell Curve" Wins

After all the heavy lifting, the result was surprisingly simple and beautiful.

No matter which "headphones" (weights) they used—whether they were listening to the orthogonal symmetry or the symplectic symmetry—the wiggling pattern (the centered moments) of the zeros always settled down into the same shape: The Bell Curve (Gaussian Distribution).

• If $n$ is odd: The wiggles cancel out perfectly, resulting in zero.
• If $n$ is even: The wiggles follow a specific pattern based on the "variance" (how wide the bell curve is).

Why This Matters

This paper confirms a deep connection between Number Theory (the study of integers) and Random Matrix Theory (the study of random matrices).

It tells us that while the type of symmetry (Orthogonal vs. Symplectic) might change depending on how you weigh the data, the statistical behavior of the fluctuations is universal. It's like saying that whether you are listening to a jazz band or a classical orchestra, if you measure how much the volume fluctuates around the average, the pattern of those fluctuations is always the same.

In short: The authors proved that even when you change the rules of the game (the weights), the underlying "dance" of the numbers still follows the same elegant, predictable steps as a random matrix ensemble. They did this by inventing a new, simpler way to untangle the complex math of mixing many numbers together.

Technical Summary

1. Problem Statement and Context

Background:
The Katz-Sarnak density conjecture posits that the statistical behavior of zeros of $L$-functions near the central point (the "low-lying zeros") mirrors the distribution of eigenvalues near 1 of specific classical compact groups (Unitary, Symplectic, or Orthogonal). While the unweighted one-level density of families of cuspidal newforms is known to follow the Orthogonal symmetry type ($WO$), Knightly and Reno (2018) demonstrated that introducing specific weights based on central $L$-values can alter this symmetry. Specifically, for families of holomorphic cuspidal newforms:

• Trivial twisting character: The weighted density exhibits Symplectic symmetry ($WSp$).
• Non-trivial twisting character: The weighted density exhibits Orthogonal symmetry ($WO$).

The Gap:
Previous work by Knightly and Reno focused on the first moment (the average) of the weighted one-level density. The authors of this paper aim to generalize this dependence of symmetry on weights to the $n$-th centered moments of the one-level density.

• Objective: Determine if the $n$-th centered moments of these weighted densities also depend on the triviality of the twisting character, or if they converge to a universal distribution (Gaussian) regardless of the weight, provided the test function support is restricted.
• Challenge: Calculating $n$-th moments involves expanding products of sums over primes. Unlike the first moment (which involves single primes), the $n$-th moment introduces cross-terms involving products of $n$ primes (or arbitrary integers), requiring a generalization of trace formulas from prime powers to arbitrary integers.

2. Methodology

The authors employ a multi-step analytical approach combining analytic number theory, combinatorics, and random matrix theory.

A. Setup and Definitions

• Families: Two families of Hecke cuspidal newforms, $\mathcal{F}_1$ (level 1) and $\mathcal{F}_2$ (prime level $N$), are defined.
• Weights: The weight assigned to a form $f$ is $w_{\chi,r}(f) = \frac{\Lambda(1/2, f \times \chi) |a_f(r)|^2}{\|f\|^2}$, where $\chi$ is a real primitive Dirichlet character and $r$ is an integer coprime to the modulus.
• Statistic: The $n$-th centered moment of the weighted one-level density:
  $$A_{w} \left[ \left( D(\cdot, \phi) - A_{w}(D(\cdot, \phi)) \right)^n \right]$$
  where $D(f, \phi)$ is the one-level density and $\phi$ is a test function with Fourier transform supported in $(-1/2n, 1/2n)$.

B. Explicit Formula and Expansion

1. Explicit Formula: The authors use the explicit formula to convert the sum over zeros into a sum over Hecke eigenvalues $\lambda_f(p)$.
2. Moment Expansion: Expanding the $n$-th power of the sum over primes generates terms involving products of eigenvalues $\prod_{i=1}^n \lambda_f(p_i)$.
3. Combinatorial Re-indexing: Using the multiplicative properties of Hecke eigenvalues, the sum over $n$-tuples of primes is re-indexed into a sum over distinct prime powers $q_j^{m_j}$. This introduces combinatorial coefficients ($c_{n,m}$) and requires averaging eigenvalues at arbitrary integers, not just primes.

C. Generalized Weighted Trace Formula

A critical technical contribution is the derivation of an asymptotic weighted trace formula for the average of Hecke eigenvalues $\lambda_f(m)$ over the families $\mathcal{F}_k(N)'$ for arbitrary integers $m$.

• The authors generalize the work of Knightly and Reno (who only handled prime powers) by utilizing a formula from Jackson and Knightly (2015) for weighted sums of Fourier coefficients.
• They adapt this to handle the specific weights $w_{\chi,r}$ and show that the main term depends on the divisor sum function and the character $\chi$, while the error term is controlled by the analytic conductor.
• Key Insight: The conditions on the families ($\mathcal{F}_1$ and $\mathcal{F}_2$) ensure that "oldforms" vanish under the specific weights, allowing the authors to work directly with the space of newforms without complex bookkeeping.

D. Combinatorial Analysis

The authors analyze the main terms arising from the trace formula expansion.

• Case 1: Non-trivial $\chi$: The cross-terms involving $\chi(q)$ oscillate and cancel out, leaving only specific diagonal terms.
• Case 2: Trivial $\chi$: The terms do not oscillate, leading to a more complex combinatorial sum involving $\phi(0)$ and $\sigma_\phi^2$.
• Elementary Combinatorial Trick: Instead of using the standard Soshnikov generating series trick (common in $n$-level density calculations), the authors employ an elementary combinatorial method. They rewrite the resulting sums using binomial coefficients and interpret them as moments of a Gaussian random variable. This allows them to "unravel" the complex number-theoretic expressions into a clean Gaussian form.

3. Key Contributions

1. Generalization to $n$-th Moments: The paper extends the Knightly-Reno result from the first moment to the $n$-th centered moment, proving that the dependence on weights persists in higher moments.
2. Generalized Trace Formula: The authors successfully derive a weighted trace formula for Hecke eigenvalues at arbitrary positive integers, overcoming the limitation of previous works that were restricted to prime powers. This is essential for handling the cross-terms in $n$-th moment calculations.
3. Elementary Combinatorial Strategy: The paper introduces a novel, elementary combinatorial technique to simplify the main terms in the trivial character case. This avoids the need for generating series and Hunt-Dyson formulas, offering a more direct path to the Gaussian limit.
4. Support Restrictions: The analysis is conducted for test functions with Fourier support in $(-1/2n, 1/2n)$. The authors explicitly note that within this range, the $n$-th moments of the Orthogonal and Symplectic ensembles are indistinguishable (both are Gaussian), which explains why the weighted and unweighted moments converge to the same limit despite the different symmetry types observed in the first moment.

4. Main Results

Theorem 1.1:
For the families $\mathcal{F}_1$ and $\mathcal{F}_2$ with weights $w_{\chi,r}$, and for any test function $\phi$ with $\text{supp}(\hat{\phi}) \subset (-1/2n, 1/2n)$, the $n$-th centered moment of the weighted one-level density converges to the $n$-th centered moment of a Gaussian distribution with variance $\sigma_\phi^2 = \int \hat{\phi}(y)^2 |y| dy$.

Specifically:

• If $n$ is odd, the limit is 0.
• If $n$ is even, the limit is $(n-1)!! \, \sigma_\phi^n$.

Crucial Observation:
This result holds regardless of whether the twisting character $\chi$ is trivial or non-trivial.

• Interpretation: While the first moment (the mean) distinguishes between Symplectic (trivial $\chi$) and Orthogonal (non-trivial $\chi$) symmetry, the centered moments (fluctuations around the mean) in the restricted support regime are identical for both cases. They both follow the Gaussian statistics predicted by Random Matrix Theory for the respective ensembles in this specific support range.

5. Significance

• Validation of Katz-Sarnak: The results provide strong evidence for the Katz-Sarnak conjecture in the context of weighted families. Even when weights alter the symmetry type of the mean, the fluctuations (centered moments) remain consistent with the corresponding Random Matrix Ensemble (Gaussian behavior).
• Methodological Advancement: The derivation of the trace formula for arbitrary integers and the development of the elementary combinatorial trick provide new tools for future research in $n$-level statistics of $L$-functions.
• Clarification of Weight Effects: The paper clarifies the scope of Knightly and Reno's findings. It shows that while weights can shift the symmetry type (changing the mean behavior), the higher-order statistical fluctuations (centered moments) within the standard support range are robust and converge to a universal Gaussian limit.
• Future Directions: The authors highlight that extending the support of the test function beyond $(-1/2n, 1/2n)$ is the next logical step. In that regime, the $n$-th moments of Orthogonal and Symplectic ensembles diverge (non-Gaussian behavior). Replicating this divergence on the number-theoretic side would be a significant breakthrough in confirming the connection between $L$-functions and Random Matrix Theory.

In summary, this paper successfully bridges the gap between weighted $L$-function statistics and Random Matrix Theory for higher moments, demonstrating that while weights change the "center" of the distribution, the "shape" of the fluctuations remains Gaussian within the analytically accessible range.