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

TL;DR

This work studies how weighting families of GL(2) cusp form L-functions influences the distribution of low-lying zeros near the central point. Building on Katz-Sarnak and Knightly-Reno, it derives a weighted explicit formula and an asymptotic weighted trace formula that incorporate cross terms from n-fold prime products and uses an elementary combinatorial rewrite to align main terms with random matrix predictions. Under test functions with support in (-1/(2n),1/(2n)), the weighted n-th centered moment across two weight families converges to that of a Gaussian with variance sigma_phi^2, yielding (n-1)!! sigma_phi^n for even n and 0 for odd n; this Gaussian behavior holds for both nontrivial and trivial twisting characters, with the authors noting that extending the support could reveal non-Gaussian regimes and symmetry distinctions. By distinguishing twisting cases and providing a framework to handle cross terms, the paper strengthens the bridge between weighted L-function statistics and random matrix theory for GL(2) families.

Abstract

Katz and Sarnak conjectured that the behavior of zeros near the central point of any family of $L$-functions is well-modeled by the behavior of eigenvalues near $1$ of some classical compact group (either the symplectic, unitary, or even, odd, or full orthogonal group). In 2018, Knightly and Reno proved that the symmetry group can vary depending on how the $L$-functions in the family are weighted. They observed both orthogonal and symplectic symmetry in the one-level densities of families of cuspidal newform $L$-functions for different choices of weights. We observe the same dependence of symmetry on weights in the $n^{\text{th}}$ centered moments of these one-level densities, for smooth test functions whose Fourier transforms are supported in $\left(-\frac{1}{2n}, \frac{1}{2n}\right)$. To treat the new terms that emerge in our $n$-level calculations when $n>1$, i.e., the cross terms that emerge from $n$-fold products of primes rather than individual primes, we generalize Knightly and Reno's weighted trace formula from primes to arbitrary positive integers. We then perform a delicate analysis of these cross terms to distinguish their contributions to the main and error terms of the $n^{\text{th}}$ centered moments. The final novelty here is an elementary combinatorial trick that we use to rewrite the main number theoretic terms arising from our analysis, facilitating comparisons with random matrix theory.

Theorems & Definitions (22)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Lemma 3.1
• proof
• Lemma 4.1
• proof