Low-lying zeros in families of Maass form L-functions: an extended density theorem

TL;DR

The paper studies the one-level density of low-lying zeros for the family of Maass form $L$-functions at prime level $N$, verifying the Katz-Sarnak orthogonal prediction in the level aspect. By combining the explicit formula with the Kuznetsov trace formula and reducing off-diagonal terms to Dirichlet polynomials, the authors bound a Dirichlet-polynomial average using Heath-Brown's identity, the large sieve, and fourth-moment bounds, obtaining an unconditional extension of the admissible Fourier support to ${\rm supp}\\(\\widehat{\\phi})\\subset(-\\tfrac{15}{8},\\tfrac{15}{8})$. They also prove a conditional extension to $(-2,2)$ under the Grand Density Conjecture and establish a parallel unconditional improvement for the holomorphic-form family with the corresponding $\Theta_k$-range. The results concretely show that the Katz-Sarnak density governs the low-lying zeros within the extended range and provide a framework that yields arithmetic consequences such as bounds on the average order of vanishing at the central point. The techniques connect trace-formula analyses with deep Dirichlet-polynomial bounds, offering a robust approach to density questions in families of automorphic $L$-functions.

Abstract

We study the one-level density of low-lying zeros in the family of Maass form $L$-functions of prime level $N$ tending to infinity. Generalizing the influential work of Iwaniec, Luo and Sarnak to this context, Alpoge et al. have proven the Katz-Sarnak prediction for test functions whose Fourier transform is supported in $(-\frac32,\frac32)$. In this paper, we extend the unconditional admissible support to $(-\frac{15}8,\frac{15}8)$. The key tools in our approach are analytic estimates for integrals appearing in the Kutznetsov trace formula, as well as a reduction to bounds on Dirichlet polynomials, which eventually are obtained from the large sieve and the fourth moment bound for Dirichlet $L$-functions. Assuming the Grand Density Conjecture, we extend the admissible support to $(-2,2)$. In addition, we show that the same techniques also allow for an unconditional improvement of the admissible support in the corresponding family of $L$-functions attached to holomorphic forms.

Theorems & Definitions (34)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4