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

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.