English

Low-lying zeros of $L$-functions for Maass forms over imaginary quadratic fields

Number Theory 2020-03-31 v2

Abstract

We study the 11- or 22-level density of families of LL-functions for Hecke--Maass forms over an imaginary quadratic field FF. For test functions whose Fourier transform is supported in (32,32)\left(-\frac 32, \frac 32\right), we prove that the 11-level density for Hecke--Maass forms over FF of square-free level q\mathfrak{q}, as N(q)\mathrm{N}(\mathfrak{q}) tends to infinity, agrees with that of the orthogonal random matrix ensembles. For Hecke--Maass forms over FF of full level, we prove similar statements for the 11- and 22-level densities, as the Laplace eigenvalues tends to infinity.

Keywords

Cite

@article{arxiv.1908.06348,
  title  = {Low-lying zeros of $L$-functions for Maass forms over imaginary quadratic fields},
  author = {Sheng-Chi Liu and Zhi Qi},
  journal= {arXiv preprint arXiv:1908.06348},
  year   = {2020}
}

Comments

27 pages. To appear in Mathematika. RH added in the assumptions of Theorem A.1

R2 v1 2026-06-23T10:49:55.443Z