Low-Lying Zeros of a Thin Family of Automorphic $L$-Functions in the Level Aspect

TL;DR

This paper investigates the distribution of low-lying zeros in a specific thin family of automorphic L-functions, providing both conditional and unconditional results on their statistical behavior and symmetry types.

Contribution

It introduces a novel approach to analyze thin subfamilies of automorphic L-functions, establishing an average Weil bound for twisted Kloosterman sums and demonstrating the impact of averaging on symmetry detection.

Findings

Support for the Riemann Hypothesis up to 1 conditional on GRH.

Unconditional support slightly below 1.

Ability to distinguish between SO(even) and SO(odd) symmetry types through averaging.

Abstract

We calculate the one-level density of thin subfamilies of a family of Hecke cuspforms formed by twisting the forms in a smaller family by a character. The result gives support up to 1, conditional on GRH, and we also find several of the lower-order main terms. In addition, we find an unconditional result that has only slightly lower support. A crucial step in doing so is the establishment of an on-average version of the Weil bound that applies to twisted Kloosterman sums. Moreover, we average over these thin subfamilies by running over the characters in a coset, and observe that any amount of averaging at all is enough to allow us to get support greater than 1 and thus distinguish between the SO(even) and SO(odd) symmetry types. Finally, we also apply our results to nonvanishing problems for the families studied.