Simple Fourier Trace Formulas of Cubic Level and Applications

TL;DR

This paper develops Fourier trace formulas for automorphic forms on PGL(2) with cubic level, enabling new results on L-value non-vanishing and Weyl's law for Maass forms.

Contribution

It introduces new Fourier trace formulas for cubic level automorphic forms using the relative trace formula and supercuspidal classification, advancing the understanding of automorphic L-functions.

Findings

Non-vanishing of central L-values for holomorphic newforms

Weighted Weyl's law for Maass newforms

New Fourier trace formulas for cubic level automorphic forms

Abstract

With the method of the relative trace formula and the classification of simple supercuspidal representations, we establish some Fourier trace formulas for automorphic forms on $PGL(2)$ of cubic level. As applications, we obtain a non-vanishing result for central $L$-values of holomorphic newforms and a weighted Weyl's law for Maass newforms.