Simple Fourier Trace Formulas of Cubic Level and Applications

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.