On Murmurations and Trace Formulas

TL;DR

This paper explores murmuration phenomena in automorphic forms using various trace formulas, highlighting new avenues for proving such behavior across different families of L-functions.

Contribution

It analyzes murmuration behavior through different trace formulas and suggests potential for proving murmuration in broader contexts.

Findings

Murmuration behavior observed in Maass and modular forms.

Different trace formulas provide varied approaches to murmuration.

Potential to extend murmuration proofs to other L-function families.

Abstract

In recent work with Bober, Booker, Lee, Seymour-Howell, and Zubrilina, we proved murmuration behavior for Maass forms in the eigenvalue aspect and for modular forms in the weight aspect. Both used an approach based on the Selberg trace formula. But different trace formulas, including those due to Kuznetsov or Petersson, offer different variations. We examine murmurations from the perspective of different trace formulas and outline several families of $L$-functions where one can likely prove additional murmuration behavior.