Distribution of local signs of modular forms and murmurations of Fourier coefficients

TL;DR

This paper investigates the distribution of local signs of modular forms, explores correlations called murmurations between Fourier coefficients and root numbers, and extends previous bias results to local root numbers.

Contribution

It proves new results on local root number biases, introduces the concept of murmurations in a broader context, and corrects a sign error in prior work.

Findings

Bias of local root numbers towards +1

Existence of murmurations in Fourier coefficients

Conjecture of murmurations in general local root number combinations

Abstract

Recently, we showed that global root numbers of modular forms are biased toward +1. Together with Pharis, we also showed an initial bias of Fourier coefficients towards the sign of the root number. First, we prove analogous results with respect to local root numbers. Second, a subtle correlation between Fourier coefficients and global root numbers, termed murmurations, was recently discovered for elliptic curves and modular forms. We conjecture murmurations in a more general context of different (possibly empty) combinations of local root numbers. Last, an appendix corrects a sign error in our joint paper with Pharis.