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

Kimball Martin

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.

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

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.
Paper Structure (32 sections, 19 theorems, 118 equations, 12 figures, 2 tables)

This paper contains 32 sections, 19 theorems, 118 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Distributions of local root numbers
  3. Correlation of initial Fourier coefficients with local signs
  4. Murmurations
  5. Additional remarks
  6. Notation and preliminaries
  7. Class numbers
  8. Other quantities arising in the trace formula
  9. Traces on newspaces
  10. $A_{1,\varepsilon}$ sums
  11. Computing $\tilde{A}_{1,\varepsilon}(r;M)$ when $4 \ell < q^{r+2\varepsilon}$
  12. Computing $\tilde{A}_{1,0}(r;M)$ when $\ell = 1$
  13. Computing $\tilde{A}_{1,1}(r;M)$ when $\ell = 1$
  14. $A_2$ sums when $\ell = 1$
  15. $A_3$ sums
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $q$ be a prime, $M \ge 1$ be coprime to $q$ and write $M = 2^e M'$ where $M'$ is odd. Let $r \ge 1$ be an odd integer. If $r \le 3$ assume $q \ge 5$, and if $r=1$ further assume that $k \ge 4$ or $M$ is not squarefree. Then $\Delta_k(q^r, M) = 0$ if and only if (i) $M'$ is not cubefree; (ii) ${-

Figures (12)

  • Figure 1: Weight 2 murmurations for $W_N$ for squarefree levels $1000 \le N \le 2000$
  • Figure 2: Weight 2 murmurations for $W_N$ for squarefree levels $2000 \le N \le 4000$
  • Figure 3: Weight 2 murmurations for $\sqrt N W_1$ for squarefree levels $2000 \le N \le 4000$
  • Figure 4: Weight 2 murmurations for $\sqrt N W_1$ for squarefree levels $4000 \le N \le 8000$
  • Figure 5: $\delta$-smoothed version of \ref{['fig:murmN-1b']} with $\delta = \frac{1}{2}$
  • ...and 7 more figures

Theorems & Definitions (32)

  • Theorem 1.1: odd exponent
  • Theorem 1.2: even exponent $\ge 4$
  • Proposition 1.3: exponent 2
  • Corollary 1.4: boundedness in $k$
  • Remark 1.5
  • Theorem 1.6: bias of initial Fourier coefficients
  • Remark 1.7
  • Conjecture 1.8: Murmurations for Atkin--Lehner operators
  • Theorem 1.9
  • Conjecture 1.10: Murmurations on Atkin--Lehner eigenspaces
  • ...and 22 more