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Sign patterns of Fourier coefficients of modular forms

Andrew R. Booker

TL;DR

This paper analyzes whether the signs of Fourier coefficients of self-dual holomorphic cusp forms determine the form up to a positive scalar. It proves that, under mild level/weight constraints, nonproportional cusp forms cannot have eventually nonnegative products of coefficients, by combining joint Sato–Tate equidistribution, Ramakrishnan lift techniques, and density arguments on prime powers. The equal-weight/level case is settled via a detailed analysis of Sato–Tate angles and a density-based construction that forces proportionality unless signs flip infinitely often. Overall, the sign pattern of Fourier coefficients uniquely determines the form (up to a positive scalar) except in some trivial proportional cases, highlighting a deep link between coefficient signs and automorphic representations.

Abstract

We give conditions under which a self-dual holomorphic cusp form is determined up to scalar multiplication by the signs of its Fourier coefficients.

Sign patterns of Fourier coefficients of modular forms

TL;DR

This paper analyzes whether the signs of Fourier coefficients of self-dual holomorphic cusp forms determine the form up to a positive scalar. It proves that, under mild level/weight constraints, nonproportional cusp forms cannot have eventually nonnegative products of coefficients, by combining joint Sato–Tate equidistribution, Ramakrishnan lift techniques, and density arguments on prime powers. The equal-weight/level case is settled via a detailed analysis of Sato–Tate angles and a density-based construction that forces proportionality unless signs flip infinitely often. Overall, the sign pattern of Fourier coefficients uniquely determines the form (up to a positive scalar) except in some trivial proportional cases, highlighting a deep link between coefficient signs and automorphic representations.

Abstract

We give conditions under which a self-dual holomorphic cusp form is determined up to scalar multiplication by the signs of its Fourier coefficients.

Paper Structure

This paper contains 3 sections, 4 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Different weight or level
  3. Equal weight and level

Key Result

Theorem 1.1

Let $M,N\in\mathbb{N}$, with $\mathop{\mathrm{lcm}}\nolimits(M,N)$ not divisible by $2^4$ or the square of an odd prime. Let $f\in S_k^{\rm new}(\Gamma_0(M))$ and $g\in S_\ell^{\rm new}(\Gamma_0(N))$ be non-zero cusp forms with real Fourier coefficients $a_f(n),a_g(n)$. Then the following are equiva Thus if $f$ and $g$ are not proportional then $\{n\in\mathbb{N}:\epsilon a_f(n)a_g(n)>0\}$ is infin

Theorems & Definitions (7)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof