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Zeros of modular forms and Faber polynomials

Zeév Rudnick

TL;DR

The paper addresses the distribution of zeros of cusp forms of large weight for the full modular group when the number of finite zeros is fixed at $D$. It introduces and exploits the Faber polynomial associated to a modular form, proving that renormalized Faber polynomials converge to the truncated exponential polynomial $\mathcal{E}_D$, which in turn yields precise asymptotics: the nontrivial zeros accumulate on $D$ vertical lines at height $\approx \frac{1}{2\pi}\log k$ in the fundamental domain. This leads to a sharp description of the zeros of both the Faber polynomial and the cusp form itself: the zeros correspond to the zeros of $\mathcal{E}_D$ and their $j$-values grow like $2k z_{D,r}$, with the zeros of the cusp form occurring at $\tau_r=\frac{i}{2\pi}\log(2k z_{D,r})+O(1/k)$. The work provides a novel, explicit mechanism for zero localization of high-weight cusp forms via Faber polynomials and connects the zero distribution to the zeros of a truncated exponential polynomial, offering new insights into modular forms beyond classical equidistribution results.

Abstract

We study the zeros of cusp forms of large weight for the modular group, which have a very large order of vanishing at infinity, so that they have a fixed number D of finite zeros in the fundamental domain. We show that for large weight the zeros of these forms cluster near D vertical lines, with the zeros of a weight k form lying at height approximately log(k). This is in contrast to previously known cases, such as Eisenstein series, where the zeros lie on the circular part of the boundary of the fundamental domain, or the case of cuspidal Hecke eigenforms where the zeros are uniformly distributed in the fundamental domain. Our method uses the Faber polynomials. We show that for our class of cusp forms, the associated Faber polynomials, suitably renormalized, converge to the truncated exponential polynomial of degree D.

Zeros of modular forms and Faber polynomials

TL;DR

The paper addresses the distribution of zeros of cusp forms of large weight for the full modular group when the number of finite zeros is fixed at . It introduces and exploits the Faber polynomial associated to a modular form, proving that renormalized Faber polynomials converge to the truncated exponential polynomial , which in turn yields precise asymptotics: the nontrivial zeros accumulate on vertical lines at height in the fundamental domain. This leads to a sharp description of the zeros of both the Faber polynomial and the cusp form itself: the zeros correspond to the zeros of and their -values grow like , with the zeros of the cusp form occurring at . The work provides a novel, explicit mechanism for zero localization of high-weight cusp forms via Faber polynomials and connects the zero distribution to the zeros of a truncated exponential polynomial, offering new insights into modular forms beyond classical equidistribution results.

Abstract

We study the zeros of cusp forms of large weight for the modular group, which have a very large order of vanishing at infinity, so that they have a fixed number D of finite zeros in the fundamental domain. We show that for large weight the zeros of these forms cluster near D vertical lines, with the zeros of a weight k form lying at height approximately log(k). This is in contrast to previously known cases, such as Eisenstein series, where the zeros lie on the circular part of the boundary of the fundamental domain, or the case of cuspidal Hecke eigenforms where the zeros are uniformly distributed in the fundamental domain. Our method uses the Faber polynomials. We show that for our class of cusp forms, the associated Faber polynomials, suitably renormalized, converge to the truncated exponential polynomial of degree D.
Paper Structure (12 sections, 3 theorems, 64 equations, 1 figure, 1 table)

This paper contains 12 sections, 3 theorems, 64 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background on modular forms
  3. Basic definitions
  4. Zeros
  5. Faber polynomials
  6. Definition
  7. Computation
  8. Examples
  9. Proof of Theorem \ref{['thm:Faber']}
  10. Back to zeros of modular forms
  11. Proof of Corollary \ref{['cor:zeros of F']}
  12. Proof of Theorem \ref{['thm:zeros of f']}

Key Result

Theorem 1.1

Fix $D\geq 1$, $C>0$, and let $f\in S_k$ be as in def of special f. Then the zeros $\tau_1,\dots, \tau_D$ of $f$ in the fundamental domain, suitably labeled, satisfy

Figures (1)

  • Figure 1: The fundamental domain $\mathcal{F}$ and the points $\frac{\sqrt{-1}}{2\pi} \log\left( 2k z_{4,r} \right)$, $r=1,\dots,4$ and $k= 1000 j$, $j=1,20$ where $z_{4,r}$ are the inverse zeros of $\mathcal{E}_4(t) = 1+t+t^2/2+t^3/6+t^4/24$.

Theorems & Definitions (4)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • proof