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On the Zeros of the Miller Basis of Cusp Forms

Roei Raveh

Abstract

We study the zeros of cusp forms in the Miller basis whose vanishing order at infinity is a fixed number $m$. We show that for sufficiently large weights, the finite zeros of such forms in the fundamental domain, all lie on the circular part of the boundary of the fundamental domain. We further show and quantify an effective bound for the weight, which is linear in terms of $m$.

On the Zeros of the Miller Basis of Cusp Forms

Abstract

We study the zeros of cusp forms in the Miller basis whose vanishing order at infinity is a fixed number . We show that for sufficiently large weights, the finite zeros of such forms in the fundamental domain, all lie on the circular part of the boundary of the fundamental domain. We further show and quantify an effective bound for the weight, which is linear in terms of .
Paper Structure (25 sections, 8 theorems, 115 equations, 9 figures)

This paper contains 25 sections, 8 theorems, 115 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Background and Preliminaries on Modular Forms
  3. Definitions
  4. The Miller Basis for Modular Forms
  5. Modular Forms on the Arc and Under Conjugation
  6. Proof of Theorem \ref{['mr']}
  7. Bounds and Properties of $\Delta$
  8. A Key Proposition
  9. Proof of Proposition \ref{['mrl']}
  10. Suppose $\frac{\pi}{2}<\theta<1.9$
  11. Suppose $1.9 \le \theta < \frac{2\pi}{3}$
  12. Back to $\theta\in\pr{\frac{\pi}{2},\frac{2\pi}{3}}$
  13. Proof of Theorem \ref{['mr']}
  14. Quantifying the Bounds
  15. Outline of the Proof of Theorem \ref{['QuanMR']}
  16. ...and 10 more sections

Key Result

lemma 1

Figures (9)

  • Figure 1: The fundamental domain $\cF$
  • Figure 2: $\delta\pr{\theta}$ on the interval $\left[\frac{\pi}{2},\frac{2\pi}{3}\right]$.
  • Figure 3: The contour of integration
  • Figure 4: The contour of integration when $\theta\in\left(\frac{\pi}{2},1.9\right)$.
  • Figure 5: The contour of integration when $\theta\in\left[1.9,\frac{2\pi}{3}\right)$.
  • ...and 4 more figures

Theorems & Definitions (24)

  • remark 1
  • remark 2
  • lemma 1
  • proof
  • lemma 2
  • proof
  • proof : Proof of Proposition \ref{['Prop: Bounds of delta on the arc']}
  • proof
  • proof
  • lemma 3
  • ...and 14 more