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Moments of the zeros of Faber polynomials of the Miller basis

Adi Zilka

TL;DR

This work analyzes zeros of Miller-basis modular forms via their Faber polynomials. It proves a linearity result for the power sums of the Faber zeros, with explicit constants $A_n$ and $B_n$ and a $C_n(k')$ term, enabling both upper bounds for when zeros leave the arc and a precise description of the limit distribution when all zeros lie on the arc in the regime $m\sim c\ell$ with $0<c<2/9$. The constants $A_n$ arise from arc averages of $j^n$, while $B_n$ comes from the coefficients of $j^n$, and $C_n(k')$ encodes dependence on the weight residue $k'$. The paper provides two independent proofs for the limiting distribution on the arc, yielding an explicit density that depends on $c$, and thereby deepens understanding of zero distributions in modular form spaces.

Abstract

We study the zeros of modular forms in the Miller basis, a natural basis for the space of modular forms. We show that the zeros of their Faber polynomials have linear moments. By analyzing the moments we can extend the known range of the forms in the Miller basis for which at least one of the zeros is not on the arc - the circular part of the boundary of the fundamental domain. Additionally, for forms in the Miller basis of an index asymptotically linear in the weight such that all zeros are on the arc, we compute the limit distribution of the zeros, which depends on the asymptotic ratio of the index to the weight.

Moments of the zeros of Faber polynomials of the Miller basis

TL;DR

This work analyzes zeros of Miller-basis modular forms via their Faber polynomials. It proves a linearity result for the power sums of the Faber zeros, with explicit constants and and a term, enabling both upper bounds for when zeros leave the arc and a precise description of the limit distribution when all zeros lie on the arc in the regime with . The constants arise from arc averages of , while comes from the coefficients of , and encodes dependence on the weight residue . The paper provides two independent proofs for the limiting distribution on the arc, yielding an explicit density that depends on , and thereby deepens understanding of zero distributions in modular form spaces.

Abstract

We study the zeros of modular forms in the Miller basis, a natural basis for the space of modular forms. We show that the zeros of their Faber polynomials have linear moments. By analyzing the moments we can extend the known range of the forms in the Miller basis for which at least one of the zeros is not on the arc - the circular part of the boundary of the fundamental domain. Additionally, for forms in the Miller basis of an index asymptotically linear in the weight such that all zeros are on the arc, we compute the limit distribution of the zeros, which depends on the asymptotic ratio of the index to the weight.

Paper Structure

This paper contains 10 sections, 19 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction and Main Results
  2. Background
  3. Computing the Faber Polynomial
  4. Additional Notation
  5. Proof of the Linearity Theorem
  6. Proof of Theorem 1
  7. Computing the Values of $A_n$
  8. Computing the Values of $B_n$
  9. Upper Bound
  10. Limit Distribution and Proof of Theorem 2

Key Result

Theorem 1

There exists $A_n, B_n, C_n(k')$ such that for every $n\le \ell - m$, where $A_n = \frac{1}{2\pi} \int_{\mathcal{A}} j^n(\theta)d\theta$, $-B_n$ is the coefficient of $q^0$ in $j^n$, $C_n(0) = C_n(4) = C_n(8) = 0$ and $C_n(6) = C_n(10) = C_n(14) = -\frac{1728^n}{2}$.

Figures (1)

  • Figure 1: The contour of integration

Theorems & Definitions (38)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • Conjecture 1
  • Conjecture 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 28 more