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Hecke Eigenvalue Equidistribution over the Newspaces with Nebentypus

Erick Ross

TL;DR

This work establishes mu_p-equidistribution for the eigenvalues of the normalized p-th Hecke operator on the nebentypus newspaces S_k^{new}(N,chi) as N+k -> infinity, excluding an explicit exceptional case where the space vanishes. The main approach adapts Serre's equidistribution framework by matching moments against Chebyshev polynomials X_n and leveraging trace formulas for T_{p^n}^{new} together with robust lower bounds on auxiliary multiplicative functions; the limiting moment matches the μ_p-density. The paper also provides an application to the arithmetic of newforms, showing that the fraction of forms with bounded coefficient-field degree tends to zero, implying that most newforms have large coefficient fields. Overall, the results extend Serre-type equidistribution to newspaces with nebentypus and clarify when a limiting distribution exists.

Abstract

Fix a prime $p$, and let $\widehat T_p^{\mathrm{new}}(N,k,χ) := χ(p)^{-1/2} p^{-(k-1)/2} T_p^{\mathrm{new}}(N,k,χ)$ denote the normalized $p$'th Hecke operator over the newspace with nenbentypus $S_k^{\mathrm{new}}(N,χ)$. In this paper, we determine the distribution of the eigenvalues of $\widehat T_p^{\mathrm{new}}(N,k,χ)$ as $N+k \to \infty$.

Hecke Eigenvalue Equidistribution over the Newspaces with Nebentypus

TL;DR

This work establishes mu_p-equidistribution for the eigenvalues of the normalized p-th Hecke operator on the nebentypus newspaces S_k^{new}(N,chi) as N+k -> infinity, excluding an explicit exceptional case where the space vanishes. The main approach adapts Serre's equidistribution framework by matching moments against Chebyshev polynomials X_n and leveraging trace formulas for T_{p^n}^{new} together with robust lower bounds on auxiliary multiplicative functions; the limiting moment matches the μ_p-density. The paper also provides an application to the arithmetic of newforms, showing that the fraction of forms with bounded coefficient-field degree tends to zero, implying that most newforms have large coefficient fields. Overall, the results extend Serre-type equidistribution to newspaces with nebentypus and clarify when a limiting distribution exists.

Abstract

Fix a prime , and let denote the normalized 'th Hecke operator over the newspace with nenbentypus . In this paper, we determine the distribution of the eigenvalues of as .

Paper Structure

This paper contains 3 sections, 5 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Proof of the Main Theorem
  3. An Application of the Main Theorem

Key Result

Theorem 1.1

Fix a prime $p$, and let $\mu_p(x) := \frac{p+1}{\pi} \frac{(1-x^2/4)^{1/2}}{(p^{1/2} + p^{-1/2})^2-x^2} dx$. Consider $N \ge 1$ coprime to $p$, $k \ge 2$, and $\chi$ Dirichlet characters modulo $N$ such that $\chi(-1) = (-1)^k$. Then assuming it is not the case that $2 \mid {\mathfrak{f}}(\chi),\ 2

Theorems & Definitions (8)

  • Theorem 1.1
  • Lemma 2.1: serre
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3: cason-et-al, ross
  • proof
  • Corollary 3.1