On the average size of the eigenvalues of the Hecke operators

TL;DR

The paper analyzes the average size of Hecke eigenvalues on $S_k(\Gamma_0(N))$ via normalized operators $T'_m$ in two viewpoints. Vertically, with $m$ fixed and $N,k$ growing, the mean square of eigenvalues tends to $\sigma_1(m)/m$, yielding $A_v_m(N,k)\to\sqrt{\sigma_1(m)/m}$; horizontally, with $N,k$ fixed and $m$ growing, the mean square of normalized Fourier coefficients tends to the residue of the associated $L$-function, giving $A_v_f(x)\to \sqrt{\frac{12\cdot(4\pi)^{k-1}}{(k-1)!}}\|f\|$, and in particular for Ramanujan's $\Delta$, $A_v_\Delta(x)=0.619745...$ . The work uses trace formulas to derive the vertical limit, and Wiener–Ikehara plus Rankin–Selberg theory to obtain the horizontal limit, connecting eigenvalue distributions to classical automorphic data. It also provides a complete finite classification for when $A_v_2(N,k)\le1$ and discusses a conjectural lower bound for composite indices analogous to Atkin–Serre. Overall, the results quantify how Hecke eigenvalues grow on average in both directions and link them to fundamental invariants like $\sigma_1(m)$ and $\langle f,f\rangle$.

Abstract

We determine the average size of the eigenvalues of the Hecke operators acting on the cuspidal modular forms space $S_k(Γ_0(N))$ in both the vertical and the horizontal perspective. The "average size" is measured via the quadratic mean.

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: ross-xue, serre
• Theorem \ref{thm:Av-vertical}
• proof
• Theorem \ref{thm:Av-vertical}
• proof
• Corollary \ref{thm:Av-vertical}